Qualitative analysis of some models of delay differential equations

TESE DE DOUTORAMENTO QUALITATIVE ANALYSIS OF SOME MODELS OF DELAY DIFFERENTIAL EQUATIONS Sebastián Buedo Fernández ESCOLA DE DOUTORAMENTO INTERNACIONAL DA UNIVERSIDADE DE SANTIAGO DE COMPOSTELA PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS SANTIAGO DE COMPOSTELA 2021 DECLARACIÓN DO AUTOR/A DA TESE D./Dna. Sebastián Buedo Fernández Título da tese: Qualitative analysis of some models of delay differential equations Presento a miña tese, seguindo o procedemento axeitado ao Regulamento, e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) De ser o caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. E comprométome a presentar o Compromiso Documental de Supervisión no caso de que o orixinal non estea na Escola. En , . Sinatura electrónica AUTORIZACIÓN DO DIRECTOR / TITOR DA TESE Qualitative analysis of some models of delay differential equations D. Eduardo Liz Marzán (director) Dna. Rosana Rodríguez López (directora e titora) INFORMAN: Que a presente tese, correspóndese co traballo realizado por D. Sebastián Buedo Fernández, baixo a miña dirección/titorización, e a utorizo a súa presentación , considerando que reúne os r equisitos esixidos no R egulamento de Estudos de Doutoramento da USC, e que como director desta non incorre nas causas de abstención establecidas na Lei 40/2015. De acordo co indicado no Regulamento de Estudos de Doutoramento, declara tamén que a presente tese de doutoramento é idónea para ser defendida en base á modalidade de Monográfica con reprodución de publicacións , nos que a participación do doutorando foi decisiva para a súa elaboración e as publicacións se axustan ao Plan de Investigación. En , de de 2021 En , de de 2021 Eduardo Liz Marzán Rosana Rodríguez López Preface/Prefacio/Prefacio [EN] This document is the Ph.D. thesis of Sebasti´an Buedo Fern´andez. It is based on a major part of the work that the author has done during the period ranging from October 2016 to June 2021 as a student of the Doctoral Programme in Mathematics, at the University of Santiago de Compostela (USC), Spain; under the supervision of Eduardo Liz Marz´an (University of Vigo) and Rosana Rodr´ıguez L´opez (USC). This thesis, presented as a monograph, is mainly written in English and gathers the contents of the articles [17, 18, 19, 20, 21, 83] of the section ‘Bibliography’. In compliance with the rules of USC, brief summaries in English (pages xiii–xxii), Galician (pages xxiii– xxxiii) and Spanish (pages xxxv–xlv) are provided and further details of the journals that published the above-mentioned articles are given at the end of the document, in the part named ‘Further information’. Finally, also in such part, there is a list of the institutions that have provided funds to the author in order to develop his project, do internships and attend conferences. [GL] Este documento ´e a tese de doutoramento de Sebasti´an Buedo Fern´andez. Est´a baseada nunha gran parte do traballo que o autor realizou durante o per´ıodo que vai desde outubro do 2016 ata xu˜no do 2021 como estudante do Programa de Doutoramento en Matem´aticas, na Universidade de Santiago de Compostela (USC), Espa˜na; baixo a direcci´on de Eduardo Liz Marz´an (Universidade de Vigo) e Rosana Rodr´ıguez L´opez (USC). Esta tese, en formato monograf´ıa, est´a principalmente escrita en ingl´es e compila os contidos dos artigos [17, 18, 19, 20, 21, 83] da secci´on “Bibliography”. En cumprimento das normativas da USC, proporci´onanse breves resumos en ingl´es (p´axinas xiii–xxii), galego (p´axinas xxiii–xxxiii) e castel´an (p´axinas xxxv–xlv) e danse m´ais detalles das revistas que publicaron os devanditos artigos ao final do documento, na parte titulada “Further information”. Por ´ultimo, tam´en nesa derradeira parte, rec´ollense as instituci´ons que financiaron ao autor para desenvolver o seu proxecto, realizar estad´ıas e asistir a congresos. [ES] Este documento es la tesis doctoral de Sebasti´an Buedo Fern´andez. Est´a basada en una gran parte del trabajo que el autor ha realizado durante el periodo que va desde octubre del 2016 hasta junio del 2021 como estudiante del Programa de Doctorado en Matem´aticas, en la Universidad de Santiago de Compostela (USC), Espa˜na; bajo la direcci´on de Eduardo Liz Marz´an (Universidad de Vigo) y Rosana Rodr´ıguez L´opez (USC). Esta tesis, en formato monograf´ıa, est´a principalmente escrita en ingl´es y compila los contenidos de los art´ıculos [17, 18, 19, 20, 21, 83] de la secci´on “Bibliography”. En cumplimiento con la normativa de la USC, se proporcionan breves res´umenes en ingl´es (p´aginas xiii–xxii), gallego (p´aginas xxiii–xxxiii) y castellano (p´aginas xxxv–xlv) y se dan m´as detalles sobre las revistas que publicaron dichos art´ıculos al final del documento, en la parte titulada “Further information”. Por ´ultimo, tambi´en en esa ´ultima parte, se enumeran las instituciones que han financiado al autor para desarrollar su proyecto, realizar estancias y asistir a congresos. Con todo mi cari˜no, dedicado a mi padre, a mi madre y a mi hermana Sebasti´ an Buedo Fern´ andez information of the item fin the triple (a, f, τ) would be ‘summarised’ in the value f0(p) and such a global asymptotic stability condition would read as f0(p)∈[−a, a).(7) It is worth highlighting that the delay τdoes not appear in condition (7), since it is obtained via (5), which is independent from the value τ. Thus, condition (7) is called a delay-independent stability condition or an absolute stability condition [124]. Afterwards, Gy˝ori and Trofimchuk [54] provided, under a negativity-type assumption on Sf, an extension of (7) by using a modified version of (5) that incorporates τas part of a coefficient, and they obtained the delay-dependent global stability condition f0(p)∈[−a, a) or f0(p)<−aand τ≤1 aln f0(p) f0(p) + a.(8) Within the quest of better global stability conditions for the case of feedbacks having a negative Schwarzian derivative, Liz et al. [86] provided an improvement of (8), which turned out to be the best possible for (2), but, in the particular case of (3), still leaves a tiny region before reaching the border of the set of parameters for which the equilibrium is locally attracting, which is given by f0(p)∈[−a, a) or  f0(p)<−aand τ < arccos a f0(p) pf0(p)2−a2 .(9) We devote some pages to summarise those efforts and to visually explain the regions of parameters (a, f0(p), τ) that satisfy the previously mentioned stability conditions (see Remark 1.64 and Figure 1.8). Having recalled the main properties of (2) (with a particular interest on (3)), we focus on the part of our project [17, 20, 83] that is devoted to the gamma-models, which, as said before, include equation (2) together with (4). In particular, we justify the use of γ∈[0,1] as a parameter that provides flexibility to many models and links several couples of different well-known equations. For instance, the two delay differential equations proposed by Mackey and Glass [96] to study a model concerning the hematopoiesis are connected under our approach. In order to study the global dynamics of the gamma-models via the tools recalled above, we provide results showing which features a general feedback of the type of (4) with γ∈[0,1] provides to (2). For instance, the existence of a unique positive equilibrium p(or a unique constant positive solution), the way in which γinfluences the value of pand crucial information about the dynamics of the delay differential equation in a neighbourhood of xvi Summary pcan be obtained through the analysis of its corresponding difference equation (Theorem 2.1). Later, we study whether the known stability conditions (7), (8) and (9) can be applied to several gamma-models that are of interest due to their applications. In the following paragraphs, we describe our main findings. The first example is based on a version of the Solow’s neoclassical model of economic growth [126], where xrepresents the capital-labour rate x, which is a quantity that relates the capital stock (e.g., machines) and the labour’s force (workers, number of hours worked, etc.). The utilisation of the delay is explained as the time required to produce a commodity, as highlighted by Matsumoto and Szidarovszky [102]. If the production of such commodity is ruled by the Cobb-Douglas function [10, 126], we have h(x) = β > 0 for the feedback (4), so equation (2) takes the form x0(t) = −ax(t) + βxγ(t−˜τ(t)).(10) In [20], we find that the unique equilibrium for (10) is globally asymptotically stable regardless of the value of the parameters a, β > 0, γ∈(0,1), and the bounded continuous delay ˜τ(t) (see Theorem 2.7). Bearing in mind the context of the previous model, the timedependent delay is harmless with respect to the eventual achievement of the steady-state capital-labour rate p. The remaining cases also find applications when it comes to the Solow’s model. They are based on the incorporation of a pollution factor in the feedback f, produced by large concentrations of capital [27, 102, 103], that is mathematically represented as a decreasing function. Hence, in the second case, we assume that h(x) = βe−δx, with β, δ > 0, in (4), so equation (2) is written as x0(t) = −ax(t) + βxγ(t−˜τ(t))e−δx(t−˜τ(t)).(11) This is referred to as the Lasota equation due to its use for modelling the production of a certain type of blood cells in a paper by Lasota [74], yet it can also be interpreted as agamma-version of the Nicholson’s blowflies equation, utilised by Gurney et al. [49] to model the amount of individuals grown in lab colonies. Regarding (11), the classical approach by Allwright and Singer does not work since the feedback fdoes not have negative Schwarzian derivative. Nevertheless, we can still apply the generalised version by El-Morshedy and Jim´enez L´opez to derive (Theorem 2.12) that the unique equilibrium of the difference equation (5) related to (11) is GAS provided it is LAS, that is, if β a≤eγ+1 γ+ 1 δ1−γ . The latter condition is the sharpest delay-independent global stability condition for (11) and it is obtained from the analysis of its corresponding difference equation, which was xvii Sebasti´ an Buedo Fern´ andez developed by Liz [81]. Moreover, if the delay is constant, such condition can be refined by using the estimates provided by Gy˝ori and Trofimchuk to obtain (see also Theorem 2.12) the delay-dependent global stability condition β a≤eγ+1 1−e−aτ γ+1 1−e−aτ δ!1−γ . Thus, we extend the study of the local dynamics of (11) by Matsumoto and Szidarovszky [103] and provide a link between the global dynamics for the cases γ= 0, provided by Gy˝ori [50], and γ= 1, described by Gy˝ori and Trofimchuk [54]. Additionally, with the support of [81], we also show that γplays a subtle role in the stability of the unique equilibrium of (11). For example, there exist cases for which increasing γmay generate a couple of stability switches (see Figure 2.1). The third one is the γ-logistic delay differential equation, related with the choice h(x) = β(1 −x), with β > 0, in (4). Hence, equation (2) is now written as x0(t) = −ax(t) + βxγ(t−˜τ(t))(1 −x(t−˜τ(t))).(12) This gamma-version of an equation with a logistic production function is a particular case of (2) that requires some preliminary remarks in order to become manageable by the techniques commented above. For instance, it turned out that one has to restrict the study to solutions with values in (0,1) and impose a consistency condition on the parameters (see (2.39)). Under such constraints, we obtain (Theorem 2.22) that the unique equilibrium is GAS provided β a≤(γ+ 1) γ+ 2 γ+ 1γ , which, in an analogous way to the situation for the Lasota equation, constitutes the sharpest delay-independent global stability condition for (12). Furthermore, if the delay is constant, then the estimates by Gy˝ori and Trofimchuk are valid and we provide (also in Theorem 2.22) a delay-dependent global stability condition that improves the last one, and it is written as β a≤γ+1 1−e−aτ  γ+1+ 1 1−e−aτ γ+1 1−e−aτ !γ . Our global stability results for γ∈[0,1] extend the study of the local dynamics of (12) by Matsumoto and Szidarovszky for γ= 1 [103]. In the fourth example, we assume that h(x) = β 1+δxm, with β, δ, m > 0, in (4). Then, one can rewrite (2) as x0(t) = −ax(t) + βxγ(t−˜τ(t)) 1 + δxm(t−˜τ(t)),(13) xviii Summary and we refer to the former equation as the γ-Mackey-Glass delay differential equation, in analogy with the couple of equations proposed by Mackey and Glass [96] to represent a model of blood cell generation. The analysis of (13) seems trickier than the former cases, since we could not find easy computations that lead to an application of the result by El-Morshedy and Jim´enez L´opez. However, in the particular case of m= 1, from the work by Liz [82] on the corresponding difference equation, where the Coppel’s criterion [24] is used, the global stability of (13) is a straightforward consequence (Theorem 2.28). Nonetheless, no proper way to adapt such reasoning to the case m6= 1 was found. Thus, a new Schwarzian-type formula was proposed with the aim of handling the property ‘LAS implies GAS’. This is done in Chapter 3, which arises as a branch from Chapter 2 at this point. Namely, if in the corresponding difference equation (5) we denote F(x) = f(x)/a = xγH(x), then our condition (see (3.3)), written in the form suggested by Jim´enez L´opez, is SH(x)<H(x)−xH0(x) 2(xH(x))2(xH(x))0,for all x > 0.(14) One of the principal strengths of formula (14) is that it does not take into account the Schwarzian derivative of the map F, but only of one of its factors, namely H, which helps simplifying the computations for several models. In fact, this formula comes from a logarithmic change of variables [78] that transforms the product xγH(x) into a sum γx +H∗(x), where H∗is written in terms of H. Additionally, whenever SH∗<0, it is possible to apply an enveloping-type result proved by Liz et al. [85] to H∗and derive, after undoing the change of variables, that the difference equation (5) related to (13) satisfies the property ‘LAS implies GAS’. In other words, we obtain (Theorem 3.1) that, under condition (14), pis GAS for (5) provided it is LAS. In particular, if F(x) = βxγ a(1+δxm)in (5), the former conclusion is obtained provided m≤1 + γor "m > 1 + γ, β a≤m m−1−γγ+ 1 δ(m−1−γ)1−γ m#.(15) Indeed, condition (15) turns out to be the sharpest delay-independent global stability condition for the equation (13) (Theorem 3.13). Furthermore, we are able to link some known estimates concerning the global dynamics of equation (13) for the particular cases of γ= 0 and γ= 1 by Gopalsamy et al. [44]. Besides, the reasoning involved in the main result of Chapter 3 (Theorem 3.1) can be slightly modified in order to provide an extension for the case γ= 0 and unimodal maps H (recall that they were originally assumed to be nonincreasing). Such result (see Theorem 3.5) is analogous to the classical ones given by Allwright and Singer, but in this case based on formula (14) instead of the inequality SH < 0. xix Sebasti´ an Buedo Fern´ andez We conclude the part related to globally attracting equilibria in gamma-models by studying the equation x0(t) = −ax(t) + s(x(t))f(x(t−˜τ(t))),(16) where f(x) = Bxγ, with B > 0, which cannot be included under the framework of (2). Such an equation also appears in the context of the Solow’s neoclassical model of economic growth by considering more general assumptions. Indeed, if the function shas certain appropriate properties coherent with its meaning (sis the so-called saving rate [10, 126]), then we are able to show (Theorem 2.34) that (16) has a globally attracting equilibrium by relating its dynamics to another difference equation, more general than the one in (5). Its proof is based on a generalisation (Theorem 2.32) of a result by Ivanov et al. [64] and, indeed, we use a simpler reasoning than that of [64] since we rely on a powerful feature of global attractivity in one-dimensional maps, namely the strong attractivity [90]. Later, in Chapter 4, we shift our focus to equation (1) in a multidimensional context. In particular, we propose the higher dimensional extension of (3) given by x0 i(t) = −xi(t) + fi(x1(t−τi1), . . . , xs(t−τis)), i = 1, . . . , s, (17) and study whether there exists a multidimensional difference equation from which we can derive information about the long-term behaviour of the solutions of (17), as it happened with the relation between (3) and (5) in the scalar case. In fact, if we denote by x·,n = (x1,n, . . . , xs,n) the general term of a sequence in Rs, then the natural candidate to be the difference equation corresponding to (17) is xi,n+1 =fi(x1,n, . . . , xs,n), i = 1, . . . , s. (18) Nonetheless, the problem is certainly subtle in higher dimensions, as shown by Liz and Ruiz-Herrera [90, 91]. In fact, if the map f= (f1, . . . , fs) in (18) has a unique fixed point p, then, assuming that pis globally attracting for fis not sufficient to ensure the analogous property for the system of delay differential equations (17) [91]. In other words, some stronger condition needs to be imposed to the multidimensional difference equation (18). The authors of [90, 91] proposed a concept, namely the strong attractor (it is recalled in Definition 4.1), which reveals the main features that turned out to be essential to prove the relation between those two equations in the scalar case s= 1. Notice that, as we remarked, this concept is also useful in the study of the scalar equation (16). The key idea is to assume the existence of certain families of nested Cartesian products of compact intervals that behave well under the application of the map fin (18). Thus, if an equilibrium of (18) is a strong attractor, which is a more restrictive notion than the global attractor if the dimension is s≥2, then the equilibrium pis globally attracting for (17) (we recall their result in Theorem 4.2). xx Summary Nevertheless, a profound look into the main reasoning shown in [90] allows us (see [18]) to extend it to a more general context. We introduce a new concept, specifically a CC-strong attractor (see Definition 4.3), which also reveals the features needed to relate the dynamics of (18) and (17) in the scalar case. Indeed, this concept turns out to be an intermediate notion between a global attractor and a strong attractor in the sense of [90], provided s≥2 (see our Remark 4.7). The main difference from the concept proposed by Liz and Ruiz-Herrera is that we ‘weaken’ the geometry of the sets involved: we do not need nested families of Cartesian product of compact intervals, it is sufficient to consider certain nested families of compact and convex sets. Therefore, the weaker assumption of (18) having a CC-strong attractor is still sufficient to prove that pis also globally attracting for (17). The latter constitutes the main result of this part of the thesis (Theorem 4.5). Furthermore, the next part corresponds to Chapter 5. Here, we consider a scalar version of (1) with ˆ fbeing linear in the second variable, but now allowing the decay coefficient aand the coefficients of ˆ fto be non-constant, but depend on t. For instance, under such assumption, one of the simplest cases is the one of a single discrete variable delay, represented by x0(t) = −a(t)x(t) + β(t)x(t−˜τ(t)),(19) with βbeing a nonnegative function. The asymptotic behaviour of equation (19) is wellknown if a, β, ˜τare constant, as it is deduced from the analysis of the characteristic equation [124]. In such a case, sufficient conditions to ensure that the origin is globally attracting are available in, e.g., [58]. Nevertheless, in the general case of (19), the techniques may change, and we could even deal with a model satisfying ˜τ(t)→ ∞ as t→ ∞, which is a case that represents a ‘long-term unboundedness’ of the delay. For example, the works by Gy˝ori and Horv´ath [52, 53] handle such kind of equations and provide sharp estimates for their solutions by working with an inequality of the type x(t)≤c+Zt σ b(s)x(s−˜τ(s)) ds. The main ingredients that they use are the so-called characteristic inequality and an estimate of Gronwall-Bellman-type. The characteristic inequality is a generalisation of the widely-known characteristic equation for the case of constant functions a, β, ˜τ(see, e.g., Theorems 2.7 and 5.1 in [53]). We show (Section 5.4) that their ideas can be adapted to equations more general than (19), such as x0(t) = −a(t)x(t) + β(t)x(ν(t)) + Zν(t) σ η(t, s)x(ν(s)) ds, (20) where ν(t) := t−˜τ(t) and ηis a nonnegative function that is nondecreasing in the first variable. By working with the equation (20), one may arrive to an integral inequality of xxi Sebasti´ an Buedo Fern´ andez the type x(t)≤c+Zt σ k(t, s)x(s−˜τ(s)) ds. (21) Similarly, the characteristic inequality needs to be generalised, as we show in [21], and the main result of this part of our work (Theorem 5.6), which also deals with Gronwall- Bellman estimates, generalises parts of the main conclusions in [52]. Additionally, it serves to prove that the origin is a globally attracting solution for some equations of the form (20). Moreover, we also extend a result by Norbury and Stuart [110] since, in the case of (21), they assume ˜τ≡0. In Chapter 6, we consider once more the scalar version of (1), but now with periodic coefficients and also allowing the solutions to have discontinuities produced by impulses at given certain times tk,k∈N. In other words, we consider the equation x0(t) = −a(t)x(t) + ˆ f(t, xt), x(t+ k)−x(tk) = Ik(x(tk)),(22) where aand ˆ fare periodic functions in t, with the same period ω > 0, the impulse functions Ik: [0,∞)→Rfollow a pattern of time length ω, and for which some other technical conditions are imposed in order to ensure that the solutions relative to nonnegative segments are nonnegative (see Section 6.2). Faria and Oliveira [40] studied the existence of positive ω-periodic solutions of (22) with finite delay and linear impulse functions. Our main result (Theorem 6.8), whose basic ingredients are a version of the Krasnoselskii Fixed Point Theorem [48, 70] and the imposition of a certain behaviour on ˆ ffor ‘close-to-zero’ and ‘sufficiently large’ solutions, yields conclusions analogous to those in [40]. In [19], we generalise their assumptions in several ways, e.g., by including more general impulse functions Ik(whose signs may change) or via the consideration of infinite delay or a wider class of feedbacks. We also provide several corollaries that are useful for applications. In the last part of Chapter 6, we apply our results to integro-differential equations with infinite or periodic distributed delay. We also establish links with several results available in the literature. As a conclusion, we provide a general overview of the results of the thesis by highlighting several lines of future work that may be of interest. xxii Resumo Esta tese c´entrase no estudo dalgunhas propiedades cualitativas das ecuaci´ons diferenciais con retardo inclu´ıdas na familia x0(t) = −a(t)x(t) + ˆ f(t, xt),(23) onde x(t) se refire ao estado dunha magnitude x(que pode ser escalar ou vectorial) no tempo te cuxa evoluci´on, determinada mediante x0(t), procuramos modelar. A notaci´on xtrepresenta formalmente a historia de x, isto ´e, os valores x(s), para certos tempos s≤t. Ademais, aeˆ fson certas funci´ons cuxo significado comentaremos deseguido. A ecuaci´on (23) foi amplamente considerada para a modelaxe de varios fen´omenos cuxo estado presente afecta ´a evoluci´on do sistema de xeito retardado. Por exemplo, foi usada en modelos de crecemento poboacionais [49, 114], de produci´on celular [7, 96, 135], de crecemento econ´omico [102, 103], etc. Para atopar m´ais detalles e referencias sobre estas e outras aplicaci´ons, recom´endanse os libros de Kolmanovskii e Myshkis [68], Kuang [72] e Smith [124]. En moitos contextos nos que a ecuaci´on (23) ´e de interese, s´o as soluci´ons x(t) de (23) que sexan non negativas te˜nen un significado real. As´ı, se asumimos que a´e unha funci´on non negativa e ˆ f´e non negativa para soluci´ons non negativas, ent´on os termos do membro dereito de (23) son co˜necidos como o termo de destruci´on −a(t)x(t), que neste caso ´e linear, e o termo de produci´on ˆ f(t, xt), que pode ser non linear e re´une aqueles efectos que infl´uen na evoluci´on do sistema de maneira retardada, como mencionamos anteriormente. Concretamente, o ´ultimo termo ´e tam´en co˜necido como o feedback retardado. A teor´ıa b´asica de existencia e unicidade de soluci´ons para as ecuaci´ons diferenciais con retardo, como a que expresamos en (23), p´odese consultar en multitude de referencias. A modo de exemplo, destacamos o libro de Smith [124] para unha introduci´on a este tema e o de Hale e Verduyn Lunel [58] para unha ollada m´ais profunda a estas ecuaci´ons. Estas referencias, entre outras, serven para dar comezo ao n´ucleo desta tese (Secci´on 1.1). Unha das principais problem´aticas das ecuaci´ons do tipo (23) ´e que non se pode obter unha descrici´on completa do comportamento a longo prazo das s´uas soluci´ons se s´o as consideramos como un problema en Rs. En particular, unha das ideas chave cara a unha xxiii Sebasti´ an Buedo Fern´ andez comprensi´on total da din´amica relativa a (23) ´e ter en conta a natureza do segmento xt. En moitos modelos, xt´e simplemente unha funci´on vectorial definida nun intervalo real compacto [−τ, 0], isto ´e, unha funci´on que compila a informaci´on da soluci´on de (23) no ´ultimo intervalo temporal de lonxitude τ; este ´e o caso das ecuaci´ons con retardo finito. Claramente, para dotar de sentido ´a ecuaci´on (23), c´ompre considerar condici´ons iniciais que incorporen aos segmentos. Por isto, en vez de considerar o problema s´o no espazo euclidiano de dimensi´on s∈N, tam´en se debe ter en conta a evoluci´on dos segmentos xt, ´e dicir, a evoluci´on do sistema nun certo espacio funcional. Deste modo, poderemos recuperar moitos feitos relevantes da teor´ıa dos sistemas din´amicos (ver Secci´on 1.2). No entanto, iso provoca que te˜namos que traballar cun espazo de dimensi´on infinita. Na primeira parte da tese (Cap´ıtulos 2 e 3), traballamos principalmente cunha familia de ecuaci´ons escalares do tipo (23) asociada a unha funci´on de deca´ıda linear constante a(t) = ae un ´unico retardo discreto, isto ´e, con ˆ f(t, xt) = f(x(t−˜τ(t))), para certas funci´ons reais f, ˜τde variable real. Baixo tales hip´oteses, a ecuaci´on (23) pode ser escrita como x0(t) = −ax(t) + f(x(t−˜τ(t))).(24) Un caso particular de (24) que adquiriu relevancia na literatura [71, 79] ´e o de retardo constante, ´e dicir, aquel no que ˜τ(t) = τ. Asumir o anterior implica que (24) toma a forma x0(t) = −ax(t) + f(x(t−τ)).(25) A familia de ecuaci´ons sobre a que recae o noso interese nesta primeira parte da tese ´e a dos chamados modelos gamma [81, 114], que son unha familia de modelos que incorporan ´a ecuaci´on (24) coa escolla f(x) = xγh(x),(26) sendo γ≥0 e h: [0,∞)→(0,∞) unha funci´on mon´otona decrecente. Alg´uns modelos ben co˜necidos p´odense inclu´ır no contexto de (24) co feedback (26) [49, 96, 135]. O significado de γdepende do modelo. Por exemplo, modula o nivel de cooperaci´on [81] no eido de crecemento de poboaci´ons, o rol que xoga o capital dispo˜nible na produci´on dunha mercador´ıa [103] no modelo neocl´asico de crecemento econ´omico, etc. O noso obxectivo ´e obter condici´ons suficientes que garantan que a ecuaci´on (24) co feedback (26) te˜na un equilibrio p∈(0,∞) que sexa globalmente atractor, o que v´en a significar, grosso modo, que todas as s´uas soluci´ons estean definidas para tempos futuros e converxan a pcando t→ ∞. Para cumprir dita meta, dedicamos algunhas p´axinas (Secci´on 1.3) a lembrar algunhas claves sobre como se pode abordar o estudo da ecuaci´on (24) sen necesariamente impo˜ner o caso particular da funci´on de produci´on (26). Por´en, por cuesti´ons de simplicidade nas explicaci´ons, fac´emolo, en primeiro lugar, para o caso de retardo constante (25) e, finalmente, explicamos que resultados se extenden directamente xxiv Resumo ao caso de (24). Ademais, tendo en conta esta meta, ´e natural impo˜ner condici´ons de existencia e unicidade de equilibrio para (25), como tam´en facemos. Os tres seguintes obxectos t´ornanse relevantes para estudar a ecuaci´on (25): o par´ametro de destruci´on a, o feedback fe o retardo τ. A relaci´on existente entre os anteriores elementos determina o comportamento a longo prazo das soluci´ons de (25). De xeito particular, a relaci´on f/a ten un papel fundamental. ´ E m´ais, o comportamento asint´otico das soluci´ons da correspondente ecuaci´on en diferenzas xn+1 =f(xn) a=: F(xn),(27) isto ´e, o comportamento a longo prazo das sucesi´ons x0, F(x0), F(F(x0)), F(F(F(x0))), . . . , (28) ten implicaci´ons directas no comportamento correspondente das soluci´ons de (25). Este feito despr´endese dos traballos de Mallet-Paret e Nussbaum [98] e de Ivanov e Sharkovsky [65]. A grandes rasgos, eles probaron que, baixo condici´ons apropiadas (ver Teorema 1.33), se (27) ten un equilibrio pglobalmente atractor, ent´on p´e tam´en un equilibrio globalmente atractor para (25). Noutras palabras, a existencia dun ptal que toda sucesi´on do tipo (28) tende a pimplica que calquera soluci´on de (25) tende a pcando t→ ∞. Este tipo de resultados son realmente cruciais para o estudo da din´amica de (25): se un ´e quen de obter condici´ons de estabilidade asint´otica global para (27), ent´on estas son tam´en v´alidas para (25). A´ında que analizar se un equilibrio ´e globalmente atractor para a ecuaci´on (27) semella m´ais doado que facelo para (25), non ´e tan trivial como, por exemplo, si ser´ıa o estudo da s´ua estabilidade local. Secas´ı, existen familias de funci´ons Fque satisfacen a propiedade chamada “LAS implica GAS” (polas s´uas siglas en ingl´es “Locally/Globally Asymptotically Stable”, ´e dicir, localmente/globalmente asint´oticamente estable). Como o seu nome indica, reflexa unha situaci´on na que, se un equilibrio ´e localmente atractor, ent´on tam´en o ´e globalmente [43]. Unha das devanditas familias, presente en moitas aplicaci´ons [79], ´e a das funci´ons que te˜nen como moito un punto cr´ıtico e derivada de Schwarz negativa (remitimos ao artigo de Jim´enez L´opez e Parre˜no [67] para unha introduci´on hist´orica deste concepto). En particular, para unha funci´on real Fde clase C3definida nun intervalo real I, a derivada de Schwarz de Fdef´ınese como SF(x) = F000(x) F0(x)−3 2F00(x) F0(x)2 , para todo x∈Ital que F0(x)6= 0. Os resultados de Allwright [2] e Singer [122] destacaron por vez primeira o papel da derivada de Schwarz negativa na ligaz´on entre as din´amicas local e global, mentres que El-Morshedy e Jim´enez L´opez [33] proporcionaron unha posterior xxv Sebasti´ an Buedo Fern´ andez as s´uas soluci´ons traballando cunha desigualdade do tipo x(t)≤c+Zt σ b(s)x(s−˜τ(s)) ds. Os ingredientes principais que eles usan son a chamada desigualdade caracter´ıstica e unha cota de tipo Gronwall-Bellman. A desigualdade caracter´ıstica ´e unha xeneralizaci´on da ecuaci´on caracter´ıstica amplamente co˜necida para o caso de funci´ons constantes a, β, ˜τ (ver, por exemplo, Teoremas 2.7 e 5.1 en [53]). Mostramos (Secci´on 5.4) que as s´uas ideas poden ser adaptadas a ecuaci´ons m´ais xerais que (41), como x0(t) = −a(t)x(t) + β(t)x(ν(t)) + Zν(t) σ η(t, s)x(ν(s)) ds, (42) onde ν(t) := t−˜τ(t) e η´e unha funci´on non negativa que ´e mon´otona crecente na primeira variable. Traballando coa ecuaci´on (42), podemos chegar a unha desigualdade integral do tipo x(t)≤c+Zt σ k(t, s)x(s−˜τ(s)) ds. (43) De xeito paralelo, a desigualdade caracter´ıstica tam´en necesita ser xeneralizada, como mostramos en [21], e o resultado principal desta parte do noso traballo (Teorema 5.6), que tam´en est´a relacionado con cotas de tipo Gronwall-Bellman, xeneraliza parte das principais conclusi´ons de [52] e s´ervenos para probar que algunhas ecuaci´ons do tipo (42) te˜nen ´a orixe como soluci´on globalmente atractora. Ademais, extendemos un resultado de Norbury e Stuart [110] xa que, en particular, en (43), asumen ˜τ≡0. No Cap´ıtulo 6, consideramos unha vez m´ais a versi´on escalar de (23), pero agora con coeficientes peri´odicos e tam´en permitindo ´as soluci´ons ter discontinuidades producidas por impulsos nuns certos instantes tk,k∈N. Noutras palabras, consideramos a ecuaci´on x0(t) = −a(t)x(t) + ˆ f(t, xt), x(t+ k)−x(tk) = Ik(x(tk)),(44) onde aeˆ fson funci´ons peri´odicas na variable tco mesmo per´ıodo ω > 0, as funci´ons de impulso Ik: [0,∞)→Rseguen un patr´on de lonxitude temporal ω, e para a que se impo˜nen outras condici´ons t´ecnicas para asegurar que as soluci´ons relativas a segmentos non negativos sexan non negativas (ver Secci´on 6.2). Faria e Oliveira [40] estudaron a existencia de soluci´ons ω-peri´odicas positivas para a ecuaci´on (44) con retardo finito e funci´ons de impulso lineares. O noso principal resultado (Teorema 6.8), cuxos ingredientes b´asicos son unha versi´on do Teorema de Punto Fixo de Krasnoselskii [48, 70] e a imposici´on dun xxxii Resumo certo comportamento de ˆ fpara soluci´ons cercanas a cero ou suficientemente grandes, proporciona conclusi´ons an´alogas ´as de [40]. Por´en, xeneralizamos (ver [19]) as s´uas hip´oteses de varias maneiras como, por exemplo, inclu´ındo funci´ons de impulso Ikm´ais xerais (permitimos que os seus signos poidan variar) ou considerando retardo infinito ou unha clase m´ais ampla de feedbacks retardados. Tam´en proporcionamos alg´uns corolarios que son ´utiles para algunhas aplicaci´ons. Na parte final do Cap´ıtulo 6, aplicamos os nosos resultados a ecuaci´ons integro-diferenciais con retardo distribu´ıdo infinito ou peri´odico. Tam´en establecemos ligaz´ons con alg´uns dos resultados dispo˜nibles na literatura. A modo de conclusi´on deste traballo, ofrecemos unha visi´on xeral dos resultados da tese destacando algunhas li˜nas de traballo futuro que poden ser de interese. xxxiii Resumen Esta tesis se centra en el estudio de algunas propiedades cualitativas de las ecuaciones diferenciales con retardo incluidas en la familia x0(t) = −a(t)x(t) + ˆ f(t, xt),(45) donde x(t) denota el estado de una magnitud x(que puede ser escalar o vectorial) en el instante ty cuya evoluci´on, determinada mediante x0(t), buscamos modelizar. La notaci´on xtrepresenta formalmente la historia de x, esto es, los valores x(s), para ciertos tiempos s≤t. Adem´as, ayˆ fson ciertas funciones cuyo significado comentaremos a continuaci´on. La ecuaci´on (45) ha sido ampliamente utilizada para la modelizaci´on de varios fen´omenos cuyo estado afecta a la evoluci´on del sistema de manera retardada. Por ejemplo, ha sido empleada en modelos de crecimiento poblacional [49, 114], de producci´on celular [7, 96, 135], de crecimiento econ´omico [102, 103], etc. Para encontrar m´as detalles y referencias sobre estas y otras aplicaciones, se recomiendan tambi´en los libros de Kolmanovskii y Myshkis [68], Kuang [72] y Smith [124]. En muchos contextos en los que la ecuaci´on (45) es de inter´es, solo las soluciones x(t) de (45) que sean no negativas poseen un significado real. As´ı, asumiendo que aes una funci´on no negativa y ˆ fes no negativa para soluciones no negativas, entonces los t´erminos del miembro derecho de (45) son conocidos como el t´ermino de destrucci´on −a(t)x(t), que en este caso es lineal, y el t´ermino de producci´on ˆ f(t, xt), que puede ser no lineal y re´une aquellos efectos que influyen en la evoluci´on del sistema de manera retardada, como mencionamos anteriormente. Concretamente, el ´ultimo t´ermino es tambi´en conocido como el feedback retardado. La teor´ıa b´asica de existencia y unicidad de soluciones para las ecuaciones diferenciales con retardo, como la que expresamos en (45), puede consultarse en multitud de referencias. A modo de ejemplo, destacamos el libro de Smith [124] para una introducci´on a este tema y el de Hale y Verduyn Lunel [58] para una mirada m´as profunda a estas ecuaciones. Estas referencias, entre otras, sirven para dar comienzo al n´ucleo de esta tesis (Secci´on 1.1). Una de las principales problem´aticas que presentan las ecuaciones del tipo (45) es que no se puede obtener una descripci´on completa del comportamiento a largo plazo de xxxv Sebasti´ an Buedo Fern´ andez sus soluciones si solo las consideramos como un problema en Rs. En particular, una de las ideas clave hacia una comprensi´on total de la din´amica relativa a (45) es tener en cuenta la naturaleza del segmento xt. En muchos modelos, xtes simplemente una funci´on vectorial definida en un intervalo real compacto [−τ, 0], esto es, una funci´on que compila la informaci´on de la soluci´on de (45) en el ´ultimo intervalo temporal de longitud τ; este es el caso de retardo finito. Claramente, para dotar de sentido a la ecuaci´on (45), uno tiene que considerar condiciones iniciales que incorporen a los segmentos. De ah´ı que, en vez de considerar el problema solamente en el espacio eucl´ıdeo de dimensi´on s∈N, tambi´en se debe tener en cuenta la evoluci´on de los segmentos xt, es decir, la evoluci´on del sistema en un cierto espacio funcional. De este modo, podremos recuperar muchos hechos relevantes de la teor´ıa de los sistemas din´amicos (ver Secci´on 1.2). Sin embargo, eso provoca que tengamos que trabajar con un espacio de dimensi´on infinita. En la primera parte de la tesis (Cap´ıtulos 2 y 3), trabajamos principalmente con una familia de ecuaciones escalares del tipo (45) con una funci´on de decaimiento lineal constante a(t) = ay un ´unico retardo discreto, esto es, con ˆ f(t, xt) = f(x(t−˜τ(t))) para ciertas funciones reales f, ˜τde variable real. Bajo tales hip´otesis, la ecuaci´on (45) puede ser escrita como x0(t) = −ax(t) + f(x(t−˜τ(t))).(46) Un caso particular de (46) que ha adquirido relevancia en la literatura [71, 79] es el de retardo constante, es decir, aquel en el que ˜τ(t) = τ. Asumir lo anterior implica que (46) toma la forma x0(t) = −ax(t) + f(x(t−τ)).(47) La familia de ecuaciones sobre la que recae nuestro inter´es en esta primera parte de la tesis es la de los llamados modelos gamma [81, 114], que son una familia de modelos que incorporan a la ecuaci´on (46) con la elecci´on f(x) = xγh(x),(48) siendo γ≥0 y h: [0,∞)→(0,∞) una funci´on mon´otona decreciente. Algunos modelos bastante conocidos pueden ser incluidos en el contexto de (46) con el feedback (48) [49, 96, 135]. El significado de γdepende del modelo. Por ejemplo, modula el nivel de cooperaci´on [81] en el campo de crecimiento de poblaciones, el rol que juega el capital disponible en la producci´on de una mercanc´ıa [103] en el modelo neocl´asico de crecimiento econ´omico, etc. Nuestro objetivo es obtener condiciones suficientes que aseguren que la ecuaci´on (46) con el feedback (48) tenga un equilibrio p∈(0,∞) que sea un atractor global, lo que significa que, grosso modo, todas sus soluciones est´an definidas para tiempos futuros y convergen a pcuando t→ ∞. Para cumplir dicho objetivo, dedicamos algunas p´aginas (Secci´on 1.3) a recordar algunas claves sobre c´omo se puede abordar el estudio de la xxxvi Resumen ecuaci´on (46) sin imponer necesariamente el caso particular de la funci´on de producci´on (48). No obstante, por cuestiones de simplicidad en las explicaciones, lo hacemos en un primer lugar para el caso de retardo constante (47) y, finalmente, explicamos qu´e resultados se extienden directamente al caso de (46). Adem´as, teniendo en cuenta esta meta, es natural imponer condiciones de existencia y unicidad de equilibrio para (47), como tambi´en hacemos. Los tres siguientes objetos se convierten en elementos relevantes para estudiar la ecuaci´on (47): el par´ametro de destrucci´on a, el feedback fy el retardo τ. La relaci´on existente entre los anteriores determina el comportamiento a largo plazo de las soluciones de (47). De modo particular, la relaci´on f/a tiene un papel fundamental. Es m´as, el comportamiento asint´otico de las soluciones de la correspondiente ecuaci´on en diferencias xn+1 =f(xn) a=: F(xn),(49) esto es, el comportamiento a largo plazo de las sucesiones x0, F(x0), F(F(x0)), F(F(F(x0))), . . . , (50) tiene implicaciones directas en el correspondiente comportamiento de las soluciones de (47). Este hecho se desprende de los trabajos de Mallet-Paret y Nussbaum [98] y de Ivanov y Sharkovsky [65]. A grandes rasgos, ellos probaron que, bajo condiciones apropiadas (ver Teorema 1.33), si (49) tiene un equilibrio pglobalmente atractor, entonces pes tambi´en un equilibrio globalmente atractor para (47). En otras palabras, la existencia de un ptal que toda sucesi´on del tipo (50) tiende a pimplica que cualquier soluci´on de (47) tiende a p cuando t→ ∞. Este tipo de resultados es realmente crucial para el estudio de la din´amica de (47): si uno es capaz de obtener condiciones de estabilidad asint´otica global para (49), entonces estas son tambi´en v´alidas para (47). Aunque analizar si un equilibrio es globalmente atractor para la ecuaci´on (49) parece m´as f´acil que hacerlo para (47), no es tan trivial como, por ejemplo, s´ı ser´ıa el estudio de su estabilidad local. A´un as´ı, existen familias de funciones Fque satisfacen la propiedad llamada “LAS implica GAS” (por sus siglas en ingl´es “Locally/Globally Asymptotically Stable”, es decir, localmente/globalmente asint´oticamente estable). Como su nombre indica, refleja la situaci´on en la que se cumple que, si un equilibrio es localmente atractor, entonces tambi´en lo es globalmente [43]. Una de dichas familias, presente en muchas aplicaciones [79], es la de las funciones que tienen a lo sumo un punto cr´ıtico y derivada de Schwarz negativa (remitimos al art´ıculo de Jim´enez L´opez y Parre˜no [67] para una introducci´on hist´orica de este concepto). En particular, para una funci´on real Fde clase C3 definida en un intervalo real I, la derivada de Schwarz de Fse define como SF(x) = F000(x) F0(x)−3 2F00(x) F0(x)2 , xxxvii Sebasti´ an Buedo Fern´ andez para todo x∈Ital que F0(x)6= 0. Los resultados de Allwright [2] y Singer [122] destacaron en un primer momento el papel de la derivada de Schwarz en la conexi´on entre las din´amicas local y global, mientras que El-Morshedy y Jim´enez L´opez [33] proporcionaron una posterior generalizaci´on (la recordamos en el Teorema 1.55), donde ellos asumen la negatividad de SF solo en un cierto subconjunto de su dominio. En dichos contextos, si una funci´on regular Fque tiene como mucho un punto cr´ıtico tiene tambi´en un ´unico punto fijo ptal que F0(p)∈[−1,1), entonces pes un atractor global para (49). Adem´as, si tenemos en cuenta la notaci´on de (47) y (49), la informaci´on del objeto fen la terna (a, f, τ) estar´ıa “resumida” en el valor f0(p) y la anterior condici´on de estabilidad asint´otica global tomar´ıa la forma f0(p)∈[−a, a).(51) Conviene remarcar que el retardo τno aparece en la condici´on (51) ya que se obtiene v´ıa (49), que es independiente del valor de τ. La condici´on (51) es, por tanto, una condici´on de estabilidad independiente del retardo o una condici´on de estabilidad absoluta [124]. Unos a˜nos m´as tarde, Gy˝ori y Trofimchuk [54] publicaron, para condiciones de negatividad de Sf, una extensi´on de (51) a partir de una modificaci´on de la ecuaci´on en diferencias (49) que inclu´ıa a τcomo parte de un coeficiente y obtuvieron la condici´on de estabilidad dependiente del retardo f0(p)∈[−a, a) o f0(p)<−ayτ≤1 aln f0(p) f0(p) + a.(52) En la b´usqueda de condiciones m´as generales de estabilidad global para el caso de feedbacks con derivada de Schwarz negativa, Liz et al. [86] proporcionaron una mejora de (52), que result´o ser la ´optima para (46), pero que, en el caso particular de (47), a´un dejaba un peque˜no margen antes de llegar a la frontera de la regi´on de par´ametros para los que el ´unico equilibrio de (47) es localmente atractor, siendo esta ´ultima f0(p)∈[−a, a) o  f0(p)<−ayτ < arccos a f0(p) pf0(p)2−a2 .(53) Dedicamos algunas p´aginas a resumir estos esfuerzos y a explicar de un modo visual las regiones de par´ametros (a, f0(p), τ) que satisfacen las condiciones de estabilidad previamente mencionadas (ver la Observaci´on 1.64 y la Figura 1.8). Despu´es de recordar las principales propiedades de la ecuaci´on (46) (con un inter´es particular en (47)), nos centramos en la parte del proyecto [17, 20, 83] que est´a dedicada a los modelos gamma, que, como dijimos antes, incluyen a la ecuaci´on (46) con (48). En particular, justificamos el uso de γ∈[0,1] como un par´ametro que proporciona flexibilidad xxxviii Resumen a muchos modelos y conecta parejas de ecuaciones que han sido ampliamente estudiadas. Por ejemplo, las dos ecuaciones diferenciales con retardo propuestas por Mackey y Glass [96] para estudiar un modelo de hematopoyesis est´an conectadas en nuestro enfoque. Para estudiar la din´amica global de los modelos gamma con las herramientas que destacamos anteriormente, proporcionamos resultados que muestran qu´e propiedades le aporta a la ecuaci´on (46) un feedback general del tipo (48) con γ∈[0,1]. A modo ilustrativo, la existencia de un ´unico equilibrio positivo p(o una ´unica soluci´on positiva constante), el modo en el que γinfluye en el valor de pe informaci´on crucial de la din´amica de la ecuaci´on diferencial con retardo en las proximidades de ppueden ser obtenidas a trav´es del an´alisis de su correspondiente ecuaci´on en diferencias (Teorema 2.1). A continuaci´on, estudiamos si las condiciones de estabilidad (51), (52) y (53) pueden ser aplicadas a varios modelos gamma que son de inter´es en las aplicaciones que mostramos. En los siguientes p´arrafos, describimos nuestras aportaciones. El primer ejemplo est´a basado en una versi´on del modelo neocl´asico de crecimiento econ´omico de Solow [126], donde xrepresenta el cociente entre el capital disponible y la mano de obra que, grosso modo, es una tasa que relaciona los bienes materiales (como m´aquinas) y la fuerza trabajadora (trabajadores, n´umero de horas trabajadas, etc.). El uso del retardo se explica por el tiempo requerido en la producci´on de un cierto bien, como destacaron Matsumoto y Szidarovszky [102]. Si la producci´on de ese bien viene dada por la funci´on de Cobb-Douglas [10, 126] tenemos que h(x) = β > 0 en el feedback (48), con lo que la ecuaci´on (46) adquiere la forma x0(t) = −ax(t) + βxγ(t−˜τ(t)).(54) En [20], probamos que el ´unico equilibrio de (54) es globalmente atractor para cualquier valor de los par´ametros a, β > 0, γ∈(0,1) y del retardo continuo y acotado ˜τ(t) (ver Teorema 2.7). Si tenemos en mente el contexto del modelo anterior, esto significa que considerar un retardo como el anterior en la producci´on de una mercanc´ıa [102] no evita que finalmente se alcance un equilibrio ppara la tasa mencionada. El resto de casos que tratamos tambi´en tienen aplicaciones en relaci´on con el modelo de Solow. Se basan en la incorporaci´on de un factor de contaminaci´on en el feedback f, provocado por grandes concentraciones de capital [27, 102, 103] y que matem´aticamente est´a representado por una funci´on estrictamente decreciente. As´ı, en el segundo caso, asumimos [20] que h(x) = βe−δx, siendo β, δ > 0, en (48), con lo que la ecuaci´on (46) se escribe como x0(t) = −ax(t) + βxγ(t−˜τ(t))e−δx(t−˜τ(t)).(55) Esta se conoce como la ecuaci´on de Lasota [74] por su uso para modelar la producci´on de un tipo fijado de c´elulas sangu´ıneas (por ejemplo, gl´obulos rojos), aunque tambi´en puede verse como una versi´on gamma de la ecuaci´on de las moscas de Nicholson, utilizada para xxxix Sebasti´ an Buedo Fern´ andez determinar la cantidad de individuos en colonias de insectos criados en laboratorio, seg´un el trabajo de Gurney et al. [49]. Para la ecuaci´on (55), el enfoque cl´asico de Allwright y Singer no funciona, ya que f no tiene derivada de Schwarz negativa. Sin embargo, s´ı se puede aplicar la generalizaci´on proporcionada por El-Morshedy y Jim´enez L´opez para concluir (Teorema 2.12) que el ´unico equilibrio de la ecuaci´on en diferencias (49) relativa a (55) es GAS siempre que este sea LAS, es decir, si β a≤eγ+1 γ+ 1 δ1−γ . Esta es la condici´on de estabilidad global independiente del retardo m´as fina que se puede dar para (55) y se puede obtener a partir del an´alisis de su correspondiente ecuaci´on en diferencias, el cual fue desarrollado por Liz [81]. Adicionalmente, si el retardo es constante, la condici´on anterior se puede mejorar utilizando las estimaciones de Gy˝ori y Trofimchuk para obtener (ver tambi´en Teorema 2.12) la condici´on de estabilidad global dependiente del retardo β a≤eγ+1 1−e−aτ γ+1 1−e−aτ δ!1−γ . Con esto extendemos el estudio de Matsumoto y Szidarovszky [103] sobre la din´amica local para la ecuaci´on (55) y conectamos el estudio de la din´amica global de los casos γ= 0, analizado por Gy˝ori [50], y γ= 1, descrito por Gy˝ori y Trofimchuk [54]. Adem´as, apoy´andonos en [81], mostramos que el papel que juega γen la estabilidad del ´unico equilibrio de (55) es sutil. Por ejemplo, existen casos para los que incrementar γpuede generar un par de cambios en la estabilidad (ver Figura 2.1). El tercer caso es la ecuaci´on diferencial γ-log´ıstica con retardo, relacionada con la elecci´on h(x) = β(1 −x), con β > 0, en (48) [17]. Con esta informaci´on, la ecuaci´on (46) puede ser escrita como x0(t) = −ax(t) + βxγ(t−˜τ(t))(1 −x(t−˜τ(t))).(56) Esta versi´on gamma de una ecuaci´on con funci´on de producci´on log´ıstica es un caso particular de (46) que requiere de algunas observaciones previas para poder ser tratada con las t´ecnicas mencionadas m´as arriba. En particular, necesitamos restringir el estudio a las soluciones con valores en (0,1) e imponer una condici´on de consistencia en los par´ametros (ver (2.39)). Bajo tales restricciones, obtenemos (Teorema 2.22) que el ´unico equilibrio es GAS si β a≤(γ+ 1) γ+ 2 γ+ 1γ , lo cual, de manera an´aloga a la situaci´on de la ecuaci´on de Lasota, constituye la condici´on de estabilidad independiente del retardo m´as fina que se puede dar para (56). Es m´as, xl Resumen si el retardo es constante, las estimaciones de Gy˝ori y Trofimchuk tambi´en son v´alidas y proporcionamos (tambi´en en el Teorema 2.22) una condici´on de estabilidad dependiente del retardo que mejora a la anterior y se escribe como β a≤γ+1 1−e−aτ  γ+1+ 1 1−e−aτ γ+1 1−e−aτ !γ . Nuestros resultados de estabilidad global para γ∈[0,1] extienden el estudio de Matsumoto y Szidarovszky [103] sobre la din´amica local de la ecuaci´on (56) para el caso γ= 1 . En el cuarto ejemplo, suponemos que h(x) = β 1+δxm, con β, δ, m > 0, en (48). Entonces, reescribimos (46) como x0(t) = −ax(t) + βxγ(t−˜τ(t)) 1 + δxm(t−˜τ(t)) (57) y nos referimos a esta ecuaci´on como la versi´on gamma de la ecuaci´on diferencial con retardo de Mackey-Glass, en analog´ıa con las dos ecuaciones propuestas por Mackey y Glass [96], relacionadas con los casos γ= 0 y γ= 1 y utilizadas para representar un modelo de generaci´on de c´elulas sangu´ıneas. El an´alisis de (57) es m´as delicado que el de los casos anteriores, ya que no conseguimos encontrar c´alculos sencillos que nos llevasen a una aplicaci´on del resultado de El-Morshedy y Jim´enez L´opez. De todos modos, en el caso particular m= 1 y a partir del trabajo de Liz [82] sobre la correspondiente ecuaci´on en diferencias, donde se usa el criterio de Coppel [24], s´ı que se puede obtener de forma directa, como mostramos en [20], la estabilidad global del ´unico equilibrio de (57) (Teorema 2.28). No obstante, no encontramos ning´un modo sencillo de adaptar los razonamientos para el caso m6= 1. Con todo esto, se propone una nueva f´ormula que involucra, una vez m´as, a la derivada de Schwarz para poder disponer de la propiedad “LAS implica GAS”. Esto se hace en el Cap´ıtulo 3, que surge como una ramificaci´on del Cap´ıtulo 2 en este punto. Concretamente, si en la correspondiente ecuaci´on en diferencias (49) denotamos F(x) = f(x)/a =xγH(x), entonces la condici´on que proporcionamos (ver (3.3)), escrita en la forma sugerida por Jim´enez L´opez, es SH(x)<H(x)−xH0(x) 2(xH(x))2(xH(x))0,para todo x > 0.(58) Una de las principales fortalezas de la expresi´on (58) es que no tiene en cuenta la derivada de Schwarz de la aplicaci´on F, sino solamente la de uno de sus factores, H, lo que ayuda a simplificar los c´alculos en muchos modelos. De hecho, esta f´ormula procede de un cambio de variables logar´ıtmico [78] que transforma el producto xγH(x) en una suma γx +H∗(x), donde H∗se escribe en funci´on de H. En aquellos casos en los que SH∗<0, se puede xli Sebasti´ an Buedo Fern´ andez 4 The role of multidimensional difference equations 109 4.1 Introduction................................... 109 4.2 A key concept regarding global attractivity . . . . . . . . . . . . . . . . . . 111 4.3 Some results concerning convex sets . . . . . . . . . . . . . . . . . . . . . . 119 4.4 ProofofTheorem4.5.............................. 121 4.5 Ashortdiscussion................................ 125 5 Gronwall-Bellman estimates for delayed inequalities of Volterra-type 127 5.1 Introduction................................... 127 5.2 Preliminaries .................................. 130 5.3 Volterra-type inequalities with delay dependence . . . . . . . . . . . . . . . 132 5.4 An application to functional differential equations . . . . . . . . . . . . . . 144 5.5 Conclusion.................................... 148 6 Periodic solutions for scalar equations with impulses and infinite delay 149 6.1 Introduction................................... 150 6.2 An abstract framework and preliminary results . . . . . . . . . . . . . . . . 152 6.3 Existence of positive periodic solutions . . . . . . . . . . . . . . . . . . . . 162 6.4 Applications to delayed integro-differential equations of Volterra-type . . . 174 6.4.1 Integro-differential equations with infinite distributed delay . . . . . 177 6.4.2 Integro-differential equations with periodic distributed delay . . . . 182 6.4.3 Effect of the impulses . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.5 Finalcomments................................. 186 7 General discussion, conclusions and future work 189 Bibliography 193 Notation 207 Index 210 xlviii Introduction The issue of mathematical modelling has acquired a huge relevance in the scientific community in the recent history. Mathematics as a whole has served as a great source for the understanding of many natural and social phenomena. In particular, differential equations have become a powerful tool to analyse how some of those phenomena evolve as time goes by. For instance, ordinary differential equations have been used to explain classical problems in physical motion, and some types of partial differential equations are used in several phenomena with spatial-diffusive properties and can be interpreted as ordinary differential equations in Banach spaces. The latter equations are written in terms of a certain rule that yields the current evolution of the phenomenon in terms of its current state. The form of such rule depends on the deep nature of the phenomena they are attempting to model. Nonetheless, there exist cases when the evolution of a certain phenomenon is not fully explained by its current state. For instance, imagine that we are interested in modelling how many adult individuals of a certain animal species live simultaneously. As the generation of new mature individuals from the adult ones usually requires a certain amount of time, the evolution of such group does not depend on the current number of adult individuals: there is a natural ‘lag’ in the effect they cause on the evolution of their group. Something similar happens with some processes inside, e.g., the human body. For instance, the generation of specific cells, which may take a relevant amount of time, can be a natural response to the current lack of such type of cells. Nevertheless, this does not restrict to biological or ecological situations. For example, one can think about a machine as an instrument to control physical variables, e.g., the temperature, if such magnitude has exceeded a certain value in the last hour. Even within social contexts one may find this kind of examples. For instance, making decisions that attempt to affect the wealth of a country, region, community, etc. does not produce instantaneous outputs. Following this line, if x(t) stands for the value of a certain magnitude at time t, then we are interested in studying the evolution of xas time goes by if the model fits within xlix Sebasti´ an Buedo Fern´ andez the expression of the differential equation x0(t) = f(t, xt),(67) where xtis a common notation to denote the history of x(roughly speaking, xtgathers the information relative to x(s), s≤t), and frepresents the rule, which may be time-varying, that tells how the magnitude evolves according to its history. Equation (67) is a delay differential equation. The term delay represents the length of the time interval for which the history of the function xneeds to be known in order for the equation to make sense. A nice book for anyone interested in starting studying delay differential equations is the one by Smith [124]. For readers that would like to delve into the theory of this type of equations, the classical book by Hale and Verduyn Lunel [58] develops it in detail, and it contains plenty of supplementary remarks and references for the different particular topics that had been arising in the past decades. Furthermore, the book by Diekmann et al. [31] is also a great source with a thorough theoretical basis, where many specific underlying ideas of these equations are developed. Moreover, the book by Kolmanovskii and Myshkis [68] is another possible option to consult results and applications to modelling phenomena. In the latter line, the book by Kuang [72] is also well-known for its emphasis in the analysis of equations that are used in population models. Regarding the current research on this field, we refer to the recent survey by Walther [133], where many active topics are shown. It is quite a natural issue to aim to obtain information about the long-term behaviour of the magnitude being modelled via the equation (67). Obviously, such task may be simplified if one is able to compute the explicit solutions of the differential equation. Nevertheless, in general, it is not possible to obtain explicit expressions for its solutions, that situation is, clearly, an excepcional case. This is an issue that also arises when working with ordinary differential equations in Rn. Nonetheless, it is here where the qualitative theory of differential equations naturally comes into consideration. In fact, when those explicit expressions are not available, one can still try to derive some relevant qualitative information with respect to the asymptotic behaviour of the solutions of the equation. For example, one might be interested to know whether the phenomenon modelled by such equation evolves towards a single state as time goes by, or even if a periodic pattern arises. This is done just by studying the properties of the rule fin (67), in an analogous way to what is done in the case of n-dimensional ordinary differential equations. However, one of the main difficulties of the qualitative analysis of delay differential equations is that the deep nature of their dynamics is, generally speaking, infinitedimensional, that is, the phase space X, the set where a complete picture of how the magnitude modelled via (67) evolves, lies in a functional Banach space. This is in fact related to the necessity of having a continuum of data about the history of the solution, in l Introduction other words, the values of the solution at an infinite number of past times. For instance, classical choices for this phase space are certain sets of continuous functions with values in Rn. Moreover, several scenarios that do not hold for ordinary differential equations in Rnmight take place for delay differential equations. For instance, even in the context of existence and uniqueness of solutions, two different solutions can ‘merge’ at a certain ‘future’ time [58, Chapter 2], which means that the semiflow may not be injective. Thus, there may not be a unique backward solution, as in the case of difference equations. Hence, if one is interested in a topological description of the long-term behaviour of the solution of a delay differential equation, an appropriate framework is to consider dynamics in metric spaces [115, 125]. Sometimes, they are also assumed to be complete [56], and some powerful particular results are obtained. When it comes to autonomous dynamics, i.e., when the rule of evolution fin (67) is not time-varying, then the behaviour may be explained via the well-known concept of the semiflow map. Otherwise, other more general concepts, like processes, need to be considered for the non-autonomous case [58]. Within this context, describing the so-called global attractor of bounded sets provides a complete picture of the long-term dynamics. This concept has been widely considered in the literature regarding the qualitative analysis of delay differential equations (see, e.g., [56, 71, 115, 125] and the references therein). However, we will not dwell into further details concerning this notion, since it exceeds by far the scope of the current work. We refer to [125, Chapter 2] for its theoretical background and to [7] for one of its applications. In spite of the former consideration, we can still say that the long-term dynamics of an equation like (67) are ‘simple’ if we ensure the existence of p∈Rnsuch that any solution x(t) of (67) satisfies lim t→∞x(t) = p, (68) for a subset of admissible initial conditions that is as large as possible. Condition (68) can be rewritten, in terms of the announced underlying functional phase space X, as the existence of a global attractor of points, as we will see later. This will be our main aim. Moreover, there exist phenomena that are modelled under equation (67) which depend on external factors that are themselves periodic; for instance, bear in mind the seasonal effects on population growth. This would mean that the rule fin (67) purely depends on the first variable, and, in fact, the rule is repeated in consecutive time intervals of a certain fixed length. Thus, it is also natural to wonder whether the magnitude whose evolution is modelled via equation (67) inherits such periodic pattern; in other words, whether there exists a periodic solution of (67). The quest for the existence of such solutions will also be one of our goals. Nevertheless, we will not seek sufficient conditions to ensure the existence of globally attracting equilibria or periodic solutions for the general equation (67). Roughly speaking, this manuscript is focused on the particular family of delay differential equations of the li Sebasti´ an Buedo Fern´ andez type of (67) that are written in the form x0(t) = −d(t, xt) + p(t, xt),(69) where pand dare nonnegative functions (on each coordinate, in case it is a multidimensional equation), respectively representing what are known as the production and destruction terms. The consideration of equation (69) comes from the work by an der Heiden and Mackey [5] and represents an appropriate framework to study many biological or economic models. In particular, positive solutions are usually the center of attention. In the case of (69), the qualitative study is related to the analysis of the interplay between the destruction and production terms, since the rule fin (67) is written as f=p−d. Since the destruction term will be generally linear and ‘non-delayed’, it is the expression of the production term that acquires a major relevance, as we will see. Aims In particular, we summarise our goals below: •To find sufficient conditions for the existence of globally attracting equilibria for equations of the form (69) by: –analysing to what extent the results available in the literature concerning global attractivity for difference equations are useful to study the role of the production function pin the attractivity properties of equation (69); –improving some known theoretical results regarding difference equations and their relation to (69), with the aim of studying the qualitative properties of broader classes of equations of the form (69); –extending the techniques concerning linear integral inequalities to obtain global attractivity results for (69) in case pand dare linear but the delay is not bounded. •To provide sufficient conditions for the existence of positive periodic solutions when the functions pand din (69) are periodic in the first variable: –in cases when the whole history of xis considered; –in cases when the solutions are allowed to have discontinuities at certain given times, corresponding to abrupt changes in the phenomena being modelled. lii Introduction Organisation To write this manuscript, as highlighted in the preface, the works [17, 18, 19, 20, 21, 83] have been considered and adapted to share a common notation and framework. Some remarks and results have been added too. The organisation of this manuscript is as follows. In Chapter 1, the basics of delay differential equations are presented. The author has decided to put a particular emphasis on its development, with the help of his two supervisors, aiming to bring together the main concepts and several basic results of this theory (Section 1.1) and to provide some detailed explanations that could potentially facilitate the reading in subsequent chapters. Although proofs are generally omitted in this chapter, some guidelines, intuitive ideas and references to consult are given to help following the main path of this work. Nevertheless, no portion of this chapter is intended to be something new nor considered as a result of the research of the author; it is just his particular view. Despite of the fact that the main goal of the present work is to obtain results concerning delay differential equations like the one in (69), one of the main mathematical tools that we will use are difference equations. Thus, there is also a relevant part of this work devoted to such equations. In order to include them in our study, in Section 1.2, a general exposition on the basics of semiflows is given (see the references cited therein) in order to save some pages due to the analogy between the theories with discrete and continuous-time. Afterwards, in Section 1.3, special attention is given to scalar delay differential equations with a linear decay and a delayed feedback function (which might be nonlinear) terms. Those equations find applications in many models [5, Table 2.1]. In particular, the equation x0(t) = −ax(t) + f∗(x(t−τ)),(70) where a, τ > 0, deserves a particular attention. Such equation has been called the Mackey- Glass-type differential equation [88] due to the particular equations that appeared in the very influential paper by Mackey and Glass [96] in the context of blood cells production. Many results regarding the latter equation are available if it has a unique equilibrium. For instance, the surveys by Liz [79] and Krisztin [71] serve as a great source that sum up the efforts that have been made with respect to that equation for the case when the equilibrium is stable or unstable, respectively. We are maily interested in global asymptotic stability conditions, so we are especially focusing on the cases related to [79]. In fact, some of those results are still true if one replaces τby a certain nonnegative function ˜τ(t), as we will see. In fact, the role of f∗as a map turns out to be highly relevant. The information that comes from the iterates of f∗somehow dominates the asymptotic behaviour of the solutions of the delay differential equation (70), as it will be shown in Subsection 1.3.1. liii Sebasti´ an Buedo Fern´ andez In fact, we will recall the known conditions to guarantee the local stability of the unique equilibrium of the differential equation in Subsection 1.3.2. In some cases, global asymptotic stability type results can be obtained through local information on the equilibrium. This is the case when the map f∗satisfies certain geometrical hypotheses. For instance, the Schwarzian derivative of a function will be one of our allies. It is defined by Sf∗(x) = (f∗)000(x) (f∗)0(x)−3 2(f∗)00(x) (f∗)0(x)2 , if xis such that (f∗)0(x)6= 0. When f∗is monotone or unimodal and Sf∗<0 on a certain domain, then that local-global link becomes a realm and the study is facilitated. Such an expression comes from the work by H. Schwarz on conformal maps and M¨obius transformations and some decades ago was found to have deep dynamical implications in the works by Allwright [2] and Singer [122]. We refer to [67, Section 1] for a great and thorough introduction of the Schwarzian derivative in terms of its historical background and applications to dynamical systems. One may wonder whether imposing that f∗has at most one point of extremum or Sf∗<0 are actually strong constraints for equation (70). However, the consideration of such ‘humped’ production functions is quite common [5, 133] and, as highlighted in [79], many models that have been widely considered also satisfy that hypothesis on the Schwarzian derivative. Thus, we have devoted a great part of Section 1.3.3 to show the main dynamical features of those maps. We also recall some results that have appeared in the literature in the last decades and recall some interesting open problems. In Chapter 2, we take advantage of a significant part of the machinery that has been introduced in Chapter 1 to obtain results regarding a family of scalar models that are called gamma-models. They have been applied to certain economic or population growth models (see the references cited therein) and their name gives great importance to a certain parameter γappearing in the production term, since it serves to connect several existing models. We provide a unified treatment for their study and give some results concerning the existence of globally attracting equilibria by using scalar difference equations as the main tool. Since some difficulties arise when trying to apply known conditions to certain particular cases, Chapter 3 is devoted to provide new theoretical results that help fixing such issues. The development of the latter couple of chapters corresponds to the articles [17, 20, 83]. Chapter 4 is devoted to analysing to what extent difference equations are still useful to study global attractivity of delay differential equations like (70), but in Rnwith n≥2. This is the unique chapter in which we consider systems of delay differential equations and it is developed from the contents in [18]. liv Introduction The four articles corresponding to Chapters 2 to 4 are related with cases of bounded finite delay and correspond to the work that the author has done under the supervision of Prof. Eduardo Liz: a couple of them being co-authored and a couple of them being single-authored. In Chapter 5, we present several results regarding inequalities of the Gronwall-Bellman- type that aim to find sharper bounds for the solutions of certain types of equations belonging to the family in (69) with a non-delayed linear destruction term. In this chapter, the production term is assumed to be a linear function too, but with the difference that we get rid of the long-term boundedness of the delay and consider what may be called a distributed dependence on past states. The contents of this chapter are based on that of [21] and it is the work that the author has developed with his other supervisor, Prof. Rosana Rodr´ıguez-L´opez. The last part of the core of this work, Chapter 6, represents the most general branch, at least compared with the rest of the thesis. Although we deal with scalar delay differential equations with a linear term and a delayed feedback, three major differences are present. Firstly, we allow the equations to have infinite delay (even more general than the situation proposed in Chapter 5), which yields the consideration of different phase spaces. Moreover, as a second comment, we assume that such spaces are constituted by certain functions that are allowed to have ‘jumps’ to include, if necessary, the so-called impulses; thus, the continuity assumption of the elements of the phase space is no longer imposed. Thirdly, unlike the previous part of the thesis, which was related with globally attracting equilibria, the main aim of this chapter is to find periodic solutions. The development of this chapter comes from the one in [19], which is the work initiated during the internships that the author enjoyed at the University of Lisbon, Portugal, under the supervision of Prof. Teresa Faria. Finally, in Chapter 7, we give a brief critical discussion of the results that we have obtained in these years. Besides, we highlight several future lines of work, some of which are currently being considered by the author of this manuscript. They are mostly related with questions that arise from the previous chapters, but we also give some comments regarding the other internship that the author has done, in this case, at the University of Szeged, Hungary, under the supervision of Prof. Gergely R¨ost. We also remark that it might be recommendable taking a look to the chosen notation (see the ending part of the work) before starting the reading of the core of the thesis, although those choices are somehow standard in the fields of Analysis and Topology. Moreover, as natural, the basic notions and results from those areas are assumed to be known. lv Sebasti´ an Buedo Fern´ andez Methodology The methodology of this project is based on the study of several well-known references in the field, e.g., [58, 68, 124], which explain the principal features concerning delay differential equations. Moreover, other works as [54, 65, 98] gather the main results regarding the study of the scalar delay differential equations with production and destruction terms [5] by the use of difference equations, as we highlighted above. Hence, acquiring knowledge regarding difference equations turns out to be a crucial step, which can be accomplished through the reading of references such as [29, 33, 119]. Furthermore, the works in [78, 81, 82, 102] serve as a starting point for the part of the project that concerns the global attractivity for the family of scalar gamma-models, while [90] is the basis for the analysis of multidimensional equations in terms of the link with difference equations that has been widely studied in the scalar case. Additionally, handling the tecnhiques in [52] is relevant in order to provide sharp estimates for the solutions of Gronwall-Bellman inequalities with the aim of applying them to several non-autonomous delay differential equations that are of interest in this thesis. When it comes to the incorporation of infinite delay and nonlinear impulses in the type of equations that we are considering, the question about the existence of a positive periodic solution provided that the coefficients are periodic may be tackled via the approach in [40]. Besides, the whole above-mentioned study has been complemented with the attendance to many seminars, courses and conferences in this and other related fields, providing knowledge that has become profitable for a better exposition of the contents of this thesis. In fact, the author has tried to balance taking care about the readability of the text and giving detailed explanations. He hopes that the reader finds some enlarged discussions both useful and finally time-saving rather than time-wasting. Indeed, the reasoning is usually supported by some key figures. Since this thesis has been typed via the wonderful tool of L A T EX, almost all of them have been depicted by the use of its package TikZ [130], while a few of them (whenever it is explicitly stated) have been obtained via numerical simulations based on the MATLAB program1dde23 [101]. 1On May 21, 2021, the link https://es.mathworks.com/help/matlab/ref/dde23.html is available to check further details. lvi Chapter 1 Preliminaries All along the next pages, the reader can find the notation, the concepts and the results that constitute the basis of this manuscript. Since the main aim of this thesis is to study certain models of delay differential equations, we will devote the first section of this chapter to introducing the basics of such type of equations, with special focus on what is needed in the subsequent chapters. Moreover, the second section deals with a dynamical approach to those equations, providing a convenient framework to study the asymptotic behaviour of the solutions in certain problems. Besides, the contents of that section are presented in a sufficiently general way to let us include difference equations too. With the aim of setting such a general framework, both sections are partially based on references [56, 58, 68, 124, 125]. Finally, taking advantage of the latter, the basic facts of a type of scalar delay differential equations with both linear decay and delayed feedback terms are presented in the third section. In particular, we show some well-known results on the relation between the solutions of such a type of delay differential equations and the solutions of a certain onedimensional difference equation when it comes to their asymptotic behaviour. We refer to the surveys [71, 79, 133], which have helped to build up this section, and the references therein for further information on this topic. 1.1 Brief introduction to delay differential equations In this section, we provide the most basic concepts and results to set the bases of the present work. Delay differential equations are a type of differential equations which take into account past states in the evolution of certain variables. The definition of what does a solution of these equations mean, how does the Cauchy problem look like, and when do they show existence, uniqueness and continuous dependence of solutions are important 1 Sebasti´ an Buedo Fern´ andez ODE, bearing in mind that the solution on [σ, σ +τ] (which is unique) is actually known from the previous step. This method can be applied until we reach the time tσ,φ. Notice that no Lipschitz-type assumption has been imposed on the third variable of f∗. Thus, one might even have uniqueness of solutions of (1.1) for continuous fthat do not satisfy Lipschitz-type assumptions on xt(see, e.g, [124, Theorem 3.2]). Example 1.7. [12, Example 3.2] To complement the explanation shown in Remark 1.6, assume, for instance, that τ= 1 and that f(t, xt) = xt(−1) = x(t−1) = f∗(t, x(t), x(t−1)); then, equation (1.1) takes the form x0(t) = x(t−1).(1.8) If the initial condition satisfies φ(θ) = 1, for every θ∈[−1,0], the solution of (1.8) through (0, φ) satisfies (see also [124]) x(t; (0, φ)) = n+1 X k=0 (t−k+ 1)k k!, t ∈[n, n + 1), n ∈Z+. The second branch regarding further hypotheses on Theorem 1.3 deals with a stronger thesis compared to the third one in that result, the one about the behaviour of maximal solutions of (1.1). To do that, we need to recall the following concepts. If (X, k · kX) and (Y, k · kY) are two real Banach spaces, we say that an operator g:M⊂X→Yis bounded if it takes bounded subsets of Minto bounded subsets of Y. Moreover, we say that gis completely continuous if it is continuous and takes bounded subsets of Minto relatively compact subsets of Y. It is clear that every completely continuous operator is also bounded since every relatively compact set is bounded in a metric space. In particular, if (Y, k · kY) is the n-dimensional Euclidean space, then a continuous operator g:M→Rnis completely continuous if and only if it is bounded, because of boundedness and relative compactness of a subset being equivalent in Rndue to the well-known Theorem of Heine-Borel [128, Theorem 5.7.1]. For instance, the function f:D→Rnin the right-hand side of (1.1) is completely continuous provided it is bounded and continuous. Theorem 1.8. [58, Chapter 2, Theorem 3.1] Let Dbe an open set of R×C,(σ, φ)∈D, and f:D→Rnbe a bounded continuous function. If x: [σ−τ, b)→Rnis a maximal solution of (1.1) through (σ, φ), then, for each closed and bounded set U⊂D, there exists tU∈Rsuch that (t, xt)6∈ U,tU≤t<b. 8 Chapter 1. Preliminaries Example 1.9. There exist simple sufficient conditions to ensure the applicability of Theorem 1.8. In fact, in the line of [38], the function fis completely continuous if it is continuous and there exist continuous functions β:R→R+and µ:R+→R+such that kf(t, φ)kRn≤β(t)µ(kφk),∀(t, φ)∈D. Remark 1.10. Under the hypotheses of Theorem 1.8, if we also assume that the domain of fis D= (a, ∞)×C, for some a∈R∪{−∞}, then any maximal solution x: [σ−τ, b)→Rn of (1.1) such that b < ∞, satisfies (see [68]) lim sup t→b−kx(t)kRn=∞, that is, xis unbounded. We can gather the new assumptions coming from both branches in the following result. Corollary 1.11. Let Dbe an open set of R×C,f:D→Rnbe a completely continuous function and locally Lipschitzian with respect to the second variable, and (σ, φ)∈D. Then the following statements hold: 1. There exists a unique solution of (1.1) through (σ, φ). 2. The equation (1.1) shows continuous dependence on initial data. 3. If x: [σ−τ, tσ,φ)→Rnis the maximal solution of (1.1) through (σ, φ), then, for each closed and bounded set U⊂D, there exists tU∈Rsuch that (t, xt)6∈ U, for tU≤t < tσ,φ. Example 1.12. In the case of discrete delay, that is, the one illustrated in (1.2), if we further assume that f∗is of class C1, then the function ffulfils all the related hypotheses in Corollary 1.11. Example 1.13. If we assume that fis continuous and Lipschitzian with respect to the second variable on every bounded set, that is, if for every bounded subset V⊂D, there exists KV≥0 such that kf(t, φ)−f(t, ψ)kRn≤KVkφ−ψk,∀(t, φ),(t, ψ)∈V, then fsatisfies the hypotheses of Corollary 1.11 (see [124, Theorem 3.7]). As in Remark 1.10, if we further assume that D= (a, ∞)×C,a∈R∪{−∞}, and that the Lipschitz constant KVis independent of V(fis globally Lipschitzian with respect to the second variable), then tσ,φ =∞for every (σ, φ)∈D. 9 Sebasti´ an Buedo Fern´ andez We have presented several basic results for the general non-autonomous equation (1.1). Nevertheless, the case in which fin (1.1) does not depend on the first variable, i.e., the autonomous case of (1.1), is widely used in applications. Hence, we will also consider the equation x0(t) = g(xt),(1.9) where g:˜ D→Rnand ˜ Dis a subset of C. Note that equation (1.9) is included in the framework of (1.1) by taking D:= Rט Dand f(t, φ) := g(φ). Since the behaviour of a solution in the autonomous case does not depend on the initial time σ, but only depends on φ, the initial time can be shifted to 0 without loss of generality. Hence, the initial condition will always be considered as (0, φ) and, therefore, we can refer to solutions of (1.9) through φ. Additionally, under uniqueness of solutions of (1.9), whenever it is clear, the solution of (1.9) with condition x(θ) = φ(θ), θ∈[−τ, 0], for a given φ∈˜ D, will be denoted by x(t;φ). Besides, the corresponding segments will be written2as xt(φ). Finally, in such a case, there exists tφ∈(0,∞] such that the maximal solution of (1.9) through any φ∈˜ Dis defined on [−τ, tφ). We could rewrite all the results coming from the non-autonomous case for the particular equation (1.9), but they would just be straightforward adaptations. Hence, we only rewrite Corollary 1.11 for such a case. Corollary 1.14. Let g:C→Rnbe a function that is locally Lipschitzian and satisfies lim sup kφk→∞ kg(φ)kRn kφk≤K. Then, the following statements hold: 1. For any φ∈C, there exists a unique solution x(t;φ)of (1.9), that is defined on [−τ, ∞)and, thus, it is maximal. 2. The equation (1.9) shows continuous dependence on initial data. 1.2 Asymptotic behaviour and semiflows If we are interested in the long-term behaviour of the solutions, we obviously need them to be defined for all future times. Hence, the corresponding segments xthave to be defined for every time t∈[σ, ∞) too. This has a straightforward consequence: a natural choice for the domain of the function fin (1.1) is D= (a, ∞)ט D, 2It is also common to find the notation xφ t(see, e.g., [71, 133]). 10 Chapter 1. Preliminaries where a∈R∪{−∞} and ˜ Dis an open subset of C[58, Page 130]. Notice that, in the autonomous case, we can choose a=−∞ and, therefore, we have D=Rט D. In fact, even for several non-autonomous cases (like the ones considered in Chapter 2), it is also possible to make such choice, as we will see later. Thus, assume that we also impose that equation (1.1) has a unique maximal solution through each initial condition (σ, φ)∈D. Then, one can associate each of those conditions to the long-term behaviour of the corresponding trajectories (t, xt(σ, φ)) as t→ ∞. Under such framework, computing the explicit expression of the solutions of a DDE is one way to obtain information with respect to its long-term behaviour. Although there exist particular cases where we can compute an explicit expression for a solution, as it happens with some DDEs with a single discrete delay (see the method of steps in, e.g., Example 1.7), we expect that, in general, such an expression is not known or it is hard to work with. Therefore, we are interested in obtaining information about the solutions without solving the equation and here is where the qualitative theory comes into action. These ideas are presented and extended in the current section. In particular, most part of this work (Chapters 2–5) is related to results that provide sufficient conditions to ensure that all the solutions of a DDE have a simple asymptotic behaviour. In fact, the following concepts will play a key role in that line. Definition 1.15. Let D= (a, ∞)ט D, where a∈R∪{−∞} and ˜ Dis an open subset of C. If there exist p∈Rnand σ > a such that x(t) = p, t ∈[σ−τ, ∞), is a solution of (1.1), then pis an equilibrium of (1.1). We will also refer to ˆp, the element of Cthat takes the constant value p, as an equilibrium of (1.1). •An equilibrium is stable provided that, for every σ > a and ε > 0, there exists δ=δ(ε, σ)>0 such that kφ−ˆpk< δ implies kxt(σ, φ)−ˆpk< ε, for every t≥σ. •An equilibrium attracts points locally or is locally attracting if there exists k(σ)>0 such that kφ−ˆpk< k(σ) implies x(t;φ)→pas t→ ∞. An equilibrium attracts points globally or is globally attracting if the latter holds for every positive k(σ). •An equilibrium is locally asymptotically stable (LAS) or globally asymptotically stable (GAS) if it is stable and, respectively, attracts points locally or globally. Definition 1.16. Let D= (a, ∞)ט D, where a∈R∪{−∞} and ˜ Dis an open subset of C. If there exist η∈R+and σ > a such that a solution of (1.1) satisfies x(t+η) = x(t), t ∈[σ−τ, ∞), then such a solution is called a η-periodic solution of (1.1). 11 Sebasti´ an Buedo Fern´ andez One can also give analogous concepts regarding stability and attractivity of a periodic solution (see, e.g., [58, Chapter 5]). Following the remarks in [58, Chapter 3], considering the solutions only as functions with values in Rndoes not constitute a fully satisfactory framework regarding the dynamical properties of a solution x(t) of (1.1) as t→ ∞, in contrast to what happens in classical ordinary differential equations. For instance, the phase space is known to be the set that contains the admissible initial conditions and, in this framework, the initial conditions require the use of segments; in fact, they are a pair formed by initial ‘time’ σ and an initial ‘state’, where such state is represented by a segment φ, that is, a function of C. Thus, it is more natural to assume that the phase space lies in Cand analyse the ‘functional’ behaviour of xt(σ, φ) as t→ ∞. For the general non-autonomous equation (1.1), a formal study of this asymptotic behaviour may be done by working with the notion of process [58, Chapter 4]. Nonetheless, a great part of the ideas that we will use, e.g., in Chapter 2, are shown in an easier way for the case of autonomous delay differential equations, for which we also recall deeper results. Additionally, the particular expression of the equations that we study in a significant part of this thesis is related to certain autonomous difference equations. Therefore, for the rest of this section, we have restricted our attention to the underlying qualitative features of the autonomous case related to (1.1), i.e., equation (1.9). For the latter equation, the initial time is irrelevant and, hence, we will work with a more restrictive concept than a process: a semiflow, which is introduced below. Intuitively, it is a tool that provides a general context to study how a ‘state’ evolves as ‘time’ goes by when following a certain rule of evolution, such as a delay differential equation or a difference equation. In order to include both cases under a general framework, we need to define what do ‘state’ and ‘time’ mathematically mean. Firstly, the term ‘state’ refers to the value that a variable takes in the space of all possible values that it can take. Therefore, this is an idea with a spatial connotation. As mentioned in the introduction, although we can recall several notions and results of the theory of dynamical systems in a very general setting, e.g., when working with dynamics in general sets, metric spaces provide a sufficiently wide framework to tackle many problems [56, 123, 125] and, in particular, the ones considered in this thesis. For a basic reference on metric spaces, we refer the reader to [128]. From now on, unless more explicit conditions are given, we will suppose that (X, d) is a metric space, that is, a set Xendowed with a metric d. Whenever the structure of the space is clear, a unique Xwill refer to both the underlying space and its structure. Secondly, in order to formally express how the elements of a set change their position over time, we also have to know what the word ‘time’ actually stands for. Since we will only deal with discrete-time and continuous-time dynamics, the classical choices of Z+or 12 Chapter 1. Preliminaries R+will be sufficient to state all the subsequent results. Therefore, the set J, that will be used to refer to possible times, will hereafter be either Z+or R+. The main reason of considering a common context written in terms of J, as it is done in [125], instead of each particular case, is that many results and concepts are completely analogous, so we can avoid repetitive arguments when showing how to include DDEs in this framework and working with an important tool: difference equations. In fact, according to that reference, many results are still valid if Jbelongs to a broader class of sets, namely the time-sets. In particular, two main and simple time-sets are Z+and R+, i.e., the ‘smallest’ and the ‘biggest’ possible ones, respectively. Once we have stated what ‘time’ and ‘space’ mean, we are ready for the following definition. Definition 1.17. Asemiflow or semidynamical system is a map ϕ:J×X→Xsatisfying the following two properties: •ϕ(0, x) = x, for every x∈X, •ϕ(t+s, x) = ϕ(t, ϕ(s, x)), for every t, s ∈Jand x∈X. The family of mappings ϕt:= ϕ(t, ·) : X→X,t∈J, together with the composition of mappings form a semigroup. Each of those mappings represent a picture of the evolution of points in Xat a certain time. In fact, the value ϕt(x) = ϕ(t, x) represents the position where xis mapped to at time t. Generally speaking, an analogous interpretation can be made for ϕt(B) for any subset B⊂X. Therefore, the following well-known concept naturally arises: for any subset B⊂X, the forward orbit [124] is defined as γ+(B) := {ϕt(B) : t∈J}, which can be thought as the ‘footprint’ of Bby following the dynamics given by ϕ. When we refer to a singleton set, we will simply write γ+(x) instead of γ+({x}). Continuity is a feature that appears in the evolution of many phenomena. Hence, we define a continuous semiflow or a continuous semidynamical system as a continuous map ϕ:J×X→Xsatisfying the properties in Definition 1.17, where the space J×Xis endowed with the product topology of the one in J(inherited from the usual topology in R) and the one in X(induced by the metric). In the rest of this work, we will only handle continuous semiflows. In particular, if J=Z+, i.e., when the discrete-time case is considered, a continuous semiflow is generated by the iterations of a certain continuous function F:X→X. In fact, that map must be F:= ϕ(1,·). In such a case, it is also common to use the notation Fninstead of ϕ(n, ·), where Fn:= n z }| { F◦···◦F. 13 Sebasti´ an Buedo Fern´ andez All along the first part of this dissertation, we will deal with difference equations and the discrete-time semiflows generated by them. By a difference equation, we refer to xn+1 =F(xn), n ∈Z+,(1.10) where F:X→Xis a certain function and we may informally say that it generates the difference equation (1.10). Every solution of (1.10) is a sequence (xn)n∈Z+in X. For instance, if we additionally assume that X=I, where Iis a real interval, it is common to find in the literature the so-called cobweb analysis [69] or graphical analysis [29], which is useful to observe the dynamical behaviour of the solutions of (1.10) in a geometrical way (see Figure 1.2). y x F Figure 1.2: Cobweb/Graphical analysis. The behaviour of the forward orbit of different points x0∈X⊂Ris represented by the paths (in red, blue or yellow) drawn by the lines connecting the points (x0, x0), (x0, F(x0)), (F(x0), F (x0)), (F(x0), F2(x0)), (F2(x0), F2(x0)),. . . When considering J=R+, i.e., the continuous-time case, differential equations may be included within this framework. For instance, an ordinary differential equation, or even a delay differential equation like (1.1) can define a semiflow provided certain suitable conditions are imposed on the phase space and the function fin the right-hand side of the equation. Note that Ccan be considered as a metric space. In particular, if dis the distance induced by the norm in C, i.e., d(φ, φ∗) = kφ−φ∗k∞= sup θ∈[−τ,0] kφ(θ)−φ∗(θ)kRn, then (C, d) is a metric space. Thus, any subset of C(e.g., ˜ D) is also a metric space endowed with the inherited metric and it may be an admissible phase space in the previously-stated 14 Chapter 1. Preliminaries theory about dynamical systems. For instance, if M⊂Rn, we define the set CM:= {φ∈C:φ(θ)∈M, θ ∈[−τ, 0]}, which is, in other words, the set of continuous functions on [−τ, 0] whose image is contained in M. In fact, under conditions of existence, uniqueness and continuation of solutions, like those in Corollary 1.14, a semiflow naturally arises as explained below. From now on and for the sake of simplicity, we asume, unless otherwise stated, that ˜ D=Cin (1.9). An interesting remark is that a continuous-time semiflow also generates a discrete-time semiflow by, e.g., considering ϕt=ϕ(t, ·) only for nonnegative integer times. Remark 1.18. Whenever we refer to a difference equation, we will mainly use capital letters that are common to represent maps in the framework of discrete dynamics, including, for instance, F,G,H,T, Λ,. . . [29, 30]. Alternatively, we may use certain lower-case letters for every function appearing in the right-hand side of a DDE (1.1): f,g,h, etc. This claim does not constitute an absolute statement for the rest of the thesis, but such difference becomes clear in results or sections where both types of equations are related. This has been chosen to highlight more clearly whether we are referring to a function appearing in the context of a DDE or in the one of a difference equation and thus help avoiding misunderstandings. Proposition 1.19. [125, Proposition 5.28] If fis continuous, there exists V⊂˜ Dsuch that there is a unique maximal solution x(t;φ)of (1.9) through each φ∈Vand xt(φ)∈V, for every t∈R+, then the map ϕ:R+×V→Vgiven by ϕ(t, φ) = xt(φ), defines a continuous semiflow on V. The proof follows from a combination of checking that the main properties of the notion of semiflow hold [124] and considering the theses of Theorem 1.3 for the autonomous case. Remark 1.20. Actually, the semiflows that can be generated by the DDE in (1.9) are very special among the semiflows on subsets of C, in the sense that they show shiftlike properties [68, Page 99]. That particular feature allows working with DDEs via the approach of the infinite-dimensional phase space Cand the one of the base space Rn(the evolution of xt(0) = x(t)∈Rnhas a major role), which is useful to prove many results. So, once we have introduced the framework of semiflows, we would like to know the behaviour of ϕt(x) as t→ ∞ for a certain x∈X. In other words, whenever it makes sense, we would like to determine the set of points to which ϕt(x) approximates as t→ ∞. The following basic concept aims to fix this intuitive idea. 15 Sebasti´ an Buedo Fern´ andez Definition 1.21. Let ϕ:J×X→Xbe a semiflow and B⊂X. The ω-limit set of B (relative to ϕ) is defined as ω(B) := \ s∈J[ t≥s ϕt(B).(1.11) Once more, if B={x}, i.e., a singleton subset, the notation ω(x) will be used instead of ω({x}). Furthermore, if we want to emphasise the dependence of the expression in (1.11) on the semiflow ϕ, we will write ωϕ(B). Due to its definition, any ω-limit set is always a closed subset of Xand it can be proven that a point y∈ω(B) if and only if there exists a sequence (tn, xn)n∈Z+in J×B, with tn→ ∞, such that ϕ(tn, xn)→y[125, Lemma 2.8]. In order to state the main properties of the ω-limit sets, the following concepts become relevant. Definition 1.22. Let ϕ:J×X→Xbe a semiflow. A set M⊂Xis said to be forward invariant if ϕt(M)⊂Mfor every t∈J. A set M⊂Xis said to be invariant if ϕt(M) = Mfor every t∈J. Clearly, an invariant set is also a forward invariant set. Additionally, it can be shown that a set is forward invariant under ϕif γ+(x)⊂M, for every x∈M. Furthermore, Mis also invariant under ϕif there exists what is called a complete orbit γ(x) through x satisfying γ(x)⊂M, for every x∈M. As only forward-in-time features of the solutions are considered in this text, we will not focus on backwards states and we refer to [125] for further details on complete orbits. Moreover, it is almost clear that forward invariant sets admit its own inherited semiflow. Corollary 1.23. If ϕ:J×X→Xis a semiflow and Mis forward invariant, then ϕ|J×M:J×M→Mis also a semiflow. Some relevant examples of invariant sets are equilibria and periodic orbits. A point c∈Xis said to be an equilibrium of the semiflow if ϕt(c) = c, for all t∈J. Moreover, if there are η∈J\{0}and x∈Xsuch that ϕt+η(x) = ϕt(x) for every t∈J, we say that the (forward) orbit γ+(x) is η-periodic, or simply, periodic. Thus, the set P={ϕt(x) : t∈[0, η]} satisfies ϕt(P) = P, for every t∈J. In the context of a semiflow regarding the DDE in (1.9), where J=R+and Xis a subset of C, only the constant functions in Ccan be equilibria [124, Proposition 5.4]. Therefore, we will identify such type of functions with the value they take, bearing in mind that we are referring to a constant function on [−τ, 0]. To be clear, and following 16 Chapter 1. Preliminaries the notation of [124], since any equilibrium of (1.9) is a certain function ˆp(θ) = p∈Rn, for every θ∈[−τ, 0], we will make the abuse of notation of saying that p∈Rnis an equilibrium of (1.9). Additionally, any periodic orbit of the corresponding semiflow comes from a periodic solution of the equation (1.9). Thus, such concepts are coherent with what was said in Definitions 1.15 and 1.16. Definition 1.24. Let x∈Xand B⊂Xbe non-empty. We define the distance from x to Bas δ(x, B) := inf y∈Bd(x, y). Moreover, if A⊂Xis non-empty and bounded, we define the distance from Ato Bas δ(A, B) := sup x∈A d(x, B) = sup x∈A inf y∈Bd(x, y). Sometimes, the underlying metric space is highlighted as a subscript, that is, by using the notation δX[115]. It is known that the previous concept does not define a distance in the usual sense since, for instance, it is not symmetric. Nevertheless, the Hausdorff distance, also known as Pompeiu-Hausdorff distance [15], fixes that problem. Since the notion of ‘approaching certain values’ will be relevant, we need to set a way to measure closeness. As usual, we will say that Wis a neighbourhood of a subset M if there exists an open subset Usuch that M⊂U⊂W. In particular, if ε > 0, the ε-neighbourhood of M⊂Xwill be the set B(M, ε) := {x∈X:δ(x, M)< ε}=x∈X: inf y∈Md(x, y)< ε. Definition 1.25. Let ϕ:J×X→Xbe a semiflow. If M, B are nonempty subsets of X, then Mis said to attract B(or Bis said to be attracted by M) if δ(ϕt(B), M)→0 as t→ ∞. In other words, a nonempty subset Mattracts3another subset Bif, for any ε > 0, there exists some t0≥0 such that ϕt(B) belongs to an ε-neighbourhood of Mif t≥t0. If there is a nonempty bounded subset Bof Xthat attracts points of X, the semiflow is known to be point dissipative. In such a case, every forward orbit is bounded. Theorem 1.26. [125, Chapter 2] Let ϕ:J×X→Xbe a continuous semiflow, K be a nonempty subset of X, and assume that γ+(K)is relatively compact. Then, ω(K) is nonempty, compact, invariant and attracts K. If, additionally, J=R+and Kis connected, then ω(K)is also connected. 3See Pages 29-30 in [125] for a brief topological discussion on the idea of ‘approaching’ a set as time goes by and the role of compact attractors. 17 Sebasti´ an Buedo Fern´ andez A singular perturbation and the role of difference equations The latter change of variables, i.e., the one that involved τ, allows us to rewrite equation (1.18) as ε˙x(t) = −ax(t) + f(x(t−1)), where ε=1 τ. Hence, it can be seen as a singular perturbation of 0 = −ax(t) + f(x(t−1)) or, equivalently, x(t) = 1 af(x(t−1)) =: F(x(t−1)),(1.19) as ε→0+, or equivalently, as τ→ ∞. Consequently, it is quite natural to expect some relation between the asymptotic behaviour of the solutions of (1.16) for sufficiently large τ and that of the solutions of (1.19). Notice that the division by a, related to the alternative change of variables, also appears in (1.19). Difference equations with continuous argument like (1.19) are similar to DDEs when it comes to the phase space (see, e.g., [65, 120]) since we need to define an initial condition on [−1,0]. Nonetheless, equation (1.19) may have solutions that are discontinuous at integer times. Such issue can be fixed by imposing the consistency condition that the initial function φbelongs to the phase space C∗ I={φ∈ C([−1,0], I) : φ(0) = F(φ(−1))}. Furthermore, the concepts of solution of (1.19) through an initial function φand segments can also be defined in a similar manner. Besides, the equation (1.19) represents an iteration by Fof every element φ(θ), for θ∈[−1,0], so its solutions are governed by the difference equation with discrete argument xn+1 =F(xn).(1.20) Additionally, if F(I)⊂I, then the solutions of equation (1.19) are unique and globally defined, that is, they exist on [−1,∞). Moreover, equation (1.20) also generates a continuous discrete-time semiflow on X=I, namely ψ:Z+×I→I, which is defined by ψ(n, x) = Fn(x). Now we provide some details regarding the comparison between the solutions of (1.16) and (1.19). One could expect some likeness between the solutions of the DDE in (1.16) with large τand the ones of (1.19), which, formally speaking, corresponds to the limit case τ=∞. Nevertheless, such limit equation (1.19) shows two major issues that make such comparison tough. 24 Chapter 1. Preliminaries Firstly, no consistent condition is needed to be imposed for the initial functions of the DDE in (1.16) to generate continuous solutions, as it is actually required for (1.19). However, one can flip the reasoning and wonder if equation (1.19) without the consistent condition (thus, having discontinuous solutions) has something to do with the DDE in (1.16) in terms of that similarity of solutions. In such case, there exist results that show the existence of certain continuity features when it comes to the transition between solutions of (1.19) and solutions of (1.16) with small ε=1 τ(see, e.g., [65, Section 3]) on bounded time intervals. Secondly, consider a solution of (1.19) through an initial condition φ∈C∗ I, namely x(t;φ). Although the solution is continuous due to the consistency condition, the corresponding segments xt(φ) might not tend to any element of the phase space as t→ ∞: the limit behaviour might be, roughly speaking, a discontinuous segment. In fact, there are examples where (1.20) has a locally attracting 2-periodic orbit and a continuous solution of (1.19) approaches a step function as t→ ∞ (known as convergence to the square wave) [98]. However, equation (1.20), the discrete-time version of (1.19), is a nice tool to derive several properties about long-term behaviour of the solutions of (1.16). In the following result, we recall that the dynamics of (1.16) are, in some sense, dominated by the dynamics of (1.20). Theorem 1.33. [98, Proposition 1.1] Let Ibe a closed interval and F:I→Ibe a continuous function. If φ∈CI, the solution x(t;φ)of (1.16) satisfies x(t;φ)∈I, t ≥0. Furthermore, the following property holds ωψ(I) = ∞ \ n=0 Fn(I),(1.21) and, if ωψ(I)6=∅, then lim t→∞d(x(t;φ), ωψ(I)) = 0, while, if ωψ(I) = ∅, then x(t;φ)tends to either −∞ or ∞as t→ ∞. Remark 1.34. The first thesis of Theorem 1.33 states that any solution of (1.16) such that the image of its initial segment is contained in the interval I(which is assumed to be forward invariant for For ψ) cannot leave the interval I, and thus the set CIis forward invariant for the semiflow ϕ. Therefore, a forward-invariance-type feature is inherited from (1.20) to (1.16). In fact, this has been called the invariance property [65, 120]. 25 Sebasti´ an Buedo Fern´ andez t x −τ φx x x(t;φ) ωψ(I) I Figure 1.3: The sharp asymptotic bounds for the solution x(t;φ) of an equation of the form (1.16), x:= lim supt→∞ x(t;φ) and x:= lim inft→∞ x(t;φ). They belong to the window ωψ(I), which is the ω-limit set of Iwith respect to the difference equation (1.20). The second one gives us more precise information about the long-term behaviour of the solutions of (1.16). Since Fn+1(I)⊂Fn(I), for every n∈Z+, the set ωψ(I) takes the simple expression in the right-hand side of (1.21) (see Definition 1.11). In fact, by the continuity of Fand an inductive argument, any set Fn(I) is an interval, so the set ωψ(I) is an intersection of convex sets, and thus, an interval. In addition, the following relation holds: ωϕ(φ)⊂Cωψ(I),∀φ∈CI.(1.22) The meaning of the second thesis, which is somehow summed in (1.22), is that the asymptotic behaviour of the solutions of the difference equation (1.20) sets the window where any solution of (1.16) starting at an initial condition with values in Imust tend to, as t→ ∞. Hence, in some sense, the dynamics of (1.16) are controlled by the ones of (1.20). Notice that the nature of ωψ(I) and ωϕ(φ) are different: ωψ(I) is a real interval whilst ωϕ(φ) is a subset of continuous real functions defined on the compact interval [−τ, 0]. Furthermore, provided that ωψ(I) is non-empty and compact, then we can also rewrite the former relations (see, e.g., [87]) as lim inf t→∞ x(t;φ),lim sup t→∞ x(t;φ)⊂ωψ(I) = ∞ \ n=0 Fn(I).(1.23) For instance, the latter happens if Iis compact (see Theorem 1.26). Figure 1.3 is a sketch of this situation. In [98], this kind of result is stated by assuming that Iis just closed. Therefore, the authors also admit the choice of a closed unbounded interval I. Nevertheless, this 26 Chapter 1. Preliminaries could cause ω(I) to be empty or unbounded, and one of those cases occurs if and only if F(I) is unbounded. Since the latter will not occur in the forthcoming chapters, we avoid that possibility, having F(I)⊂K, for a certain compact subset K⊂I, and thus work with F|K:K→Kbearing in mind that every initial condition in Ienters Kand hence, ω(I) = ω(K)(⊂K) is always a nonempty compact set by means of Theorem 1.26. So, in this case, (1.22) is always valid. When the above-mentioned window ωψ(I) is a singleton subset, we obtain the following result. In particular, if Iis compact and pis the global attractor (of points) in I, then ωψ(I) = {p}(see, e.g., [98, Proposition 1.2]). Corollary 1.35. [98, Corollary 1.2] Let Ibe a compact interval and F:I→Ibe a continuous function. If pis a global attractor for xn+1 =F(xn), then pis a global attractor for x0(t) = −ax(t) + f(x(t−τ)). Hereafter, we will say that (1.16) and (1.20) are corresponding equations to one another, based on the strong relation regarding the long-term dynamics that we have recalled above. We will further assume that, in the interval of interest I, the function is of class C1 on the interior of its domain to facilitate the stability analysis of the unique equilibrium of its interior. In particular, this is trivial if Iis open. Having shown the key properties of (1.16) to simplify its study, we will then focus on the set of triples (a, f, τ) that satisfy the following set of hypotheses, which will be jointly called (T): (T1) a, τ ∈R+, f ∈ C1(I, R), I⊂Ris a non-empty open interval, (T2) f a:I→Iis well-defined, (T3) f(x) = ax has a unique solution on I, (T4) K⊂Iis a non-degenerate compact interval that is globally attracting for f a. Notice that, while the first two parameters are strongly related via the function f/a, there is ‘freedom’ on τ. Condition (T4) is a technical condition to ensure that the solutions starting close to the boundary of Ieventually go away from it, so one avoids solutions ‘escaping’ from I. Moreover, if (a, f, τ) satisfies the hypotheses (T) then f0(p)≤a. Otherwise, if f0(p)> a, a simple cobweb analysis of f/a shows that no set Kas in (T4) could exist. In fact, the limit case f0(p) = acan only take place under (T) in particular circumstances. For instance, with higher smoothness of f,f00(p) = 0 must hold. 27 Sebasti´ an Buedo Fern´ andez Relative position of the graph of the feedback regarding the unique equilibrium When it comes to the qualitative study of (1.16) with (a, f, τ) satisfying (T), it is common to classify the feedback fin terms of the position of its graph with respect to the unique equilibrium pof (1.10). In particular, this turns out to be relevant, as we will see at the very end of this chapter. We can make the change ˜x(t) = x(t)−pand ˜ f(y) = f(y+p)−ap to assume that 0 is the unique equilibrium of the DDE. In fact, ˜x0(t) = x0(t) = −ax(t) + f(x(t−τ)) = −ax(t) + ap +f(x(t−τ)) −ap =−a˜x(t) + f(˜x(t−τ) + p)−ap =−a˜x(t) + ˜ f(˜x(t−τ)). Therefore, the origin, which is an equilibrium of the equation ˜x0(t) = −a˜x(t) + ˜ f(˜x(t−τ)),(1.24) shares the attractivity properties with the equilibrium pof equation (1.16). In particular, we have pbeing stable or globally attracting for (1.16) if and only if we have 0 being respectively stable or globally attracting for (1.24). y x ˜y ˜x p f(p) = ap f(x)ax Figure 1.4: Change of variables to the origin in (1.16). From the previous comments, there is no loss of generality in assuming that f(0) = 0. In fact, such condition is actually considered in the literature; see, e.g., the references [71, 94, 97, 113]. We will say that the function ˜ fis the shifted function of fand its graph is represented in the shifted variables. Regarding the latter, the relative position of the graph with respect to its unique equilibrium pis a key feature to define relevant classes of feedbacks. Definition 1.36. Let a > 0, f∈ C1(I, R) and assume that p∈Iis the unique solution of f(x) = ax on I. If ˜ fdenotes the shifted function of f, then, fis a positive feedback for 28 Chapter 1. Preliminaries (1.16) provided ˜x˜ f(˜x) = (x−p)(f(x)−ap)>0,∀x∈I\{p}(∀˜x∈(I−p)\{0}),(1.25) or a negative feedback for (1.16) provided ˜x˜ f(˜x) = (x−p)(f(x)−ap)<0,∀x∈I\{p}(∀˜x∈(I−p)\{0}).(1.26) Remark 1.37. Notice that, if fbelongs to some of the families named above, then so are ˜ f,f/a and ˜ f/a; hence, the parameter a > 0 does not influence the above-mentioned features. The role of the latter two functions will be clarified later, but we can just highlight that conditions (1.25) and (1.26) can be respectively rewritten as ˜x˜ f(˜x) a= (x−p)f(x) a−p>0,∀x∈I\{p}(∀˜x∈(I−p)\{0}), ˜x˜ f(˜x) a= (x−p)f(x) a−p<0,∀x∈I\{p}(∀˜x∈(I−p)\{0}). Remark 1.38. If (a, f, τ) satisfies (T), then (a, ˜ f, τ) also satisfies (T). In particular, if f is defined on the non-empty open interval I⊂Rand p∈Iis the unique root of f(x) = ax, then ˜ fis defined on the shifted interval I−pand 0 ∈I−pis the unique root of f(x+p)−ap =˜ f(x) = ax. 1.3.2 Some aspects about the local dynamics For the purposes of this subsection, assume that (a, f, τ) satisfies (T) and that pis the unique root of f(x) = ax. At a first glance, we could use Corollary 1.35 to derive a sufficient condition for an equilibrium pto be LAS for (1.16). If we take Ipas the immediate basin of attraction of pfor F, that is, the connected component of the set {x∈X:Fn(x)→pas t→ ∞} that contains p, then we can apply Corollary 1.35 to F|Ipand obtain that, if x(t;φ) is the solution of (1.16) through φ∈CIp, then x(t;φ)→pas t→ ∞. Corollary 1.39. If p∈Iis LAS for the difference equation (1.20), then pis LAS for the DDE (1.16). An easy-to-check well-known sufficient condition for a regular function Fto have a locally asymptotically stable equilibrium is the following one (see, e.g., [29, Section 1.4] or [30, Chapter 5]). 29 Sebasti´ an Buedo Fern´ andez Proposition 1.40. Let F:I→Ibe a function of class C1and p∈Ibe a fixed point of F. The following statements hold: •If |F0(p)|<1, then pis LAS for F, •If pis LAS for F, then |F0(p)| ≤ 1. The previous result is ‘almost’ an equivalence regarding local asymptotic stability when working with a sufficiently regular F. The case |F0(p)|= 1, also known as nonhyperbolic case, in contrast with the hyperbolic case of |F0(p)| 6= 1, is special and it highly depends on further properties of F. Furthermore, if |F0(p)|>1 then the equilibrium is called a repelling point [29, 30]. Remark 1.41. Bear in mind that, in case we decide not to normalise the parameter a regarding the linear decay of (1.16), we have F0(p) = f0(p) a. Nevertheless, the condition |f0(p)|<|a|is not the sharpest one, it is only a sufficient condition for an equilibrium to be LAS for (1.16). In order to obtain a refined condition, we have to go deeper than Corollary 1.39 actually does. It is here where the interesting particular case of (1.16) with f(x) = bx, for some b∈R, appears. The dynamics of this linear case are well understood and, additionally, their study provides some useful information for the general equation (1.16). In fact, the local asymptotic stability of the equilibrium pin (1.16) can be studied through its linearised equation at the point p x0(t) = −ax(t) + f0(p)x(t−τ),(1.27) for which the origin is an equilibrium. Specifically, the following Poincar´e-Liapunov-type result, which is a particular case of [12, Theorem 11.2], holds. Theorem 1.42. Assume that the origin is a global attractor for u0(t) = −au(t) + bu(t−τ),(1.28) and let ˜g:B0⊂R→Rbe a continuous function defined on a neighbourhood B0of the origin and such that lim |u|→0|˜g(u)| |u|= 0. Then the origin is LAS for u0(t) = −au(t) + bu(t−τ) + ˜g(u(t−τ)). 30 Chapter 1. Preliminaries We remark that, regarding (1.28), the origin is GAS if and only if it is a global attractor, that is, for the linear equation (1.28), the stability is derived from global attractivity of the origin. We can take advantage of the statement of Theorem 1.42 for u(t) = ˜x(t), where ˜x is the shifted variable of x, defined in the last subsection. We can write the following computations: ˜x0(t) = −a˜x(t) + ˜ f(˜x(t−τ)) =−a˜x(t) + ˜ f0(p)˜x(t−τ) + ˜ f(˜x(t−τ)) −˜ f0(p)˜x(t−τ) =−a˜x(t) + ˜ f0(p)˜x(t−τ) + ˜g(˜x(t−τ)), where ˜g(˜x) := ˜ f(˜x)−˜ f0(p)˜x. Now, since ˜ fis C1and ˜ f(0) = 0 , it is a basic fact from Taylor polynomials that lim |˜x|→0|˜g(˜x)| |˜x|= 0. Consequently, studying the dynamics of the solutions of (1.27) provides us direct information about the local dynamics around the unique equilibrium of the DDE (1.16). Thus, let us focus on equations that are written as x0(t) = −ax(t) + bx(t−τ).(1.29) The explicit region of parameters (a, b, τ) for which the origin is globally attracting for (1.29) comes from the study of an equation that, in general, is trascendental. In fact, if we look for solutions of (1.29) of the form eλt, with formally allowing λto take complex values, we would arrive to the equation λ+a−be−λτ = 0.(1.30) The equation (1.30) is called the characteristic equation of the DDE in (1.29). Bearing in mind the previously given appetiser regarding the exponential-type solutions, it is natural to expect that the roots of (1.30) induce certain asymptotic behaviour in the solutions of (1.29). In fact, via the use of the Laplace transform, the following property holds (see, e.g., [12, 58, 68]). Theorem 1.43. If all the roots of (1.30) have negative real parts, then the origin is GAS for (1.29). Hence, the study of the attractivity/stability of the origin for (1.16) is reduced to finding the location of the roots of the entire function w(λ) = λ+a−be−λτ (also called the characteristic function [127]) for a particular triple of parameters (a, b, τ). Finally, the answer to that issue is known and we can write it in the form of the following criterion. 31 Sebasti´ an Buedo Fern´ andez Theorem 1.44. [12, Theorem 13.8] Let a, τ > 0. Then, all the roots of (1.30) have negative real parts if and only if −ps2+ (aτ)2< bτ < aτ, (1.31) where sis the unique solution of −x tan(x)=aτ in (π 2, π). Notice that Theorem 1.44 is sometimes given allowing triples with a≤0 too [58, Appendix], but we have omitted the triples with that condition since a > 0 in all the models that we will study (remember the hypotheses (T)). In particular the limit condition as a→0+is −π 2< bτ < 0. Moreover, the condition on the limit case τ= 0 is trivial: b−a < 0. Remark 1.45. If b∈[−a, a), then (1.31) holds. Nevertheless, there are triples (a, b, τ) that also satisfy (1.31) and for which b < −a. For the latter ones, there is an upper bound in the delay τfor the origin of (1.29) to be GAS. Thus, for those triples, it is also useful to handle a condition derived from (1.31) that explicitly shows such bound on the delay. In fact, the lower boundary for bτ for any aτ > 0 is represented by the curve (aτ, bτ) = −s tan(s),−s sin(s), s ∈(π/2, π).(1.32) From equation (1.32), we can obtain the equalities s= arccos a b,(bτ)2−(aτ)2=s2, which can be used to derive that the above-mentioned upper bound for the delay is ¯τ:= arccos(a b) √b2−a2. More detailed information about the location of the roots of (1.30) is known (see, for instance, [31, 124]). We have only recalled a result excluding roots with nonnegative real part, but one can give analogous conditions to ensure that a certain number of roots lie on the half-right plane, which, for instance, has implications in the dimension of the unstable manifold of the equilibrium of (1.16) [71]. As a consequence, Theorem 1.44 in combination with the sufficient condition of Theorem 1.43 provides a way to compute the stability of the origin regarding the linear equation (1.29) in terms of their parameters. Corollary 1.46. If condition (1.31) holds, then the origin is GAS for the linear DDE (1.29). 32 Chapter 1. Preliminaries Finally, due to Theorem 1.42, the previous corollary has a direct implication in the asymptotic behaviour of the solutions of (1.16) that are close to the equilibrium p. Corollary 1.47. If condition (1.31) holds with b=f0(p), then the origin is LAS for the equation (1.16). For further references to the previous results, we define Las the region of triples (a, b, τ)∈R3for which the unique equilibrium pof (1.16) is LAS; in particular, those satisfying condition (1.31) with b=f0(p). In Figure 1.5, we can find a couple of twodimensional slices of the region L. They provide a nice picture of the set Ldue to the reduction of parameters via the alternative changes of variables previously given. Besides, notice that the dashed orange lines in Figure 1.5 represent the set of parameters in equation (1.16) for which either λ= 0 or a couple of conjugated imaginary numbers are roots of the characteristic equation (1.30). In such case, since the origin in equation (1.16) is non-hyperbolic, other techniques need to be considered. f0(p) a (0,−π 2) Not in our framework Unstable LAS f0(p) τ (0,−1) (0,1) Not in our framework Unstable LAS Figure 1.5: Both graphics represent the restriction of the region of parameters Lfor which the unique equilibrium pof (1.16) is LAS (yellow), for the cases τ= 1 (left) and a= 1 (right). The thick dashed orange curves represent the limit cases where either zero (upper case) or a couple of conjugated imaginary numbers (lower case) are roots of the linearised equation and do not fall under the sufficient condition in Corollary 1.47. Gray dashed lines represent the asymptotic behaviour of the nonlinear curves when the first coordinate tends to ∞. 33 Sebasti´ an Buedo Fern´ andez function ˜ F) with F0(0) <0, then SF < 0 implies that Fis enveloped by a rational function R(x) = αx 1+γx such that 0 = F(0) = R(0), F0(0) = R0(0) = αand F00(0) = R00(0). Some conditions for the map Rto have the origin as a globally attracting equilibrium can be easily obtained via Theorem 1.49. The results on enveloping from [25, 33], such as Theorem 1.51, thus yield the global stability result for F. There exist equivalent conditions to SF < 0 that would also make sense with less regularity on F[119, Chapter 5, Section 3]. Nevertheless, such conditions may become quite technical and hard to work with. Thus, since the functions Fwe are handling are at least C3, we have assumed such regularity in Theorem 1.55. Therefore, after having exposed the importance of the Schwarzian derivative for the study of the dynamics of certain maps, we introduce the following concepts. Definition 1.58. Let F:I→Ibe a function of class C3. •If Fis an M-map and SF < 0 on I, then Fis an SM-map. •If Fis a U-map and SF < 0 on I, then Fis an SU-map. •Any SM-map or SU-map will be generally called an S-map. •If Fis a U-map and SF < 0 on the subinterval of Iwhere Fdecreases, then Fis an SU∗-map. •Any SM-map or SU∗-map will be generally called an S∗-map. This type of notation is commonly used in the literature. For instance, see [43, 67, 88]. Obviously, every SU-map is an SU∗-map. Notice that there is no need to define SM∗- maps since the Schwarzian derivative is only used when F0(p)<0; thus, provided Fis an M-map, that would mean is decreasing on the whole I. Recall that, if (a, f, τ) satisfies the hypotheses (T), then F=f/a satisfies ∆F 1<0 on I\{p}. Notice that if ∆F 1were positive at any side of p, then no Kas in (T4) would exist, as a simple graphical analysis shows. Hence, one can combine Theorem 1.55 with Corollary 1.48 to obtain the following result. Corollary 1.59. Let (a, f, τ)satisfy (T), pbe the unique root of f(x) = ax, and assume that fis an S∗-map. If f0(p)∈[−a, a), then pis GAS for the DDE (1.16). 40 Chapter 1. Preliminaries Let us clarify and sum up the features of the procedure that we have followed. Up to this point, we have already seen that Corollary 1.48 together with the criterion in Theorem 1.49 or Theorem 1.55 provide sufficient conditions in terms of the parameters a and fto deduce the global attraction properties of the equilibrium pin (1.16). Since those conditions come from analysing the difference equation (1.20) and in such an equation the delay τdoes not appear, the conditions obtained are then called delay-independent stability conditions or absolute stability conditions. For instance, the applicability of Corollary 1.59 is independent from the value τ > 0. Based on these remarks, if we try to go further and obtain sharper conditions, they must be delay-dependent stability conditions. In this line, the results in [54], which were extended in [94] in a more general context, are enlightening. Theorem 1.60. [54] Let (a, f, τ)satisfy (T) and pbe the unique root of f(x) = ax. Assume that pis GAS for xn+1 =e−aτ p+ (1 −e−aτ )F(xn),(1.36) where F=f/a. Then, pis GAS for (1.16). The hypothesis of Theorem 1.60 is related with a difference equation that clearly depends on τand, thus, a first delay-dependent condition naturally appears, as explained below. Corollary 1.61. Let (a, f, τ)satisfy (T), pbe the unique root of f(x) = ax, and assume that fis an S∗-map. If f0(p)∈[−a, a)or f0(p)<−aand e−aτ ≥f0(p) + a f0(p),(1.37) then the origin is GAS for (1.16). Theorem 1.51 provides another way to interpret condition (1.37) in terms of enveloping. In fact, if we define the function Fτas in the right-hand side of (1.36), that is, Fτ(x) := e−aτ p+ (1 −e−aτ )F(x), and assuming that there exists τ∗>0 such that the hypotheses in Theorem 1.60 hold, then, since the function Fτ∗envelops, in the sense of [33], every Fτof a lesser τ, the result also holds for every τ∈[0, τ∗]. In particular, τ∗can be chosen to make the delay-dependent inequality in (1.37) an equation. This situation is depicted in Figure 1.7, where we have to bear in mind that F0 τ∗(p) = −1. 41 Sebasti´ an Buedo Fern´ andez ˜y ˜x y xp p F Fτ∗ Fˆτ Figure 1.7: For I= (0,∞), the graph of the map Fis drawn in blue. The maps Fτ(x) are nested in terms of the enveloping in [33]. The thick blue graph is the one of Fτ∗, whereas the black one stands for the graph of a certain Fˆτ, with 0 <ˆτ < τ∗. Remark 1.62. Despite the geometrical significance of the above-mentioned condition, it is possible to go further [54]. In fact, Theorem 1.60 is sharpened by switching global attraction hypothesis for (1.36) to global attraction for another more complicated difference equation. In particular, in case fwere further an S-map, a sharper version of Corollary 1.61 could also be given in case f0(p)<−aby using the relation e−aτ ≥f0(p)2+af0(p) f0(p)2+a2.(1.38) However, if we directly assume that Sf < 0 on the whole I, then the deeper analysis of equation (1.16) in [94], which takes into account the enveloping of funcions with negative Schwarzian derivative by rational functions, in the line of Remark 1.57, actually yields a sharper result that the ones in Corollary 1.61 and Remark 1.62. Theorem 1.63. [86, Proposition 1.1] Let (a, f, τ)satisfy (T), pbe the unique root of f(x) = ax, and assume that fis an S-map. If f0(p)∈[−a, a)or f0(p)<−aand e−aτ >−f0(p) aln f0(p)2−af0(p) f0(p)2+a2,(1.39) then the origin is GAS for (1.16). As remarked in [86], condition (1.39) is sharper than the one in (1.37) due to the linear 42 Chapter 1. Preliminaries bound x > ln(1 + x), for every x > 0. In fact, provided f0(p)<−a < 0, we have e−aτ ≥f0(p) + a f0(p)=−f0(p) a−af0(p)−a2 f0(p)2>−f0(p) a−af0(p)−a2 f0(p)2+a2 >−f0(p) aln 1 + −af0(p)−a2 f0(p)2+a2=−f0(p) aln f0(p)2−af0(p) f0(p)2+a2>0. Thus, condition (1.37) is more restrictive when fis an S-map and (1.39) opens a larger window of parameters for which the origin is GAS for (1.16). In fact, (1.38) appears between both (see the last expression of the first line in the chain of inequalities above). The interest of (1.37) is that it is related to easier explicit conditions in applications, while (1.39) might generate implicit conditions and (1.38) involves tougher explicit expressions. Moreover, condition (1.37) is the estimate that we can clearly use when fis only an S∗-map. Moreover, it is important to bear in mind that all these conditions imply that the origin is LAS and thus they must belong to the region Lof local asymptotic stability, which was defined at the end of Subsection 1.3.2 and depicted in Figure 1.5. We are close to the end of this section and the current chapter, so we sum up the information that we have obtained with respect to the qualitative analysis of (1.16) in the remark below. We recall that the negativity condition on the Schwarzian derivative of a map allows us to focus on local dynamics to deduce information regarding global behaviour of solutions and that F0(p) and f0(p) are strongly related (Remark 1.41). In fact, fis an S-map if and only if Fis an S-map. Moreover, to write and better understand the following remark, it is useful to define Gas the region of triples (a, f0(p), τ) for which the origin is GAS for (1.16). The conditions on global and local stability that have appeared in the current chapter, namely (1.37), (1.39) and (1.31), are a motivation to define the following functions. Regarding estimates for the delay τ, we respectively define ρ1(a, f0(p)) := 1 aln f0(p) f0(p) + a, ρ2(a, f0(p)) := −1 aln −f0(p) aln f0(p)2−af0(p) f0(p)2+a2, ρ3(a, f0(p)) := arccos a f0(p) pf0(p)2−a2, 43 Sebasti´ an Buedo Fern´ andez and, with respect to estimates for f0(p), we respectively define ξ1(a, τ) := −a 1−e−aτ , ξ2(a, τ) := −aM−1 2(e−aτ ), ξ3(a, τ) := −ra2+(M−1 3(aτ))2 τ2, where M2: (1,∞)→(0,1) and M3: (π/2, π)→(0,∞) are the bijective functions defined by M2(x) := xln x2+x x2+ 1, M3(x) := −x tan(x). The following Remark 1.64 summarises the previous results with respect to the value of the delay and it is graphically supported by Figure 1.8. Remark 1.64. Let (a, f, τ) satisfy (T), fbe an S∗-map, F=f/a and pbe the unique fixed point of F. We have the following cases: •If f0(p) a=F0(p)>1, then this case does not fall under our framework. •If f0(p) a=F0(p) = 1, then pis non-hyperbolic for both equations (1.16) and (1.20). However, f00(p) = 0 must hold and pis GAS for (1.16) (see [119, Lemma 5.10]). •If f0(p) a=F0(p)∈[−1,1), then pis GAS for F, and, thus, for (1.16) too. •If f0(p) a=F0(p)<−1, and τ≤ρ1(a, f0(p)), then pis GAS for (1.16). •If f0(p) a=F0(p)<−1, fis an S-map and ρ1(a, f0(p)) < τ < ρ2(a, f0(p)), then pis GAS for (1.16). •If f0(p) a=F0(p)<−1 and ρ2(a, f0(p)) ≤τ < ρ3(a, f0(p)), then pis LAS for (1.16) and a general result about the global attractivity of pfor such an equation is, as far as we know, not available yet. 44 Chapter 1. Preliminaries •If f0(p) a=F0(p)<−1 and τ=ρ3(a, f0(p)), then pis non-hyperbolic for (1.16) and other techniques should be considered. •If f0(p) a=F0(p)<−1 and ρ3(a, f0(p)) < τ, then pis unstable for (1.16). f0(p) a Not in our framework Unstable GAS (0,−1) (0,−3 2) (0,−π 2) f0(p) τ (0,−1) (0,1) Not in our framework Unstable GAS Figure 1.8: These pictures sketch the restriction of the region Gthat comes from Remark 1.64 for the cases τ= 1 (on the left) and a= 1 (on the right). The darker green filling colour represents such region for S∗-maps. It is respectively bounded by the solid green curves ρ1(a, f0(p)) = 1 and ξ1(1, τ) = f0(p). The lighter green filling colour represents the extension of Gto S-maps and is respectively bounded by the dashed green curves ρ2(a, f0(p)) = 1 and ξ2(1, τ) = f0(p). The yellow zone represents the region of Lfor which the global dynamics of (1.16) are not completely understood and it is respectively bounded by the dashed orange curves ρ3(a, f0(p)) = 1 and ξ3(1, τ) = f0(p). Remark 1.65. Notice that, for each i∈ {1,2,3}, any inequality τ < ρi(a, f0(p)) (or its non-strict version, alternatively) appearing in Remark 1.64 could have been substituted by f0(p)> ξi(a, τ) (or by its non-strict version, respectively) to produce an analogous version of Remark 1.64. Remark 1.64 and Figure 1.8 are provided in a general sense. The particular choices of the feedback fmay imply the introduction of feedback inner parameters, which have 45 Sebasti´ an Buedo Fern´ andez a certain meaning depending on what one is attempting to model. For instance, if we take f(x) = βxe−δx, where β, δ ∈R, then βand δwould be feedback inner parameters. Specifically, f0(p) may depend on some of those inner parameters. Therefore, in the case f0(p) acquires major relevance when it comes to global asymptotic stability of the equilibrium (e.g., with S-maps), we can provide analogous conditions and figures that highlight the role of any of those parameters compared to, e.g., the delay τ. This will be done afterwards, in Chapters 2 and 3, where we analyse some particular models. Remark 1.66. We have seen that there is an underlying relation with discrete dynamics provided that a > 0. Although we only focus on the latter case, it is worth recalling what happens with a= 0 and, furthermore, whether it coincides with the limit version of the given conditions as a→0+. The works [63, 85] provide the basis towards a sharp result regarding the global dynamics of equation (1.15) with a= 0 and an S-map feedback. Such general result is given in [95, Theorem 2]. The latter work generalises both the well-known work by Yorke [142] and the one by Wright [136] via the use of rational functions (1.34) (recall Remark 1.57). In our framework, such sharp condition from [95] shall be interpreted as 0> τf0(p)>−3 2, also including the case τf0(p) = −3 2if f00(p)6= 0. Through some computations, it can be seen that this condition is consistent with the mentioned limit of (1.39) as a→0+. Moreover, the limit form of the LAS-type condition given in (1.31) coincides with the one of the case a= 0, which is 0 > τf0(p)>−π/2. In fact, by continuity, we can extend the previous functions to a= 0 by its limit form as a→0+and define ρ1(0, f0(p)) := −1 f0(p), ρ2(0, f0(p)) := −3 2f0(p), ρ3(0, f0(p)) := −π 2f0(p). When the unique equilibrium pof the difference equation (1.20) is unstable and also f0(p)/a =F0(p)<−1 (repelling point), the terms of any sequence with x0sufficiently close to pgo away from the equilibrium. In such case, the instability is inherited by the corresponding DDE (1.16) for large values of the delay τ. In fact, if the function F=f/a is of negative feedback on the set Krelative to (T4), then the loss of stability implies the loss of global attractivity [55, 71]. That is one relevant reason for saying at the beginning of this subsection that we were restricting our study to the region Lof local asymptotic stability. As a summary of these comments, we write Remark 1.67 and provide Figure 1.9. Remark 1.67. Consider the assumptions of Remark 1.64 together with F=f/a being of negative feedback on the set Krelative to (T4). If f0(p)<−a, then the value of the delay affects the stability of the unique equilibrium in the following particular way: 46 Chapter 1. Preliminaries •there exists τ∗∈(0,∞) such that the origin is GAS for (1.16) and any τ∈[0, τ∗). •there exists ¯τ∈(0,∞) such that the origin is unstable for (1.16) and any τ∈(¯τ, ∞). τ∗¯τ ODE τ= 0 Difference equation ‘τ=∞’ DDE Figure 1.9: Global picture of the role of the delay τin the dynamics of (1.16) when the equilibrium is unstable for the corresponding difference equation (1.20) in the conditions of Remark 1.67. The unique equilibrium pof (1.16) is stable (green, left zone) for small values of the delay, and unstable (orange, right zone) for large values of it. Notice that, for general feedbacks, it is not known whether τ∗= ¯τ. In other words, the cases of S-maps that fall into the region between the curves ρ2= 1 and ρ3= 1 in the (a, f0(p))-plane are not completely studied. Conjecture 1.68. [94, Conjecture 2.1] Let (a, f, τ)satisfy (T), pbe the unique root of f(x) = ax, and assume that fis an S-map. If pis LAS for the DDE (1.16), then it is also GAS. Nevertheless, in some particular cases, Conjecture 1.68 is true and the folklore statement ‘LAS implies GAS’ holds. See [79, Section 6.2] and the references therein to find a case with a > 0 where there is an affirmative answer to the above-mentioned conjecture. Moreover, there is also a very well-known particular case for a= 0 that has acquired great attention in the last decades, which is related with the cases in this section. We refer to the famous Wright’s equation x0(t) = βx(t−1)(1 + x(t)).(1.40) Equation (1.40) is not included in the framework of the current Section 1.3, but, after a change of variables, it can be transformed into x0(t) = β(e−x(t−1) −1) =: f(x(t−1)),(1.41) which is just an example of the equations we have been treating in the current section with τ= 1 and the limit case a= 0. Notice that the unique equilibrium of (1.41) on I=Ris p= 0 and the linearised equation of (1.41) at p= 0 is x0(t) = f0(0)x(t−1) = −βx(t−1), 47 Sebasti´ an Buedo Fern´ andez so that the unique equilibrium of (1.41) is LAS provided that β∈(0, π/2). In fact, fis an S-map. Wright proved [136] that the unique equilibrium of (1.41) is GAS provided that the condition β∈(0,3/2] holds and the problem of extending such a result up to β=π/2 was later named as the Wright’s conjecture [71, 79, 132] (compare with Remark 1.64 and Figure 1.8). This conjecture remained unproven until few years ago, when a combination of the works by B´anhelyi et al. [9] and by van der Berg and Jaquette [132] finally achieved this landmark. Certainly, the issue ‘LAS implies GAS’ is not only restricted to feedbacks satisfying certain hypotheses regarding the sign of its Schwarzian derivative. Following [79, Section 6.2] and some references therein, there are different kind of maps that satisfy certain a relation between its derivative f0(x) and the quotient f(x)/x for which the property ‘LAS implies GAS’ is also true. We also refer to the works [43], in the context of difference equations, and [41], regarding its application to DDEs, to consult recent results showing how the negative condition on the Schwarzian derivative can be overcome, and to complement the results exposed above when it comes to the quest of such property. We conclude this chapter with one important issue that we remarked before. In fact, we have been considering the DDE in (1.16), which has a constant delay. This equation is generalised by the non-autonomous one (of variable delay) in (1.15). It is also natural to wonder whether the unique equilibrium of (1.15) has some attractivity properties. Since the DDE in (1.15) is non-autonomous, it does only fit with the very beginning of Section 1.2; nevertheless, it is still possible to extend the above-stated results. We refer to the works [64, 86] for further details on this adaptation. While [64] is related to the extension of Corollary 1.48, one can observe that Lemmas 4.5 and 4.7 and Theorem 2.1 in [86] serve as a source to check the validity of the estimates in Theorem 1.60, Remark 1.62 and Theorem 1.63 to the non-autonomous case (1.15). We sum up these cases in the last result of this chapter. Theorem 1.69. Let ˜τ:R→R+be a bounded continuous function and τbe its supremum. Assume that (a, f, τ)satisfy (T) and let pbe the unique equilibrium of F=f/a. The following assertions hold: •If pis a global attractor for F, then pis a global attractor for equation (1.15). •If fis an S-map and f0(p)∈[−a, a)or [f0(p)<−a and τ < ρ2(a, f0(p)) ] ,(1.42) then the unique equilibrium pof (1.15) is GAS. 48 Chapter 1. Preliminaries In fact, regarding a proof of the first point in the last theorem, we anticipate that a more general result, in the line of [64], will be needed in Section 2.2.2, namely Theorem 2.32. Therefore, the reader can also check the proof there. The situation shown in Theorem 1.69 seems to provide some analogy with the constant delay case. Nevertheless, there is a major issue that deserves to be highlighted: as it is recognised in [86], condition (1.42) is sharp: in case τwere at least ρ2(a, f0(p)), it would be possible to find a certain delay function ˜τ(t) such that τis its supremum and the equilibrium pis not GAS. This means that the result is optimal as it provides a common estimate for every feedback f, parameter aand bounded continuous delay ˜τ(t). However, as it has been mentioned before, there exist some particular cases, including some of constant delay for which the ‘stability limit’ is even larger and the issue ‘LAS implies GAS’ remains open (see Conjecture 1.68 and the comments below). 49 Sebasti´ an Buedo Fern´ andez Since we are mostly interested in globally attracting equilibria, we avoid the possibility of having more than one equilibrium by focusing on the case 0 ≤γ≤1. For these values of γ, the use of equation (2.13) in population dynamics is intended to gain flexibility to fit population data. For instance, when H(x) = βe−δx,β, δ > 0, the equation (2.13) provides the gamma-model included in the list of spawner-recruit models in the book by Quinn and Deriso [114]3. For example, Zheng and Kruse [148] found that this model fits well the stock-recruitment data for three Alaskan crab stocks; for more references, see [81]. Moreover, the difference equation corresponding to (2.11), generated by the γ-logistic map (in analogy to the so called γ-Ricker map [81]) has been used by Avil´es [6] in the framework of cooperative interaction in a group of individuals (for 1 ≤γ≤2), and by Eskola and Parvinen [35] in the context of populations with Allee effects (for γ= 2). Finally, we refer to the work [5], which deals in a general way with delay differential equations having destruction and production terms like in (2.7). Once more, we recall the reader the existence of [5, Table 1], where many references regarding applications of equation (2.5) to several fields are gathered. Our main results in this chapter, which can be found in Section 2.2, are organised according to the application on Economics that we have recalled above. Nonetheless, they do not lose their meaning in population dynamics; it is just a matter of the name of parameters. In fact, bearing in mind the meaning of γon both applications, we focus on the case γ∈(0,1), which allows us to work with a single positive equilibrium and links the limit cases γ∈ {0,1}. Those limit cases can also be studied under our approach, yet we warn the reader that some further special considerations should be given. 2.2 Global dynamics of several gamma-models To start with, we provide a common framework that includes all the mentioned delaydifferential gamma-models. Within the latter framework, we say that the delay differential equation x0(t) = −ax(t) + s(x(t))xγ(t−˜τ(t))k(x(t−˜τ(t))) (2.14) is an example of a delay differential gamma-model. In this chapter, we will assume the following set of hypotheses regarding equation (2.14): (A) a > 0, γ∈(0,1), ˜τ:R→[0,∞) is a bounded continuous function and τ > 0 is its supremum. (B) s, k : [0,∞)→(0,∞) are nonincreasing functions of class C1. 3This is why we use the name gamma-model for, e.g., equation (2.13). 56 Chapter 2. Gamma-models We could have defined sand kon (0,∞), since we are only interested in the dynamics of positive solutions, but having a certain regularity at x= 0 and thus the bounded character of such functions and their derivatives facilitates the study. Moreover, equation (2.14) includes the most general model (2.7) that we have mentioned in Section 2.1 by considering kas the product of the constant Band the pollution function p2. In fact, it is more general than (2.7) since we allow the delay to be variable. In the following subsections, the reader can find different choices for the functions involved in equation (2.14). We split the section into the following two parts. Firstly, in Subsection 2.2.1, we assume a constant saving rate term in (2.14). This equation belongs to the well-known family of differential equations with instantaneous linear decay and delayed feedback that has been introduced in Section 1.3, which were partially studied via the corresponding difference equation. Then, we consider different choices of pollution function, yielding equations that have already been considered in the literature for the particular cases γ= 0 and/or γ= 1. We provide sharp delay-independent and some delay-dependent conditions for the positive equilibrium to be GAS for each of those choices, which highlights the key role of the pollution function (or the per capita production function in the context of population dynamics) in the stability properties of neoclassical models with delay. Moreover, we analyse how the parameter γinfluences both the value and the stability of the equilibrium, sometimes in a subtle way depending on the other parameters of the model, a fact that may lead to stability windows in the bifurcation diagram. Secondly, in Subsection 2.2.2, we deal with equation (2.14) with variable saving rate term. A suitable framework to look for such case comes from the family of delay differential equations considered by Ivanov et al. in [64], namely, x0(t) = −g1(x(t))f2(x(t−τ)) + f1(x(t−τ))g2(x(t)),(2.15) where f1, f2, g1, g2are positive functions. We shall give further details about the analysis of equation (2.15). Nevertheless, we will also relate it with a particular difference equation, using analogous techniques to those in Section 1.3. In fact, we generalise a known global stability result for equation (2.15), allowing variable delays, and apply it to some examples of (2.14). 2.2.1 Constant saving rate In this subcase, where s(x) = ¯s > 0, the main equation (2.14) reads as x0(t) = −ax(t) + xγ(t−˜τ(t)) h(x(t−˜τ(t))),(2.16) 57 Sebasti´ an Buedo Fern´ andez with h(x) := ¯sk(x). The equation (2.16) is actually included in the setting of equation (1.15) by simply assuming f(x) = xγh(x). In fact, notice that assumption (B) directly holds for (2.14) since h: [0,∞)→(0,∞) is a nonincreasing function of class C1. In order to study the global dynamics of (2.16) as in Section 1.3, we now relate the framework of (2.16) to the hypotheses in (T) (see page 27). Hence, we have to analyse the main features of F:= f aor the difference equation corresponding to (2.16), namely xn+1 =F(xn) = xγ nH(xn), H := h a,(2.17) which may be known as an example of a discrete gamma-model. Note that F(or f) has been split into a term xγand the remaining part H(respectively, h), which is a nonincreasing function. The shape of the nonlinearity H(or h) will lead to different behaviours. To start with, we remark that the function F:I→I, with the choice I= (0,∞), is well-defined and of class C1on I. In fact, one has more information, since F(x)→0 and F0(x)→ ∞ as x→0+; and F(x)→0 as x→ ∞. Moreover, a, τ > 0 will be provided under condition (A). The existence of a unique positive root pof F(x) = xis shown in the following result. Besides, Theorem 2.1 also shows the role of the parameter γon the value of p and reunites some common and underlying analytical features of those gamma-models shown in [17, 20, 81, 82, 83], which will be useful in the subsequent pages. Particularly, it especially resembles the spirit and the ideas of the proofs for the particular cases in [81, Theorem 2], [82, Theorem 5.1] together with the general flavour of [83, Theorem 1]. Theorem 2.1. Let a > 0and h: [0,∞)→(0,∞)be a nonincreasing function of class C1. Then, for each γ∈(0,1), there is a unique positive fixed point of the map Fin (2.17), namely p:= p(γ)∈(0,∞), which is the unique root of equation p1−γ=H(p). Moreover, we obtain ∆F 1(x) = (F(x)−x)(x−p)<0, x ∈(0,∞)\{p}. Besides, one of the following cases holds: •p(γ)<1is decreasing on (0,1), or, equivalently, H(1) <1; •p(γ) = 1, for every γ∈(0,1), or, equivalently H(1) = 1; •p(γ)>1is increasing on (0,1), or, equivalently, H(1) >1. 58 Chapter 2. Gamma-models Finally, we have F0(p) = γ+pγH0(p).(2.18) Proof. Let γ∈(0,1). If pis a fixed point of F, then p=F(p) = pγH(p) (2.19) which is equivalent to p1−γ=H(p), Since 1 −γ > 0, the function p1−γis increasing, which combined with the fact of Hbeing nonincreasing, leads to the existence of a unique positive root of (2.19). From the fact that there is a unique positive fixed point pof F: (0,∞)→(0,∞), we deduce that (F(x)−x)(x−p)6= 0 on (0,∞)\ {p}. Nevertheless, one can obtain more information from lim sup x→∞ F(x) x= lim sup x→∞ xγ−1H(x)≤H(1) lim sup x→∞ xγ−1= 0, lim inf x→0+ F(x) x= lim inf x→0+xγ−1H(x)≥H(1) lim inf x→0+xγ−1=∞, where we have used that His nonincreasing. Clearly, this can only be possible if ∆F 1<0 on (0,∞)\{p}. Consider the auxiliary function ζ: (0,∞)×(0,1) of class C1defined by ζ(p, γ) := pγ−1−1 H(p). Clearly, every positive fixed point of Fsatisfies the equation ζ(p, γ) = 0. Moreover, the inequality H0≤0 yields ∂1ζ(p, γ) = (γ−1)pγ−2+H0(p) H(p)2<0,for every (p, γ)∈(0,∞)×(0,1).(2.20) Therefore, we may apply the Implicit Function Theorem to ζand derive that there exists a function p(γ) of class C1on (0,1), which satisfies ζ(p(γ), γ) = 0 and p0(γ) = −∂2ζ(p(γ), γ) ∂1ζ(p(γ), γ)=pγ−1ln(p) −∂1ζ(p, γ), where we have finally written p=p(γ) for simplicity. Since the denominator in the last expression is positive from (2.20), so as pγ−1, then the sign of p0(γ) coincides with the sign of ln(p), and thus, on the relative position of pwith respect to the value 1. 59 Sebasti´ an Buedo Fern´ andez Besides, p(γ) = 1 for every γ∈(0,1) is equivalent to H(1) = 1 via (2.19). Otherwise, if p(γ) is not the constant function 1, we can finish this part of the proof by using the condition (F(x)−x)(x−p(γ)) <0, x ∈(0,∞)\{p(γ)} evaluated at x= 1, that is, 0>(F(1) −1)(1 −p(γ)) = (H(1) −1)(1 −p(γ)). In other words, the relative position of pwith respect to 1 is known from the relative position of H(1) with respect to 1. Finally, F0(p) = γpγ−1H(p) + pγH0(p) = γ+pγH0(p), where expression (2.19) has been used once more. Remark 2.2. Notice that, regarding equations (2.16) and (2.17), we consider F=f/a. Thus, f0(p) may be computed as f0(p) = aγ +γ+h0(p). Additionally, it is important to highlight that (2.18) implies F0(p)<1, f0(p)< a, which will be relevant from the results given in Subsection 1.3.3 (see, in particular, Remark 1.64 and Theorem 1.69). Regarding (T4) (see page 27), there exists a non-degenerate compact set K⊂I= (0,∞), which is a globally attracting set for the difference equation (2.17). This can be shown via standard arguments, supported by a cobweb analysis. In fact, let q= supx∈(0,p)F(x). If q=p, then we can choose K={p}. If q > p, then our choice could be K= [w, q], where w= minx∈[p,q]F(x)>0. All the hypotheses in (T) have been translated into the framework of gamma-models. Thus, we actually know what do we need to impose to equation (2.16) in order to apply the general results of existence and uniqueness of solutions and their continuous dependence on initial data from Section 1.1. Following [86, Theorem 4.1] and allowing an increasing function ftoo, we adapt the next result from [84, Theorem 3.1]. Theorem 2.3. Assume that (A) holds and h: [0,∞)→(0,∞)is a nonincreasing function of class C1. If φ∈C(0,∞), then there is a unique solution x(t; (0, φ)) of (2.16) through (0, φ), which is defined on [−τ, ∞), positive and lim inf t→∞ x(t; (0, φ)),lim sup t→∞ x(t; (0, φ))⊂K, (2.21) 60 Chapter 2. Gamma-models for a certain non-degenerate compact set K⊂(0,∞)that can be chosen independently of φ. Proof. Global existence and uniqueness of solutions come from the results in Chapter 1. The asymptotic behaviour displayed by (2.21) is obtained via [86, Theorem 4.1] and the comments before this result regarding the non-degenerate compact set K, which is globally attracting for the corresponding difference equation (2.17). The case of increasing f(x) = xγh(x), can be handled in a similar way. Remark 2.4. We have restricted our attention to γ∈(0,1), yet many of the results are also directly adaptable to the limit case γ= 0; in particular, it is the case of Theorem 2.1. The another limit case, γ= 1, would require the additional assumption lim x→∞h(x)< a < h(0) to ensure that there is a unique equilibrium for (2.23). As a last general comment, we remark that we will follow a common path for the analysis of the given models, which would include the particular expression of the functions f, F, h, H, stability properties about the unique equilibrium and the role of varying γwith respect those features. In particular, we provide a summary in Table 2.1 at the end of the section to facilitate the comparison between them. No pollution effects The first very particular case is the one of no-pollution effects. For instance, if one recalls what was explained in the introduction, we fall into the case p2(x) = 1 and, by virtue of the comments below (2.14), the function kappearing in such general delay-differential gamma-model is constant. Thus, if we rename the product of constants, we can work with (2.16), where the function h: [0,∞)→(0,∞) is defined by h(x) = β > 0, that is, we deal with equation x0(t) = −ax(t) + βxγ(t−˜τ(t)).(2.22) In this context, the difference equation corresponding to (2.22) is xn+1 =β axγ n=xγ nH(xn) = F(xn),(2.23) where H(x) = β/a, a constant function, so, by applying Theorem 2.1, the positive equilibrium is the unique root of p1−γ=H(p) or, in other words, p=p(γ) = β a1 1−γ .(2.24) 61 Sebasti´ an Buedo Fern´ andez Moreover, according to Theorem 2.1, we can view pas a function of γin (2.24) and analyse the role of γin the value of the equilibrium. Corollary 2.5. Let a, β > 0. If we denote by p(γ)the unique positive equilibrium of (2.23), given by (2.24) for γ∈(0,1), it satisfies the following properties: •If β a>1, then p(γ)>1for every γ∈(0,1) and it is increasing. •If β a= 1, then p(γ)=1for every γ∈(0,1). •If β a<1, then p(γ)<1for every γ∈(0,1) and it is decreasing. Under global stability conditions, the dependence of the unique equilibrium pon γ can also be seen as the role of γin the so-called ‘population abundance’ [81]. Moreover, the global attractivity of the unique equilibrium of (2.23) is easy to obtain. Theorem 2.6. Let a, β > 0,γ∈(0,1). Then the equilibrium pgiven by (2.24) is GAS for the difference equation (2.23). Proof. Clearly, the function F: (0,∞)→(0,∞) in (2.23) is increasing. By the first part of Theorem 1.55, we conclude that pis a global attractor for (2.23) for the set X=I= (0,∞). The asymptotic behaviour of the solutions of the DDE (2.22) is also quite simple. Theorem 2.7. Assume (A) and β > 0. Denote by x(t; (0, φ)) the unique solution of (2.22) through (0, φ), with φ∈C(0,∞)and by pthe unique equilibrium of (2.23) given by (2.24). Then, limt→∞ x(t; (0, φ)) = pfor any φ∈C(0,∞). Proof. It is a straightforward conclusion from Theorem 2.6 and the first part of Theorem 1.69. Remark 2.8. While the limit case γ= 0 of (2.22) is a simple ODE that has a unique equilibrium which is GAS, the one of γ= 1 cannot be considered under our framework (notice that kwould be constant; compare this situation with the contents of Remark 2.4). Pollution effects Unlike the previous subcase, where we considered (2.16) with k(x) being constant (so as the saving rate), we now consider a pure pollution effect, that is, a decreasing function k. Therefore, h= ¯sk in (2.16) and H=h/a in (2.23) will also be decreasing. 62 Chapter 2. Gamma-models The choice of the pollution function may lead to stability switches, as we will see in the first two choices, or may not, as we consider in the third example. Thus, it might be a critical choice. One relevant difference with respect to the no-pollution subcase is that the equilibrium may now lie on an interval of decrease of F, which profoundly affects not only the dynamics of the difference equation (2.17), but also the one of the corresponding DDE (2.16), which is the equation we are eventually interested in. As we have seen at the final part of Subsection 1.3.3, when it comes to stability-related results, we need to work with lower bounds for f0(p) or F0(p) by using the functions ξi, i∈ {1,2,3}. Since fand Ffunctions will now have particular expressions, we can wonder whether it is possible to find equivalent conditions in terms of their inner parameters to the global asymptotic stability of its unique equilibrium. Nevertheless, one great issue is that the expression of F0(p) includes the value p(see equation (2.18)), so, at a first glance, it seems that the computation of pis unavoidable. However, assume that we would like to know when does the inequality κ≤F0(p) hold, for a certain real number κ < 1. Then, from (2.18), such condition is equivalent to W(p) := −pγH0(p)≤γ−κ. Hence, whenever Whas an inverse (which would be strictly monotone), the condition κ≤F0(p) will be equivalent to a certain inequality concerning pand η:= W−1(γ−κ). Then, the condition (F(x)−x)(x−p)<0, x6=p, can be applied to ηto yield an equivalent inequality between ηand F(η) which avoids the computation of the equilibrium p. Instead of fully formalising this fact here in an abstract way, we introduce the reader to the particular cases that we treat, in which we handle an explicit expression of the auxiliary functions. Lasota equation Firstly, assume that the pollution function is selected as p2(x) = e−δx, with δ > 0. Then, by gathering the constant factors in the feedback under the notation β > 0, equation (2.16) takes the form x0(t) = −ax(t) + βxγ(t−˜τ(t))e−δx(t−˜τ(t)),(2.25) which will be referred to as the Lasota delay differential equation or, simply, the Lasota equation. Notice that equation (2.25) may also be considered as a gamma-version of the Nicholson’s blowflies equation. In this case, the function h: [0,∞)→(0,∞) appearing in (2.16) is defined by h(x) = βe−δx and, thus, it is a decreasing function of class C1. The corresponding difference equation of (2.25) is xn+1 =β axγ ne−δxn,(2.26) 63 Sebasti´ an Buedo Fern´ andez which will be called the gamma-Ricker difference equation. It is generated by the gamma- Ricker map F(x) = β axγe−δx.(2.27) We list some properties of the γ-Ricker map in the following result. It is based on Propositions 1 and 2, and Theorem 1 in [81]. As usual, we consider the notation H=h/a. Proposition 2.9. The map F: (0,∞)→(0,∞)defined by F(x) = β axγe−δx;a, β, δ > 0; γ∈(0,1); satisfies the following properties: (i) Fis of class C∞and lim x→0+F(x) = 0 = lim x→∞F(x),lim x→0+F0(x) = ∞. (ii) Fis a U-map, with a unique critical point at c=γ/δ, where Fattains its global maximum. (iii) There is a unique p=p(γ)∈(0,∞)such that F(p) = p, which is given by the unique root of p1−γeδp =β a. Besides, (F(x)−x)(x−p)<0for every x∈(0,∞)\{p}. (iv) SF(x)<0, for all x > c. (v) F0(p) = γ−δp. (vi) For any κ < 1, the inequalities κ≤F0(p)<1hold if and only if the following inequality is satisfied: β a≤eγ−κγ−κ δ1−γ =: Tκ,δ(γ).(2.28) Proof. Assertions (i)-(v) are mainly proven in [81], while some of them can also be seen as direct conclusions of the general Theorem 2.1, e.g., Assertion (iii). We also recall the proof of (v) to show how our notation works. In fact, notice that, by virtue of (2.18), we obtain F0(p) = γ+pγH0(p) = γ−δpγβ ae−δp =γ−δpγp1−γ=γ−δp, 64 Chapter 2. Gamma-models which is always less than 1. Moreover, regarding the proof of (vi), if κ < 1, then, κ≤F0(p)⇐⇒ δp ≤γ−κ⇐⇒ p≤γ−κ δ⇐⇒ Fγ−κ δ≤γ−κ δ ⇐⇒ β aγ−κ δγ e−δ(γ−κ δ)≤γ−κ δ⇐⇒ β a≤eγ−κγ−κ δ1−γ , where we have used the condition (F(x)−x)(x−p)<0 on (0,∞)\{p}and F(p) = pin the last equivalence in the first line. This obviously gives us condition (2.28). In Assertion (vi) in the last result, we obtained that F0(p) being bounded from below by a certain constant is equivalent to a particular condition that is independent from p, as we announced before starting the current part devoted to Lasota equation. Bear in mind that, in this particular case, the computation of W−1has been easy, since W(p) = δp, while η=W−1(γ−κ) = γ−κ δ. By an application of Theorem 2.1 to the function Fin (2.27), the role of γin the value of the equilibrium for (2.26) or (2.25) is as follows. Corollary 2.10. [81, Theorem 2] Let a, β, δ > 0. If we denote by p(γ)the unique positive fixed point of the γ-Ricker map (2.27) with γ∈(0,1), then p(γ)satisfies the following: •If β a> eδ, then p(γ)∈(1,∞)is increasing. •If β a=eδ, then p(γ)=1, for every γ∈(0,1). •If β a< eδ, then p(γ)∈(0,1) is decreasing. According to the properties of the γ-Ricker map shown in Proposition 2.9, including that Fis a S∗-map and that (F(x)−x)(x−p)<0 on (0,∞)\{p}, it is possible to apply Theorem 1.55 to the γ-Ricker difference equation. Theorem 2.11. [81, Theorem 1(A)] Let a, β, δ > 0,γ∈(0,1) and denote by pthe unique positive fixed point of the γ-Ricker map. Then pis GAS for the γ-Ricker difference equation (2.26) if and only if β a≤eγ+1 γ+ 1 δ1−γ .(2.29) After having recalled the main features of the function Fand the role of the value of γon them, we are in a position to analyse the corresponding DDE, namely the Lasota equation in (2.25). The following result provides sufficient conditions for the positive equilibrium pof (2.25) to be GAS. 65 Sebasti´ an Buedo Fern´ andez 1 1 0x y Figure 2.3: The γ-logistic map for β a= 1.55 and γ= 0.2,0.5,0.925 with increasing darkness in γ. analogously called the gamma-logistic map. To ensure that Fmaps (0,1) into (0,1), and hence to deal with a well-defined difference equation in (2.37), we need to impose a consistency condition. Once we have ensured the mentioned property, then, no matter how we extend hin (2.36), the globally attracting compact set K⊂(0,∞) for (2.37) would in fact be a subset of (0,1) (recall the introductory part of Subsection 2.2.1). Therefore, all solutions of the modified γ-logistic DDE in the sense of (2.36) through an initial condition (0, φ), φ∈C(0,∞)will eventually enter and remain in (0,1). Unlike the Lasota equation (2.25), for which we had the work by Liz [81] to support us with many analytical features of its corresponding difference equation (2.26), we now need to study the properties of the γ-logistic map (2.38). In Proposition 2.16, we show some basic features of the function Fsuch as the existence of a unique fixed point. In Theorem 2.22, a global stability condition for equation (2.34) is provided. Finally, the influence of the parameter γon the stability and the value of the equilibrium is respectively analysed in Proposition 2.20 and Theorem 2.18. Once more, whenever it is needed, H=h/a. Proposition 2.16. The map F: (0,1) →(0,∞)defined by F(x) = β axγ(1 −x); a, β > 0; γ∈(0,1); satisfies the following properties: (i) Fis of class C∞and lim x→0+F(x) = 0 = lim x→1−F(x),lim x→0+F0(x) = ∞. 72 Chapter 2. Gamma-models (ii) Fis a concave U-map, with a unique critical point at c:= γ γ+1, where Fattains its global maximum. (iii) Condition β a<(γ+ 1) γ+ 1 γγ =: Γ(γ),(2.39) implies Im F⊂(0,1). (iv) There is a unique p=p(γ)∈(0,1) such that F(p) = p, which is given by the unique root of p1−γ 1−p=β a. Additionally, (F(x)−x)(x−p)<0for every x∈(0,1) \{p}. (v) SF(x)<0, for all x∈(0,1),x > c. (vi) F0(p) = γ−β apγ. (vii) For any κ < 1, the inequalities κ≤F0(p)<1hold if and only if the following inequality is satisfied β a≤(γ−κ)γ+ 1 −κ γ−κγ =: Tκ(γ).(2.40) Proof. Assertion (i) is trivial from the definition of F(2.38). The first, second and third derivatives of Fare, respectively, F0(x) = β axγ−1[γ−(γ+ 1)x], F00(x) = β aγxγ−2[γ−1−(γ+ 1)x], F000(x) = β aγ(γ−1)xγ−3[γ−2−(γ+ 1)x]. It is easy to check that F00 is negative using that 0 < γ < 1. It is also trivial that F0(c) = 0 is equivalent to c=γ γ+1. By using that F0(x)>0 for x < c and F0(x)<0 for x > c, one can affirm that cis a global maximum. This proves Assertion (ii). As cis a global maximum and Fis positive, F((0,1)) ⊂(0,1) is equivalent to 1> F(c) = β aγ γ+ 1γ1 γ+ 1, 73 Sebasti´ an Buedo Fern´ andez which proves Assertion (iii). Let z(x) := F(x)−x. Then, z(0) = 0, z0(0+) = ∞and z(1) = −1, so there exists at least one fixed point of Fin (0,1). Take pas the least positive fixed point of F, which exists because z0(0+) = ∞, which also implies that F(x)−x > 0, for 0 < x < p. Moreover z0(p)≤0 or, equivalently, F0(p)≤1. As F00(x)<0, pis the unique fixed point of Fin (0,1) and the proof of Assertion (iv) finishes. By doing some calculations, one can reach SF(x) = γ(γ+ 1)q(x) 2x2[γ−(γ+ 1)x]2, x ∈(0,1), x 6=c, where qis a polynomial of degree two defined by q(x) := −(γ+ 1)(γ+ 2)x2+ 2(γ+ 2)(γ−1)x−γ(γ−1). Since q00(x) = −2(γ+1)(γ+2) <0, for all x∈(0,1), q(0+)>0 and q0(0+)<0, then there is at most one root of qin (0,1). By checking q(c) = −3γ γ+1 <0, we conclude that there is a unique positive root x∗of q(x) on (0,1) and it satisfies x∗<c<1. Then q(x)<0, for every x > c. This implies SF(x)<0, for all x∈(0,1), x>c, proving Assertion (v). By applying Theorem 2.1, we have that Assertion (vi) follows from F0(p) = γ+pγH0(p) = γ−β apγ, which is, additionally, less than 1. Finally, if κ < 1, then κ≤F0(p)⇐⇒ β apγ≤γ−κ⇐⇒ p≤γ−κ β/a 1 γ ⇐⇒ F γ−κ β/a 1 γ!≤γ−κ β/a 1 γ ⇐⇒ β a γ−κ β/a 1−γ−κ β/a 1 γ!≤γ−κ β/a 1 γ ⇐⇒ γ−κ≤(γ+ 1 −κ)γ−κ β/a 1 γ , which yields the desired result regarding Assertion (vii). Once more, the computation of W−1and ηhas not been difficult since W(p) = βpγ/a. 74 Chapter 2. Gamma-models Remark 2.17. Condition (2.39) becomes relevant to continue the study because it ensures that the solutions of the difference equation (2.37) are well defined for all x0∈(0,1) and n≥0. Hence, in the following, we have to be careful since (2.39) shall hold. Although we have only supposed that a, β > 0, condition (2.39) implies that β a<4. In fact, it can be checked that the function Γ in the inequality (2.39) is increasing on (0,1), tends to 1 as γ→0+and tends to 4 as γ→1−. In other words, for any fixed a, β > 0 such that β/a ∈(0,4), if γ∈(0,1) is such that condition (2.39) is fulfilled, then such condition is also satisfied for those γ∈(γ, 1). Once more, Theorem 2.1 provides us with the role of γin the value of the equilibrium, which we state in the following result. Corollary 2.18. Let a, β > 0and γ∈(0,1) be such that the consistency condition (2.39) is fulfilled. If we denote by p(γ)the unique fixed point of the γ-logistic map (2.37) with γ∈(0,1), then the function p(γ)∈(0,1) is decreasing. Proof. It is trivial from Theorem 2.1 and lim x→1−H(x) = 0 <1. Notice that we have obtained that the value of the equilibrium pdecreases as the production function tends to be logistic in the classical sense (γ→1−). Theorem 2.19. Let a, β > 0and γ∈(0,1) be such that the consistency condition (2.39) is fulfilled, and denote by p∈(0,1) the unique fixed point of the γ-logistic map. Then, p is GAS for the γ-logistic difference equation (2.37) if and only if β a≤(γ+ 1) γ+ 2 γ+ 1γ .(2.41) Proof. By using Theorem 1.55, we can ensure that pis GAS for (2.37) if −1≤F0(p)<1, since SF(x)<0, for any x > c and ∆F 1<0 on (0,1) \{p}(see Assertions (iv) and (v) in Proposition 2.16). Moreover, −1≤F0(p)<1 is equivalent to (2.40) with κ=−1. By using Theorem 2.19 and its key condition (2.41), we can write a result about the influence of γon the stability of pfor (2.37) and each fixed a, β with β/a ∈(0,4). In fact, we obtain the values of β/a for which γproduces a stability switch in p. Proposition 2.20. Let a, β > 0such that β/a < 4. For any γ∈(0,1), let p:= p(γ)be the unique fixed point of Fon (0,1). The following assertions are valid: •If β≤a, then pis GAS for any γ∈(0,1). •If β∈(a, 3a), for those γ∈(0,1) for which condition (2.39) is fulfilled, the increment of γswitches the type of stability of p: from being unstable to being GAS. 75 Sebasti´ an Buedo Fern´ andez •If β≥3a, then pis unstable for any γ∈(0,1) for which (2.39) is fulfilled. Proof. We analyse the properties of the function T−1: (0,1) →R T−1(γ) := (γ+ 1) γ+ 2 γ+ 1γ , which is a particular case of (2.40) considered in the proof of Theorem 2.19. First of all, the function T−1is trivially positive. By taking logarithms, one can check that T0 −1(γ) = T−1(γ)2 (γ+ 1)(γ+ 2) + ln γ+ 2 γ+ 1>0. Then, lim γ→0+T−1(γ)=1,lim γ→1−T−1(γ) = 3. Since T−1(γ) is the right-hand side of (2.41), which provides the stability condition for p, the assertions in the statement of the theorem hold (see Figure 2.4). 1 1 3 4 0 Unstable GAS T−1 Γ γ β/a Figure 2.4: The stability diagram of the γ-logistic difference equation (2.37). The graph of the function Γ delimits from above the region of parameters that satisfy the consistency condition (2.39) (solid colours) and from below the ones that do not (filled with blue lines). The graph of the function T−1provides the curve of stability switch for its unique equilibrium p, below which we can find the region of parameters for which pis GAS (in green). Notice that the β/a axis has been rescaled. Remark 2.21. The sharp estimate for the unique equilibrium to be GAS for γ= 0 is β < a, while the limit condition (2.42) as γ→0+is β≤a. Notice that, while SF < 0 if 76 Chapter 2. Gamma-models γ∈(0,1], Fis a rational map if γ= 0 and, thus, SF = 0. One has to be aware that the limit case F0(p) = −1 might cause F2=Id [25, Page 995], which is a fact that does not happen with SF < 0, so that a tiny difference between conditions is somehow justified. Moreover, the limit condition of (2.42) as γ→1−is β/a ≤3 which coincides with the classical estimate for global asymptotic stability of the logistic map. We have studied the function Fin some detail and we are now ready to move into the stability properties of the corresponding delay differential equation (2.34). Theorem 2.22. Assume that (A) holds together with β > 0and that the consistency condition (2.39) is valid. Denote by x(t; (0, φ)) the unique solution of (2.34) through (0, φ), with φ∈C(0,1), and by pthe unique positive fixed point of the γ-logistic map (2.38). If β a≤(γ+ 1) γ+ 2 γ+ 1γ ,(2.42) then limt→∞ x(t; (0, φ)) = p, for every φ∈C(0,1). Moreover, condition (2.42) is the sharpest absolute stability condition. If we further assume that ˜τ(t) = τ > 0, for all t∈R, condition β a≤γ+1 1−e−ατ  γ+ 1 + 1 1−e−ατ γ+1 1−e−ατ !γ (2.43) is also sufficient to ensure that pis GAS for the autonomous DDE (2.34). Proof. The proof is completely analogous to the one in Theorem 2.12 but with the new definition of the function T−1(γ) from the expression in (2.40) instead of T−1,δ(γ) from the one in (2.28). The ideas in Remark 2.13 also apply for the previous proof. Moreover, Remark 2.14 is easily adapted to the γ-logistic DDE; in fact, such link between the absolute stability conditions for γ= 0 and γ= 1 comes from Remark 2.21. Moreover, we also include the corresponding version of Remark 2.15. Remark 2.23. If we assume that ˜τ(t) = τ > 0, for every t∈R, then pis LAS provided β a< γ+r1 + s2 a2τ2!  γ+1+q1 + s2 a2τ2 γ+q1 + s2 a2τ2  γ ,(2.44) 77 Sebasti´ an Buedo Fern´ andez where sis the unique root of −x tan(x)=aτ in (π 2, π). Condition (2.44) comes from the estimate F0(p)> ξ3(a, τ)/a; so an analogous condition is τ < ρ3(a, f0(p)) = ρ3(a, aγ −βpγ) = arccos a aγ−βpγ p(aγ −βpγ)2−a2.(2.45) Remark 2.24. The limit version of the local stability condition (2.45) as γ→1−coincides with the one in [102, Theorem 3], which is τ < arccos a 2a−β p(2a−β)2−a2. To see the latter, notice that lim γ→1−f0(p(γ)) = lim γ→1−(aγ −βp(γ)γ) = a−βp(1) = a−(β−a) = 2a−β, where we have used that the unique fixed point of the 1-logistic map (2.38) is p=β−a β. From the stability issues of the corresponding difference equation, one can give analogous conditions for equation (2.34) provided the consistency condition (2.39) is fulfilled. Since the case of constant delay allows us to deduce more information, we consider such assumption in the remaining part devoted to equation (2.34). We have seen that, if β/a ≤1, then the unique equilibrium pof the γ-logistic DDE (2.34) is GAS for any γ∈(0,1) and τ > 0. Moreover, if β/a > 1, then increasing γforces the equilibrium pto switch from instability to stability. In particular, if β/a ≥3, then there exists no absolute stability condition, that is, for any admissible fixed γ∈(0,1), there is a sufficient large τsuch that, if τ > τ, then pis unstable. Analogously to the Lasota equation, we give a stability diagram for a particular set of parameters that is of interest. In particular, we depict in Figure 2.5 the stability diagram of the γ-logistic DDE with constant delay, for β= 2.7 and a= 1. First of all, one has to get rid of all the cases that do not satisfy the consistency condition (2.39), that is, the ones with Γ(γ)≥β/a = 2.7. Thus, we only consider the cases of γ > γ, where γ≈0.5358 is the unique root of Γ(γ) = 2.7. For those cases, we can affirm that pis GAS regardless of the value of τ > 0 if γ≥γ∗, with γ∗≈0.8687. Moreover, if τ≤τ, with τ≈1.6451, then pis stable independently of the value of γ∈(0,1). In particular, the sufficient conditions that we have used ensure that pis GAS regardless of the value of γ∈(0,1) if τ≤τ∗, with τ∗≈0.9089. In Figure 2.6, we plot a couple of numerical situations corresponding to the case depicted in Figure 2.5 with τ= 3. They correspond to different values of γ(0.6 and 0.9), which have been chosen to show the mentioned stability switch (see also Figure 2.5). 78 Chapter 2. Gamma-models γ τ 4 γ∗ γ τ∗ 4 τ 4 1 GAS Figure 2.5: Stability diagram for equation (2.34) with constant delay in the plane (γ, τ/4), with a= 1 and β= 2.7. The region with a lined blue pattern represents the values of the parameters that do not satisfy the consistency condition (2.39). The green solid line represents the boundary of the region where global asymptotic stability is guaranteed. The orange dashed curve shows the boundary of local asymptotic stability. Dashed gray lines correspond to threshold values of γand τ (see the text). Gamma-Mackey-Glass DDE Now, assume that the pollution function is p2(x) = 1 1+δxm, with δ, m > 0. With this choice, equation (2.16) takes the form x0(t) = −ax(t) + βxγ(t−˜τ(t)) 1 + δxm(t−˜τ(t)),(2.46) which will be called the gamma-Mackey-Glass DDE. This choice means that the function h: [0,∞)→(0,∞), defined by h(x) = β 1+δxm, is decreasing and of class C1. Analogously, the map F(x) = βxγ a(1 + δxm)(2.47) will be referred to as the gamma-version of the Maynard Smith and Slatkin map, which generates the difference equation corresponding to (2.46), xn+1 =βxγ n a(1 + δxm n),(2.48) from now on called the gamma-version of the Maynard Smith and Slatkin difference equation in analogy with the model of Maynard Smith and Slatkin [104, 121], which assumes γ= 1. 79 Sebasti´ an Buedo Fern´ andez 0 20 40 60 80 100 120 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 t x 0 10 20 30 40 50 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 t x Figure 2.6: Numerical simulations for the solutions of equation (2.34), with α= 1, β= 2.7, τ= 3, and different values of γ. For γ= 0.6 (left), the equilibrium pis unstable, and there are sustained oscillations; for γ= 0.9 (right), pis globally asymptotically stable. These pictures have been obtained via the use of MATLAB and, in particular, its solver dde23. Concerning the classical map of Maynard Smith and Slatkin, a well-known particular case is the Beverton-Holt map, for which m= 1 (see [82, Section 1] for further information). Hence, the map in (2.47) with m= 1 can be called the gamma-Beverton-Holt map and, respectively, the difference equation it generates, the gamma-Beverton-Holt difference equation. The latter equation and its corresponding DDE (equation (2.46) with m= 1) are simpler to study. Thus, let us first state the following two results regarding this choice and then explain the general case. The following result summarises the main properties of the γ-Beverton-Holt map, and can be derived from Proposition 1 and Theorem 3.1 in [82]. Its third point can also be seen as a corollary of Theorem 2.1. Proposition 2.25. The map F: (0,∞)→(0,∞)defined by F(x) = βxγ a(1 + δx);a, β, δ > 0; 0 < γ < 1; (2.49) satisfies the following properties: (i) Fis of class C∞and lim x→0+F(x) = 0 = lim x→∞F(x),lim x→0+F0(x) = ∞. (ii) Fis a U-map, with a unique critical point at c=γ δ(1−γ), where Fattains its global maximum. 80 Chapter 2. Gamma-models (iii) There is a unique p=p(γ)∈(0,∞)such that F(p) = p, which is given by the unique root of p1−γ(1 + δp) = β a. Moreover, (F(x)−x)(x−p)<0for every x∈(0,∞)\{p}. (iv) Equation F2(x) = xhas no positive solutions different from p. In fact, we have ∆F 2(x) = (F2(x)−x)(x−p)<0, x ∈(0,∞)\{p}. The role of γon the value of the unique positive fixed point pof the γ-Beverton-Holt map is as follows. Corollary 2.26. Let m= 1 and a, β, δ > 0. If we denote by p(γ)the unique positive fixed point of the γ-Beverton-Holt map (2.49) with γ∈(0,1), then p(γ)satisfies the following: •If β a>1 + δ, then p(γ)∈(1,∞)is increasing. •If β a= 1 + δ, then p(γ)=1, for every γ∈(0,1). •If β a<1 + δ, then p(γ)∈(0,1) is decreasing. Proof. It is a straightforward conclusion from Theorem 2.1, since H(1) = β a(1+δ). Notice that we could have provided Corollary 2.26 without imposing m= 1, since H(1) would also be equal to β a(1+δ)for any m > 0. Hence, this fact turns out to be useful for the case m6= 1. The next result follows as a direct consequence of Proposition 2.25 and the criterion in Theorem 1.49. Theorem 2.27. [82, Theorem 3.1] Let m= 1,a, b, δ > 0,γ∈(0,1), and denote by pthe unique positive fixed point of the γ-Beverton-Holt map. Then pis GAS for the γ-Beverton-Holt difference equation. Now, Theorem 2.27, combined with Theorem 1.69, yields the natural result concerning the long-term behaviour of the solutions of the DDE (2.46) with m= 1. Theorem 2.28. Let m= 1. Assume that (A) holds together with β, δ > 0. Denote by x(t; (0, φ)) the unique solution of (2.46) through (0, φ),φ∈C(0,∞), and by pthe unique positive fixed point of the γ-Beverton-Holt map. Then, limt→∞ x(t; (0, φ)) = p, for every φ∈C(0,∞). 81 Sebasti´ an Buedo Fern´ andez 2.34), so the generalisation of Solow’s equation given by (2.54) still predicts convergence to the steady-state capital-labour ratio. Second, even if a negative factor is introduced in the production function, the particular form of this factor may prevent instabilities due to large delays (see, e.g., Theorem 2.28). Roughly speaking, the ‘pollution function’ p2(x) needs to have a fast rate of convergence to zero as xtends to infinity. In contrast with other papers, our stability analysis focuses on global results and allows variable delays. In particular, we establish sharp delay-independent global stability conditions, and also prove that small constant delays cannot destroy the global stability of the equilibrium. The mathematical approach we use to get global stability results for (2.7) is not new. However, the use of the notion of strong attractor introduced in [90] has some advantages: on the one hand, the proofs are much simpler than in previous papers [64, 65]; on the other hand, it is easy to consider variable delays, providing more general results. Another interesting novelty of our results is that we use a generalisation of Allwright- Singer’s theory for maps with negative Schwarzian derivative [122], due to El-Morshedy and Jim´enez L´opez [33], that we have recalled in Theorem 1.55. This result is crucial in the analysis of equations (2.9) and (2.11) because, in contrast with other cases (γ= 0, γ≥1), the respective feedbacks f(x) = βxγe−δx and f(x) = βxγ(1 −x) do not have negative Schwarzian derivative everywhere if 0 < γ < 1: they are what we called S∗-maps. In this way, our results fill a gap in the stability theory of equation (2.9). As far as we know, equation (2.9) has been introduced for the first time by Lasota in 1977 [74], to model blood cell production (erythropoiesis). Lasota formulated a conjecture concerning the ergodic properties of (2.9), see [107]. An interesting biological interpretation of the parameter γin the model has been given by Mitkowski in his Ph.D. thesis [106]. It is related to disturbed erythropoiesis (dyserythropoiesis), when the feedback loop that regulates the production of cells in the red bone marrow does not work properly. Roughly speaking, γrepresents the degree of disturbance of the normal erythropoietic response. When γ= 0, the answer is correct, but, when γ > 0, the response is inhibited and the greater the inhibition is, the greater the value of γis. The authors of [102] have dealt with the study of local stability of some particular cases of the logistic delay differential equation with a linear decay term (2.11). In a similar manner to what was shown for Lasota equation (2.25), we have extended the study in [102] by providing global stability results for the delay-differential gamma-model (2.34). Notwithstanding the advances that we have been able to provide, we have also realised that not all the γ-versions of well-known delay-differential models can be studied in a clear manner via the Allwright-Singer-type of results. For instance, the study of the γ-Mackey- Glass DDE is based on that of the γ-version of the Maynard Smith and Slatkin map (2.48), for which, as far as we know, only results regarding its global stability for m= 1 88 Chapter 2. Gamma-models are available [82]; they are based on the Coppel-type criterion recalled in Theorem 1.49. If m6= 1, no simple way to apply the known tools was found, a fact that suggested starting a quest of new sufficient conditions to check when an equilibrium is GAS for a difference equation. Generally speaking, some further work needs to be done in order to extend the global stability conditions and, in particular, whether the shown models satisfy the ‘LAS implies GAS’ property. Yet another interesting problem would be to study equation (2.14) with state-dependent delay. It is remarkable that the same equation with different values of γhas been used for different mathematical models governed by delay differential equations: •For γ= 0, it is a model for blood-cell production, proposed by Wa˙zewska-Czy˙zewska and Lasota [135], and later modified by Lasota [74], allowing positive values of γ. •For 0 < γ < 1, it is a model in Economics, proposed by Matsumoto and Szidarovszky [103], as a generalisation of the fundamental Solow’s equation [126]. •For γ= 1, it is the famous equation introduced by Gurney, Blythe and Nisbet [49] to explain some qualitative aspects of Nicholson’s classical experiments on laboratory cultures of sheep blowflies. •For γ > 1, equation (2.9) has been proposed to study the dynamics of single-species populations subject to Allee effects [60, 92]. Finally, it is worth mentioning that the Lasota equation (2.9) can be seen as a continuous version of the γ-Ricker map, which has been used as a flexible discrete model for animal populations, and in the context of cooperative interaction in a group of individuals [81]. 89 Chapter 3 A Schwarzian-type formula for sharp global stability criteria We present a new formula that makes it possible to get sharp global stability results for one-dimensional discrete-time models in an easy way. In particular, it allows to show that the local asymptotic stability of a positive equilibrium implies its global asymptotic stability for a new family of difference equations that finds many applications in population dynamics, economic models, and also in physiological processes governed by delay differential equations, which have been named gamma-models and which have been studied in Chapter 2. The main ingredients to prove our results are the Schwarzian derivative and some dominance arguments. Finally, we derive some relevant conclusions concerning the dynamics of the so-called gamma-Mackey-Glass models. This chapter explains and extends a great part of the work developed in [83] by Eduardo Liz1and the author of this thesis (S. Buedo-Fern´andez2). Further details of the former article may be found in its complete reference below. E. Liz, S. Buedo-Fern´andez. A new formula to get sharp global stability criteria for onedimensional discrete-time models. Qualitative Theory of Dynamical Systems 18, 813–824 (2019). Springer, ISSN 1662-3592 (Electronic) 1575-5460 (Print). 3.1 Introduction Sufficient conditions to ensure the global attractivity of an equilibrium for a scalar difference equation are facilitated if the function being iterated has negative Schwarzian 1Departamento de Matem´atica Aplicada II, Campus Marcosende, Universidade de Vigo, 36310 Vigo, Spain. 2Departamento de Estat´ıstica, An´alise Matem´atica e Optimizaci´on, Facultade de Matem´aticas, Campus Vida, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. 91 Sebasti´ an Buedo Fern´ andez derivative in a certain subinterval of its domain. In such case, as we have seen before, the global attractivity study is moved to the easier task of checking the local asymptotic stability of the equilibrium. Many well-known models and their variants satisfy those types of assumptions concerning the Schwarzian derivative. However, not all maps in population or economic models satisfy such conditions, as shown in Chapter 2. Thus, one may be interested in searching for alternative conditions that preserve the ‘LAS implies GAS’ property [43]. Another useful tool, enveloping, has been recalled in Theorem 1.51, which, generally speaking, involves a comparison of a map with another one that has the same equilibrium and it is known to be globally attracting. In fact, the approaches in [33] and [78] combine both tools (Schwarzian derivative and dominance). Our main aim in this chapter consists in proving sharp global stability results for the family of difference gamma-models given in Section 2.2.1. We recall their general expression xn+1 =xγ nH(xn) =: F(xn), n ∈Z+,(3.1) where γ≥0, H: [0,∞)→(0,∞) is a smooth function, and any solution of equation (3.1) is a sequence (xn)n∈Z+starting at any initial condition x0>0. The main finding shown in this chapter is a new formula – stated as the forthcoming equation (3.3) – which provides sharp global stability results in an easy way. This formula is applicable in many cases when the criterion of the Schwarzian derivative does not work. In this way, it extends the classical Allwright-Singer result [2, 122]. To illustrate this fact, consider the following equivalent form of (3.3) suggested by V´ıctor Jim´enez L´opez: SH(x)<H(x)−xH0(x) 2(xH(x))2(xH(x))0,∀x > 0.(3.2) According to our main result, Theorem 3.1, condition (3.2) ensures the global asymptotic stability of the unique equilibrium pof (3.1) if H: [0,∞)→(0,∞) is decreasing, γ∈[0,1], and pis locally asymptotically stable. For instance, if γ= 0, condition (3.2) is strictly weaker than the classical condition SH(x)<0 when His decreasing but xH(x) is an increasing function. In fact, in such a case, we provide a positive (functional) upper bound for SF. For example, consider the map F(x) = a/(1 + x)m, with a > 0, m > 0. Such map is of the form (3.1), with γ= 0 (i.e., H=F) and satisfies SF(x) = (1 −m2)/(2(1 + x)2). If m > 1, then SF(x)<0 for all x > 0, and therefore the Allwright-Singer result applies (see also Theorem 1.55). However, if 0 < m ≤1, then SF(x)≥0 for all x > 0, so this classical result cannot be applied. Nonetheless, as remarked above, since xF(x) is increasing, condition (3.2) is sharper than the classical estimate, and, in fact, this turns out to be useful, since SF satisfies (3.2). Therefore, our results ensure the global stability 92 Chapter 3. A Schwarzian-type formula for sharp global stability criteria of the unique positive equilibrium p(notice that 0 > F0(p) = −mp/(1 + p)>−1 if 0< m ≤1). The principal idea to ensure that (3.3) yields a ‘LAS implies GAS’ type result regarding (3.1) is to deal with certain consecutive changes of variables that transform such difference equation into one that is easier to handle. Moreover, in Section 1.3, the simple topological conjugacy Θ(x) = x−pwas also proposed to shift the unique equilibrium pto the origin and, thus, to provide a unifying treatment for all the maps considered therein. Bearing in mind such aims, we define the change of variables Λ(x) = −ln(x/p), which shifts the unique equilibrium pto the origin but also becomes essential since it modifies the form of the right-hand side of the discrete gamma-model in (3.1): the product xγH(x) is transformed into a certain sum γx +G(x), where the function Gis defined in terms of H. Thus, by using some known tools as enveloping or functions with negative Schwarzian derivative, any condition on Gthat is obtained regarding the asymptotic behaviour of the solutions of the difference equation xn+1 =γxn+G(xn) would translate into conditions on Hin the original gamma-model (3.1). Therefore, the term H, which profoundly affected the dynamics, as we have seen in Chapter 2, can be somehow isolated; this was not the case in that chapter, where we have studied the function F(x) = xγH(x) as a whole. Two important features of the central result in this chapter are the following. On the one hand, it is quite general, allowing to recover some known results in a unified and simpler way, and to prove some new relevant results for applications. On the other hand, it is easily verifiable, as we show in the applications. Therefore, it may be a useful tool for researchers interested in the global stability of difference equations and delay differential equations. 3.2 Main result and examples For the main result in this section, we consider equation (3.1) with 0 ≤γ≤1. Next, we state the main assumptions for the map H. (H) H: [0,∞)→(0,∞) is of class C3and H0(x)<0 for all x > 0. For γ= 1, we also assume that lim x→∞H(x)<1< H(0). 93 Sebasti´ an Buedo Fern´ andez Note that, in case γ= 0, we are assuming that F=H, and thus, we assume that F is a decreasing M-map, a constraint that does not appear in the related Theorem 1.55, where unimodal maps were allowed too. We will make further comments regarding this particular case in Subsection 3.3. Besides, most of the reasoning is analogous for the case of a nonincreasing function H(i.e., H0≤0) if γ∈[0,1). Furthermore, if (H) holds, then there is a unique p > 0 such that H(p) = p1−γ, that is, a unique equilibrium pof the difference equation (3.1) (see Theorem 2.1 and recall that we usually denote H=h/a). Theorem 3.1. Assume that 0≤γ≤1,Hsatisfies (H), and the following condition holds: x2 2SH(x) + H0(x) H(x)2!<1,∀x > 0.(3.3) Then the local asymptotic stability of the unique positive equilibrium pin equation (3.1) implies its global stability. This stability condition is W(p) = −pγH0(p)≤1 + γ. (3.4) Remark 3.2. We notice that an application of Theorem 2.1 yields that condition (3.4) is equivalent to −1≤γ+pγH0(p) = F0(p). Moreover, in our framework, F0(p) is always less than 1 because 0 ≤γ≤1 and H0(p)<0. In this way, the stability condition (3.4) is equivalent to the local asymptotic stability of the equilibrium p. We give the proof of Theorem 3.1 in Section 3.4. Now we show its applicability by providing two classical examples from the literature about population dynamics. The second one gives absolute stability results for a generalisation of the Mackey-Glass equation [96]. Example 3.3. Consider the gamma-Ricker difference equation [81], previously considered in (2.26), xn+1 =β axγ ne−δxn, n ∈Z+,(3.5) where a, β, δ > 0 and 0 ≤γ≤1. In this case, we easily get that H(x) = β ae−δx, SH(x) = −δ2 2,H0(x) H(x)=−δ. 94 Chapter 3. A Schwarzian-type formula for sharp global stability criteria Thus, 2SH(x) + H0(x) H(x)2 = 0,∀x > 0, and (3.3) trivially holds. Next, condition (3.4) is equivalent to pδ ≤1 + γ, which in turn is equivalent to the global stability condition (see [81] or Assertion (vi) in Theorem 1.3): β a≤eγ+1 γ+ 1 δ1−γ . Therefore, likewise Theorem 1.55, an application of Theorem 3.1 also provides the global stability criterion for the γ-Ricker map shown in Theorem 2.11. For γ= 1, we need the extra condition β > a, so one gets, once more, the well-known global stability criterion 1< β/a ≤e2for the classical Ricker map. Example 3.4. Consider xn+1 = β axγ n 1 + δxm n , where a, β, δ, m > 0 and 0 ≤γ≤1. We recall that the latter equation is a generalisation of the population model proposed by Maynard Smith and Slatkin [104, 121] (they consider the particular case of γ= 1), which was introduced in Section 2.2.1. We now have H(x) = β/a 1 + δxm, SH(x) = 1−m2 2x2,H0(x) H(x)=−δmxm−1 1 + δxm. Thus, x2 2SH(x) + H0(x) H(x)2!= 1 −m2+m2δxm 1 + δxm2 <1,∀x > 0, and condition (3.3) is satisfied. We leave further comments regarding the global stability to Section 3.5, since the difference equation of this example is related with the DDE that remained unstudied in Chapter 2. 3.3 The case γ= 0 The case γ= 0 deserves special attention. Firstly, since our assumptions yield F=H, Theorem 3.1 with γ= 0 provides a new formula to prove global stability when the sign of the Schwarzian derivative of Fis non-constant. Nevertheless, since F=H, it would only 95 Sebasti´ an Buedo Fern´ andez be valid for a decreasing function F, letting aside the unimodal case that was actually considered in results like Theorem 1.55. Hence, it is worth formulating a more general result in order to include U-maps. Its proof is also given in Section 3.4. Theorem 3.5. Assume that the C3map F: (0,∞)→(0,∞)is either decreasing or unimodal (with a unique critical point c, which is a local extremum), and has a unique positive fixed point p, such that F(x)> x if x < p and F(x)< x if x > p. If F0(p)≥ −1 and (3.3) holds for all x > 0such that F0(x)6= 0, then pis a global attractor for xn+1 =F(xn), n ∈Z+.(3.6) We give several examples that show the applicability of Theorem 3.5. Example 3.6. In [89], the study of the absolute global stability of the positive equilibrium of a commodity market model governed by a delay differential equation was reduced to the global stability of the positive equilibrium of the following difference equation: xn+1 =−b+a(d+xm n) cxm n1/k =: F(xn),(3.7) where a, b, c, d, m > 0, k≥1, and bc < a. The map F: (0,∞)→(0,∞) is decreasing, and [89, Lemma 2.5] ensures that (SF)(x)<0 holds for all x > 0 if and only if m > k. However, Fsatisfies (3.3) also in the case m≤k, so a direct application of Theorem 3.5 proves that the local asymptotic stability of the equilibrium implies its global stability. Indeed, direct computations lead to x2 2(SF)(x) + F0(x) F(x)2!−1 = −(a−bc)m2xm(2ad + (a−bc)xm) (ad + (a−bc)xm)2<0, since, by hypothesis, a−bc > 0. In [89], the case m≤kwas solved using the Coppel-type result in Theorem 1.49. The previous example corresponds to a monotone map F. The following one involves a unimodal map. Example 3.7. Consider F(x) = β a√xe−x, which is a particular case of (3.5) with γ= 1/2 and δ= 1. We recall that SF does not have constant sign on (0,∞) (see [81]). However, x2 2(SF)(x) + F0(x) F(x)2!=1−8x (1 −2x)2<1,∀x > 0, x 6= 1/2. Hence, Theorem 3.5 guarantees that the condition β2 a2≤3e3/2 for the local asymptotic stability of the positive equilibrium implies its global asymptotic stability. 96 Chapter 3. A Schwarzian-type formula for sharp global stability criteria An important remark is that, for some difference equations, both Theorem 3.1 and Theorem 3.5 can be applied, as it happens with the γ-Ricker difference equation (compare Examples 3.3 and 3.7). In these cases, Theorem 3.1 is usually easier to use. An additional example showing that Theorem 3.1 works in some situations which do not fall into the scope of Theorem 3.5 is the following one. Example 3.8. Consider the generalised gamma-Ricker difference equation (see, e.g., [78] for the case γ= 1): xn+1 =xγ nα+ (1 −α)er(1−xn)=: xγ nH(xn), n ∈Z+,(3.8) where r > 0, 0 ≤γ≤1, 0 ≤α < 1. Since the map F(x) = xγH(x) can have two critical points, Theorem 3.5 does not always apply. However, one can check that 2SH(x) + H0(x) H(x)2 =−r2+r2(1 −α)er(1−x) α+ (1 −α)er(1−x)2 ≤0,∀x > 0, and therefore (3.3) holds. 3.4 Proofs The main ingredient to prove Theorem 3.1 is the following generalisation of Theorem 2.3 in [78], devoted to the particular case γ= 1. Theorem 3.9. Assume that 0< γ ≤1and that there exist constants α > 0,k≥0such that the continuous function G:R→Rsatisfies Rα,k(y)< G(y)<0,∀y > 0and 0< G(y)< Rα,k(y),∀y∈(−1/k, 0) ,(3.9) where Rα,k(y) = −αy/(1 + ky), and in case k= 0, by −1/k we mean −∞. If γ−α≥ −1, then the origin is GAS for equation yn+1 =γyn+G(yn) =: T(yn).(3.10) Proof. Assume first that k > 0. This case will be proven in several steps that are organised in the following way. We start by highlighting that it is enough to prove the result for k= 1. Afterwards, for such case, we will check that the map on the right-hand side of (3.10) is enveloped, in the sense of Theorem (1.51), by another map (involving Rα,1) which has a globally attracting equilibrium. 97 Sebasti´ an Buedo Fern´ andez [83] E. Liz and S. Buedo-Fern´andez. A new formula to get sharp global stability criteria for one-dimensional discrete-time models. Qualitative Theory of Dynamical Systems, 18(3):813–824, 2019. [84] E. Liz, C. Mart´ınez, and S. Trofimchuk. Attractivity properties of infinite delay Mackey-Glass type equations. Differential and Integral Equations, 15(7):875–896, 2002. [85] E. Liz, M. Pinto, G. Robledo, S. Trofimchuk, and V. Tkachenko. 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Fisheries Research, 65(1-3):103–121, 2003. 205 Notation Zinteger numbers Z+nonnegative integer numbers Npositive integer numbers Rreal numbers R+nonnegative real numbers R+positive real numbers R−nonpositive real numbers P(X) parts of a set X Dom fdomain of a function f Im fimage of a function f f◦gcomposition of functions fand g C(X, Y ) set of continuous functions from Xto Y Cn(X, Y ) set of functions from Xto Ywith continuous n-th derivative ∂jf j-th partial derivative of a function with multivariable domain Bclosure of a set B Int(B) interior of a set B ∂B boundary of a set B Mn×n(R) space of real n×nmatrices 207 Index attractor CC-strong, 114 global, 19 local, 19 strong, interval-strong, I-strong, 84, 113 characteristic equation, 31 characteristic inequality, 128, 132 cobweb analysis, 14 delay constant, 3 discrete, 3 distributed, 3 infinite, 153 variable, 3 delay differential equation, 3 corresponding, 27 gamma-logistic, γ-logistic, 71 gamma-Mackey-Glass, 79 Lasota, 63 Mackey-Glass-type, 22 Wright-type, 22 difference equation, 14 corresponding, 27 gamma-Beverton-Holt, 80 gamma-logistic, γ-logistic, 71 gamma-Ricker, γ-Ricker, 64 Maynard Smith and Slatkin γ-, 79 enveloping, 35 equilibrium, 11, 16 globally asymptotically stable, GAS, 11 globally attracting, 11 hyperbolic, 30 locally asymptotically stable, LAS, 11 locally attracting, 11 non-hyperbolic, 30 stable, 11 feedback negative, 29 positive, 28 feedback inner parameters, 45 gamma-model, γ-model delay differential, 56 discrete, 58 graphical analysis, 14 map S-, 40 S∗-, 40 absolutely continuous, 131 decreasing, 37 gamma-Beverton-Holt, 80 209