Dynamics of Holomorphic and Stochastic Differential Equations

Acknowledgments First of all I wish to thank my supervisors Professor Helena Reis and Professor Alberto Adrego Pinto. I have learned a lot from their deep knowledge. Their support and advice have been essential to the completion of this thesis. I would like to thank Professor Julio Rebelo for his hospitality on every visit I have made to the Equipe Emile Picard at Universite Paul Sabatier, Toulouse, France. I would like to thank Professor Athanasis Yannacopoulos for his hospitality during visits to the Athens University of Economics and Business, Athens, Greece. I would like to express special thanks to my husband Diogo for his love, huge patience, support and encouragement during the last 4 years. I am very grateful to my Father and my Mother for all their endless encouragement and comprehension throughout my life. Thank you very much for everything. I also would like to thank my Brother for his constant presence in my life and, most of all, for his friendship. I would like thank all my friends, specially Laura, Electra, Miguel, Nuno A., Sara, José FCUP iv Carlos, Luís, Rita, Nuno V. and João, for giving me the moral strength to proceed with the work I proposed myself to do. To FCT – Fundação para a Ciência e a Tecnologia for the financial support given in the form of the PhD scholarship SFRH / BD / 61547 / 2009, co-funded by POPH – Programa Operacional do Potencial Humano. Abstract This thesis is divided in two main topics. The first one concerns a problem on complex differential equations that has been open for thirty years. More precisely, it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C3,0) is not a topological invariant. We provide an example of two topologically equivalent foliations such that only one of them admits two independent holomorphic first integrals. The existence of invariant surfaces over which the induced foliation possesses infinitely many separatrices possibly constitutes the sole obstruction for the topological invariance of complete integrability and a characterization of foliations admitting this type of invariant surfaces is also given. The second main topic of this thesis is connected with real stochastic differential equations. In particular, we consider a logistic growth model with a predation term given by a Holling type-nfunctional response, with n≥2integer, and a stochastic perturbation driven by a one-dimensional Brownian motion with a power-type diffusion coefficient. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, namely, linear growth and the Lipschitz condition. Nevertheless, under a weak set of assumptions, we are able to prove that solutions with a positive initial condition exist and are unique up to the first instant of time at which zero is reached. Additionally, we provide criteria for population extinction, persistence and for the existence of a stationary measure. Moreover, we provide a detailed characterization for the asymptotic stationary measure density in the former case. FCUP vi We then move on to the study of an optimal harvesting problem associated with a population whose size evolves according to the stochastic logistic growth model described above. Since the stochastic differential equation associated with this model does not always fit the standard assumptions in the stochastic optimal control literature, namely Lipschitz continuity and linear growth, we develop a dynamic programming principle for a family of stochastic optimal control problems that contains the one we are interested in. We then use these results to provide a description of the optimal harvesting policies, as well as some qualitative properties of the corresponding value function. Keywords: Holomorphic foliations, First integrals, Topological invariance, Singularities, Population dynamics, Stochastic differential equations, Stationary measures, Stochastic Optimal Control, Optimal harvesting Resumo Esta tese está dividida em dois temas principais. O primeiro diz respeito a um problema em equações diferenciais complexas, o qual está em aberto há cerca de trinta anos. Mais precisamente, é mostrado que a existência de dois integrais primeiros holomorfos independentes para folheações definidas por curvas em (C3,0) não é um invariante topológico. É construído um exemplo de duas folheações topologicamente equivalentes e tais que apenas uma admite dois integrais primeiros holomorfos independentes. A existência de superfícies invariantes sobre as quais a folheação induzida possui uma infinidade de separatrizes constitui uma obstrução à invariância topológica de integrabilidade completa sendo, possivelmente, a única obstrução. É também dada uma caracterização das folheações que admitem este tipo de superfícies invariantes. O segundo tópico principal desta tese está relacionado com equações diferenciais estocásticas reais. Em particular, é considerado um modelo logístico com um termo de predação dado por um funcional Holling tipo-n, com n≥2inteiro, e uma perturbação estocástica gerada por um movimento Browniano de dimensão um com coeficiente de difusão do tipo potência. Apesar de tal equação diferencial estocástica não satisfazer as condições usuais de existência e unicidade de soluções, nomeadamente, crescimento linear e condição de Lipschitz, é provado, sob um conjunto de hipóteses mais fracas, que as soluções de tal equação com condição inicial positiva existem e são únicas até ao instante de tempo em que atingem zero. Adicionalmente, são dados critérios para extinção da população, persistência e para a existência de uma medida estacionária. Além FCUP viii do mais, é dada uma caracterização detalhada para a densidade da medida estacionária neste último caso. De seguida passa-se ao estudo de um problema de colheita óptima associado a uma população cujo tamanho evolui de acordo com o modelo logístico estocástico descrito acima. Uma vez que a equação diferencial estocástica associada ao modelo nem sempre satisfaz as condições usuais na literatura de controlo óptimo estocástico, nomeadamente, continuidade Lipschitz e crescimento linear, é desenvolvido um princípio de programação dinâmica para uma família de problemas de controlo óptimo estocástico da qual o problema de colheita óptimo aqui considerado é um exemplo particular. De seguida, estes resultados são usados para fornecer a descrição de uma política de colheita óptima, assim como algumas propriedades qualitativas da função valor que lhe está associada. Palavras-Chave: Folheações Holomorfas, Integrais primeiros, Invariância topológica, Singularidades, Dinâmica de populações, Equações diferenciais estocásticas, Medidas estacionárias, Controlo óptimo estocástico, Colheita óptima Contents Acknowledgments iii Abstract v Resumo vii 1 Introduction 1 2 Topological aspects of completely integrable foliations 13 2.1 Conjugated foliations and first integrals . . . . . . . . . . . . . . . . . . . . . 15 2.2 On dicritical invariant surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Furthercomments ................................ 40 3 On a stochastic logistic growth model with predation 43 3.1 Problem formulation and motivation . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 The deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.2 The stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Existence of Global Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Criteria for Population Extinction . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Persistence.................................... 76 3.5 Existence and uniqueness of a stationary measure . . . . . . . . . . . . . . 80 FCUP 4 foliation by curves is a (germ of) analytic curve passing through the origin and invariant by the flow of a vector field germ defining the foliation. The notion of eigenvalues for a singular holomorphic foliation Fwill also be needed in the sequel. To define the eigenvalues of F(at a given singular point), we first choose a holomorphic vector field Xtangent to Fand possessing a singular set of codimension at least 2. The eigenvalues of Fat the singular point pare defined as the eigenvalues of the linear part of Xat p. Since Xis well-defined only up to an invertible factor, it follows that the eigenvalues of Fare well-defined up to a multiplicative constant. Let us now come back to the role played by isolated singular points. This has to do with the interaction between isolated singularities and the existence of dicritical invariant surfaces. Curiously, modulo very mild generic assumptions, Theorem 1.0.3 below tells us that these surfaces always exist provided that the foliation in question has an isolated singularity at the origin. In other words, in view of the preceding conjecture, the “correct" generalization of Mattei-Moussu’s theorem may involve, in an intrinsic way, foliations possessing curves of singular points. To make the discussion more accurate, let us now state Theorem 1.0.3. Theorem 1.0.3 Let Fbe a foliation by curves on (C3,0) having an isolated singularity at the origin and admitting two independent holomorphic first integrals. Suppose that e F, the transform of Fby the punctual blow-up centred at the origin, has only isolated singularities which, in addition, are simple. Then Fpossesses an invariant surface over which the induced foliation is dicritical. In the above statement, by a simple singularity, it is meant a singular point of Fwith at least one eigenvalue different from zero. Ultimately the statement of Theorem 1.0.3 should be understood as a statement about pairs F, G of holomorphic functions defined about the origin in C3. This point of view is however tricky since it is not very amenable to standard techniques of “analytic geometry”. Indeed, the natural approach to it involves foliations as somehow suggested by the very 5FCUP Chapter 1. Introduction assumption that Fhas isolated singularities. This assumption can be formulated in terms of the functions F, G by considering the vector field Xgiven as the vector product of the gradient vector fields of F, G. To state that Fhas isolated singularity amounts to saying that, after eliminating all common factors being the components of X, the resulting vector field Yhas an isolated singularity at the origin. Hence, from a technical point of view, this condition can be thought of as a condition on the functions F, G. It turns out, however, that the geometric meaning of the condition in question is naturally captured by the standard notions from foliation theory while it may be quite elusive to other geometric or algebraic techniques. Also, the “generic” assumptions made in Theorem 1.0.3 are probably not indispensable and a general idea on how this theorem can be proved in full generality will be given below. However, note that for the immediate purpose of substantiating our claim that an assumption of type “isolated singularity” will not lead to the “correct” generalization of Mattei-Moussu’s result, Theorem 1.0.3 is more than enough. Naturally, in the course of the proof of Theorem 1.0.3, some additional insight into the structure of the set of foliations possessing dicritical invariant surfaces will be obtained. The information collected there might also be seen as some preliminary steps towards the conjecture stated above. Let us now shift our focus to the second main topic in this thesis. The aim of Chapter 3 is to provide a detailed study of a stochastic logistic growth model with a predation term given by a Holling type-nfunctional, for some integer n≥2, and a stochastic part driven by a one-dimensional standard Brownian motion with a diffusion coefficient of power-type, i.e. proportional to xαdWt, where xis the population size, αis some positive constant and Wis a standard one-dimensional Brownian Motion. Such choice of diffusion coefficient endows the model under consideration here with the particular feature of having constant elasticity of variance, a family of stochastic perturbation that has deserved careful consideration by the mathematical finance community for a long time already (see, e.g. [21,22]). However, this does not seem to be the case in what concerns population dynamics. Indeed, except FCUP 6 for Ji et al. paper [32] where the case α∈(1,3/2) is considered, we are not aware of any other stochastic population dynamics model with a diffusion coefficient such as the one considered here. This one parameter family of stochastic perturbations produces a wide range of distinct interesting qualitative phenomena since different choices for the value of αlead to different shapes for the stochastic perturbation coefficient (concave when α < 1, linear if α= 1 and convex for α > 1). We should also remark that these different shapes of the diffusion coefficient correspond to very distinct modeling point of views. More precisely, the variance of the instantaneous rate of growth of the population (given its present size), equal to σ2x2α−2, increases with the size of the population when α > 1and decreases when α < 1, remaining constant and equal to σ2in the case of a linear diffusion coefficient. This is particularly relevant since it enables us to provide a qualitative description covering different setups in real problems. Having this point in mind, the stochastic term in this model represents the fluctuation of the population size due to external factors as distinct as, for instance, changes in weather, climate, influence of diseases and competition with other species. We start by proving the existence and uniqueness of global solutions for the Stochastic Differential Equation (SDE) dx(t) = ρx(t)1−x(t) K−ε(x(t))n−1 1+(x(t))n−1dt+σ(x(t))αdWt,(1.1) with positive initial condition and up to the first instant of time where such solution reaches zero. Given the setup under consideration here, the only meaningful and relevant continuation for a solution reaching zero, is for such solution to remain constant and equal to zero afterwards. There are two key difficulties to be addressed. The first one is that the coefficients of (1.1) do not satisfy the linear growth condition, so that solutions may explode in finite time. The second issue that needs to be addressed is that whenever α < 1the diffusion coefficient of (1.1) is not Lipschitz continuous in any neighbourhood of zero, which may lead to problems concerning uniqueness of solutions. We overcome such 7FCUP Chapter 1. Introduction problems by introducing a modified locally Lipschitz condition, which turns out to be particularly suitable for the problem under consideration here. Moreover, we use Lyapunov functions techniques to prove that the solutions of (1.1) do not blow up to infinity in finite time. In addition to proving the existence of solutions to the logistic stochastic model with predation term (1.1), we provide conditions under which extinction of the population occurs, respectively, with positive probability and full probability. More precisely, if one of the following sets of conditions is satisfied . α < 1 . α = 1,n= 2 and ρ−ε<σ2/2 . α = 1,n > 2and ρ<σ2/2 . α > 1,n= 2 and ρ<ε then, there exists a set of initial conditions (with positive Lebesgue measure) for which the population becomes extinct with positive probability. Additionally, in the last three cases we provide extra conditions under which the population becomes extinct exponentially fast with full probability. More importantly, in the special case where α < 1, we prove that extinction occurs in finite time with full probability. From an intuitive point of view, population extinction with full probability when α < 1, regardless of every other parameter values, can be explained by the combination of two factors. First, notice that in the absence of any randomness the population size tends to be below the (finite) carrying capacity of the drift term of (1.1). Second, the variance of the instantaneous rate of growth of the population increases with decreasing population size, becoming arbitrarily large when the population size approaches zero. Thus, since the population size can not escape to infinity, the population eventually becomes extinct in finite time whenever α < 1due to the large variance of its instantaneous rate of growth. FCUP 8 After obtaining criteria for population extinction, we shift our focus to the subject of population persistence and existence of a (absolutely continuous) stationary measure under the dynamics of the SDE (1.1). Indeed, such measures play an analogue role to stable equilibria in the corresponding deterministic models. Specifically, we prove that if one of the following set of conditions holds . α = 1,n= 2 and ρ−ε>σ2/2 . α = 1,n > 2and ρ>σ2/2 . α > 1,n= 2 and ρ>ε . α > 1and n > 2 then, the population persists, i.e. the population size rises to or above a positive threshold infinitely often with probability one. Furthermore, under the same set of conditions, we prove that a unique stationary distribution exists. Finally, we use the Forward Kolmogorov equation to provide a rather detailed characterization for the densities of the stationary measures of (1.1), summarizing all the information in a “stochastic” bifurcation diagram. We remark that such bifurcation diagram is complete in the sense that the subset of parameter space where persistence of population occurs (and an absolutely continuous stationary measure exists) is the complement in parameter space of the subset where population extinction is guaranteed to occur (up to the measure zero subset corresponding to the bifurcation thresholds). Previous related work includes Roberts and Saha paper [62] concerning the asymptotic behaviour of logistic epidemic models. We should also mention that Gary et al. [29] extend the classical SIS epidemic model from a deterministic framework to a stochastic one, yielding a stochastic perturbation of a logistic differential equation whose diffusion coefficient is (a quadratic polynomial) proportional to the population deterministic growth rate. The authors prove that the resulting SDE has a unique global positive solution and 9FCUP Chapter 1. Introduction establish conditions for the extinction and persistence of the epidemic. In the case of persistence, the authors show the existence of a stationary distribution and derive expressions for its mean and variance. In [33,34] Jiang et al. consider a randomized logistic equation with coefficients given by periodic functions and a diffusive coefficient depending linearly on the population size. Somewhat closer to our model, in [32] Ji et al. consider a stochastic logistic equation, but with no predation term and the additional assumption that the parameter αis restricted to the interval (1,3/2). Conditions under which the positive solution of the stochastic differential equation under consideration does not explode at any finite time are provided. Moreover, existence, uniqueness, boundedness, stochastic persistence and global stability of such positive solution is considered. In [11,12] Braumann considers a large family of stochastic differential equations modelling the growth of populations subjected to harvesting (referred to as fishing on the first of these papers and predation in the present work) in randomly fluctuating environments. Conditions for non-extinction and for the existence of stationary distributions are provided for these models. However, the SDE under consideration here does not fit into the assumptions used there. More precisely, in [11] Braumann considers two specific forms for the diffusion coefficient: a first one where such coefficient is proportional to the population growth rate (similar to the choice of [46] and quite distinct from the choice under consideration here) and a second one where the diffusion coefficient is proportional to the population size (corresponding to the special case α= 1 in the present work). In the more recent paper [12], Braumann considers a wider family of diffusion coefficients. However, due to the wide range of systems considered in [11,12], the conditions imposed on the functions representing the growth rate of the population, the harvesting, and the diffusion coefficient, are rather restrictive. For instance, the result in [12] guaranteeing the existence of solutions to the SDE describing the evolution of the population size with time applies only to our case α≤1, and provides no information concerning the fact that solu- FCUP 10 tions reach zero in finite time with full probability as we do here. In what concerns criteria for population extinction, the results in [12] apply only to the special case where α= 1 and n= 2. Finally, the result in [12] concerning the existence of stationary measures applies only to our special case α= 1. Additionally, we remark that the techniques used for the proof of the main results of [11, 12] are of a different nature than the ones used here. This is mainly due to the fact that the results in [11,12] rely on assumptions concerning the asymptotic behaviour of the functions determining the SDE under consideration there in the limits where the population size approaches either zero or infinity. On the other hand, in the current work we employ a more explicit treatment (closer to some extent to the ones of [29,32]), which is mainly possible because of the explicit knowledge of every term in (1.1). As a by-product we obtain a complete parametric description of the asymptotic properties of the dynamics of (1.1), which is hard to perform under the conditions of [11,12] due to the lack of particular information on the functions determining the SDE under consideration there. Similar results can be found in [5, 28, 51] and the references therein, for specific density-dependent natural growth functions and harvesting models. Given the ever increasing social, cultural and economical relevance of being able to continuously harvest natural populations (e.g. plantations and fisheries) while preserving its long term sustainability, the topic of optimal harvesting has long been considered to be worthy of attention by the scientific community. Naturally, early developments of the subject were related with deterministic population dynamics models, for both discrete-time and continuous-time systems (see, e.g. [20] and references therein for further details on the subject). With the development of the stochastic calculus from the 1960’s onwards, renewed interest was devoted to the topic of optimal harvesting for population dynamics models under the influence of random noise. Earlier approaches to this topic consisted in finding harvesting strategies maximizing sustained yields [5,51], under the working assumption that a (absolutely continuous) stationary distribution exists for the population size, ignoring the risk of population extinction. The com- 11 FCUP Chapter 1. Introduction bined influence of extinction from demographic and environmental stochasticity, as well as from harvesting strategies, was studied in [42], where the criteria used to find the optimal harvesting strategies was the maximization of the cumulative harvest subject to some prescribed risk of extinction. Some more recent works on the topic of optimal harvesting have focused mainly on the use of stochastic optimal control techniques to maximize the expected total discounted amount of the harvested population under the assumption that the population size evolves according to some form of the stochastic logistic growth model [1,45]. The approach followed here uses modern techniques from stochastic optimal control theory to find harvesting policies that jointly maximize the utility derived from continuously harvesting part of the population over a finite interval of time and the utility obtained from reaching the final instant in that interval with the largest possible population size. Thus, our problem combines the point of view of maximizing utility from harvesting, while also aiming at long-term population preservation. We assume that, in the absence of harvesting, the population size evolution with time follows a continuous-time logistic growth model with a predation term and a diffusion coefficient of power-type driven by a one-dimensional Brownian motion. Such random dynamical system presents a rich variety of distinct asymptotic qualitative behaviours as discussed in Chapter 3. We will use dynamic programming techniques to address the stochastic optimal control problem just described above. The reason why such choice is particularly suitable is related with the fact that the solutions of the stochastic logistic growth model under consideration here have the Markov property. Dynamic Programming was firstly developed by Bellman in the 1950’s [6–9], and further extended by Florentin [26,27] and Kushner [39]. The key goal in the dynamic programming methodology is to obtain a backwards recursive relation for the value function associated with a given optimal control problem. If additional regularity conditions are satisfied, such recursive relation can be written as a boundary FCUP 12 value problem associated with a second order partial differential equation known as the Hamilton-Jacobi-Bellman (HJB) equation. Rather complete discussions of this subject may be found in the monographs [25,65]. A few difficulties arise when trying to implement dynamic programming techniques to address the optimal harvesting problem associated with the stochastic logistic growth model described earlier. The main problems are caused by the fact that the linear growth condition does not hold for the drift part of our stochastic logistic growth SDE and the fact that the diffusion coefficient is not always Lipschitz continuous. Nevertheless, under a weaker set of assumptions, we are able to obtain a dynamic programming principle and the corresponding HJB equation for a family of stochastic optimal control problems which includes the one we were initially interested in. We then apply these abstract results to find the optimal harvesting strategies in feedback form and to provide a couple of qualitative properties for the value function, namely in what concerns its monotonicity with respect to the state variable and some relevant model parameters. We conclude with a static analysis for the solution of this optimal harvesting problem for the case of constant absolute risk aversion utility functions. Chapter 2 Topological aspects of completely integrable foliations This chapter concerns certain aspects of higher dimensional generalizations of MatteiMoussu’s celebrated topological characterization of integrable holomorphic foliations/vector fields in dimension 2, cf. [50]. In the early 80’s, Mattei and Moussu presented a topological characterization of singular foliations by curves defined on a neighbourhood of the origin of C2and admitting a non-constant holomorphic first integral. More precisely, they proved that a singular foliation admits a non-constant holomorphic first integral if and only if its leaves are closed in U\(0,0) and only a finite number of these leaves accumulate at the origin. This implies that the existence of non-constant holomorphic first integrals for holomorphic foliations defined about (0,0) ∈C2can be read off natural necessary topological conditions. In particular, it follows that the existence of a non-constant holomorphic first integral for singular foliations on (C2,0) is a topological invariant. In other words, consider two holomorphic foliations F1,F2defined on a neighbourhood of the origin of C2and assume that F1,F2are topological equivalent (i.e. there exists a homeomorphism hdefined about (0,0) ∈C2and taking the leaves of F1to the leaves of F2). Then F1admits a non-constant holomorphic first integral if and only if so does F2. FCUP 20 2.1. Conjugated foliations and first integrals an extension (as homeomorphism) to a neighbourhood of the entire exceptional divisor. Therefore the transform of the foliations F|{z=0}and D|{z=0}are topologically conjugate on a neighbourhood of the exceptional divisor. By collapsing this exceptional divisor a topological conjugacy between the original foliations F|{z=0}and D|{z=0}in a neighbourhood of the origin is then obtained. In our case, we need to construct a topological conjugacy between F,Don a neighbourhood of the origin, i.e. a homeomorphism defined about the origin of C3and taking leaves of Fto leaves of D. For this, let us first note that the singular sets of F,Dare not reduced to an isolated point. In fact, both sets coincide with the z-axis. In turn, since the z-axis is invariant by both foliations F,D, it is natural to consider the blow-up map whose center is the z-axis. Let us now proceed to the details. Denote by e C3the blow-up of C3centred on the z-axis. Again e C3possesses natural affine coordinates (x, t, z)and (u, y, z)with the identification (2.2). The exceptional divisor, denoted by E, is given in the corresponding affine coordinates by {x= 0}and {y= 0}. Similarly the blow-up map πzhas local expressions given by πz(x, t, z) = (x, tx, z) (resp. πz(u, y, z) = (uy, y, z)). Note that the pre-image in e C3of a relatively compact neighbourhood of the origin is naturally isomorphic to f C2×D, where f C2denotes the blowup of C2by the punctual blow-up at the origin and Dstands for the unit disc of C. In the affine coordinates (x, t, z)the disc Dis naturally equipped with the coordinate z. Denote by e F(resp. e D) the transform of F(resp. D) by this blow-up map. The resulting exceptional divisor Eis now invariant by e F,e D, although both foliations possess leaves intersecting Etransversely. In fact, all leaves contained in the invariant plane {z= 0}intersect Etransversely with exception of a single leaf, which is tangent to E. The tangency point of e F(resp. e D) with Eis given, in the coordinates (x, t, z), by (0,0,0) (resp. (0,1,0)). Let V1denote a small neighbourhood of the point (0,1,0). We begin by presenting a homeomorphism from V1into V0, a small neighbourhood of (0,0,0), taking the leaves of e Dto the leaves of e F. It will later be shown that this homeomorphism admit an extension 21 FCUP Chapter 2. Topological aspects of completely integrable foliations (as homeomorphism) to a neighbourhood of the “entire" exceptional divisor. The foliation Fis completely characterized by the two independent holomorphic first integrals FF, GFor, equivalently, by the two independent meromorphic first integrals GF, HF. Thus its transform e Fmust be completely characterized by the two independent functions e GF(x, t, z) = GF◦πz(x, t, z) = xz , e HF(x, t, z) = HF◦πz(x, t, z) = t2−x . Analogously, the transformed foliation e Dalso admits two independent first integrals, namely e GD(x, t, z) = −txetz , e HD(x, t, z) = 1 tet2x+t. Following [17], let ϕ: (e C2,(0,1)) →(e C2,(0,0)),ϕ= (ϕ1, ϕ2), be the homeomorphism, actually diffeomorphism, given by ϕ1(x, t) = e HD(0, t, 0) −e HD(x, t, 0) , ϕ2(x, t) = V(t), where V(t)is a square root of e HD(0, t, 0) −e HD(0,1,0), i.e. where Vsatisfies (V(T))2= e HD(0, t, 0) −e HD(0,1,0). Geometrically, this homeomorphism sends straight lines through the origin into straight lines through the origin, being the correspondence on each line made through ϕ1noticing that, for a fixed line, distinct values of xcorrespond to distinct leaves. Now, let e C2be identified to e C2× {0} ⊆ e C2×D≃e C3. With this identification, ϕnaturally satisfies e HF◦ϕ=e HD. In other words, the homeomorphism ϕtake leaves of e D|{z=0}to leaves of e F|{z=0}. Since e HFand e HDare still independent first integrals for e F,e D, respectively, to construct a homeomorphism Φ : V1→V0taking leaves of e Dto FCUP 22 2.1. Conjugated foliations and first integrals leaves of e F, it suffices to impose that Φmust satisfy the following three conditions: (a) The restriction of Φto {z= 0}coincides with ϕ (b) The first two components of Φdo not depend on z (c) e GF◦Φ = e GD where Φ = (Φ1,Φ2,Φ3). Conditions (a) and (b) imply that Φ1,Φ2coincide with ϕ1, ϕ2, respectively. Therefore e HF◦Φ = e HDeverywhere. As already mentioned, we have e HF◦Φ = e HDas long as Φ1=ϕ1and Φ2=ϕ2. It remains to check the existence of a continuous map Φ3such that the condition e GF◦Φ = e GDis satisfied as well. In particular, Φ3must admit a continuous extension to the part of the exceptional divisor contained in V1. Since e GF(x, t, z) = xz, the product between Φ1 and Φ3must coincide with e GD. In other words, Φ3satisfies the equation (e HD(0, t, 0) −e HD(x, t, 0))Φ3=−txetz . Therefore Φ3(x, t, z) = −t2x 1−et2xz . (2.3) The function Φ3is clearly continuous in V1\E. Furthermore lim x→0φ3(x, t, z) = lim x→0−t2x 1−et2xz=z . This means that Φ3admits a continuous extension to V1∩E. The extension is, in fact, holomorphic as follows from Riemann extension theorem. Note that the geometric conditions used in the construction of Fand Dare fulfilled by many foliations. Our particular choice of Fand Dwas intended to yield the following extra advantage: the above mentioned extension of Φ3to the exceptional divisor Ecoincides with the identity in E. 23 FCUP Chapter 2. Topological aspects of completely integrable foliations Let us now show that the homeomorphism Φcan be extended to a homeomorphism defined on a neighbourhood of a compact part of the exceptional divisor. Denote by E0the pre-image of the origin by πz, i.e. E0=π−1 z(0) so that E0is isomorphic to CP(1). Fix r > 0sufficiently small such that Φis continuous on a small neighbourhood Wof DD,r, where DD,r ⊆E0represents the disc of radius rcentred at the tangency point t= 1. Denote by W◦a neighbourhood of S=E0\DD,r0, where 0< r0< r. The restriction of e Dto W◦can be viewed as a fibration over L=E0∩W◦. To check this claim, a suitable projection ψ:W◦→L needs to be defined. This is done as follows. Since the leaves of e Dcontained in the invariant plane {z= 0}are transverse to Eon W◦, to every point a∈W◦∩{z= 0}it is associated the unique intersection point ψ(a)of the leaf through awith L. Consider now a point a∈W◦\{z= 0}. The leaf through adoes not intersect the exceptional divisor. Nonetheless, it can be projected, through the projection map pr2, to a leaf of e D|{z=0} which is, in turn, transverse to L. Thus, assuming that in local coordinates ais given by (xa, ta, za), we define the fiber ψ(a)as ψ(a) =    taif xa= 0 ψ(xa, ta,0) otherwise . In particular, it follows from the definition of ψthat (xa, ta, za)and (xa, ta,0) belong to the same fiber of the fibration in question. Consider, for simplicity, the change of variable T=t−1that essentially serves to “moving” the tangency point between Dand Eto the origin. In this coordinate, the disc DD,r is characterized by the condition |T|< r. Given ρ, ε > 0, consider the loop in {z= 0} defined by γ:   x(θ) = ρe−iθ T(θ) = εeiθ . FCUP 24 2.1. Conjugated foliations and first integrals for θ∈[0,2π]. In coordinates u=y−xand T0= 1/T, this same loop becomes γ:   u=ρε |T0|=1 ε . Fix ρ, ε so that the image of γis contained in W∩W◦(for example, εcan be chosen so that r0< ε < r). Then consider in E0(≃CP(1)) the image Γ = ψ◦γof γby the fibration map ψ. Also, denote by Γthe image of Γby the change of coordinates mentioned above (T=t−1). Clearly Γ(resp. Γ) is a loop of index 1around T= 0 (resp. T0= 0). Furthermore, from the expression of T(θ)in the definition of γ, it follows that Γcorresponds to the boundary of two complementary open discs in CP(1), namely Dε={T:|T|< ε} and D0 ε={T0:|T0|<1/ε}. It is also clear that Dε⊆Wwhile D0 ε⊆W◦. Now consider the compact set K=ψ−1(D0ε)∩{|u| ≤ ρε , |z| ≤ ε} and let Ube its complement in W∩W0so that U⊆W. The restriction of ψto K, denoted by ψ0, is a fibration with base D0εand whose fibre is isomorphic to Dρε ×Dε. Since the base of the fibration is a disc, and hence contractible, this fibration must be topologically trivial. In particular, the diagram below commutes. Kξ −→D0ε×Dρε ×Dε D0ε pr1 ∨ ψ0 > Note that, in local coordinates, the map pr1is nothing but the projection on the first component (i.e. pr1(T, Y, Z) = T) whereas ξis given by ξ= (ψ0, U, z). A fibration following the leaves of e Dhas just been defined. A similar fibration following now the leaves of e Fcan also be defined. Let DF,r (resp. DF,r0) be the image of DD,r 25 FCUP Chapter 2. Topological aspects of completely integrable foliations (resp. DD,r0) by the conjugating homeomorphism Φdescribed above. These sets are naturally contained in E0and they are both topological discs. Let Vbe the image of Wby Φ. Fix now a small neighbourhood of E0\DF,r0and denote it by V◦. Naturally V◦can be chosen so that V◦∩Vcorresponds to Φ(W◦∩W). Then, setting I=E0∩V◦, it follows that e Fcan be viewed as a fibration over I ψ1:W0 1→I . The fibration ψ1is defined in perfect analogy with the definition of ψ. Let γ1(resp. γ1,Γ1,Γ1) be the image of γ(resp. γ,Γ,Γ) by Φ. Now Γ1corresponds to the boundary of two complementary open sets that are topological discs. Let us denote by D1,ε (resp. D0 1,ε) the “disc" containing t= 0 (resp. t=∞). Finally, let K1be a compact neighbourhood of D0 1,ε so that K1∩ψ−1 1(Γ1) = U1∩ψ−1 1(Γ1), where U1= Φ(U). Furthermore, modulo reducing K1, there is no loss of generality in supposing that K1is a compact neighbourhood of D0 1,ε such that the restriction of ψ1to K1 is still a fibration with fiber isomorphic to Dρε ×Dε. This restriction is again a topologically trivial fibration since the base is a disc. The homeomorphism Φinduces a bundle morphism Φ0between (ψ0)−1(Γ) and (ψ0 1)−1(Γ1). To conclude that e Fand e Dare topologically equivalent, it suffices to show that this homeomorphisms admits a fibration-preserving continuous extension. Both D0 εand D0 1,ε are homeomorphic to the unit disc D. Denote by λ(resp. λ1) a homeomorphism from D0 ε(resp. D0 1,ε) onto D. The homeomorphism Φ0induces a homeomorphism vbetween the boundary of the unit discs, v:∂D→∂D. In turn, vcan naturally be extended to all of Dby using the radial lines, i.e. by sending the line of angle FCUP 26 2.1. Conjugated foliations and first integrals θonto the line of angle v(θ). In polar coordinates, this extension is given by V(r, θ)=(r, v(θ)) . Let us now consider V=λ−1 1◦V◦λ. It represents a continuous extension of Φ0to D0 ε and this extension yields the following commutative diagram D0 ε V > D0 1,ε D λ∨V>D λ1 ∨ In other words, it has been established a continuous correspondence between the bases of the trivial fibrations ψand ψ1. The next step is to set up a correspondence between the fibers of the fibrations in question. The bundle morphism Φ0gives us, already, a correspondence between the fibers over the boundaries of the base, i.e. over ∂D0 εand ∂D0 1,ε, leading to the commutative diagram below ∂D0 ε×Dρε ×Dε Φ0 > ∂D0 1,ε ×Dρε ×Dε ∂D ∨v> ∂D ∨ Let Diff0(Dρε ×Dε,0) denote the group of homeomorphisms of Dρε ×Dεpreserving the origin. Consider also the parametrization of ∂D0 εby the polar coordinate θ. Since the fibers are all isomorphic to Dρε ×Dε, the bundle morphism Φ0induces a continuous 1-parameter family {hθ}θ∈[0,2π]of elements of Diff0(Dρε ×Dε,0). Lemma 2.1.3 The homotopy class of {hθ}θ∈[0,2π]in Diff0(Dρε ×Dε,0) is trivial. Before proving this lemma, note that h= Φ0admits an extension as homeomorphism to D0 ε×Dρε ×Dε. To check this assertion, assume that the homotopy class of {hθ}θ∈[0,2π] 27 FCUP Chapter 2. Topological aspects of completely integrable foliations is trivial. Fix θ∈[0,2π]and let Ξθ:Dρε ×Dε×[0,1] →D1,ρε ×D1,ε be the homotopy map joining h0to hθ, i.e. let Ξθbe the map such that Ξ(u, z, 0) = h0(u, z), Ξ(u, z, 1) = hθ(u, z)and Ξ(0,0, s) = (0,0) for all s∈[0,1]. Then the map defined by H((r, θ), u, z) = V(r, θ),Ξθ(u, z, r), is a continuous extension of hto D0 ε×Dρε ×Dε. This homeomorphism induces a natural homeomorphism from Konto K1and this ends the proof of the theorem.  It remains however to prove Lemma 2.1.3. Proof: [Proof of Lemma 2.1.3] The elements hθ,θ∈[0,2π], correspond to the restriction of hto the fiber over θ∈∂D0 ε. More precisely, hθ=h|(θ×Dρε×Dε). In particular, hθ(0) = 0 for all θ∈[0,2π]. Since the projection map pr2takes leaves of F (resp. D) to leaves of F(resp. D), it follows that hθhas the form hθ(u, z)=(h1,θ(u), h2,θ(u, z)) , where h1,θ (resp. h2,θ) stands for the restriction of Φ1(resp. Φ3) to a fixed θ∈∂D0 ε. Therefore, {h1,θ}coincides with the corresponding 1-parameter family of homeomorphism constructed in [17]. Furthermore, this 1-parameter family has trivial homotopy class in Diff0(Dρε,0) (cf. [17]). Let ℵθdenote a homotopy map joining h1,0to h1,θ, i.e. let ℵθbe a map such that ℵ(u, 0) = h1,0(u),ℵ(u, 1) = h1,θ(u)and ℵ(0, s) = 0 for all s∈[0,1]. Then a homotopy map Ξθjoining h0to hθcan easily be constructed. In fact, recalling the FCUP 28 2.2. On dicritical invariant surfaces expression of Φ3,Ξθis explicitly given by Ξθ(u, z, s) = ℵθ(u, s),−(1 −eisθ)2ℵθ(u, s) 1−e(1−eisθ)2ℵθ(u,s)z.  2.2 On dicritical invariant surfaces Note that both foliations Fand Dconsidered in the preceding possess an invariant surface over which the induced foliation is dicritical, i.e. they share a common dicritical invariant surface. Furthermore, the singular sets of Fand Dare not reduced to isolated singular points. Examples of vector fields admitting two independent holomorphic first integrals and such that their associated foliations possess either (a) an isolated singular point and dicritical invariant surfaces or (b) a curve of singular points but no dicritical invariant surfaces can easily be constructed. For example, the vector field X=x∂/∂x −y∂/∂y −z∂/∂z of Example 2.1.1 clearly defines a foliation of type (a). In turn, the vector field Y= x∂/∂x −y∂/∂y, which admits F(x, y, z) = xy and G(x, y, z) = zas holomorphic first integrals, induces a foliation of type (b). In this section, it will be proved that under “generic" conditions there is no completely integrable foliation with an isolated singular point and without dicritical invariant surfaces. This is the contents of Theorem 1.0.3. Before proving Theorem 1.0.3, some comments should be made. First of all, it should be noted that the existence of dicritical invariant surfaces depends on the existence of non-trivial common factors between the decomposition into irreducible factors of two independent holomorphic first integrals. In fact, let Fand Gbe two independent holomorphic first integrals for a foliation Fand let fi, gjbe their irreducible factors. The leaves accumulating on the origin and, in particular, the separatrices of Fare all contained in the union 29 FCUP Chapter 2. Topological aspects of completely integrable foliations of the 2-dimensional invariant varieties given by {fi= 0}and by {gj= 0}. Assume now that F, G admit only trivial common factors, modulo multiplication by a nowhere vanishing function. Then the restriction of F(resp. G) to each irreducible component {gj= 0}(resp. {fi= 0}) provides a non-constant holomorphic first integral for the restriction of Fto the same surface. Therefore, the restriction of the foliation Fto all invariant surfaces through the origin admits a finite number of separatrices. A complete characterization, in terms of common irreducible factors, of the foliations admitting dicritical invariant surfaces can be obtained. More precisely, let F=hk1 1···hkp pfα1 1···fαq q, G=hl1 1···hlp pgβ1 1···gβr r. be the decomposition of two independent holomorphic first integrals F, G into irreducible factors, where h1, . . . , hprepresent the non-trivial common factors. Naturally it is not excluded the case of having p= 0, i.e. Fand Gshare only trivial common factors. Similarly, it may occur that all non-trivial factors of F, G are, indeed, common (i.e. q=r= 0). To abridge notations, these cases will be referred to by saying that p= 0 or that q= 0,r= 0, respectively. Note that, when q=r= 0, we necessarily have p≥2, otherwise Fand G would be dependent. Without loss of generality, we can assume that h1, . . . , hpare ordered so that k1 l1≤ ··· ≤ kp lp .(2.4) Furthermore, if all the inequalities above are, in fact, equalities, then both q, r should be assumed greater than or equal to 1. Indeed, at least one between q, r must be greater than 1, otherwise F, G would be again dependent. The case where only one between q, r is strictly positive can easily be transformed into the case where F, G do not admit common irreducible factors. From now on, we shall assume that F, G are independent FCUP 36 2.2. On dicritical invariant surfaces that the first non-trivial homogeneous component of e G(1) is a power of Fk. In particular, we obtain that Gj=α1Fjfor all k≤j < 2k. Thus Gcan be written as G=α1F+G(1) with G(1) as described above. Note that the multiplicative constant α2might be equal to zero. Proof of Claim (2) Suppose now that G=α1F+···+αiFi+G(i)with G(i)as above. Let G(i+1) =G(i)−αi+1Fi+1 . The order pof G(i+1) is greater than (i+ 1)k. Again the foliation FEcannot be defined by the restriction of e Fp/(e G(i+1))kto {z= 0}if and only if the first non-zero homogeneous component of G(i+1) is a power of Fk. Therefore G(i+1) has order at least (i+ 2)kand G(i+1) (i+2)k=αi+2Fi+2 kfor some αi+2 ∈C. Furthermore G(i) j=αi+1Fi+1 j for all (i+ 1)k≤j < (i+ 2)k. The result follows. The proof of Proposition 2.2.4 is over.  Recall that a singular foliation Fdefined on (C2,0) is said to define a saddle-node singularity if Fhas exactly one eigenvalue different from zero. It is well-known that saddlenode singularities possess only constant meromorphic first integral, cf. for example [48]. In view of it, an immediate consequence of the previous proposition is the following. Corollary 2.2.5 Let p∈Ebe a singular point for e Fand let λ, µ denote the eigenvalues of e FEat p. Then λ6= 0 if and only if µ6= 0. Furthermore, if λ, µ are distinct from zero then λ/µ ∈Q. Next we have: 37 FCUP Chapter 2. Topological aspects of completely integrable foliations Lemma 2.2.6 Let p∈Ebe a simple singular point for e F. Then none of the eigenvalues of e Fat pis equal to zero. Proof: Let λ1, λ2, λ3denote the three eigenvalues of e Fat pand assume that λ3 corresponds to the eigenvalue associated to the direction transverse to E. By assumption, not all the eigenvalues λ1, λ2, λ3are equal to zero. To prove the lemma, let us assume for a contradiction that at least one of them is equal to zero. By using Corollary 2.2.5, suppose first that λ1=λ2= 0. Thus λ36= 0 and it follows that e Fpossesses a separatrix Stransverse to E. Since Fadmits two independent holomorphic first integrals, we conclude that the separatrix Smust be contained in an invariant surface Mfor e Fwhich is, in addition, transverse to E. Naturally the intersection of M with the exceptional divisor is a (local) analytic curve which is invariant by the foliation induced on Eand this curve clearly contains the singular point in question. Besides, the mentioned curve is not contained in the singular set of either foliations since these have isolated singularities. It then follows that the intersection of Mand Edefines a separatrix for the induced foliation on E. Consider now the restriction e FMof e Fto M. The point p belongs to the singular set of e FM. More precisely, pis a singular point of saddle-node type for e FMwhat immediately yields a contradiction since Fis completely integrable and thus e FMpossesses a non-constant first integral. Assume now that λ3= 0. Then λ1.λ26= 0. The standard Poincaré-Dulac Theorem guarantees that e Fadmits a formal separatrix ˆ Sthat is, in addition, transverse to E. Modulo performing finitely many blow-ups, this formal separatrix admits a (formal) parametrization through the variable z. In view of Ramis-Sibuya theorem, [60], there must exist an actual leaf Sof e Faccumulating at pand admitting an analytic parametrization defined on a certain open sector Vwith coordinate z. Furthermore, it admits ˆ Sas its asymptotic expansion. Since e Fis completely integrable, it follows that Sis, indeed, an analytic separatrix for e Fat p. Moreover, this separatrix is transverse to Esince it is asymptotic to ˆ Swhich, in turn, is (formally) transverse to E. The proof now follows as in the previous case.  FCUP 38 2.2. On dicritical invariant surfaces We are now able to prove Theorem 1.0.3. Proof: [Proof of Theorem 1.0.3] Let Fbe a foliation as stated in Theorem 1.0.3 and assume for a contradiction that Fdoes not admit an invariant surface over which the induced foliation is dicritical. Fix a singular point p∈Eand denote by λ1, λ2, λ3the eigenvalues of e Fat p. From Lemma 2.2.6, we know that none of these eigenvalues is equal to zero. Through the point p, let us fix a separatrix contained in the exceptional divisor Eand consider the corresponding holonomy map. The eigenvalues of the linear part of the holonomy map are given, up to a relabelling of the eigenvalues, by e2πiλ2/λ1and e2πiλ3/λ1. Since Fis completely integrable, the holonomy map is periodic which, in turn, implies that λi/λj∈Q∗ for all i6=j. Claim: Let λ3denote the eigenvalue associated to the direction transverse to the component of the exceptional divisor. Then λ1, λ2have the same sign which, moreover, is opposite to the sign of λ3. Proof: [Proof of the Claim] Let us first observe that λ1, λ2, λ3cannot all have the same sign. Indeed, if this were the case, then one of the two possibilities below would hold for e F •e Fis locally linearisable about p •e Fpossesses a Poincaré-Dulac normal form about p, [3]. In the first case, it can immediately be checked by direct integration that all leaves nearby paccumulate on p, contradicting the complete integrability of the foliation. In the second case, a suitable finite sequence of punctual blow-ups would yield a singular point of saddle-node type for the corresponding transform of F. Since the existence of this type of singularity is not compatible with complete integrability, as seen in the proof of Lemma 2.2.6, it follows that λ1, λ2, λ3cannot all have the same sign as desired. Thus there exists isuch that λihas opposite sign to the other eigenvalues. 39 FCUP Chapter 2. Topological aspects of completely integrable foliations Assume for a contradiction that λ1, λ2, the eigenvalues associated to the foliation induced on E, have opposite signs. Then e FEadmits two separatrices S1, S2. Suppose that λ1and λ3have the same sign and assume that S1is the separatrix associated to λ1. Since the vector field is completely integrable, there exists an invariant surface M, transverse to Eand containing S1, whose associated foliation has eigenvalues λ1, λ3at p. In particular its eigenvalues at phave the same sign, implying that all leaves on Maccumulate on p. This contradicts Lemma 2.2.3. The claim is proved.  A corollary of the above claim is that all singular points p∈Eof e Fare dicritical singular points for e FE. In other words, they are singular points for e FEat which e FEadmits infinitely many separatrices. In particular, the Baum-Bott index of e FEat p, which is given by BB(e FE, p) = λ1 λ2 +λ2 λ1 + 2 , is greater than or equal to 4. Since e FEhas only non-degenerate isolated singular points over E, the number of singular points of e FEis given by 1 + k+k2, where kdenotes the degree of the foliation e FE. Thus n X i=1 BB(e FE, pi)≥4(k2+k+ 1) , where p1, . . . , pnare all the singularities of e FE. Nonetheless, the Baum-Bott Theorem says that the sum of the Baum-Bott indexes for all singular points should be n X i=1 BB(e FE, pi)=(k+ 2)2. The resulting contradiction ends the proof of Theorem 1.0.3.  FCUP 40 2.3. Further comments 2.3 Further comments Along the same ideas discussed in the previous section, it might be useful to remove the “generic" condition from the statement of Theorem 1.0.3. Let us then close this Chapter by sketching an argument that might be enough to show that every completely integrable foliation about the origin of C3should possess a dicritical invariant surface. This goes as follows. Recall that the Seidenberg desingularization theorem plays a major role in the study of singular foliations in dimension 2and, in particular, it is used in the topological characterization of integrable foliations. A completely faithful generalization of the Seidenberg result for foliations on 3-manifolds cannot exist, since some non-simple singularities are persistent under blow-ups (cf. [55] for details). Nonetheless final models on a desingularization process of foliations on 3-manifolds have been described on different papers such as [16], [52], [55]. For example, in [55] it is proved that simple singularities can always be obtained by means of a finite sequence of “permissible” blow-ups provided that blow-ups transformations with weights are allowed. Concerning the standard blow-up transformations, Cano et al. proved the existence of a finite sequence of “permissible" blow-ups leading to a new foliation all of whose singular points either are simple or are singular points of a certain “special type” with order at most 2, [16]. In a first moment, the idea to remove the generic condition from Theorem 1.0.3 consists of showing that this “special type” of singular points cannot appear in the desingularization procedure of Fprovided that Fis completely integrable. This assertion can probably be established by building on the material of [52]. Once this result is available, we shall have a foliation e Fpossessing only simple singular points contained in a certain exceptional divisor (where a singularity is said to be simple if the foliation has at least one eigenvalue different from zero). Summarizing, we must be able to obtain a foliation e Fleaving invariant a more complicated (not necessarily irreducible) exceptional divisor. This foliation e Fshould, nonetheless, have only singularities that are “well-behaved” since 41 FCUP Chapter 2. Topological aspects of completely integrable foliations they are simple. However these singularities are, indeed, much better than general simple singularities: at isolated singular points, they must have 3eigenvalues different from zero as a consequence of Lemma 2.2.6 in Section 2.2 (note that this lemma has a local nature so that it remains valid in this more general context). When the singularity is not isolated, then a variant of the proof of Lemma 2.2.6 should still imply that the corresponding singular points have 2-eigenvalues different from zero. At this point, the desired conclusion is likely to be achieved by conducting a careful study of the possible arrangements by means of standard index formula. Namely we have the Baum-Bott formula and the index formula of Lehmann (cf. [43]) generalizing to dimension 3an earlier formula by Camacho-Sad ( [14]). Chapter 3 On a stochastic logistic growth model with predation We will now move to the discussion of real stochastic differential equations. More precisely, we consider a logistic growth model with a predation term given by a Holling type-nfunctional response, with n≥2integer, and a stochastic perturbation driven by a one-dimensional Brownian motion with a power-type diffusion coefficient. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions. Nevertheless, under a weak set of assumptions, we are able to prove that solutions with a positive initial condition exist and are unique up to the first instant of time at which zero is reached. Additionally, we provide criteria for population extinction, persistence and for the existence of a stationary measure. Moreover, we provide a detailed characterization for the asymptotic stationary measure density in the former case. The contents of this Chapter are currently submitted for publication in an international peer-reviewed journal [56]. FCUP 44 3.1. Problem formulation and motivation 3.1 Problem formulation and motivation In this section we provide a description of the deterministic logistic growth model with a predation term, followed by a simple analysis of its equilibria. We then motivate the introduction of a stochastic perturbation, leading to a new model that we will study carefully throughout this Chapter. 3.1.1 The deterministic model Our starting point is the well known deterministic logistic equation modelling the evolution with time of the size of a given population ˙u=ρu 1−u κ,(3.1) where ˙udenotes the derivative of uwith respect to time tand ρ, κ are positive constants representing, respectively, the per capita birth rate of the population and the environment carrying capacity. Note that the carrying capacity κdetermines the size of the population non-zero equilibrium, whereas ρis the exponential growth rate of the population when no constraints imposed by the environment are in play. We will consider predation terms given by Holling type functional responses of the form Ha,b,n(u) = aun−1 b+un−1,(3.2) where n≥2is an integer and a, b are real positive constants related with the efficiency and the average time taken by predators when attacking, killing and devouring their preys (see [31, 61]). Such functional responses represent the pressure caused on the growth of the population under consideration by predation by other species sharing the same habitat. It is also very common to refer to a proportional harvesting as a Holling type-I functional response. However, we will only consider here the case of non-linear functional 45 FCUP Chapter 3. On a stochastic logistic growth model with predation responses of the form (3.2) with n≥2. The logistic growth model with a Holling type-n functional response is then given by ˙u=ρu 1−u K−Hn,a,b(u). Introducing the non-dimensional quantities x=u b1/(n−1) , K =κ b1/(n−1) , ε =a b1/(n−1) , we arrive at the dimensionless ordinary differential equation ˙x=ρx 1−x K−εxn−1 1 + xn−1,(3.3) where ρ, K, ε are positive non-dimensional quantities and n≥2is an integer. We will now provide a brief overview of the equilibria of (3.3) and its stability. The biologically relevant equilibria are the non-negative real numbers satisfying the equation ρx 1−x K=εxn−1 1 + xn−1. Clearly, x= 0 is one such solution. Moreover, it is possible to check that x= 0 is stable if n= 2 and ρ < ε or if n= 2,ρ=εand K≤1, being unstable otherwise. To find any remaining positive equilibria, we consider the equation ρ ε1−x K=xn−2 1 + xn−1.(3.4) Let us denote by hn(x)the right hand side of (3.4), that is hn(x) = xn−2 1 + xn−1, n ∈N, n ≥2. It is clear that equilibria of (3.3) correspond to intersections of the graph of the function in FCUP 52 3.2. Existence of Global Solutions with xk(0) = x0. Let us define the stopping time σk=T∧inf {t∈[0, T] : xk/∈Ak}. Note that xk(t) = xk+1(t)if 0≤t≤σk. This implies that σkis increasing and bounded, and thus the limit σ∞= lim k→∞ σk exists. Define {x(t)}0≤t<σ∞through x(t) = xk(t), t ∈[[σk−1, σk[[ , k ≥1, where σ0= 0 and [[σk−1, σk[[ is the random interval [[σk−1, σk[[ = (t, ω)∈R+×Ω : σk−1(ω)≤t < σk(ω). Noting that x(t∧σk) = xk(t∧σk), it follows from (3.9) that x(t∧σk) = xk(t∧σk) =x0+Zt∧σk 0 fk(xk(s)) ds+Zt∧σk 0 gk(xk(s)) dWs =x0+Zt∧σk 0 f(x(s)) ds+Zt∧σk 0 g(x(s)) dWs, for any t∈[0, T]. Moreover, if σ∞< T, then either lim sup t→σ∞|x(t)|=∞or lim inf t→σ∞|x(t)|= 0 . To prove uniqueness of the solution, assume that {¯x(t)}0≤t<¯σ∞is another maximal solution and define ¯σk= ¯σ∞∧inf {t∈[[0,¯σ∞[[ : |¯x(t)|/∈Ak}. 53 FCUP Chapter 3. On a stochastic logistic growth model with predation Then, we have that ¯σk→¯σ∞a.s. and P{x(t) = ¯x(t),for all t∈[[0, σk∧¯σk]]}= 1 , for all k≥1. Letting k→ ∞ yields that P{x(t) = ¯x(t),for all t∈[[0, σ∞∧¯σ∞[[}= 1 . To complete the proof we need to show that σ∞= ¯σ∞a.s.. In fact, for almost any ω∈ {σ∞<¯σ∞}, we have |¯x(σ∞, ω)|= lim k→∞ |¯x(σk, ω)|= lim k→∞ |x(σk, ω)|∈{0,∞} which contradicts the fact that ¯x(t, ω)is continuous on t∈[0,¯σ∞(ω)). We must hence have σ∞≥¯σ∞a.s.. Similarly, we prove that σ∞≤¯σ∞a.s.. Therefore, we must have σ∞= ¯σ∞a.s..  Thus, as a consequence of Proposition 3.2.1 we obtain that for every T > 0and for every α > 0, the stochastic differential equation (3.6) admits a unique maximal local solution in R+,x(t), defined on [0, σ∞). If σ∞=T, then x(t)is finite and strictly positive for all t∈[0, T]. Otherwise, x(t)becomes zero or blows up to infinity at the instant σ∞< T. We will see next that the solutions of (3.6) never blow up to infinity. Moreover, if α≥1 the solutions remain positive for every t≥0. However, if α < 1the solutions of (3.6) may reach zero in finite time. From such point onwards, Proposition 3.2.1 can not be applied and uniqueness of solutions may not hold [19]. Nevertheless, from the point of view of the problem under consideration here, the only meaningful continuation for a solution reaching zero in finite time is to have it constant and equal to zero from that instant of time onwards. More precisely, let τ0be the stopping time defined as τ0= inf{t≥0 : x(t)=0}, FCUP 54 3.2. Existence of Global Solutions and let x(t)be such that x(t) =    x(0) + Rt∧τ0 0f(x(s)) + Rt∧τ0 0g(x(s))dsif t<τ0 0if t≥τ0 .(3.10) From now on we restrict our attention to the set Mof solutions of (3.6) of the form (3.10), i.e. with the property that if there exists t∗>0such that x(t∗) = 0, then x(t) = 0 for every t>t∗. We will now prove the statements outline above. We consider the case α < 1first and deal with the case α≥1afterwards. Theorem 3.2.2 For any given positive initial condition x(0) = x0and any 0< α < 1, the SDE (3.6) has a unique global non-negative solution in M. Proof: Since (3.6) is a SDE with locally Lipschitz continuous coefficients, Proposition 3.2.1 ensures that for any initial condition x(0) >0there exists in Ma unique maximal solution x(t)for t∈[0, τ], where τrepresents the explosion time of the corresponding solution. Notice that the restriction to Mis needed to ensure uniqueness when a given solution hits zero in the case where α < 1. Since the SDE (3.6) is autonomous, then by [47, Thm. 3.18] it is enough to check that there exists K > 0such that for every x∈R+ xf(x) + 1 2g2(x)≤K1 + x2 to ensure that the solutions x(t)to (3.6) in Mdo not explode to infinity. Due to the particular form of gand the assumption α < 1, such condition is easily verifiable, concluding the proof.  The proof of the next theorem uses a technique introduced for the proof of a similar result in [32]. 55 FCUP Chapter 3. On a stochastic logistic growth model with predation Theorem 3.2.3 For any given positive initial condition x(0) = x0and any α≥1, the SDE (3.6) has a unique global positive solution x(t). Proof: Since (3.6) is a SDE with locally Lipschitz continuous coefficients, Proposition 3.2.1 ensures that for any initial condition x(0) >0there exists a unique maximal solution x(t)for t∈[0, τ], where τrepresents either the explosion time of the corresponding solution or the instant of time where the solution first reaches zero. We will prove that τ=∞ a.s.. Let M > 0be large enough so that the following inequalities hold 1 M< x0< M . For each integer N≥M, define the stopping time τN= inf t∈[0, τ] : x(t)/∈1 N, N, where we set inf ∅=∞. Note that {τN}N≥Mis an increasing sequence and set τ∞= limN→∞ τN. Clearly, τ∞≤τa.s.. We will now show that τ∞=∞a.s.. Suppose this is not the case. Then, there exist constants T > 0and δ∈(0,1) such that P{τ∞≤T}> δ . Hence, there is an integer N1≥Msuch that P{τN≤T} ≥ δ, for all N≥N1.(3.11) Consider the function V:R+→R+given by V(x) = √x−1−1 2log x . FCUP 56 3.2. Existence of Global Solutions By Itô’s formula, we have that for any u∈[0, T]and N≥N1 V(x(u)) = V(x(0)) + Zu 0 Vx(x(s))f(x(s)) + 1 2Vxx(x(s))g2(x(s)) ds +Zu 0 Vx(x(s))g(x(s)) dWs, where f(x)and g(x)are as in (3.7) and (3.8), respectively. Making u=t∧τNand computing the expected value of V(x(t∧τN)), we obtain E[V(x(t∧τN))] = E[V(x0)] + EZt∧τN 0 Vx(x(s))f(x(s)) + 1 2Vxx(x(s))(g(x(s)))2ds+EZt∧τN 0 Vx(x(s))g(x(s)) dWs. Thus, we get that E[V(x(t∧τN))] = V(x0) + EZt∧τN 0 LV (x(s)) ds,(3.12) where LV (x(s)) = Vx(x(s))f(x(s)) + 1 2Vxx(x(s))(g(x(s)))2 =F(x(s)) + G(x(s)) , and Fand Gare the functions given by F(x) = 1 2"ρ x1/2−1−x3/2 K+x K!+σ2 2−1 2x2α−3/2+x2α−2# and G(x) = −ε 2 xn−3/2−xn−2 1 + xn−1!. It is clear that F(0) = −ρ/2if α > 1and F(0) = −ρ+σ2/2if α= 1. Moreover, 57 FCUP Chapter 3. On a stochastic logistic growth model with predation F(x)tends to −∞ as xtends to +∞. Thus, there exists x∗>0such that F(x)is strictly negative for every x>x∗. Since F(x)is continuous in [0, x∗], we obtain that Fis bounded above. In what concerns the function G, we have that G(0) = 0 and that G(x)tends to 0 as xtends to +∞. Hence, for every ε > 0there exists x∗>0such that G(x)< ε for every x>x∗. Since G(x)is continuous in every interval [0, x∗], we obtain that Gis also bounded above. We conclude that LV (x(s)) is upper bounded by some ∆>0, i.e. LV (x(s)) <∆. Combining (3.12) with the upper bound obtained above, and noting that x(t∧τN)∈R+, we get that E[V(x(t∧τN))] ≤V(x0)+∆E(t∧τN)≤V(x0)+∆T . (3.13) Set ΩN={τN≤T}for N≥N1. By (3.11) we have that P(ΩN)≥δ. Note that for every ω∈ΩN,x(τN, ω)equals either Nor 1/N, and hence V(x(τN, ω)) ≥√N−1−1 2log N∧ r1 N−1 + 1 2log N!. It then follows from (3.13) that V(x0)+∆T≥E[1ΩN(ω)V(x(τN, ω))] ≥δ"√N−1−1 2log N∧ r1 N−1 + 1 2log N!#, where 1ΩNis the indicator function of ΩN. Letting N→ ∞ we obtain the contradiction ∞ ≤ V(x0)+∆T . Therefore, we must have τ∞=∞a.s., concluding the proof of the theorem.  The next result follows from combining Theorems 3.2.2 and 3.2.3 with a stochastic dominance argument. FCUP 58 3.2. Existence of Global Solutions Corollary 3.2.4 Consider the family of stochastic differential equations of the form dx(t) = f(x(t))dt+σ(x(t))αdWtt > 0,(3.14) where σand αare positive constants. Assume that fis locally Lipschitz in R+and that the following additional conditions hold: i) f(0) = 0 ii) there exist A, B > 0such that f(x)≤Ax(B−x)for every x > 0. Then, for any positive initial condition x(0) = x0, the SDE (3.14) has a unique non-negative solution in M. Proof: Existence and uniqueness of local solutions is a consequence of Proposition 3.2.1. Extension to global solutions follows from Theorems 3.2.2 and 3.2.3 by noticing that a solution of (3.14) is bounded above by a solution of an equation of the form dx(t) = Ax(t)(B−x(t))dt+σ(x(t))αdWt,(3.15) for some positive constants A, B for which the assumption ii) holds, and below by the zero solution. Therefore, all that remains to be checked is that solutions of (3.15) always dominate solutions of (3.14) with the same initial conditions, i.e. if x1(t)is a solution of (3.14) with initial condition x0and x2(t)is a solution of (3.15) with the same initial condition, then P{x1(t)≤x2(t),for all t≥0}= 1 . But this is a consequence of the a.s. continuity of x1(t)and x2(t)with respect to t≥0 and item ii) in the statement above. Indeed, since x1(t)and x2(t)have the same initial condition, item ii) implies that there exists some ε > 0such that x1(t)< x2(t)for every t∈(0, ε). If there exists any t∗≥εsuch that x1(t∗) = x2(t∗)then, the same argument 59 FCUP Chapter 3. On a stochastic logistic growth model with predation can be applied to ensure the existence of ε0such that x1(t)< x2(t)a.s. Pfor every t∈(t∗, t∗+ε), completing the proof.  3.3 Criteria for Population Extinction In this section we will provide criteria for the extinction of a population whose size evolves according to the SDE (3.6). Before moving on to the statement and proof of the main results, we recall the definition of local time for a given stochastic process and the statement of the Itô-Tanaka formula, which will be useful for the proof of our first criteria. See [19] for further details related with these concepts. A right continuous adapted process M={Mt}t≥0is called a local martingale if there exists a non decreasing sequence {τk}k≥1of stopping times with τk↑ ∞a.s. such that every {Mτk∧t−M0}t≥0is a martingale. We denote by Mc,loc(Ft,P)the space of continuous local martingales, i.e. the space of equivalence classes of local martingales Mtwhose sample paths are continuous almost surely and which satisfy M0= 0. Let FV denote the space of processes Atfor which almost all paths are of finite variation on each compact interval of R+, i.e. for a.e. sample path and for every interval of the form [a, b]⊂R+, we have that the supremum of n X i=1 Ati−Ati−1 over the set P([a, b]) of all partitions of [a, b]with elements determined by sequences a=t0< t1<··· < tn=bis finite. A process {Wt}t≥0is a continuous semimartingale process on the filtered probability space Ω,F,{Ft}t≥0,Pif Wtis Ft-adapted and there exist Mt∈ Mc,loc(Ft,P)and At∈ FV such that Wt=W0+Mt+At. Given a continuous semimartingale, for every a∈R there exists an increasing continuous process La(W) = {La t(W)}t≥0, FCUP 60 3.3. Criteria for Population Extinction called the local time of Wat the point a, satisfying (Wt−a)+= (W0−a)++Zt 0 I{Ws>a}dWs+1 2La t(W), t ≥0, where X+denotes max{0, X}. Let I⊂Rbe some interval and suppose that f:I→Ris continuous and equal to the difference of two convex functions. Then, for each xon the interior of I, there exist a left derivative f0 −(x)and a right derivative f0 +(x). Moreover, there exists a second derivative f00 in the sense of distributions. This is a signed measure on Isuch that, for any a,bon the interior of I, we have f00([a, b)) = f0 −(b)−f0 −(a). If the left endpoint `of Ibelongs to I, then we define f0 −(`)=0and assume that the measure f00 has an atom of mass f0 +(`)at the point `. Let {Wt}t≥0be a continuous semimartingale and let fbe a function such as described above. Assume also that f00 is finite on any compact subset of I. Then, the Itô-Tanaka formula states that f(Wt) = f(W0) + Zt 0 f0(Ws)dWs+1 2ZI Lx(W)f00(dx), t ≥0.(3.16) The Itô-Tanaka formula above is a generalization of Itô’s Lemma to include functions which are not necessarily C2, such as the absolute value function. In the next theorem we prove that for every α < 1the population will become extinct with probability one. Theorem 3.3.1 If α < 1and x0>0, then P{∃t < ∞:x(t)=0}= 1 , that is, x(t)reaches zero a.s.. In other words, the population goes extinct with probability one. 61 FCUP Chapter 3. On a stochastic logistic growth model with predation Proof: Let α < 1and consider again the SDE (3.6). By Theorem 3.2.1, we have that a unique solution of (3.6) exists up to the instant of time where it reaches zero. We will prove that such event occurs with probability one. The proof is broken into the following sequence of steps. We start by checking that our analysis can be reduced to the case where no predation term is present, i.e. ε= 0. The second step consists of finding an alternative representation for the solution of SDE (3.6) when ε= 0. This step in based on the construction of a weak solution of (3.6) and follows closely the approach described in [19] to deal with singular SDEs. The third step uses these weak solutions representation to obtain an estimate for the expected time for a solution to reach either zero or a given fixed positive level. The final step of the proof combines Dynkin’s inequality with the estimate obtained in the third step to check that the probability of a solution reaching zero in finite time is equal to one. We start by reducing our analysis to the special case where ε= 0, i.e. no predation term is present in the SDE (3.6). Let xε x0(t)denote the solution of the SDE (3.6) for some positive initial condition x0and ε > 0, and let x0 x0(t)be the solution of the SDE dx(t) = ρx(t)1−x(t) Kdt+σ(x(t))αdWt,(3.17) with the same initial condition (but ε= 0). Since ε > 0, we get that xε x0(t)≤x0 x0(t) for all t≥0P-a.s. by a stochastic dominance argument similar to the one of Corollary 3.2.4. Furthermore, we get that if x0 x0(t)=0, then xε x0(t)=0P-a.s.. Thus, we will restrict ourselves to the case ε= 0 for the remaining of this proof. The second step of this proof is to provide an alternative representation for the solution of (3.6) when ε= 0. For the construction of such representation, we follow the strategy described in [19, Thm. 2.11]. We provide an overview here for the sake of completeness. Given a stochastic process X={Xt}t≥0, let T0,`(X) = inf {t≥0 : Xt= 0 or Xt=`} FCUP 68 3.3. Criteria for Population Extinction Since limx→0+r1(x)=1, we have lim x→0+H0 1(x) = 1 + 2 lim x→0+ s1(x)f(x) g2(x),(3.34) provided that the limit on the right hand side exists. We will now show that such limit exists, computing its value in the process. We separate our analysis into the following two cases: 0< α ≤1/2and 1/2< α < 1. In the first case, we have that lim x→0+ f(x) g2(x)=ρ σ2lim x→0+x1−2α1−x K=   ρ/σ2if α= 1/2 0if α < 1/2 . Since limx→0+s1(x)=0, we obtain that lim x→0+ s1(x)f(x) g2(x)= 0 .(3.35) We now consider the case where α > 1/2. We start by noting that lim x→0+ g2(x) f(x)=σ2 ρlim x→0+ x2α−1 1−x K = 0 . Thus, using L’Hôpital rule, we obtain lim x→0+s1(x)f(x) g2(x)= lim x→0+ s1(x) g2(x)/f(x) = lim x→0+ s0 1(x)f2(x) 2g0(x)g(x)f(x)−g2(x)f0(x)(3.36) = 0 , where the last equality follows from the fact that s0 1(x) = r1(x)and the form of fand g 69 FCUP Chapter 3. On a stochastic logistic growth model with predation given in (3.19). Therefore, combining (3.35) and (3.36) with (3.34), we conclude that lim x→0+H0 1(x)=1.(3.37) Note also that lim x→+∞r1(x)=+∞, and, consequently, limx→+∞s1(x)=+∞. Thus, to evaluate lim x→+∞H1(x) = lim x→+∞ s1(x) r1(x), we apply L’Hôpital rule once again to obtain lim x→+∞H1(x) = s0 1(x) r0 1(x), that is, lim x→+∞H1(x) = lim x→+∞ r1(x) −2f(x) g2(x)r1(x)=−lim x→+∞ g2(x) 2f(x). Substituting fand gby the expressions given in (3.17) and rearranging terms, we obtain that lim x→+∞H1(x) = −Kσ2 2ρlim x→+∞ x2α−1 K−x= 0 (3.38) by noticing that 2α−1<1for every α∈(0,1). Combining the fact that H1(0) = 0 and H1(x)>0for every x > 0with (3.37) and (3.38), we conclude that there exists M > 0such that the global maximizer of H1is in [0, M]. Hence, we obtain that for every 0< β < 1there exists C:= C(β)>0such that H1(x)≤Cxβ,for every x > 0. Since the function His the restriction of H1to [0, a], we obtain that for every 0< β < 1 FCUP 70 3.3. Criteria for Population Extinction there exists C > 0such that H(x)≤Cxβ(3.39) for every 0≤x≤a. Choose 0< β < 1such that max {0,2α−1}< β < min {2α, 1} and combine the estimate (3.39) with (3.32) to obtain E[T0,a(Z)] ≤CZa 0 xβ−2αdx . Evaluating the integral on the right hand side of the inequality above, we conclude that for every βsuch that max{0,2α−1}< β < min{2α, 1}there exists C0:= C0(β)>0such that E[T0,a(Z)] ≤C0aβ−2α+1 .(3.40) To complete the proof, we resort to Dynkin’s formula [54, Thm. 7.4.1] applied to the function V(x) = x. Let x(t)be the solution of (3.17) with initial condition x0∈(0, a)and consider the stopping time T0,a = inf {t≥0 : x(t)=0or x(t) = a}. Since the stochastic processes xand Zhave the same law, then (3.40) holds for T0,a. Define p0=P(x(T0,a) = 0) and pa=P(x(T0,a) = a). Then, we have that E[V(x(T0,a))] = V(x) + EZT0,a 0 f(x(t))dt. Recalling that fis bounded above, we obtain that there exists M > 0such that apa≤V(x) + MEZT0,a 0 1 dt, 71 FCUP Chapter 3. On a stochastic logistic growth model with predation or equivalently, pa≤V(x) + ME[T0,a] a. But we have seen before that there exists C > 0independent of aand γ∈(0,1) such that E[T0,a]≤Caγ. Thus, we conclude that pa≤V(x) + MCaγ a. Noting that p0+pa= 1 and using the inequality above, we obtain that P{∃t < ∞:x(t) = 0}= lim a→+∞P{x(T0,a)=0} = 1 −lim a→+∞P{x(T0,a) = a} = 1 −lim a→+∞pa= 1 completing the proof.  The previous result states that extinction occurs with probability one whenever α < 1. The next results in this section cover the cases α= 1 and α > 1, providing additional conditions under which extinction occurs. Before moving forward, let us recall some notation and terminology. We denote by LV :R+→Rthe infinitesimal generator associated with the SDE (3.6) acting on a function V(x)∈C2(R+,R), given by LV (x) = f(x)Vx(x) + 1 2(g(x))2Vxx(x),(3.41) where fand gare, respectively, the drift and diffusion coefficients of (3.6) and Vxand Vxx denote the first and second derivatives of Vwith respect to x. The trivial solution x(t) = 0 of (3.6) is said to be stochastically stable (or stable in FCUP 72 3.3. Criteria for Population Extinction probability) if for every pair of 1∈(0,1) and r > 0, there exists δ1=δ1(1, r)such that P{x(t)< r for all t≥0} ≥ 1−1 whenever the corresponding initial condition x0is such that x0< δ1. The trivial solution x(t) = 0 of (3.6) is said to be stochastically asymptotically stable if it is stochastically stable and, additionally, for every 2∈(0,1) there exists δ2=δ2(2)such that Plim t→+∞x(t)=0≥1−2 whenever the corresponding initial condition x0is such that x0< δ2. Theorem 3.3.2 Assume that one of the following conditions holds . α = 1,n= 2 and ρ−ε<σ2/2 . α = 1,n > 2and ρ<σ2/2 . α > 1,n= 2 and ρ<ε. Then, the trivial solution x(t)=0of (3.6) is stochastically asymptotically stable, i.e. there exists a set of positive Lebesgue measure A⊂R+such that for any initial condition x0∈A, the solution of (3.6) through x0becomes extinct with positive probability. Proof: Let Kdenote the family of continuous nondecreasing functions µ:R+→R+ such that µ(0) = 0 and µ(x)>0if x > 0. According to [46, Thm. 4.2.3], to prove the asymptotic stochastic stability of the the trivial solution x(t) = 0 it is enough to find a function V:R+→R+, a constant δ > 0and three functions µ1, µ2, µ3∈ K such that the following inequalities µ1(x)≤V(x)≤µ2(x)and LV (x)≤ −µ3(x) 73 FCUP Chapter 3. On a stochastic logistic growth model with predation hold for every x∈(0, δ). We consider the case where α= 1 and n= 2 first. Since we are assuming that ρ−ε<σ2/2, there exists γ∗>1such that ρ−ε=σ2/(2γ∗). Take γ∈(1, γ∗)and let Vbe given by V(x) = γ γ−1x(γ−1)/γ .(3.42) Then, using (3.41), we obtain that LV (x) = ρ−ε−1 2γσ2x(γ−1)/γ −ρ Kx2−1/γ +εx2−1/γ 1 + x. The result follows from taking µ1(x) = µ2(x) = V(x)and µ3(x) = Cx(γ−1)/γ, where Cis such that 0< C < −(ρ−ε−σ2/(2γ)). Finally, we note that the constant Cin µ3is picked in such a way that LV (x)<−µ3(x)in a sufficiently small right neighbourhood of zero. We define δto be the smallest positive real number such that LV (x) = −µ3(x). In the case where LV (x)<−µ3(x)for every x∈R, the constant δcan be taken to be any positive real number. The proof in the case where α= 1 and n > 2is similar to the previous case. Since in this case we are assuming that ρ < σ2/2, there exists γ∗>1such that ρ=σ2/(2γ∗). Taking γ∈(1, γ∗)and letting Vbe as given in 3.42, we obtain that in this case LV (x) = ρ−1 2γσ2x(γ−1)/γ −ρ Kx2−1/γ −εxn−1−1/γ 1 + xn−1. The result then follows from taking µ1(x) = µ2(x) = V(x)and µ3(x) = Cx(γ−1)/γ, where Cis such that 0< C < −(ρ−σ2/(2γ)). The constant δis obtained as in the previous case. Finally, if α > 1and n= 2, we let Vbe given by V(x) = 1 2x2 FCUP 74 3.3. Criteria for Population Extinction so that LV (x) = (ρ−ε)x2−ρ Kx3+εx3 1 + x+1 2σ2x2α.(3.43) We conclude the proof by taking µ1(x) = µ2(x) = V(x),µ3(x) = Cx2with C∈(0,−(ρ−ε)), and selecting δas in the previous two cases.  We will now provide conditions under which the trivial solution x(t)=0is almost surely exponentially stable. As we will see below, such conditions are strongly related with the existence of a negative upper bound for the infinitesimal generator associated with the SDE (3.6) acting on the logarithm function ln x. To fix notation, let F:R+→Rdenote the aforementioned infinitesimal generator: F(x) = ρ1−x K−εxn−2 1 + xn−1−1 2σ2x2α−2.(3.44) We note that, given some choice of parameters, exactly one of the following three alternative descriptions holds for F: i) Fis strictly decreasing for every x > 0; ii) there exists x∗>0such that Fis strictly increasing in [0, x∗)and strictly decreasing for every x>x∗; iii) there exist 0< x1< x2such that Fis strictly decreasing in [0, x1), strictly increasing in (x1, x2)and strictly decreasing for every x>x2. Moreover, we remark that each one of the three alternative behaviours above holds in a subset of parameter space with positive Lebesgue measure. Theorem 3.3.3 Assume that one of the conditions of Theorem 3.3.2 holds and that, additionally, the function Fin (3.44) admits a strictly negative upper bound. Then, for any given positive initial condition, the solution of the SDE (3.6) obeys lim sup t→∞ 1 tlog(x(t)) <0a.s. , 75 FCUP Chapter 3. On a stochastic logistic growth model with predation namely, the trivial solution x(t)=0is almost surely exponentially stable. In other words, the population goes extinct with probability one, exponentially fast. Proof: We start by noting that the conditions of Theorem 3.3.2 guarantee that F(0) is strictly negative. Applying Itô formula to log(x), we obtain log x(t) = log x(0) + Zt 0 F(x(s))ds+Zt 0 σ(x(s))α−1dWs, where F(x)is as given by (3.44). By assumption, we have that Fhas a negative upper bound. Thus, there exists M > 0such that 1 tlog x(t)≤1 tlog x(0) −M+1 tZt 0 σ(x(s))α−1dWs. To complete the proof, note that the Itô integral on the right hand side of the inequality above is a martingale. As such, by the large number theorem for martingales [46, Thm. 1.3.4], we have that 1 tZt 0 σ(x(s))α−1dWs→0,(3.45) as t→+∞. The proof is completed by combining (3.45) with the fact that 1 tlog x(0) →0 as t→+∞. Remark Under the conditions of Theorem 3.3.2, there exists a subset of parameter space with positive Lebesgue measure and such that the map Fin (3.44) has a negative upper bound, thus ensuring that the trivial solution x(t)=0is almost surely exponentially stable for a set of positive Lebesgue measure of parameters. Indeed, it is possible to check that: 1) If α= 1,n= 2,ρ−ε<σ2/2, and ρ≥εK, then Fis strictly decreasing and thus x(t)=0is almost surely exponentially stable by Theorem 3.3.3. 2) If α= 1,n= 2,ρ−ε<σ2/2, and ρ < εK, then Fhas a unique global maximum FCUP 76 3.4. Persistence equal to ρ+ρ K−2ρε K1/2−1 2σ2. Hence, if the quantity above is strictly negative, Theorem 3.3.3 implies that x(t)=0 is almost surely exponentially stable. 3) If α= 1,n= 2 and ρ < σ2/2, then x(t) = 0 is almost surely exponentially stable. To see that this statement is true, start by noting that solutions of the SDE (3.6) with ε= 0 dominate solutions of (3.6) with positive ε(with the remaining parameters kept fixed). Then, the argument of the proof of Theorem 3.3.3 can be applied and the result follows by noting that in the case where ε= 0 the function Fin (3.44) is bounded above by ρ−σ2/2. 4) If α= 1 and n > 2the function Fin (3.44) is again bounded above by ρ−σ2/2. Thus, whenever ρ < σ2/2the trivial solution x(t) = 0 is almost surely exponentially stable. 3.4 Persistence In this section we will provide a criteria for the persistence of a population whose size evolves according to the SDE (3.6). The proof of the next theorem uses a technique introduced for the proof of a similar result in [29]. Theorem 3.4.1 Assume that one of the following sets of conditions hold: . α = 1,n= 2 and ρ−ε>σ2/2; . α = 1,n > 2and ρ>σ2/2; . α > 1,n= 2 and ρ>ε; . α > 1and n > 2. 77 FCUP Chapter 3. On a stochastic logistic growth model with predation Then, for any given positive initial condition x0the solution of the SDE (3.6) satisfies lim sup t→+∞ x(t)≥ξ−a.s. (3.46) and lim inf t→+∞x(t)≤ξ+a.s. ,(3.47) where ξ−and ξ+are, respectively, the smaller and the larger positive roots of the function F:R+→Rin (3.44). In other words, x(t)will rise to or above ξ−>0infinitely often with probability one. Proof: Start by recalling that applying Itô formula to the function log(x), we obtain log x(t) = log x(0) + Zt 0 F(x(s))ds+Zt 0 σ(x(s))α−1dWs,(3.48) where F:R+→Ris as given in (3.44). Under each set of conditions provided in the statement, we have that 1) F(0) >0 2) Fhas one of the following three monotonic behaviours: 2a) Fis strictly decreasing for every x > 0; 2b) there exists x∗>0such that Fis strictly increasing in [0, x∗)and strictly decreasing for every x>x∗; 2c) there exist 0< x1< x2such that Fis strictly decreasing in [0, x1), strictly increasing in (x1, x2)and strictly decreasing for every x>x2. 3) limx→+∞F(x) = −∞. Notice that conditions 1) to 3) above ensure that Fhas exactly one positive root if its monotonicity is as described in 2a) and 2b) and at least one but at most three positive roots if its monotonicity is as in 2c). FCUP 84 3.5. Existence and uniqueness of a stationary measure Then, the SDE (3.6) has a unique stationary distribution P∞(·)in (R+,B(R+)). Proof: Let F(x)be given by (3.44) and let ξ−and ξ+be as given in the statement of Theorem 3.4.1. Fix any aand bsuch that 0< a < ξ−≤ξ+< b. Then, by the qualitative description of the monotonicity of F(x)provided in item 2) of the proof of Theorem 3.4.1, we have that i) F(x)≥F(a)∧F(0) if 0< x ≤a ii) F(x)≤F(b)if x≥b. Define τas in Theorem 3.5.2. For any x0∈(0, a), using (3.48) we obtain that log x(t∧τ) = log x0+Zt∧τ 0 F(x(s))ds+Zt∧τ 0 σ(x(s))α−1dWs, that is E[log x(t∧τ)] = log x0+EZt∧τ 0 F(x(s))ds+EZt∧τ 0 σ(x(s))α−1dWs. Thus, using item i) above we conclude that log a≥E[log x(t∧τ)] ≥log x0+ (F(a)∧F(0)) E[t∧τ] for every t≥0. Taking the limit t→ ∞, yields E[τ]≤log(a/x0) F(a)∧F(0) (3.56) for every x0∈(0, a). Similarly, for x0> b, we obtain log b≤E[log x(t∧τ)] ≤log x0−|F(b)|E[t∧τ] 85 FCUP Chapter 3. On a stochastic logistic growth model with predation for all t≥0. Letting t→+∞, we conclude that E[τ]≤log(x0/b) |F(b)|(3.57) for every x0> b. The conditions of Theorem 3.5.2 follow from inequalities (3.56) and (3.57). Thus, a unique stationary measure associated with the Markov process defined by the solution of (3.6) exists.  The probability measure Px0,t(·)induced by the solution x(t)of (3.6) with positive initial condition x0has a density px0(t, x)with respect to the Lebesgue measure in R+. It is known that px0(t, x)satisfies the forward Kolmogorov equation ∂ ∂tp(t, x) = −∂ ∂x (f(x)p(t, x)) + 1 2 ∂2 ∂x2g2(x)p(t, x),(3.58) where f(x)and g(x)are as given in (3.7) and (3.8), respectively (see [36,38] for further details). Moreover, the unique steady state pstat(x)of (3.58) satisfying the constraints that supp(pstat(x)) ⊆R+,pstat(x)≥0for every x∈R+and ZR+ pstat(x)dx= 1 , is the density of the stationary measure P∞(·), whenever such measure exists. 3.6 Asymptotic Behaviour We have seen in Section 3.3 that the population becomes extinct with full probability whenever α < 1. In this section we will restrict ourselves to the case α≥1and we will study the steady states of the Forward Kolmogorov equation associated with the SDE (3.6), namely ∂ ∂tp(x, t) = −∂ ∂x (f(x)p(x, t)) + 1 2 ∂2 ∂x2g2(x)p(x, t),(3.59) FCUP 86 3.6. Asymptotic Behaviour where f(x)and g(x)are as given in (3.7) and (3.8), respectively. As we have already seen in the previous section, if P∞(·)is a stationary measure in (R+,B(R+)) for the Markov process defined by the solutions of (3.6), then its density with respect to the Lebesgue measure in R+is a steady state of (3.59). 3.6.1 The general form of the steady states of the Forward Kolmogorov Equation Before moving on to the analysis of the steady states of (3.59), we need to introduce some notation. Let eα,n :R+→Rbe the function given by eα,n(x) = xn−1−2α 1 + xn−1,(3.60) and let Eα,n :R+→Rbe defined as Eα,n(x) = Zx 1 eα,n(y)dy, x > 0.(3.61) Lemma 3.6.1 For every α≥1and n∈Nsuch that n≥2, the function Eα,n :R+→Ris of class C∞in R+. Moreover, we have that lim x→0+Eα,n(x) =    finite if n > 2α −∞ if n≤2α , and that limx→+∞Eα,n(x)is finite. Proof: The first part of the statement is a straightforward consequence of the Fundamental Theorem of Calculus. For the second part, start by considering the integral Z1 0 eα,n(x)dx . (3.62) 87 FCUP Chapter 3. On a stochastic logistic growth model with predation It is easy to check that if n≥1+2α, then (3.62) is finite since eα,n(x)tends to a finite value as x→0+. If n < 1+2α, then we rewrite eα,n(x)as eα,n(x) = 1 x2α−(n−1) ·1 1 + xn−1. Thus, for every x∈(0,1) the following inequalities hold 1 2 1 x2α−(n−1) ≤eα,n(x)≤1 x2α−(n−1) . Hence, the integral in (3.62) is finite if and only if the integral Z1 0 1 x2α−(n−1) dx(3.63) is convergent. But it is well known that (3.63) is convergent if and only if n > 2α, being divergent otherwise. We conclude that (3.62) is convergent if n > 2αand divergent if n≤2α. Finally, we note that by definition of the function Eα,n, we have that Eα,n(x)<0 for every 0<x<1. Thus, we conclude that lim x→0+Eα,n(x) =   −∞ if n≤2α finite if n > 2α . We will now compute lim x→+∞Eα,n(x) by studying the convergence of the integral Z+∞ 1 eα,n(x)dx . (3.64) Writing eα,n(x)as eα,n(x) = 1 x2α·xn−1 1 + xn−1, FCUP 88 3.6. Asymptotic Behaviour we obtain that for every x > 1the following inequalities hold 1 2 1 x2α< eα,n(x)<1 x2α. Thus, (3.64) converges if and only if the integral Z+∞ 1 1 x2αdx , (3.65) converges, being divergent otherwise. The result follows by noticing that the integral (3.65) is convergent if and only if 2α > 1and that Eα,n(x)>0for every x > 1. The next result provides the general form for the solutions of (3.59), uniquely determined up to a constant. Lemma 3.6.2 Up to a multiplicative constant, the steady states of (3.59) are of the form pε,n(x;ρ, σ, K, α) = p0(x;ρ, σ, K, α) exp −2ε σ2Eα,n(x), where p0(x;ρ, σ, K, α) = x−2αexp −2ρ σ2Kx3−2α 3−2α−Kx2−2α 2−2α, for α∈(1,3/2) ∪(3/2,+∞), p0(x;ρ, σ, K, 1) = x−(2−2ρ/σ2)exp −2xρ σ2K, and p0x;ρ, σ, K, 3 2=x−(3+2ρ/(σ2K)) exp −2ρ σ2x. Proof: The steady states of (3.59) are determined by the ordinary differential equation d2 dx2g2(x)p(x)−2d dx(f(x)p(x)) = 0 . 89 FCUP Chapter 3. On a stochastic logistic growth model with predation Integrating the equation above once with respect to x, we obtain the first order equation d dxg2(x)p(x)−2f(x)p(x) = C , which we rewrite as g2(x)d dxp(x)+2g(x)g0(x)−f(x)p(x) = C . Since α≥1and g(0) = f(0) = g(0)g0(0) = 0, we get that the constant Cmust be equal to zero. Therefore, we obtain that p(x)is a solution of g2(x)d dxp(x)+2g(x)g0(x)−f(x)p(x)=0.(3.66) We rewrite (3.66) as d dxp(x) + Fε(x)p(x)=0,(3.67) where Fε(x) = 2 (g(x)g0(x)−f(x)) g2(x) and compute the corresponding solutions. Note that (3.67) is a first order linear ODE. Its general solution is easily found by computing the corresponding integrating factor, being given by pε,n(x) = Ce−RFε(u)du ,(3.68) where Cis another integration constant. We consider the case ε= 0 first and extend to ε > 0afterwards. If ε= 0, we get that F0(x) = 2α x−2ρ σ2x1−2α+2ρ σ2Kx2−2α. FCUP 90 3.6. Asymptotic Behaviour Clearly, we need to distinguish among the following three cases: α= 1,α= 3/2and α∈(1,3/2) ∪(3/2,∞). Up to an additive constant, if α= 1 we obtain ZF0(x)dx=2−2ρ σ2ln x+2ρ σ2Kx , (3.69) if α= 3/2we have ZF0(x)dx=3 + 2ρ σ2Kln x+2ρ σ2x,(3.70) and finally, in the case where α > 1and α6= 3/2, we get ZF0(x)dx= 2αln x−2ρ σ2 x2−2α 2−2α+2ρ σ2 x3−2α K(3 −2α).(3.71) Combining (3.69), (3.70) and (3.71) with (3.68), we conclude that, up to a multiplicative constant, we have that p0(x;ρ, σ, K, α) = x−2αexp −2ρ σ2Kx3−2α 3−2α−Kx2−2α 2−2α if α∈(1,3/2) ∪(3/2,∞), that p0(x;ρ, σ, K, 1) = x−(2−2ρ/σ2)exp −2xρ σ2K, and that p0x;ρ, σ, K, 3 2=x−(3+2ρ/(σ2K))exp −2ρ σ2x. We will now extend to the case ε > 0. Start by writing Fε(x)in the form Fε(x) = F0(x) + 2ε σ2eα,n(x), where eα,n(x)is as given in (3.60). Recalling the form of the function Eα,n(x)introduced at the beginning of Section 3.6.1, we obtain that, up to a multiplicative constant, 91 FCUP Chapter 3. On a stochastic logistic growth model with predation pε(x;ρ, σ, K, α)is given by pε,n(x;ρ, σ, K, α) = p0(x;ρ, σ, K, α) exp −2ε σ2Eα,n(x), as required.  We will now study the steady states of the forward Kolmogorov equation (3.59) yielding a probability density function. As remarked before, we will look for a steady state pε,n(x;ρ, σ, K, α)satisfying the three following requirements i) supp(pε,n(x;ρ, σ, K, α)) ⊆R+; ii) pε,n(x;ρ, σ, K, α)≥0for every x∈R+; iii) RR+pε,n(x;ρ, σ, K, α)dx= 1. Then, if all these conditions hold, from the results in Section 3.5, the steady state pε,n(x;ρ, σ, K, α) is the density of the stationary measure associated with the dynamics of (3.6). Note that conditions i) and ii) are trivially satisfied by the functions in the statement of Lemma 3.6.2. Thus, we will focus our attention in condition iii). This last condition can be reduced to checking the existence of some positive constant normalizing the integral of pε,n(x;ρ, σ, K, α)to one. Let us introduce the notation I+ α,ε(a) = Z+∞ a pε,n(x;ρ, σ, K, α)dx and I− α,ε(a) = Za 0 pε,n(x;ρ, σ, K, α)dx . Then, the integral of pε,n(x;ρ, σ, K, α)normalizes to one if and only if, for every a > 0, the two integrals I+ α,ε(a)and I− α,ε(a)are finite. According to the strategy used for the proof of Lemma 3.6.2, we will break our analysis into the cases ε= 0 and ε > 0, and henceforth into further particular cases. FCUP 92 3.6. Asymptotic Behaviour 3.6.2 Case ε= 0 Recall the form of p0(x;ρ, σ, K, α)from Lemma 3.6.2. We will need to separately consider the cases α= 1,α= 3/2and α∈(1,3/2) ∪(3/2,+∞). Lemma 3.6.3 Let α= 1. Then, the following statements hold: i) I+ 1,0(a)is finite for every a > 0; ii) we have that lim x→0+p0(x;ρ, σ, K, α) =    finite if ρ≥σ2 +∞if ρ<σ2; iii) for every a > 0,I− 1,0(a)is finite if ρ > σ2 2and diverges if ρ≤σ2 2. Proof: Recall from Lemma 3.6.2 that for α= 1 we have p0(x;ρ, σ, K, 1) = x−(2−2ρ/σ2)exp −2xρ σ2K. Then it is clear that lim x→+∞p0(x;ρ, σ, K, 1) = 0 and I+ 1,0(a)<+∞for every a > 0, completing the proof of item i). To prove items ii) and iii), we note that for every a > 0there exist constants C1, C2>0such that C2x−(2−2ρ/σ2)< p0(x;ρ, σ, K, 1) < C1x−(2−2ρ/σ2) for every 0< x < a. We obtain that lim x→0+p0(x;ρ, σ, K, 1) =          +∞if ρ<σ2 1if ρ=σ2 0if ρ>σ2 , 93 FCUP Chapter 3. On a stochastic logistic growth model with predation and that I− 1,0(a)is finite if ρ > σ2/2and diverges otherwise.  Lemma 3.6.4 If α > 1we have that i) I+ α,0(a)is finite for every a > 0; ii) limx→0+p0(x;ρ, σ, K, α)=0, and thus I− α,0(a)is finite for every a > 0. Proof: Recall from Lemma 3.6.2 that, for α= 3/2, we have p0x;ρ, σ, K, 3 2=x−(3+2ρ/(σ2K)) exp −2ρ σ2x,(3.72) and that for α∈(1,3/2) ∪(3/2,+∞), we have p0(x;ρ, σ, K, α) = x−2αexp −2ρ σ2Kx3−2α 3−2α−Kx2−2α 2−2α.(3.73) We consider only the case α6= 3/2, the proof of the case α= 3/2being similar. To prove item i), we note that for every a > 0there exists a constant C > 0such that p0(x;ρ, σ, K, α)≤Cx−2α for every x > a. Hence, since α > 1,I+ α,0(a)is finite for every a > 0. For the proof of item ii), it is enough to notice that due to the exponential term in (3.73) one must have lim x→0+p0(x;ρ, σ, K, α)=0. Thus, I− α,0(a)is convergent for any a > 0for every α > 1. The following couple of results build on Lemmas 3.6.3 and 3.6.4 to characterize the probability density functions arising from the steady states of the Forward Kolmogorov equation (3.59). FCUP 100 3.6. Asymptotic Behaviour for every x∈(0,1). Using the representation for pε,n(x;ρ, σ, K, α)provided in Lemma 3.6.2, we obtain that there exist C1, C2>0such that C1L(x)≤pε,n(x;ρ, σ, K, α)≤C2U(x), for every x∈(0,1), where L(x)and U(x)are given by L(x) = p0(x;ρ, σ, K, α) exp 2ε σ2 xn−2α 2α−n U(x) = p0(x;ρ, σ, K, α) exp ε σ2 xn−2α 2α−n. We need to consider two further subcases: n > 2and n= 2. If n > 2, replacing p0(x;ρ, σ, K, α)in the definition of L(x)and U(x)above by the corresponding expressions provided in Lemma 3.6.2 (being careful to distinguish between the case α= 3/2and the case α∈(1,3/2) ∪(3/2,+∞)), and noticing that n−2α > 2−2αif n > 2(n−2α > −1in the particular case where α= 3/2), we obtain that I− α,ε(1) is finite by the argument used in the proof of Lemma 3.6.4 item ii). Thus, we have proved that (C) if n < 2αand n > 2then the integral of pε,n(x;ρ, σ, K, α)over R+is convergent. If n= 2, a sharper estimate is required. Recall again the form of eα,n from (3.60) and rewrite it as eα,2(x) = x1−2α1 1 + x. Using the Taylor series representation for the second factor, we obtain that eα,2(x) = ∞ X k=0 (−1)kx1−2α+k(3.74) for every x∈(0,1). We now split the analysis into the following two alternative cases: 1) 1−2α+k6=−1for every non-negative integer k; 101 FCUP Chapter 3. On a stochastic logistic growth model with predation 2) 1−2α+k∗=−1for some non-negative integer k∗. We start by case i). From (3.74) we obtain that Eα,2(x) = C0+ ∞ X k=0 (−1)kx2−2α+k 2−2α+k, where, since Eα,2(1) = 0,C0is given by C0=− ∞ X k=0 (−1)k 2−2α+k. Thus, if α∈(1,3/2) ∪(3/2,+∞)is such that 1−2α+k6=−1for every non-negative integer k, we get that (up to a multiplicative constant) pε,2(x;ρ, σ, K, α)is given by pε,2(x;ρ, σ, K, α) = x−2αexp (aε(x;ρ, σ, K, α)) , where aε(x;ρ, σ, K, α) = 2ρ σ2−2ε σ2x2−2α 2−2α−2ρ σ2K−2ε σ2x3−2α 3−2α −2ε σ2 ∞ X k=2 (−1)kx2−2α+k 2−2α+k. We conclude that I− α,ε(1) is finite if and only if one of the following sets of conditions hold (D1) ρ > ε (and item 1) holds) (D2) ρ=ε,K > 1and α > 3/2(and item 1) holds). We now deal with case ii). From (3.74) we obtain that Eα,2(x) = C0+ (−1)k∗ln(x) + ∞ X k=0 k6=k∗ (−1)kx2−2α+k 2−2α+k, FCUP 102 3.6. Asymptotic Behaviour where, since Eα,2(1) = 0,C0is given by C0=− ∞ X k=0 k6=k∗ (−1)k 2−2α+k. Thus, if α= 3/2, we have that k∗= 1 and consequently we obtain that (up to a multiplicative constant) pε,n(x;ρ, σ, K, 3/2) is given by pε,n(x;ρ, σ, K, 3/2) = x−(3+2ρ/(σ2K))x2ε/σ2exp (bε(x;ρ, σ, K, α)) , where bε(x;ρ, σ, K, α) = −2ρ σ2−2ε σ21 x−2ε σ2 ∞ X k=2 (−1)kx−1+k −1 + k. We conclude that I− 3/2,ε(1) is finite if and only if (E1) ρ>εand α= 3/2(and item 2) holds) (E2) ρ=ε,α= 3/2,K > 1and ρ>σ2K/(K−1) (and item 2) holds). On the other hand, if α > 1and α6= 3/2, we get that (up to a multiplicative constant) pε,n(x;ρ, σ, K, α)is given by pε,n(x;ρ, σ, K, α) = x−2αx(−1)k∗+12ε/σ2exp (cε(x;ρ, σ, K, α)) , where cε(x;ρ, σ, K, α) = 2ρ σ2−2ε σ2x2−2α 2−2α−2ρ σ2K−2ε σ2x3−2α 3−2α −2ε σ2 ∞ X k=2 k6=k∗ (−1)kx2−2α+k 2−2α+k. 103 FCUP Chapter 3. On a stochastic logistic growth model with predation Note that if k∗= 0, then α= 1, and if k∗= 1, then α= 3/2. Thus, one must have that k∗≥2(or equivalently α≥2). We obtain that I− α,ε(1) is finite if and only if (F1) ρ > ε and α≥2(and item 2) holds) (F2) ρ=ε,α≥2and K > 1(and item 2) holds). To conclude the proof we note that: item i) in the statement of the Lemma is the combination of (A),(B) and (C); item ii) corresponds to the combination of (D1),(E1) and (F1); item iii) corresponds to (E2); and item iv) corresponds to the combination of (D2) and (F2). Since the list of cases provided in this proof is constructive and exhaustive, for every other choice of parameters the integral of pε,n(x;ρ, σ, K, α)over R+diverges.  The following two Propositions provide a characterization of the density of the stationary measure of the Markov process defined by the solutions of (3.6) with positive initial condition. Proposition 3.6.9 Let ε > 0and α= 1. Assume that one of the following two sets of conditions holds . n = 2 and ρ−ε>σ2/2 . n > 2and ρ>σ2/2. Then, the function pstat ε,n (x;ρ, σ, K, 1) = 1 I+ 1,ε(1) + I− 1,ε(1)p0(x;ρ, σ, K, 1) exp −2ε σ2E1,n(x) is a probability density function in (0,+∞). Moreover, the following qualitative properties hold: i) if n= 2 and σ2/2< ρ −ε<σ2, then pstat ε,n is a strictly decreasing function of xand lim x→0+pstat ε,n (x;ρ, σ, K, 1) = +∞ FCUP 104 3.6. Asymptotic Behaviour ii) if n= 2 and ρ−ε>σ2, then pstat ε,n is an unimodal function and lim x→0+pstat ε,n (x;ρ, σ, K, 1) = 0 iii) if n > 2and σ2/2< ρ < σ2, then pstat ε,n is a strictly decreasing function of xand lim x→0+pstat ε,n (x;ρ, σ, K, 1) = +∞ iv) if n > 2and ρ>σ2, then pstat ε,n is an unimodal function and lim x→0+pstat ε,n (x;ρ, σ, K, 1) = 0 . Proof: Follows from Lemma 3.6.7 and its proof.  Proposition 3.6.10 Let ε > 0and α > 1. Assume that one of the following two sets of conditions holds . n > 2 . n = 2 and ρ>ε. Then, the function pstat ε,n (x;ρ, σ, K, α) = 1 I+ α,ε(1) + I− α,ε(1)p0(x;ρ, σ, K, α) exp −2ε σ2Eα,n(x) is a probability density function in (0,+∞). Moreover, we have that pstat ε,n is an unimodal function and lim x→0+pstat ε,n (x;ρ, σ, K, α)=0. Proof: Follows from Lemma 3.6.8 and its proof.  105 FCUP Chapter 3. On a stochastic logistic growth model with predation Remark In the case where α > 1and ρ=εwe have not proved that a stationary measure exists for the Markov process defined by the solution of (3.6). Indeed, we believe that in such case the existence of a stationary measure may be a subtle issue. However, the existence of a steady state solution of the Forward Kolmogorov equation (3.59) provides some support in favour of the existence of such stationary measure. Using Lemma 3.6.8 proof, if such a measure exists, its density pstat ε,n should have the following properties: i) if n= 2,ρ=ε,α= 3/2,K > 1and σ2K/(K−1) <ρ<3σ2K/(2(K−1)), then pstat ε,n is a strictly decreasing function of xand limx→0+pstat ε,n (x;ρ, σ, K, 3/2) = +∞ ii) if n= 2,ρ=ε,α= 3/2,K > 1and ρ > 3σ2K/(2(K−1)), then pstat ε,n is an unimodal function and limx→0+pstat ε,n (x;ρ, σ, K, 3/2) = 0 iii) if n= 2,ρ=ε,α > 3/2and K > 1, then pstat ε,n is an unimodal function and limx→0+pstat ε,n (x;ρ, σ, K, 3/2) = 0. 3.7 Stochastic Bifurcation Diagram In this section we will combine our previous results to construct what we refer to as a stochastic bifurcation diagram (see [2] for further details on stochastic bifurcation theory). Notice that within the stochastic dynamics framework, stationary measures play an analogue role to that played by invariant sets such as equilibria and periodic orbits in the deterministic framework. Thus, we interpret changes in the qualitative properties of the stationary measures of (3.6) as a sign that a stochastic bifurcation as occurred. Let us start by discussing how the stationary measures change while the parameter α > 0changes (see Figure 3.4a). From Theorem 3.3.1, we know that whenever α < 1, the solutions of (3.6) reach zero in finite time with full probability. Thus, there can be no absolutely continuous stationary measure when α < 1. Instead, the Dirac measure based at zero is the unique possible stationary measure. If α= 1, a very rich picture containing FCUP 106 3.7. Stochastic Bifurcation Diagram several distinct qualitative behaviours emerges. We will discuss this particular case with detail in a moment. When α > 1, two distinct situations may occur, depending on whether n= 2 or n > 2. If α > 1and n > 2, then according to Theorem 3.5.2 and Proposition 3.6.10, there exists an absolutely continuous stationary measure whose density is a unimodal map. On the other hand, if α > 1and n= 2, then two distinct behaviours may occur. If ρ<ε, then by Theorem 3.3.2 the population becomes extinct with positive probability for a set of initial conditions with positive Lebesgue measure, i.e. an (eventual) stationary measure must contain a Dirac mass at zero. If however ρ > ε then, according to Theorem 3.5.2 and Proposition 3.6.10, there exists an absolutely continuous stationary measure with an unimodal density. In what concerns the bifurcation threshold ρ=ε, a rich behaviour can again be found as described in the remark after the proof of Proposition 3.6.10. Let us now consider the special case where α= 1, i.e. the diffusion coefficient depends linearly on the population size (see Figures 3.4a and 3.4b). The cases n= 2 and n > 2are again somewhat different. If n > 2(resp. n= 2) and ρ < σ2/2(resp. ρ<ε+σ2/2) then, according to Theorem 3.3.3, the population becomes extinct with positive probability for a set of initial conditions with positive Lebesgue measure and an (eventual) stationary measure must contain a Dirac mass at zero. At the bifurcation value ρ=σ2/2(resp. ρ=ε+σ2/2), based on the divergence of the integral of pε,n over R+, we conjecture that no absolute continuous stationary measure exists. However, for ρ > σ2/2 (resp. ρ>ε+σ2/2) an absolute continuous stationary measure exists by Theorem 3.5.2. According to Proposition 3.6.9, the stationary measure density is a strictly decreasing function if σ2/2< ρ < σ2(resp. σ2/2< ρ −ε<σ2) and a unimodal function with limit zero when xtends to zero if ρ>σ2(resp. ρ>ε+σ2/2). Finally, at the bifurcation value ρ=σ2 (resp. ρ=ε+σ2) the density pstat ε,n is such that its limits as xtends to zero is finite and strictly positive. A few final comments are due in what concerns the connection between the results 107 FCUP Chapter 3. On a stochastic logistic growth model with predation obtained here and the qualitative properties of the deterministic system discussed in Section 3.1.1. First of all, we notice that when α < 1the asymptotic qualitative behaviour of our stochastic model is in striking contrast with the behaviour of the deterministic model – the stochastic model has the origin as an “attracting equilibrium” whereas this is not always the case for the deterministic system. The case α > 1is the one that presents the stronger similarities between the stochastic and the deterministic models – for both extinction occurs if n= 2 and ρ < ε, while persistence of the population is guaranteed to occur whenever n > 2or n= 2 and ρ>ε. Finally, we consider the case α= 1. If n= 2, the two models have similar behaviours, i.e. the population persists for large enough values of ρ−ε(with the threshold being determined by the “volatility” coefficient σ), and becomes extinct otherwise. If n > 2, then the two models show different behaviours again: while for the deterministic model, the origin is always an unstable equilibrium, for the stochastic model population extinction occurs (with positive probability) for sufficient small values of the natural growth rate ρ. 3.8 Conclusion We have provided a detailed description of the dynamics of a stochastic logistic growth model with a predation term and a diffusion coefficient of a power type. The nonlinearity introduced by the diffusion coefficient leads to interesting distinct asymptotic behaviours depending on the convexity of such coefficient. Another key ingredient of the stochastic logistic growth model under consideration here is the presence of a predation term given by a Holling type-nfunctional response. The combined influence of the population natural growth rate and the size of the predation term turn out to be responsible for some of the different qualitative behaviours described here. Further extensions of this work may take into consideration different types of stochastic perturbations. Another possible direction of future research is to consider higher dimen- FCUP 108 3.8. Conclusion sional population dynamics models subject to analogue stochastic perturbations. 109 FCUP Chapter 3. On a stochastic logistic growth model with predation (a) Bifurcation Diagram for varying values of α > 0 (b) Bifurcation Diagram for varying values of ρand σin the case where α= 1 and n > 2 (c) Bifurcation Diagram for varying values of ρand σin the case where α= 1 and n= 2 Figure 3.4: Bifurcation Diagrams for the stationary measure associated with the SDE (3.6) FCUP 116 4.2. Dynamic programming principle and HJB equation (iii) z: [s, T]×Ω→Zis a {Fs t}t∈[s,T]-adapted process on (Ω,F,P); (iv) under z(·), for any y∈R+ 0, the stochastic differential equation (4.6) admits a unique solution x(·)on Ω,F,{Fs t}t∈[s,T],Pwithin the set Mof solutions that remain constant after hitting zero. We call Aw[s, T]the set of weak admissible controls. Whenever the meaning is clear from the context, we will use the shorter notation z(·)∈ Aw[s, T]for the tuple Ω,F,{Fs t}t∈[s,T],P, W, z∈ Aw[s, T]. Under the assumptions (A1)–(A3), for any (s, y)∈[0, T)×R+ 0and z∈ Aw[s, T], the SDE (4.6) admits a unique solution x(·) = xs,y(·;z(·)) within the set of solutions Mthat remain constant after hitting zero. Hence, the objective functional (4.7) is well-defined. Thus, the value function V: [0, T ]×R+ 0→Ris well-defined through    V(s, y) = supz(·)∈Aw[s,y]J(s, y;z(·)) V(T, y) = Ψ(T, y) .(4.8) The rest of this section is devoted to obtaining a couple of implicit descriptions for the value function Vdefined above: the dynamic programming principal and the corresponding HJB equation. 4.2.1 Auxiliary Lemma Before moving forward we provide some results that will be useful for the proof of the dynamic programming principle. The proof of the next result uses a technique introduced by Fang and Zhang [23,24] for the study of stochastic differential equations with continuous non-Lipschitz coefficients. Lemma 4.2.1 Let conditions (A1)–(A3)hold. Then, for any s∈[0, T),y∈R+ 0and z(·)∈ 117 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Aw[s, T], the following identity holds lim y0→yE"sup t∈[s,T]|xs,y(t;z(·)) −xs,y0(t;z(·))|#= 0 . Proof: We need to distinguish among the following two cases: α≥1and α < 1. We address the former case first. If α≥1, then the diffusion coefficient of (4.3) is locally Lipschitz continuous and standard results from the stochastic differential equations literature (see, e.g. [46,54]) ensure that for any s∈[0, T)and any z(·)∈ Aω[s, T]there exists K > 0such that E"sup t∈[s,T]|xs,y(t;z(·)) −xs,y0(t;z(·))|#≤K|y−y0|. In the case where α < 1, due to the fact that the diffusion coefficient σxαis not Lipschitz in any neighbourhood of zero, we are not able to provide an estimate as strong as the one above for the case α≥1. Instead, we need to adjust our strategy to the fact that the derivative of xαbecomes arbitrarily large as xapproaches zero. Notice that under assumptions (A1)–(A3), there exists C > 0such that |f(x, z)−f(y, z)| ≤ C|x−y|2α−1(4.9) |xα−yα| ≤ C|x−y|α, for every x, y ≥0such that |x−y|<1and every z∈Z. The second inequality follows from the fact that xαis C0,α-Hölder continuous. In what concerns the first inequality, this is clearly not sharp. However, we should remark that it holds for every quadratic polynomial of the form Ax(B−x), where Aand Bare positive constants. As a consequence, it holds for every map f:R×Z→Rsatisfying assumption (A2). FCUP 118 4.2. Dynamic programming principle and HJB equation Let abe some positive real number and define the function ζa: (0,1) →Ras ζa(λ) = Zλ 0 1 xα+adx . Then, for any λ∈(0,1), we have that ζa(λ)↑ζ0(λ)as a→0+. Moreover, we have that ζ0(λ) = Zλ 0 1 xαdx diverges for every λ∈(0,1). Consider also the function φa: (0,1) →Rgiven by φa(λ) = exp(ζa(λ)) .(4.10) Then, it is clear that φ0 a(λ)(λα+a) = φa(λ)(4.11) and φ00 a(λ) = 1−αλα−1 (λα+a)2φa(λ). Since α∈(0,1), there exists δ > 0such that φ00 a(λ)≤0for every λ∈(0, δ). Fix y≥0, let δ > 0be as given above, and let y0≥0be such that |y−y0|<<δ. Define the processes ηand ξas η(t) = xs,y(t;z(·)) −xs,y0(t;z(·)), t ∈[s, T ] and ξ(t)=(η(t))2, t ∈[s, T ] and let τbe the stopping time given by τ= inf t≥s:η(t)≥2. 119 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Recalling that η(t) = y−y0+Zt s f(xs,y(r;z(·)), z(r)) −f(xs,y0(r;z(·)), z(r))dr +σZt s (xs,y(r, z(·)))α−(xs,y0(r;z(·)))αdWr and using Itô’s formula, we obtain φa(ξ(t∧τ)) = φa(ξ(s)) + 2 Zt∧τ s φ0 a(ξ(r))η(r) (f(xs,y(r;z(·)), z(r))) −f(xs,y0(r;z(·)), z(r))dr + 2σ2Zt∧τ s φ00 a(ξ(r))(η(r))2(xs,y(r;z(·)))α−xs,y0(r;z(·))α2dr +σ2Zt∧τ s φ0 a(ξ(r)) (xs,y(r;z(·)))α−xs,y0(r;z(·))α2dr + 2σ2Zt∧τ s φ0 a(ξ(r))η(r)(xs,y(r;z(·)))α−xs,y0(r;z(·))αdWr. Taking the expected value on both sides of the identity above, and combining the inequalities (4.9) with the fact that φ00 a(λ)≤0for every λ∈(0, δ), we obtain that there exists C > 0such that E[φa(ξ(t∧τ))] ≤φa(ξ(s)) + 2CEZt∧τ s φ0 a(ξ(r))(ξ(r))αdr. Using (4.11), we get that E[φa(ξ(t∧τ))] ≤φa(ξ(s)) + 2CZt∧τ s E[φa(ξ(t∧τ))] dr . Hence, applying Gronwall’s inequality, we get E[φa(ξ(t∧τ))] ≤φa(ξ(s))e2C(t−s). FCUP 120 4.2. Dynamic programming principle and HJB equation Combining the inequality above with Markov’s inequality, we obtain P(φa(ξ(t∧τ)) ≥φa()) ≤E[φa(ξ(t∧τ))] φa() ≤φa(ξ(s))e2C(t−s) φa().(4.12) We will now provide an estimate for the right hand side of the inequality above. Using identity (4.11) once again, we get that the inequality aφ0 a(λ)≤φa(λ) holds for every λ∈(0,1). Combining Gronwall’s inequality with the inequality above, we obtain φa(λ)≤eλ/a . Thus, putting together the inequality above and the definition of ξ(·), we obtain that φa(ξ(s)) ≤e(y−y0)2/a .(4.13) We now take a=|y−y0|and combine the definition of the map φain (4.10) with the inequalities (4.12) and (4.13) to obtain P(φa(ξ(t∧τ)) ≥φa()) ≤E[φa(ξ(t∧τ))] φa() ≤e−ζa()eae2C(t−s). Hence, we get that P sup s≤t≤Txs,y(t;z(·)) −xs,y0(t;z(·))≥!=P(τ < T) ≤e−ζa()eae2C(T−s). 121 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Letting a=|y−y0| → 0, we obtain that the right hand side of the inequality above tends to zero. We conclude that, lim y0→yP sup s≤t≤Txs,y(t;z(·)) −xs,y0(t;z(·))≥!= 0 , that is, the family of (non-negative) random variables Xy(y0) = sup s≤t≤Txs,y(t;z(·)) −xs,y0(t;z(·)),|y−y0|<, (4.14) converges to zero in probability as y0tends to y. To complete the proof of the lemma, we just need to check that the family of random variables Xy(y0)in (4.14) is uniformly integrable. This is the case if Xy(y0)is dominated by an integrable non-negative random variable. Hence, since Xy(y0)≤sup s≤t≤T|xs,y(t;z(·))| for every y0such that |y−y0|< , it is enough to check that the (non-negative) random variable Xy= sup s≤t≤T|xs,y(t;z(·))| is integrable. Using a classical inequality due to Berman [10, Thm 2.1], we obtain the estimate P(Xy≥x)≤exp yx−α−x1−α+1 2(T−s). Integrability of the random variable Xyfollows as a consequence of the estimate above and we obtain that lim y0→yE"sup t∈[s,T]|xs,y(t;z(·)) −xs,y0(t;z(·))|#= 0 as required.  FCUP 122 4.2. Dynamic programming principle and HJB equation Combining assumptions (A1)–(A3), the triangle inequality and the previous Lemma, we obtain that lim y0→y|J(s, y;z(·)) −J(s, y0;z(·))|= 0 ,(4.15) for every z(·)∈ Aw[s, T]. The next Lemma follows from taking the supremum over z(·)∈ Aw[s, T]. Lemma 4.2.2 Let conditions (A1)–(A3)hold. Then, for every s∈[0, T)and y∈R+ 0, we have lim y0→y|V(s, y)−V(s, y0)|= 0 . For any s∈[0, T), let {Fs t}t≥sbe the filtration generated by the Brownian motion W over the time interval [s, t]augmented by all the P-null sets in F. Take s0∈[s, T), fix z(·)∈ Aw[s, T]and z0(·)∈ Aw[s0, T]and define z(t) =      z(t)if t∈[s, s0) z0(t)if t∈[s0, T] . Since for every s0∈[s, T)we have that xs,y(s0;z(·)) is Fs s0-measurable, the solutions xs,y(t;z(·)) and xs0,xs,y(s0;z(·))(t;z(·)) of (4.6) agree a.s. for every t∈[s0, T]. The next Lemma follows as a consequence. Lemma 4.2.3 Let (s, y)∈[0, T)×R+ 0and z(·)∈ Aw[s, T]. For any s0∈[s, T)and z0(·)∈ Aω[s0, T], the following equality holds P-a.s. ω J(s0, xs,y(s0;z(·)); z0(·)) = EZT s0 U(t, xs0,xs,y(s0;z(·))(t;z0(·)), z0(t))dt +Ψ T, xs0,xs,y (s0;z(·))(T;z0(·))Fs s0i(ω). 4.2.2 The Dynamic Programming Principle and the HJB equation We are now able to state and prove the main results in this section. 123 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Theorem 4.2.4 (Dynamic Programming Principle) Assume that conditions (A1)–(A3) hold. Then, for any (s, y)∈[0, T)×R+ 0and s≤s0≤Twe have that V(s, y) = sup z(·)∈Aw[s,T] E"V(s0, xs,y(s0;z(·))) (4.16) +Zs0 s U(t, xs,y(t;z(·)), z(t))dt#. Proof: Denote the right-hand side of (4.16) by V(s, y). Start by noting that for any  > 0there exists z(·)∈ Aw[s, T]such that V(s, y)− < J(s, y;z(·)) . Recalling the definition of the objective functional in (4.7) and letting s0∈[s, T], we obtain V(s, y)− < E"Zs0 s U(t, xs,y(t;z(·)), z(t))dt +EZT s0 U(t, xs,y(t;z(·)), z(t))dt+ Ψ (T, xs,y(T;z(·))) Fs s0#. Resorting to the Markov property of the solutions of (4.6), we get V(s, y)− < E"Zs0 s U(t, xs,y(t;z(·)), z(t))dt +EZT s0 U(t, xs0,xs,y(s0;z(·))(t;z(·)), z(t))dt +Ψ T, xs0,xs,y (s0;z(·))(T;z(·))Fs s0#. FCUP 124 4.2. Dynamic programming principle and HJB equation Using the representation provided by Lemma 4.2.3, we can rewrite the previous inequality as V(s, y)− < EZs0 s U(t, xs,y(t;z(·)), z(t))dt(4.17) +J(s0, xs,y(s0;z(·)); z(·)). Combining the definition of the value function in (4.8) with (4.17), we obtain V(s, y)− < E"Zs0 s U(t, xs,y(t;z(·)), z(t))dt+Vs0, xs,y(s0;z(·))# ≤V(s, y). To prove the converse inequality, let (s, y)∈[0, T)×R+ 0and notice that by Lemma 4.2.2 for any sufficient small  > 0there is δ > 0such that if |y0−y|< δ, then J(s0, y;z(·)) −J(s0, y0;z(·))+V(s0, y)−V(s0, y0)≤(4.18) for every z(·)∈ Aω[s0, T]. Let {Bj}j≥1be a Borel partition of R+ 0(i.e. Bj∈ B(R+ 0),∪j≥1Bj= R+ 0and Bi∩Bj=∅, if i6=j) with diam(Bj)< δ. Take xj∈Bjand notice that for each j≥1there exists zj(·)∈ Aω[s0, T]such that V(s0, xj)−≤J(s0, xj;zj(·)) .(4.19) Hence, combining (4.18) and (4.19), we obtain that J(s0, x;zj(·)) ≥J(s0, xj;zj(·)) −≥V(s0, xj)−2≥V(s0, x)−3 . (4.20) 125 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Define the process z(t, ω) =      z(t, ω)if t∈[s, s0) zj(t, ω)if t∈[s0, T]and xs,y(t;z(·)) ∈Bj . From the definition of the value function and the weak admissible control z(·), we get V(s, y)≥J(s, y;z(·)) =E"Zs0 s U(t, xs,y(t;z(·)), z(t)) dt +EZT s0 Ut, xs0,xs,y(s0;z(·))(t;z(·)), z(t)dt +Ψ T, xs0,xs,y (s0;z(·))(T;z(·))Fs s0#. Combining Lemma 4.2.3 with the last inequality, we obtain V(s, y)≥E"Zs0 s U(t, xs,y(t;z(·)), z(t)) dt+Js0, xs,y(s0;z(·)); z(·)#. We can thus conclude from inequality (4.20) that V(s, y)≥E"Zs0 s U(t, xs,y(t;z(·)), z(t)) dt+V(s0, xs,y(s0;z(·))) −3#. The proof is completed by taking the supremum over z(·)∈ Aw[s, T]. Proposition 4.2.5 Assume that conditions (A1)–(A3)hold. If the pair (¯x(·), z(·)) is optimal for (4.6)–(4.7), then V(t, x(t)) = EZT t U(r, x(r), z(r)) dr+ Ψ (T, x(T)) Fs tP−a.s. for every t∈[s, T]. FCUP 132 4.3. The optimal harvesting problem Recalling that Ψ(T, x)is increasing with respect to x, we obtain J(t, xs,y0(t;h(·)); h(·)) ≤J(t, xs,y(t;h(·)); h(·)) , for all h(·)∈ Aω[s, T]. Taking the supremum over h(·)∈ Aω[s, T], we obtain that V(s, y0)≤V(s, y), concluding the proof.  Proposition 4.3.2 Assume that the SDE (4.26) is of the form dx(t) = fλ(x(t), h(t))dt+σ(x(t))αdWt(4.31) where σand αare positive and fλ:R+ 0×R+ 0→Rdepends on a parameter λ∈Λ⊂R. Assume also that i) Ψ(T, x)is increasing with x ii) for every (x, h)∈R+ 0×R+ 0, if λ1≤λ2then fλ1(x, h)≤fλ2(x, h). Then, the value function is increasing with λ∈Λ, that is, for every (t, x)∈[0, T]×R+ 0if λ1≤λ2, then Vλ1(t, x)≤Vλ2(t, x). Proof: Fix (s, y)∈[0, T)×R+ 0and h(·)∈ Aω[s, T]. Let xλ s,y(t;h(·)) denote the solution of (4.31) associated with the parameter λ∈Λ. Using uniqueness and a.s. continuity of the solution of (4.31), combined with item ii) in the statement of the Proposition, we obtain that for any λ1, λ2∈Λsuch that λ1≤λ2, we have xλ1(t;h(·)) ≤xλ2(t;h(·)) for every t∈[s, T]P-a.s.. Combining the remark above with item i), we obtain that Jλ1(s, y;h(·)) ≤Jλ2(s, y;h(·)) 133 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model for every h(·)∈ Aω[s, T], where the superscripts denote the dependence on the parameter λ∈Λ. The result then follows by taking the supremum over all h(·)∈ Aω[s, T]. 4.3.2 The case of Constant Absolute Risk Aversion Utilities We will now specialize our analysis to the following class of discounted exponential utility functions U(t, h)=e−θt 1−e−γh γ,Ψ(T, x)=e−θT 1−e−βx β,(4.32) where the risk aversion parameters γand β, as well as the discount rate θ, are strictly positive constants. The family of utility functions in (4.32) has the property of having a constant ArrowPratt coefficient of absolute risk aversion (firstly introduced in [4,59]), making these utility functions key examples for the modelling of preference relations in Economic Theory [49]. Alternative choices could include, for instance, families of utility functions with constant coefficients of relative risk aversion, also widely used in Game Theory, Economics and Finance. Combining (4.30) with (4.32), we obtain that h∗(t, x) = −1 γ(θt + ln(xVx)) .(4.33) Substituting the optimal harvesting h∗(t, x)above in the HJB equation (4.28), we arrive at the partial differential equation Vt+1 γe−θt −(1 −θt −ln(xVx))xVx(4.34) +ρx 1−x K−εxn−1 1 + xn−1Vx+1 2σ2x2αVxx = 0 FCUP 134 4.3. The optimal harvesting problem with terminal condition is given by V(T, x)=e−θT 1−e−βx β.(4.35) We will now proceed with a static analysis for the optimal harvesting strategies (4.33) and the corresponding value function, i.e. the solution of the PDE (4.34) with boundary condition (4.35). By Proposition 4.3.1, we have that the value function Vis increasing with respect to the state variable x. This is clearly observable in Figure 4.1. Note also that Vis strictly concave with respect to the state variable x, inheriting such property from the utility functions Uand Ψ. Indeed, we conjecture that this property is robust within a larger class of concave utility functions. In what concerns the time dependence of V, we observe that Vis concave and decreasing for the class of exponential utility functions (4.32). Finally, we remark that all the properties of the value function described above seem to hold for a arbitrary choice of (reasonable) parameter values. In what concerns the dependence of the value function Von the model parameters, it follows from Proposition 4.3.2 and the form of the drift term in (4.26) that, provided all other parameters remain constant, for every (t, x)∈[0, T]×R+ 0,V(t, x)increases with Kand decreases with ε. Additionally, we notice that using the strategy employed in the proof of Propositions 4.3.1 and 4.3.2, we are able to prove that if the utility functions depend monotonically on some extra parameter then, provided all other parameters remain fixed, the value function Vinherits such monotonic behaviour. As a consequence, we obtain that for every (t, x)∈[0, T]×R+ 0the value function decreases with increasing risk aversion parameters γand βand discount rate θ. We will now shift our attention to the optimal harvesting strategies (4.33). For concreteness of exposition, we fix T= 1 and consider the following choice of parameters, that we take as a benchmark in the discussion that follows: 135 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model (P) ρ= 1, K = 10, ε = 0.5, n = 3, σ = 0.1, α = 1.5, γ = 0.5, β = 0.5, θ = 0.04. See Figure 4.2 for a plot of the optimal strategy h∗(t, x)given in (4.33). To make the interpretation of this figure clearer, we provide in Figures 4.3 and 4.4 sections of the graph of h∗(t, x)for fixed values of xand fixed values of t, respectively. Notice that the optimal harvesting is increasing as a function of both variables, being also convex with respect to time. Figure 4.1: Value Function Vfor the set of parameters (P) Figure 4.2: Optimal harvesting h∗in feedback form for the set of parameter values (P) In what concerns the optimal harvesting strategy dependence on the remaining model parameters, the experiments performed by numerical integration of the PDE (4.34)–(4.35) indicate that h∗increases with: FCUP 136 4.3. The optimal harvesting problem Figure 4.3: Sections of the graph of the optimal harvesting h∗for the set of parameter values (P) and fixed values of population size x∈ {5,10,15} Figure 4.4: Sections of the graph of the optimal harvesting h∗for the set of parameter values (P) and fixed instants of time t∈ {0.75; 0.95; 1} i) increasing values of natural growth rate ρ(see Figure 4.5) ii) decreasing values of the predation size εand decreasing values of the Holling functional parameter n iii) increasing values of the volatility coefficient σand of the convexity parameter α iv) decreasing values of the risk aversion parameters γand β(see Figure 4.6) and of the discount rate θ. In what concerns the parameters σ,αand n, we should also add that these seem to have very little influence on the feedback form of the optimal strategies. 137 FCUP Chapter 4. Optimal harvesting for a stochastic logistic growth model Figure 4.5: Optimal harvesting strategy for different values of the population natural growth rate ρand fixed t= 0.5for x∈[5,20]. All the parameters are as those in (P) except for ρthat takes values ρ= 1.25 (green), ρ= 1 (blue), ρ= 0.5 (red) Figure 4.6: Optimal harvesting strategy for different values of the risk aversion parameter γand fixed t= 0.5for x∈[5,20]. All the parameters are as those in (P) except for γthat takes values γ= 0.3(green), γ= 0.4(blue), γ= 0.5(red) 4.4 Conclusions We have studied an optimal harvesting problem associated with a stochastic logistic growth model with a predation term given by a Holling type-nfunctional response and a diffusion coefficient of power type. 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