Contributions to mathematical analysis of non-linear models with applications in population dynamics

TESE DE DOUTORAMENTO CONTRIBUTIONS TO MATHEMATICAL ANALYSIS OF NON-LINEAR MODELS WITH APPLICATIONS IN POPULATION DYNAMICS Cristina Lois Prados ESCOLA DE DOUTORAMENTO INTERNACIONAL DA UNIVERSIDADE DE SANTIAGO DE COMPOSTELA PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS SANTIAGO DE COMPOSTELA 2021 DECLARACIÓN DA AUTORA DA TESE CONTRIBUTIONS TO MATHEMATICAL ANALYSIS OF NON-LINEAR MODELS WITH APPLICATIONS IN POPULATION DYNAMICS Dna. Cristina Lois Prados Presento a miña tese, seguindo o procedemento axeitado ao Regulamento, e declaro que: 1. A tese abarca os resultados da elaboración do meu traballo. 2. De ser o caso, na tese faise referencia ás colaboracións que tivo este traballo. 3. Confirmo que a tese non incorre en ningún tipo de plaxio doutros/as autores/as nin de traballos presentados por min para a obtención doutros títulos. 4. A tese é a versión definitiva presentada para a súa defensa e coincide a versión impresa coa presentada en formato electrónico. E comprométome a presentar o Compromiso Documental de Supervisión no caso de que o orixinal non estea na Escola. En Santiago de Compostela, a 21 de xuño de 2021. Asdo. Cristina Lois Prados AUTORIZACIÓN DOS DIRECTORES/TITORA DA TESE CONTRIBUTIONS TO MATHEMATICAL ANALYSIS OF NON-LINEAR MODELS WITH APPLICATIONS IN POPULATION DYNAMICS D. Eduardo Liz Marzán, Dna. Rosana Rodríguez López INFORMAN: Que a presente tese se corresponde co traballo realizado por Dna. Cristina Lois Prados, baixo a nosa dirección/titorización, e autorizamos a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como directores/titora desta non incorren nas causas de abstención establecidas na Lei 40/2015. De acordo co indicado no Regulamento de Estudos de Doutoramento da USC, declaramos tamén que a presente tese de doutoramento é idónea para ser defendida en base á modalidade Monográfica con reprodución de publicacións, nas que a participación da doutoranda foi decisiva para a súa elaboración e as publicacións axústanse ao Plan de Investigación. En Vigo/Santiago de Compostela, a 21 de xuño de 2021. Asdo. Eduardo Liz Marzán Asdo. Rosana Rodríguez López Aos meus pais, que, ensinándome e apoiándome, conseguiron que chegara ata aquí. ACKNOWLEDGEMENTS/AGRADECEMENTOS personal/persoal Esta tese de doutoramento non é só o resultado do esforzo realizado durante estes catro últimos anos, eu tamén a considero o reflexo do camiño percorrido da man dos que me foron acompañando dende os meus inicios e, por suposto, daqueles que se foron unindo, e son eles aos que van dedicadas as seguintes liñas. Xa fai moito tempo que as miñas primas saben do meu gusto polas Matemáticas, pois parece que cando era unha nena recitaba a táboa de multiplicar en voz alta e tamén lles pedía que me puxeran exercicios de cálculo (seguro que isto non é o peor que tiñan que aguantar...). Sen embargo, a miña primeira vocación foi ser profesora de Educación Física. Tal vez o motivo era que nesa materia non tiñamos deberes. A verdade é que non me gustaba moito traer tarefas para a casa, seguro que os meus pais o poden corroborar, pois teñen pasado horas e horas sentados comigo para que as fixera. A eles teño que agradecerlles toda a dedicación que me prestaron, coa que conseguiron que mellorara as miñas calidades como estudante. Foi en Bacharelato cando me decatei de que a miña paixón eran as Matemáticas. No primeiro curso, a profesora trataba de ensinarnos a esencia desta ciencia exacta, conseguindo así espertar o meu gusto por ela. No segundo ano mudou o profesor, pero non o meu interese, que se pode dicir que se viu reforzado. A Juan Antonio e a Jose Jorge (Jota) teño que darlles as grazas por orientarme e apoiarme nos meus últimos anos de instituto, pero tamén por acompañarme na miña etapa universitaria. Agora vén un dos piares fundamentais que sustentan a miña etapa na universidade, sen o que estou segura de que moitos dos éxitos académicos conseguidos non serían posibles. Son os compañeiros que estiveron ao meu carón nos primeiros anos que pasamos na nosa segunda casa, a Facultade de Matemáticas. Estou falando dos integrantes do “Grupo Abierto” ou “Grupo Haveliano” e as rapazas do grupo “Aplicadas”. Mención especial para Pilar, Ángela e Uzal. Este piar aumentou de tamaño no ecuador da carreira, cando a promoción 2012/2016 se converteu nunha piña que atopou a maneira de desfrutar do ano máis duro destes estudos. Que bos recordos! Non me esquezo de Rosana e Fernando Costal, os profesores que fixeron que me interesara pola Análise Matemática e as Ecuacións Diferenciais. Tamén quero destacar neste momento o apoio dos meus pais e a miña madriña, que sempre confiaron na miña elección e fixeron todo o que estaba nas súas mans para que esta etapa fora posible. Tras o Grao en Matemáticas, estudei o Máster en Matemáticas e o que veu despois supoño que o podedes imaxinar. Teño que agradecer a todas esas perix developed by Cristina Lois Prados during her Master degree studies, since they are the starting point for new achievements in fixed point theory, see the initial paragraphs of the corresponding sections for further details. In this chapter, we have also added some extra information to the Introduction and to Sections 2.3and 2.4. In Chapter 3, we have included additional background in the Introductory section and we have provided more insight on the problems found concerning the applicability when working with some well-known fixed point results and the model into consideration. The Discussion section is new in both Chapters 2and 3. The structure of Chapters 2and 3is similar, since both of them start with an Introduction and conclude with a Discussion; but differs in the central sections. In the introductory sections, we show the mathematical interest of our research and, in Chapter 3, we also include the mathematical formulation of the non-autonomous Lotka-Volterra type system that we will investigate. In the Discussion section, we highlight some relevant aspects of our study and we compare it with other related research works. The intermediate sections/subsections are devoted to show the contributions of our research work. In Chapter 2, we give or improve Krasnosel’skii type compression-expansion results for set contractions and conical domains determined by balls or star convex sets. In Chapter 3, we focus on the existence of positive periodic solutions to Lotka-Volterra systems with general prey growth and functional response of predators. We use an operator approach based on the homotopy version of Krasnosel’skii expansion fixed point theorem. Chapter 1: Background definitions and results I We start by presenting the mathematical framework where we will develop or apply fixed point theory techniques. We first consider an initial value problem for a simple first order differential equation, and we illustrate the procedure needed to apply the classical Krasnosel’skii compression-expansion fixed point result, showing the hypothesis that should be satisfied by the associated nonlinear operator. Then, we provide other examples of initial or boundary value problems that justify the necessity to generalize or replace some of the classical hypotheses owed to Krasnosel’skii. We continue by stating the notions and results in which we have based our research work on fixed point theory. In Section 1.1, we recall some concepts and results related to compact maps and set contractions, that is, the regularity conditions required to the mappings involved. In Section 1.2, we include the Krasnosel’skii compression-expansion fixed point theorem and some of its generalizations. They are divided in two subsections, one for the generalizations in terms of the mapping hypotheses, and the other for those replacing the compression and expansion conditions. xvi Chapter 2: Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets In the framework of fixed point theory, many generalizations of the classical result due to Krasnosel’skii are known, so we begin with an introductory section, where we recall some extensions of this result in different directions, and we also compare two common approaches used in their proofs: direct arguments or topological degree. One of the extensions consists in relaxing the conditions imposed on the mapping, working with set contractions instead of continuous and compact mappings. These results work for conical domains determined by balls or, equivalently, by the norm. Therefore, they are not useful to distinguish fixed points with the same norm. To overcome this necessity, we generalize these results to star convex sets that can be determined by functionals more general than a norm. The first step for the generalization is given in Section 2.2, where we determine the conditions that we will require to the star convex sets. Then, in Section 2.3, we prove the main results. We first deal with the compressive case, where we adapt to this more general framework the proof developed by Potter. Then, the expansive case is reduced to the compressive one by means of a change of variable. We improve the existing result for balls and provide a new theorem for star convex sets. Finally, in Section 2.4, we look for functionals that define an admissible star convex set and, consequently, are appropriate replace the norm. To illustrate the theory, we give an application to the initial value problem for a system of implicit first order differential equations. We conclude the chapter with a discussion of our findings in the context of fixed point theory, with special attention to Krasnosel’skii type fixed point results. We begin explaining the relevance of the obtained generalizations of compression-expansion type results, we mention the complexity that may appear when dealing with set contractions and star convex sets and, we comment the utility of defining the localization domains by means of functionals. Chapter 3: Applications of fixed point theory to periodic predator-prey differential equations In the literature, there exist many several papers devoted to the study of the classical Lotka-Volterra equations and its generalizations. For instance, Tsvetkov (1996) considers the classical system with periodic coefficients and proves the existence of periodic solutions via the topological method of fixed point index. A different approach was developed in (Teixeira Alves and Hilker, 2017), where they include logistic prey growth and hunting cooperation between predators in the autonomous model and, by means of a qualitative study, they observe the presence of oscillations as a consequence of cooperation. Inspired by the xvii general formulation proposed by Teixeira Alves and Hilker (2017) and the periodicity of the model studied in (Tsvetkov, 1996), we consider the following non-autonomous Lotka-Volterra population model    x0=a(t)xg(x) − ϕ(t,x,y)xy; y0= −b(t)y+c(t)ϕ(t,x,y)xy; for which we require some conditions to the functions a,b,c,gand ϕ, including periodicity in the time variable. For particular expressions of the prey growth gand the functional response ϕ, we recover the models consider by Teixeira Alves and Hilker (2017) and Tsvetkov (1996). As a continuation of their research work, we are also interested in the existence of periodic solutions. To that purpose, we use an operator approach as in (Tsvetkov, 1996), but instead of applying topological methods, we use the homotopy version of Krasnosel’skii fixed point theorem. To our knowledge, for these particular predatorprey models, the application of fixed point results, whose hypotheses are given directly in terms of the mapping, has not been considered. Other more popular versions of Krasnosel’skii fixed point theorem have been applied to similar models, but they do not work for our particular formulation. Thus, we contribute to fill this gap in the literature. In addition, we study some interesting properties of the localized solutions that can be seen as a small contribution to the qualitative study of this type of systems. We first state sufficient conditions to ensure that the periodic solution does not reduce to a steady state. Then, under uniqueness conditions, we prove that the nonconstant solutions are positive, therefore the prey and predator populations do not end in extinction. To conclude, we summarize our main contributions, we compare our fixed point theory approach with other related research work and we comment some aspects of the preliminary qualitative study. research line ii The research within this part of the thesis corresponds to that in the joint works with Eduardo Liz Marzán and Frank M. Hilker: (Liz and Lois-Prados, 2020b), (Lois-Prados and Hilker, submitted) (Chapter 5) and (Liz and Lois-Prados, 2020a) (Chapter 6). The contents and results which had been previously developed by other authors are mainly comprised in Chapter 4and small portions of Chapters 5(Subsections 5.2.1and 5.2.2) and 6(Section 6.1). The preliminary contents in Chapter 4were compiled from the literature and most of them do not appear in the three mentioned research articles. By contrast, the majority of Chapters 5and 6uses the contents of these three research works; in Chapter 5, we have included some extra information in the initial paragraphs of the Introduction section, and we have reorganized the distinct parts of the involved articles in order to get a similar structure for both of them; in Chapter 6, there is xviii a new subsection, where we describe in detail the different smooth bifurcations of fixed points. We notice that Chapters 5and 6follow a similar structure: we start with an Introduction where we show several mathematical and biological motivations of the research study and, in this section of Chapter 6, we also include the mathematical formulation of the blood cell production model proposed by Lasota, while in Chapter 5we just provide the ecological description of some fisheries management policies and shift their mathematical formalization to a separate section. The subsequent sections/subsections are devoted to determine the asymptotic dynamics of the models considered. For that purpose, we complement an analytical approach by means of the qualitative theory of dynamical systems with some numerical tools such as 1-parameter bifurcation diagrams. By using this information, we plot 2-parameter bifurcation diagrams, which give a global picture of the long-term dynamics with respect to the variation of two parameters. We conclude the chapters with a discussion section where we compare our study with other related research works. It is worth mentioning that, in Chapter 5, we consider general functions satisfying some typical conditions of discrete-time one-dimensional population maps, and we also work with some particular cases in order to obtain more information from analytical results and numerical simulations. However, in Chapter 6, we just deal with a case study, which is flexible enough to fulfill different typical conditions for population maps, depending on the choice of the parameters. For instance, the associated map can be monotone, unimodal or even bimodal. Chapter 4: Background definitions and results II We begin by introducing, in a mathematical and ecological context, the types of one-dimensional difference equations to be considered. In the mathematical setting, we distinguish between smooth and piecewise-smooth continuous or discontinuous equations. In the ecology context, particularly that of unmanaged single-species populations with discrete-time reproductive seasons, we recall different stock-recruitment relationships and we state some typical mathematical conditions fulfilled by the maps describing these relations. Then, we present the notions and results we use to carry out the qualitative study of asymptotic dynamics. In Section 4.2, we recall some basic concepts that are useful when determining long-term dynamics of smooth and piecewise-smooth maps, then we state some auxiliary results which ensure global stability of an equilibrium. We use some prototypes of stock-recruitment models to illustrate their applicability. In Sections 4.3and 4.4, we describe in detail a number of bifurcations and features of 1-parameter bifurcation diagrams which appear throughout the thesis. The bifurcations are local or global and the variety or complexity of local bifurcations increases when the regularity of the piecewise-smooth map is decreased. The features one can find in the analxix ysis of bifurcation diagrams suggest unexpected dynamic behaviors which can have important consequences for the management of ecological systems. Chapter 5: Combinations of constant quota and threshold based harvesting strategies We study two discrete-time models for single-species populations subject to different harvesting rules. Both strategies combine constant catches to obtain predictable yield and a threshold or minimum biomass level to protect the population. In Section 5.1, we introduce these control rules in the context of fisheries management. From this point of view, the simplest rule allows to harvest a maximum annual quota Hif the population size after reproduction is above the threshold T; and, if it is below the threshold, no harvesting is applied. We refer to this strategy as threshold constant catch harvesting (TCC). If we require the additional condition that a minimum biomass level Tmust remain after harvesting, then the strategy becomes more protective and difficult to apply, we refer to it as precautionary threshold constant catch (PTCC). Discrete-time mathematical models for these strategies lead in a natural way to piecewise-smooth maps, whose dynamics are challenging because multiple non-smooth bifurcations may appear. As the map associated to the TCC rule is discontinuous and that corresponding to PTCC is continuous, the qualitative study of TCC dynamics is more complex. Thus, we first study the PTCC rule in Section 5.3; then the TCC rule in Section 5.4. In both cases, we combine analytical and numerical results to provide a comprehensive overview of the dynamics, which depend on the two relevant harvesting parameters Hand T. In Section 5.3, we provide a thorough analytical description of dynamics and bifurcations for general compensatory population models. In the overcompensatory case, where we found more complicated dynamics, we explain the dynamical behavior in some regions on the 2-parameter plane (H,T), but we choose the Ricker model as a case study to give a global picture of the dynamics. In Section 5.4, the discontinuity of the map associated to TCC induces complex dynamical behavior for the strictly increasing compensatory case. Thus, we restrict our study to this type of stock-recruitment relations, for which we use the Beverton-Holt model as a particular example. For the TCC rule, we devote an additional subsection to the study of two frequently considered management objectives: average yield and harvest frequency. We conclude the chapter with a discussion of our findings in the context of piecewise-smooth difference equations and harvesting control rules. xx Chapter 6: Lasota discrete model for blood cell production In an attempt to explain some experimental evidences of chaotic behavior in blood cell populations, Andrzej Lasota proposed in (Lasota, 1977) the following discrete-time one-dimensional model: xn+1= (1−σ)xn+ (cxn)γe−xn,n∈N∪{0}. This model includes a destruction rate parameter σ∈(0,1)and a gamma- Ricker function as a representation of the quantity of cells produced in the bone marrow. The gamma-Ricker map incorporates two other parameters γ,c > 0. With the aim of explaining the influence of the destruction rate on the dynamics, Lasota fixed the parameter values as c=0.47 and γ=8, and chose some values for σ. These values allowed him to describe the dynamics in some relevant clinical cases: normal conditions (σ=0.1: depending on the initial condition, solutions either go to extinction or converge to a positive equilibrium), non-severe disease (σ=0.4: the positive equilibria are unstable and there is a 2-periodic attractor), and severe disease (σ=0.8, where Lasota observed the presence of a 3-periodic orbit and, therefore, chaotic behavior). In this chapter, we study the model in detail, by means of an analytical approach combined with some numerical simulations. In particular, we revisit the results which appear in the original paper, but we also discover new interesting phenomena. In the analytical part, we find sufficient conditions for extinction and stability (including a sharp global stability condition for γ⩽1) depending on the involved parameters. Then, we show some 1-parameter bifurcation diagrams using either γor σas bifurcation parameters, while keeping c=0.47 as in (Lasota, 1977). These diagrams allow us to discover the rich dynamics of the model, which exhibits features such as stability switches (bubbles), extinction windows, hydra effects and sudden collapses, among others. We also present a 2-parameter bifurcation diagram as an illustration that summarizes the long-term dynamics. Finally, we compare our results with those in the previous work carried out by Lasota, and we additionally interpret them in the context of population dynamics. The considered equation is also suitable to model the dynamics of populations with discrete reproductive seasons, adult survivorship, overcompensatory density dependence and Allee effects. In this context, our results show the rich dynamics of this type of models and point out the subtle interplay between adult survivorship rates and strength of density dependence (including Allee effects). xxi RESUMO ESTENDIDO Os contidos desta tese de doutoramento son o resultado do traballo realizado pola autora Cristina Lois Prados durante os seus estudos correspondentes ao Programa de Doutoramento en Matemáticas da Universidade de Santiago de Compostela, en colaboración cos seus directores Eduardo Liz Marzán (Universidade de Vigo) e Rosana Rodríguez López (Universidade de Santiago de Compostela); así como cos profesores responsables das dúas estadías de investigación realizadas, Frank M. Hilker (Osnabrück University, Alemaña) e Radu Precup (Babe¸s-Bolyai University, Romanía). Este documento comprende dúas liñas de investigación diferentes que se desenvolven no ámbito da análise matemática dos modelos non lineais. As maiores diferenzas obsérvanse no tipo de ecuacións que se consideran e na metodoloxía utilizada. Na Liña de Investigación I (Research Line I), mediante a xeneralización dalgúns resultados da teoría de punto fixo, mellórase a localización das solucións a problemas de valor inicial ou problemas de fronteira para diferentes tipos de ecuacións diferenciais, en particular, para as ordinarias; ademais contribúese á aplicación da teoría de punto fixo a modelos de poboación para esta clase de ecuacións. Na Liña de Investigación II (Research Line II), o noso obxectivo principal é describir o comportamento asintótico e as bifurcacións de algúns sistemas dinámicos discretos unidimensionais. Séguese unha metodoloxía máis aplicada, traballando con modelos de poboación que xorden na xestión de recursos pesqueiros ou no modelado da produción de glóbulos vermellos. Dado que os contidos das liñas de investigación mencionadas son bastante diferentes, decidimos dividir este manuscrito en dúas partes autocontidas, que inclúen a súa propia lista de referencias. Sen embargo, seguen unha estrutura similar: comezan con algúns resultados preliminares que serven para sentar as bases matemáticas de cada parte da tese doutoral e rematan cunhas conclusións e algunhas ideas para continuar as liñas de investigación deste manuscrito. Os capítulos intermedios recollen os contidos centrais da investigación. No que segue, damos unha breve descrición dos contidos dos capítulos correspondentes a cada liña de investigación. liña de investigación i A investigación levada a cabo nesta liña correspóndese cos traballos realizados conxuntamente con Rosana Rodríguez López e Radu Precup: (Lois-Prados, Precup e Rodríguez-López, 2020; Lois-Prados e Rodríguez-López, 2020) (Capítulo 2) e (Lois-Prados e Precup, 2020) (Capítulo 3). A maioría dos contidos desta parte da tese foron recompilados dos tres artigos mencionados e redistribuídos xxiii nos diferentes capítulos. As nocións e os resultados preliminares previamente establecidos por outros autores están maioritariamente recollidos no Capítulo 1. Ademais, no Capítulo 2(Sección 2.2e Subsección 2.4), aparecen algúns contidos que Cristina Lois Prados desenvolveu durante os seus estudos de mestrado, dado que son a punto de partida de novas achegas no ámbito da teoría de punto fixo, para máis detalles véxanse os parágrafos iniciais das seccións correspondentes. Por outra banda, na Introdución e nas Seccións 2.3e2.4deste capítulo, incluíuse información adicional á que aparece nos artigos. No Capítulo 3, tamén se amplía a descrición do contexto no que se desenvolve o traballo realizado (Sección 3.1) e ofrécense máis detalles sobre os problemas que aparecen ao aplicar versións coñecidas do resultado de punto fixo de Krasnosel’skii ao modelo que estamos a considerar. A Discusión é nova en ambos Capítulos 2e3. A estrutura dos Capítulos 2e3é similar, xa que ambos comezan cunha Introdución e rematan cunha Discusión, pero diferéncianse nas seccións centrais. Nas seccións introdutorias, abordamos o interese da nosa investigación dende un punto de vista matemático, no Capítulo 3tamén se inclúe a formulación matemática do modelo non-autónomo de Lotka-Volterra co que imos traballar. Nas seccións de Discusión, destacamos os aspectos máis relevantes do estudo realizado e comparámolo con outras investigacións relacionadas. As seccións/subseccións intermedias amosan a investigación que se levou a cabo. No Capítulo 2, probamos ou melloramos resultados de punto fixo de tipo compresivo-expansivo de Krasnosel’skii para aplicacións contractivas e dominios cónicos determinados por bolas ou conxuntos estrelados. No Capítulo 3, centrámonos na existencia de solucións periódicas para modelos de tipo Lotka- Volterra con termo xenérico de predación. Facemos uso do operador non lineal asociado, que se atopa nas condicións do caso expansivo da versión homotópica do resultado clásico de Krasnosel’skii. Capítulo 1: Definicións e resultados preliminares I Comezamos presentando o contexto matemático no que se desenvolverán e aplicarán as técnicas de punto fixo. En primeira instancia, consideramos un problema de valor inicial para unha ecuación diferencial ordinaria sinxela e ilustramos o procedemento a seguir para aplicar o resultado clásico de Krasnosel’skii de tipo compresivo-expansivo, mostrando as hipóteses que debe verificar o operador non lineal asociado. Despois, engadimos outros exemplos de problemas de valor inicial ou de fronteira, que xustifican a necesidade de xeneralizar ou cambiar algunhas das hipóteses clásicas requiridas por Krasnosel’skii. A continuación lembramos algunhas nocións e resultados nos que baseamos a nosa investigación no eido da teoría de punto fixo. Na Sección 1.1, establecemos conceptos e resultados relacionados con aplicacións compactas e contractivas, que son as condicións de regularidade que lles esiximos aos operadores. xxiv Na Sección 1.2, incluímos o resultado de Krasnosel’skii de tipo compresivoexpansivo e algunhas das súas xeneralizacións. Dividímolas en dúas subseccións, unha para as xeneralizacións en termos das hipóteses sobre as aplicacións e outra para aquelas que modifican as condicións de compresión e expansión. Capítulo 2: O teorema de punto fixo de Krasnosel’skii de tipo compresivo-expansivo para aplicacións contractivas e conxuntos estrelados No ámbito da teoría de punto fixo, existen moitas xeneralizacións do resultado clásico de Krasnosel’skii, polo que comezamos cunha sección introdutoria, onde lembramos algunhas das extensións existentes en varias direccións, e tamén comparamos dúas técnicas comunmente utilizadas nas súas demostracións: argumentos directos e teoría do grado topolóxico. Unha das extensións consiste en relaxar as condicións que se lle impoñen á aplicación, traballando con aplicacións contractivas en lugar de operadores continuos e compactos. Estes resultados son válidos para dominios cónicos determinados por bólas, ou de xeito equivalente, por normas. En consecuencia, estes resultados non serven para localizar puntos fixos coa mesma norma. Para paliar esta necesidade, obtivemos unha xeneralización destes resultados a conxuntos estrelados que poden vir dados por funcionais máis xerais que a norma. O primeiro paso para dita xeneralización dáse na Sección 2.2, onde se determinan as condicións que lles imos esixir aos conxuntos estrelados. A continuación, na Sección 2.3, faise a demostración dos resultados principais. Trabállase en primeiro lugar co caso compresivo, no que se adapta a demostración de Potter a este ámbito máis xeral. Despois, na proba do caso expansivo faise unha redución ao caso compresivo mediante un cambio de variable. Neste caso, mellórase o resultado xa coñecido para bólas e conséguese un resultado novo para conxuntos estrelados. Finalmente, na Sección 2.4, búscanse funcionais que definan un destes conxuntos estrelados, e que en consecuencia, poidan ser usados para substituír a norma. Para ilustrar esta teoría, aplicamos os resultados obtidos a un problema de valor inicial para un sistema de ecuacións diferenciais implícitas de primeira orde. Rematamos este capítulo cunha discusión sobre as nosas achegas á teoría de punto fixo, con especial atención aos resultados do tipo dos de Krasnosel’skii. Comezamos explicando a relevancia que pode ter esta xeneralización dos resultados de tipo compresivo-expansivo, explicamos tamén cal é a complexidade que se presenta ao traballar con aplicacións e conxuntos máis xerais e, por último, poñemos en valor a posibilidade de definir os dominios de localización mediante funcionais. xxv on the dynamics of discrete time population models (see Liz, 2010b). Here, we try to obtain similar results for intervention rules that consider a threshold value or minimum biomass level under which no harvesting is allowed. More precisely, we work in the following goals: G3 Dynamical study of one-dimensional piecewise-smooth population models applied to threshold harvesting: We start by studying the dynamics of different sustainable intervention strategies applied to stricly increasing or unimodal maps with at most one positive fixed point, such as the classical Beverton- Holt and Ricker models. It is interesting to study the influence of the underlying harvesting parameters on the dynamics, for that purpose we look for regions of global stability, bistability, chaos, or parameter values where there is a change on the population asymptotic dynamics (bifurcations). Once we finish with the qualitative study, we find it significant to compare the results with those obtained for some related traditional strategies. G4 More flexible population models: A similar but more ambitious objective is to describe the dynamics of the same intervention strategies applied to more general maps, not necessarily monotone nor unimodal and with 0,1or 2positive fixed points (see, for example, Liz, 2018a). As the model flexibility usually increases the difficulty of the dynamical analysis, we can start studying the dynamics of these models without harvesting, which implies a reduction in the number of parameters. xxxii METHODOLOGY This chapter of the thesis is devoted to describe the methods or approaches used in the development of the goals G1-G4. In general, the research begins with a detailed review of some related classical and recent bibliographical references in order to acquire some necessary knowledge about relevant results and techniques in the related field. After that, we start to work on the underlying problem by following these approaches and, sometimes, we also need to look for or to develop other statements and methods. In particular, in relation with the objective G1, we provide a generalization of the compression-expansion Krasnosel’skii fixed point theorem by using classical arguments of this theory, following the approach used by Krasnosel’skii and other authors. For the rest of the goals, unless we use distinct methods, somehow we follow a similar approach in which we formulate the model and then we look for the proper results and techniques that can be used for its analysis. In the following, we provide more details about the methodology that we use to attain each of the objectives. We use specific results and techniques for goals G1 and G2. In the case of goals G3 and G4, some methods are applied to both of them, so we explain the similarities and differences in M3. M1 For the generalization of the compression-expansion Krasnosel’skii fixed point theorem to set contractions and star-convex sets, we first observe that, in the compressive case, the approach carried out by Potter (1974) for set contractions can be adapted to more general domains determined by star-convex sets instead of balls. Thus, we start by determining the properties we have to impose on the star-convex sets in order to reproduce the proof given by Potter in this more general framework and, then, we adapt his main results. After that, we observe that the approach due to Các and Gatica (1979), in the expansive case, cannot be properly adapted to our framework. So, we look for other methodologies and we finally use the idea in (Precup, 2006), that is, a change of variable to reduce the expansive case to the compressive one. It is worth mentioning that we do not use topological degree techniques. M2 In the application of fixed point theory to periodic predator-prey models, we start by transforming the ordinary differential equation into its equivalent integral formulation, so we obtain an operator Twhose fixed points are the solutions of the predator-prey model. Then, we look for a special set where we intend to localize a periodic solution, and for a proper fixed point theorem that could be applied to this type of problems. Finally, we also use some basic arguments in the analysis of ordinary differential equations to provide sufficient conditions that ensure the non-constancy and positiveness of the localized periodic solution. xxxiii M3 In order to realize a detailed study of the long-term dynamics of the onedimensional difference equations that we consider in Chapters 5and 6, we first determine the number of positive fixed points of the associated map depending on the parameter values, and we also study its local and global stability. By using this information, one can classify some of the bifurcations in which fixed points are involved. Then, we continue with the analytical study of other interesting phenomena, such as, the bifurcations of 2-cycles, the boundary-collision bifurcations, the regions of bistability or the sudden collapses. Once we conclude with the analytical study, we use 2-parameter BD to provide a global picture of the long term dynamics, and 1-parameter BD to illustrate the most relevant dynamical features. This is the general procedure that we follow to attain goals G3 and G4, however, we use different results and arguments to study the dynamics. On the one hand, the intervention strategies that we have mentioned in objective G3 produce a piecewise-smooth map whose long-term behavior presents some features that cannot occur for smooth models. On the other hand, the map in goal G4 is smooth but more flexible, which in some sense increases the difficulty of the dynamical study. xxxiv CONTENTS acknowledgements ix abstract/resumo xiii preface xv resumo estendido xxiii aims and objectives xxxi methodology xxxiii i research line i 1 1 background definitions and results i 3 1.1Conditions for integral mappings . . . . . . . . . . . . . . . . . . . 5 1.2Krasnosel’skii type fixed point results . . . . . . . . . . . . . . . . 8 1.2.1Generalizations to set contractions . . . . . . . . . . . . . . 10 1.2.2Norm type, vector and homotopy versions . . . . . . . . . 10 2 krasnosel’skii type results in star convex sets 13 2.1Introduction ............................... 14 2.2Generalframework........................... 17 2.3Mainresults ............................... 20 2.3.1Compressivecase........................ 22 2.3.2Expansivecase ......................... 31 2.4Admissible sets defined by functionals . . . . . . . . . . . . . . . . 35 2.5Application to a first order implicit differential system . . . . . . 41 2.6Discussion ................................ 48 3 applications to predator-prey differential equations 51 3.1Introduction and model description . . . . . . . . . . . . . . . . . 52 3.2Integral version of the system and related notations . . . . . . . . 54 3.3Existence, localization and other properties of solutions . . . . . . 56 3.3.1Existence and localization . . . . . . . . . . . . . . . . . . . 56 3.3.2Properties of solutions . . . . . . . . . . . . . . . . . . . . . 66 3.3.3Time independent predators functional response ϕ.... 71 3.4Discussion ................................ 75 conclusions and future prospects i 77 references i 81 ii research line ii 85 4 background definitions and results ii 87 4.1Conditions for single-species population models . . . . . . . . . . 88 4.2Stability concepts and results . . . . . . . . . . . . . . . . . . . . . 91 4.3Bifurcationtypes ............................ 96 xxxv 4.41-parameter bifurcation diagrams features . . . . . . . . . . . . . 100 5 combinations of cc and th harvesting strategies 103 5.1Introduction ............................... 104 5.2Modelsdescription ........................... 110 5.2.1Constantcatchrule....................... 110 5.2.2Threshold harvesting rule . . . . . . . . . . . . . . . . . . . 111 5.2.3Precautionary threshold constant catch rule . . . . . . . . 112 5.2.4Threshold constant catch rule . . . . . . . . . . . . . . . . . 113 5.3Precautionary threshold constant catch (PTCC) . . . . . . . . . . . 115 5.3.1Fixed points: location, stability and local bifurcations . . . 115 5.3.2Complex dynamics: essential attraction and chaotic behavior............................... 127 5.3.3Impact of harvest parameters on population dynamics . . 129 5.4Threshold constant catch (TCC) . . . . . . . . . . . . . . . . . . . . 139 5.4.1Fixed points: location, stability and related bifurcations . 139 5.4.2Complex dynamics: absorbing intervals . . . . . . . . . . . 144 5.4.32-cyclesSBsandBCBs ..................... 146 5.4.4Impact of harvest parameters on population dynamics . . 148 5.4.5Average yield and harvest frequency . . . . . . . . . . . . . 156 5.5Discussion ................................ 158 6 lasota discrete model for blood cell production 165 6.1Introduction and model description . . . . . . . . . . . . . . . . . 166 6.2Fixed points: location, stability and bifurcations . . . . . . . . . . 167 6.2.1Preliminaryresults....................... 167 6.2.2Existence and stability . . . . . . . . . . . . . . . . . . . . . 169 6.2.3Bifurcations of fixed points . . . . . . . . . . . . . . . . . . 172 6.3Impact of parameters variation on long-term dynamics . . . . . . 173 6.4Discussion ................................ 177 conclusions and future prospects ii 181 references ii 186 glossary 193 transfer of copyright/publish agreement 197 xxxvi Part I RESEARCH LINE I 1 BACKGROUND DEFINITIONS AND RESULTS I For the sake of completeness, in this chapter we provide some notions and results on fixed point theory. We begin introducing the type of problems we will deal with. In general, given an initial or boundary value problem for differential equations, we are interested in its integral characterization or, in obtaining a mapping Twhose fixed points are in correspondence with solutions of the initial or boundary value problem. Once we get the associated mapping, fixed point results constitute a useful tool to prove the existence and to localize the fixed points of T. Throughout this part of the thesis, this procedure to find solutions of differential equations is referred to as (classical) operator approach. We illustrate the process with a simple initial value problem:    x0(t) = f(t,x(t)),t∈[0,1]; x(0) = 0;(1) where f: [0,1]×R−→ Ris bounded and continuous. For some particular expressions of the map f, there exist techniques that allow to find the solutions of (1). In general, by using the continuity of f, we can assert that there exists at least a solution, but it may not be easy to find procedures to solve the problem explicitly. Thus, it is interesting to get the equivalent problem to which we can apply some existence and localization fixed point results. For that purpose, let us consider the non-linear operator T:C([0,1],R)−→ C([0,1],R)given by T(x)(t) = Zt 0 f(s,x(s))ds,x∈C([0,1],R),t∈[0,1]. (2) Each solution of the initial value problem (1) in C([0,1],R)is in correspondence with a fixed point of the operator Tgiven by (2), that is, an element x∈C1([0,1],R)such that x(t) = T(x)(t)for all t∈[0,1]. The simple expression of the map T, together with the regularity of f, allows us to use, for instance, the classical compression-expansion Theorem of Krasnosel’skii (see Krasnosel’skii, 1964, Chapter 4) to localize fixed points of T. However, the above-mentioned fixed point result cannot be applied directly to the map Tin (2) without the establishment of additional structures and the imposition of certain restrictions on the mapping f, as we explain below. We first should equip the set C([0,1],R)with a norm in order to obtain a Banach space. For instance, the pair (C([0,1],R),kxk∞)is a Banach space, where kxk∞:= max{|x(t)|,t∈[0,1]}and |·|denotes the absolute value in R. Secondly, the result does not work in the whole space C([0,1],R), so we have to consider a particular subset of the Banach space which is called a cone, for 3 4 background definitions and results i example, C([0,1],R+). Thus, if we require that f([0,1]×R+)⊂R+, then it is guaranteed that Tmaps C([0,1],R+)into itself. Thirdly, the restriction of Tto the cone C([0,1],R+)should be a compact mapping. This is indeed the case since fis bounded and continuous. Finally, the compression or expansion conditions should be fulfilled. They depend on the mapping Tand the elements of the cone with kxk∞=rand kxk∞=R, where r < R are two positive real numbers. Once we have checked that all the hypotheses are fulfilled, the result provides the existence of a fixed point x∈C([0,1],R+)with r⩽kxk∞⩽R, and, therefore, a localization for this fixed point is given. The mentioned classical results due to Krasnosel’skii can be applied to a huge amount of problems. However, if we relax or replace some of the required hypotheses, we can improve the localization of fixed points or even apply the results to a wider range of initial or boundary value problems. The generalizations of Krasnosel’skii fixed point theorem can be stated by relaxing different hypotheses: the properties of the domain of T; the compactness of the mapping and the compression-expansion conditions. In Sections 1.1and 1.2, we state some generalizations for two of these conditions, but for those regarding the domain of Twe refer the reader to Chapter 2. In Section 1.1, we start describing the concept of compact mapping and recalling the Arzelà-Ascoli characterization of relatively compact sets in the Banach space (C(I,Rn),k·k∞), where (I,d)is a compact metric space, that will be useful in applications. Then, we review some concepts that generalize the previous ones: the measure of noncompactness and α-Lipschitz maps. We also provide some results that state useful properties of both concepts. Section 1.2is devoted to recall some Krasnosel’skii type fixed point theorems that we use in this part of the monograph. We first review the concept of cone and the classical Krasnosel’skii compression-expansion result. Then, we organize the generalizations in two different subsections. In Subsection 1.2.1, we recall Krasnosel’skii type results for set contractions. These generalizations allow to localize solutions for implicit first order initial value problems of the form:    x0(t) = f(t,x(t)) + g(t,x0(t)),t∈[0,1]; x(0) = 0;(3) where f,g: [0,1]×R−→ Rare continuous and gsatisfies some Lipschitz type condition. It is worth mentioning that the non-linear operator Tassociated to this problem is not necessarily compact, but it can be a set contraction, see Chapter 2for further details. In Subsection 1.2.2, we state three fixed point results preserving the compactness hypothesis of the mapping T, but considering different compression-expansion conditions. In Chapter 3, we study their applicability to a class of Lotka-Volterra type equations. The first result is a generalization due to Güo and Lakshmikantham (1998), with compression-expansion conditions of norm type, which localizes the solutions in more general domains 1.1 conditions for integral mappings 5 determined by a cone and two open sets. This generalization works for some particular Lotka-Volterra models with time-delays, but it cannot be applied to simpler formulations of these predator-prey systems. The second result is the vector version given in (Precup, 2007), which was applied to localize positive periodic solutions of a differential system with linear and non-linear terms. That system is quite similar to Lotka-Volterra type equations; however, it seems that the vector version presents some difficulties in its applicability to simple formulations of these population models, as it happens with the original result owed by Krasnosel’skii. Besides, the homotopy version of the original Theorem of Krasnosel’skii, stated at the end of Subsection 1.2.2, works for these type of predator-prey equations. 1.1 conditions for integral mappings In this section, we state some basic notions and results related with the regularity of the mapping. These concepts will appear in the hypothesis of our fixed point results. We first recall the concepts of (relatively) compact set and compact mapping. We also provide the Arzèla-Ascoli theorem that gives a characterization of relatively compact sets in (C(I,Rn),k·k∞)where (I,d)is a compact metric space. This result will help to prove the compactness of mappings in applications, in particular, we will use Corollary 1.1.4. For further details, see (Precup, 2002, Chapter 1). Definition 1.1.1.Let (X,d)be a metric space. We say that Xis a compact set if, for each ε > 0,Xadmits a finite covering by open balls of radius ε. More precisely, for each ε > 0, there exist Nε∈Nand xi∈X,i=1,...,Nε, such that X⊂ Nε [ i=1 B(xi,ε), where B(xi,ε) = {y∈X:d(xi,y)< ε}. A subset Dof Xis relatively compact if its closure Dis a compact set (as a metric subspace of X). Definition 1.1.2.Let X,Ybe metric spaces and D⊂X. The map T:D⊂X−→ Y is compact if, for all A⊂Dbounded, T(A)is a compact set. Theorem 1.1.3(Arzelà-Ascoli).Let (I,d)be a compact metric space and consider the Banach space (C(I,Rn),k·k∞). A set D⊂C(I,Rn)is relatively compact if and only if the following conditions are satisfied: 1.Dis bounded, that is, there exists M > 0 such that kxk∞< M for all x∈D; 2.Dis uniformly equicontinuous, that is, for all ε > 0, there exists δ > 0 such that for every x∈D, |x(t) − x(s)|< ε,for all t,s∈I,with d(t,s)< δ. 2 KRASNOSEL’SKII TYPE COMPRESSION-EXPANSION FIXED POINT THEOREM FOR SET CONTRACTIONS AND STAR CONVEX SETS The contents of this chapter are comprised in the research articles (Lois-Prados and Rodríguez-López, 2020)1and (Lois-Prados, Precup, and Rodríguez-López, 2020)2. In both manuscripts, we deal with generalizations of Krasnosel’skii type compression-expansion results to set contractions and domains determined by a cone and two star convex sets; and we show the applicability of the results to initial or boundary value problems. We organize the contents of the chapter as follows. We begin with the Introduction section, where we give a contextualization of our study in the framework of fixed point theory. In particular, we review some generalizations of the classical Theorem of Krasnosel’skii and we explain how we contribute to them. We also justify the use of a direct approach by means of classical fixed point arguments rather than topological degree techniques. The theoretical results we have developed are contained in Sections 2.2-2.4. In Section 2.2we describe the type of star convex sets we use to localize the 1Cristina Lois-Prados (Instituto de Matemáticas, Universidade de Santiago de Compostela, Spain) & Rosana Rodríguez-López (Instituto de Matemáticas, Universidade de Santiago de Compostela, Spain), “A generalization of Krasnosel’skii compression fixed point theorem by using star convex sets”, Proceedings of the Royal Society of Edinburgh - A (ISSN: 14737124, 03082105),150, pp. 277-303,2020. The final authenticated version is available online at: https://doi.org/10.1017/prm.2018.119. JCR 2019 (category; impact factor; relative position; quartile): Mathematics; 1.009; Q2;111/325. SJR 2019 (category; impact factor; quartile; H index): Mathematics (miscellaneous); 1.08; Q1; 52. PhD student contributions: The research idea was conceived by my supervisor R. Rodríguez- López, then C. Lois-Prados developed the majority of contents for her Master degree final dissertation. At the beginning of her PhD studies, she elaborated Section 3.1within the published manuscript. My supervisor R. Rodríguez-López provided help, support and ideas through all the elaboration and publication process. 2Cristina Lois-Prados & Radu Precup (Department of Mathematics, Babe¸s Bolyai University, Romania) & Rosana Rodríguez-López, “Krasnosel’skii type compression-expansion fixed point theorem for set-contractions and star convex sets”, Journal of Fixed Point Theory and Applications (ISSN: 16617738,16617746),22,2020. The final authenticated version is available online at: https://doi.org/10.1007/s11784-020-00799-0. JCR 2019 (category; impact factor; relative position; quartile): Mathematics; 1.741;29/325; Q1. PhD student contributions: In this manuscript, we improved the two main results obtained in the Master degree final dissertation of C. Lois-Prados (supervised by R. Rodríguez-López). C. Lois-Prados developed almost all the contents of the submitted version of the article, following some suggestions from her supervisor (improvement of the compression result) and professor R. Precup (ideas for the expansion result and the application section). Asked by a reviewer, professor R. Precup enriched the application results, now working for systems rather than for a singular equation. 13 14 krasnosel’skii type results in star convex sets fixed points and we also derive some useful properties. Then, we are in a position to prove the main fixed point results in Section 2.3. Finally, in Section 2.4, we show that we can use functionals to describe the underlying star convex sets, and these functionals are more general than a norm. In Section 2.5, we show the applicability of the compression type result to an initial value problem for a system of first order differential equations. The associated integral type mapping is noncompact but a set contraction; and the outer boundary of the localization domain is defined by several functionals more general than a norm. We conclude with the Discussion section, where we summarize our main achievements and compare our research work with other related studies in the framework of fixed point theory. 2.1 introduction In this section, we introduce the research work developed within this chapter in the framework of fixed point theory. We present the contents divided in blocks. Applicability of Krasnosel’skii type fixed point results Krasnosel’skii type compression-expansion fixed point theorems are a powerful tool to prove the existence of positive solutions to several classes of problems and also to obtain multiple solutions. Erbe and Wang (1994), Torres (2003), and Zima (2004) have applied a generalization of Theorem 1.2.5written in terms of the norm to different second order boundary value problems. For instance, Torres (2003) considers a second order equation with periodic boundary conditions and a Caratheodory non-linear term. He also shows several applications, such as one to equations with jumping nonlinearities. O’Regan and Precup (2005) have applied to Hammerstein integral equations a generalization of Theorem 1.2.5with compression-expansion conditions given in terms of two norms. Other authors have applied this type of results to first order differential systems. In (Wang, 2011), it is proved the existence and multiplicity of ω-periodic solutions for a non-autonomous singular system; while Bolojan and Precup (2014) have studied an implicit system with nonlocal conditions. Generalizations of the Krasnosel’skii compression-expansion fixed point theorem We already mentioned, in the introductory paragraphs of Chapter 1, that the classical Krasnosel’skii compression-expansion fixed point theorem can be generalized by relaxing different hypothesis. In Subsection 1.2.1, we recall some extensions of the Theorem of Krasnosel’skii working for more general mappings, more precisely, for set contractions. In Subsection 1.2.2, we consider the norm type, vector and homotopy versions in which the compression-expansion condi- 2.1 introduction 15 tions differ from the original ones. Moreover, one of these extensions considers more general domains. This is the result due to Güo and Lakshmikantham (1998) which localizes the fixed point in the more general region C∩(Ω2\Ω1), where Cis a cone in a Banach space (X,k·k)and Ω1,Ω2are bounded open sets. Another generalization in this direction can be found in (Precup, 2006), where the fixed point is located in the region {x∈C:r < kxk1,kxk2< R}, which is determined by the cone and two norms. These two generalizations work for continuous and compact mappings, but change the compression-expansion conditions and consider more general domains. This is also the case of Theorem 4.1in (Anderson, Avery, and Henderson, 2010), which considers continuous concave/convex functionals that are involved in the Legget-Williams type conditions and the localization domains. In the work by Kwong (2008), we find several extensions of Theorem 1.2.5. In this chapter, we deal with a generalization that preserves the compressionexpansion type conditions of the Krasnosel’skii original result, but works for set contractions and more general domains, which are determined by a cone and two star convex sets. The motivation for working with star convex sets comes from the necessity to distinguish between two solutions in case that they have the same norm, see Figure 1. It is worth mentioning that the results owed by Precup (2006) could also overcome this problem. However, our results work for more general mappings and we claim that the star convex sets can provide a better localization than the sets determined by two norms. Moreover, in Section 2.4, the statements of Corollary 2.4.4is given in terms of two norms and it is similar to Theorem 3in (Precup, 2006), with the difference that our result works for more general mappings, but the other does not require the completeness of the space. Direct approach vs degree theory Krasnosel’skii proved his theorem directly, using only classical arguments of fixed point theory, particularly Schauder’s fixed point theorem (Krasnosel’skii, 1960,1964). However, it is well-known that this result can also be deduced as a consequence of the topological degree theory (see Granas and Dugundji, 2013). Moreover, many of the generalizations of the classical results due to Krasnosel’skii have been proved using topological degree theory, such as those in (Anderson, Avery, and Henderson, 2010; Güo and Lakshmikantham, 1998). Nevertheless, from a theoretical perspective, a direct approach without using degree arguments could be useful when trying to extend the results from compact mappings to more general ones. Such a possibility is shown in (O’Regan and Precup, 2001, Chapter 10), where some compression-expansion results are established for a family of mappings for which the topological degree has not been developed. Also, for applications, it seems more convenient to use the Theorem of Krasnosel’skii rather than the degree theory, because the fixed 16 krasnosel’skii type results in star convex sets (A) SR Sr • • (B) C∩F2 C∩F1 • • Figure 1: Illustration of the regions where the Krasnosel’skii compression-expansion fixed point theorem (A)and our main results (B)locate the fixed points. The cone C={(x,y)∈R2:x,y⩾0}is represented in gray color; the black dots are the fixed points and the black arc represents the points with the same norm as the fixed points. The magenta and blue curves are the sets where the compression or expansion conditions are satisfied. The regions in between these curves correspond to those where the results localize the fixed points. In (A)both fixed points are inside this region, while in (B)the region isolates one of them. point result offers directly the compression-expansion conditions that have to be fulfilled. The direct approach owed by Krasnosel’skii was also followed by Potter (1974) and Các and Gatica (1979), who respectively extended the compression and expansion result from continuous and compact mappings to set contractions (see Theorems 1.2.7,1.2.8). Notice that both Krasnosel’skii and Potter obtained a solution localized in conical annular sets determined by the norm, but the result obtained by Các and Gatica provides a less refined localization, since they can only assert that there exists a positive fixed point in the cone. Our main results can be seen as generalizations of the results due to Potter (1974) and Các and Gatica (1979) to more general domains; and our proofs also follow a direct approach without using topological degree techniques. Overview of the research development Our study is focused on the generalization of the compression-expansion results for set contractions to conical domains determined by two star convex sets. In this regard, we first require some additional conditions to these non necessarily convex sets. An essential property is the following one: each ray traveling from zero and passing through any other point in the star convex set meets a unique element in its boundary. We notice that, for the theoretical arguments, we use an equivalent condition given in terms of a continuous functional. Once we have determined the framework where we will develop 2.2 general framework 17 our research, to prove the compression result we adapt Potter’s ideas to star convex sets. Nevertheless, for the expansive case we do not follow the proof given by Các and Gatica, but we use the idea of reducing the expansive case to the compressive one, as shown in (Precup, 2006) for compact mappings. In this way, we improve the localization of fixed points provided by the existing theorem for balls and state a new result for star convex sets. The possibility to use star convex sets instead of balls allows to determine the conical domain by functionals more general than a norm. The use of such functionals seems to be very useful for applications, so we derive some sufficient conditions that they shall satisfy. Finally, it is important to show the applicability of the new results. For that purpose, we consider an initial value problem for a system of first order implicit differential equations, we apply the compression result to the associated noncompact integral mapping, and we use functionals to define a star convex set. 2.2 general framework We have already mentioned that our results improve the localization of fixed points in Theorem 1.2.5by using star convex sets. Thus, we begin this section by recalling this concept and by stating some additional conditions we will require to the sets considered. We provide two characterizations of the underlying star convex sets, one easier for visualization and the other more useful for the theoretical aspects. Then we prove the equivalence between these characterizations and we deduce some helpful properties of these sets. We notice that most of the section contents were developed for the Master degree dissertation of Cristina Lois Prados. However, it was during the PhD period when the equivalence between the characterizations in Condition 1was completed with Proposition 2.2.2. Definition 2.2.1.Let (X,k·k)be a Banach space, E⊂Xand x0∈E. We say that Eis an x0-star convex set if λx0+ (1−λ)x∈E, for all λ∈[0,1]and x∈E. In case that x0=0,Eis simply called a star convex set. Condition 1.In the following, we consider a Banach space (X,k·k),Ca cone in Xand E⊂Xa star convex set (which trivially satisfy E∩C6=∅). In addition, the star convex set shall fulfill: 1.Eis bounded and closed; 2. If Fis the boundary of Ein X, then 0 /∈F; and one of the following equivalent hypotheses: 3. for every x∈E\{0}, there exists a unique βx> 0 with βxx∈F; 18 krasnosel’skii type results in star convex sets 4. there exists a continuous mapping ∂:E\{0}−→ F,x7−→ ∂(x), such that    ∂(x) = ∂(λx),∀x∈E,∀λ∈(0,1]; ∂(x) = x,∀x∈F. The following result proves that hypothesis 3in Condition 1implies statement 4, but in the general context of an x0-star convex set. Proposition 2.2.2.Let (X,k·k)be a Banach space, x0∈Xand E⊂Xbe a bounded closed x0-star convex set such that its boundary Fdoes not contain x0,and for all x∈E\ {x0},there is a unique λx> 0 with λxx+ (1−λx)x0∈F. (6) Then there exists a continuous mapping ∂:E\ {x0}→Fsuch that ∂(x) = ∂(λx + (1−λ)x0),for all x∈E\ {x0},λ∈(0,1]; ∂(x) = x,for all x∈F.(7) Proof. As x0∈E\F, there exists γ > 0 such that B:= B(x0,γ)⊂E\F. Let Sbe the boundary of B, i.e., S:= {x∈X:kx−x0k=γ}. We define the mapping ∂ as the composition η◦η0, where η0is the radial projection η0:E\ {x0}→S,η0(x)=γ kx−x0k(x−x0), and η:S→F,η(x)=λxx+(1−λx)x0. From (6), the mapping ηis well-defined. Also, it is easy to see that condition (7) is satisfied. Clearly, η0is continuous, so it remains to prove the continuity of η. To this aim, it suffices to prove the continuity of the function λ:S−→ R+,λ(x)=λx. For that purpose, let {yn}n∈Nbe any sequence in Sconverging to some y∈S. Since Eis bounded and λ(S)⊂[0,+∞), then there exists m∈R+such that λ(S)⊂[0,m]. Therefore, the sequence {λ(yn)}n∈Nis included in the compact interval [0,m], so any of its limit points is finite. Let lbe any limit point of {λ(yn)}n∈N. From η(yn) = λ(yn)yn+ (1−λ(yn))x0∈F, we find that ly +(1−l)x0∈F. This, in view of (6), gives l=λ(y). Hence λ(yn)→λ(y)as n→+∞. Therefore, λis continuous as wished. 2.2 general framework 19 The next result proves the remaining implication. Proposition 2.2.3.Let Cbe a cone a Banach space (X,k·k)and Ebe a star convex set satisfying Condition 1with hypothesis 4. Then, Ealso satisfies hypothesis 3, that is, for all x∈E\{0}, there exists a unique real number βx> 0 such that βxx∈F. In particular, if x∈C∩(E\{0}), then βxx∈C∩F. Proof. The existence of such a number is clear since Cis a cone and Eis closed, bounded, with non-empty interior, and a star convex set. Suppose that there exist β1 x,β2 x∈R+such that β1 x6=β2 xand β1 xx,β2 xx∈F. Without loss of generality, we assume that β2 x> β1 x, then 0 < β1 x β2 x< 1 and, therefore, ∂(β1 xx) = ∂β1 x β2 x β2 xx=∂(β2 xx) = β2 xx∈F. (8) Moreover, since β1 xx∈F, we have ∂(β1 xx) = β1 xx∈F. By using (8), we obtain β1 xx=β2 xx. Taking the norm, we get β1 xkxk=β2 xkxk, and, since x6=0, we conclude that β1 x=β2 x, i.e., the element βxin the statement is unique. Besides, if x∈Cand βx> 0, then it is clear that βxx∈Cby using the definition of cone. Apart from the continuous map ∂given by hypothesis 4in Condition 1, the map defined by the scalar number referred to in the equivalent hypothesis 3 also plays an important role in the theoretical development. Thus, it is interesting to deduce some of its properties. Proposition 2.2.4.Let Cbe a cone and Ebe a star convex set satisfying Condition 1. Then the mapping β:C∩(E\{0})−→ [1,+∞) x7−→ β(x) := βx; where βxis the unique positive real number such that βxx∈F∩C, is continuous and β(x)→+∞as x→0. Proof. First of all, we prove that the image of βis a subset of [1,+∞). Let x∈C∩(E\{0}), then it is satisfied that βxx=∂(x)∈C∩F. Thus, by using that Eis a star convex set satisfying Condition 1, we get βx=k∂(x)k kxk⩾1. The continuity of βfollows easily since it can be expressed as a composition of continuous functions: β:C∩(E\{0})−→ [1,+∞) x7−→ β(x) = d(0,∂(x)) d(0,x)=(d0◦∂)(x) d0(x); 20 krasnosel’skii type results in star convex sets where ∂is continuous by hypothesis and d0(x) := d(0,x) = kxk,x∈X, is continuous because of the distance properties. Finally, it remains to prove that β(x)tends to infinity as xgoes to 0. Thus, for all M∈R+, we look for δ∈R+such that β(x)> M, for all x∈C∩(E\{0})with ||x|| < δ. If M∈(0,1), then β(x)> M is trivially satisfied for all x∈C∩(E\{0}), since β∈[1,+∞). If M⩾1, we take 0 < δ := d(0,F)/M < +∞, with d(0,F) := inf{d(0,y) : y∈F}. Let x∈C∩(E\{0})with ||x|| < δ, then we prove that β(x)> M: β(x) = d(0,∂(x)) d(0,x)⩾d(0,F) d(0,x)=d(0,F) ||x|| >d(0,F) δ=M. We conclude this section by showing another useful property of both mappings ∂and β, the fact that they can be continuously extended to C\{0}. Remark 2.Let Cbe a cone and Ebe a star convex set satisfying Condition 1. As Eis closed, a star convex set and 0∈˚ E, then the restriction of the mapping ∂to C∩(E\{0})can be continuously extended to C\{0}as follows ∂C:C\{0}−→ F x7−→ ∂C(x) :=    ∂(x),x∈E\{0}; ∂d(0,F) ||x|| x,x∈C\(C∩˚ E). Figure 2illustrates the behavior of ∂Cin a particular case. By using ∂C, we can also continuously extend βto C\{0}as: βC(x) = d(0,∂C(x)) d(0,x), for all x∈C\{0}. 2.3 main results In this section, we extend the Krasnosel’skii compression-expansion fixed point theorem to set contractions and star convex sets. For that purpose, the first step will be to reformulate the compression-expansion conditions in (4)-(5) to star convex sets satisfying Condition 1. After that, we are in a position to prove the more general results. Since we use different approaches to prove the compressive and expansive cases, we devote one subsection to each one. Let us consider Ca cone and Ei,i=1,2, star convex sets satisfying Condition 1. We establish the following notations for i=1,2: •˚ Eiand Fiare the interior and the boundary of Ei, respectively. 2.3 main results 21 C∩F •(x1,y1) • ∂(x1,y1) • (x2,y2) •∂C(x2,y2) Figure 2: Illustration of mappings ∂and ∂Cfor the cone C={(x,y)∈R2:x,y⩾0}. The blue curve represents the intersection of the cone with the boundary of the star convex set. The black segment lines are rays traveling from (0,0)to the points (xi,yi)∈C\ {0}, for i=1,2. •∂i:Ei\ {0}−→ Fiis the continuous mapping given in Condition 1and ∂C i the continuous extension to C\ {0}provided in Remark 2. •βi:C∩(Ei\ {0})−→ [1,+∞)is the continuous map defined in Proposition 2.2.4and βC iis the corresponding continuous extension to C\ {0}provided in Remark 2. Next, we assume that E1⊂E2,F1∩F2=∅, and consider a mapping T:C∩E2˚ \E1→C. We say that Tis a compression of the set C∩E2\˚ E1(see Figure 3(A)) if: (C1)x−T(x)/∈C, for all x∈C∩F1; (C2)T(x) − (1+ε)x /∈C, for all ε > 0 and x∈C∩F2. We say that Tis an expansion of the set C∩E2\˚ E1(see Figure 3(B)) if: (E1)T(x) − (1+ε)x /∈C, for all ε > 0 and x∈C∩F1; (E2)x−T(x)/∈C, for all x∈C∩F2. Before going through the details of the proof of the generalized results, we provide some arguments to justify that, unless there exists a bounded homeomophism hthat transforms a star convex set Ei(i=1,2) satisfying Condition 1 in a bounded closed ball Bi, we cannot adapt Theorems 1.2.7and 1.2.8to these star convex sets by simply using the classical idea of composing the mapping Twith the mentioned homeomorphic transformation. The main reasons are: •Tis a k-set contraction: while if we first apply a continuous and compact mapping (0-set contraction) and then a continuous and bounded map, the composition inherits the compactness property, if k∈(0,1), then the corresponding condition is not generally preserved. 28 krasnosel’skii type results in star convex sets • If x∈C∩(E2\˚ E1), as it is a bounded set and Tis a k-set contraction, there exists M2∈R+such that  T(x) =kT(x)k⩽M2, for all x∈C∩(E2\˚ E1). Therefore, we conclude that there exists R1∈R+with the above property. In addition, there exists R2∈R+such that R2=sup{d(0,x) : x∈C∩E2}, since C∩E2is a bounded set. Thus, we consider R=max{R1,R2}∈R+and B≡BR={x∈C:d(0,x)⩽R}. We have chosen R∈R+such that T(BR)⊆BR. Besides, the set BRis bounded, closed and convex, because it is the intersection of the closed and convex set C with the bounded, closed and convex set B(0,R) = {x∈X:d(0,x)⩽R}. Step 2:We prove that T|BRis a k-set contraction. We first show that Tis continuous. It is clearly continuous on C\{0}and the continuity at x=0follows from similar arguments to those in Lemma 2.3.1. Then, we shall prove that there exists k<1such that α(T(A)) ⩽kα(A), for all A⊂BR. In order to prove it, we define some helpful auxiliary mappings: T1:BR∩E1−→ BRis given by T1(x) :=          δh,x=0; 1 β1(x)T(β1(x)x) + δh,x∈BR∩Eδ 1\{0}; 1 β1(x)T(β1(x)x) + d(x,F1)h,x∈BR∩E1\˚ Eδ 1. T2:BR\BR∩˚ E1−→ BRis given by T2(x) :=    T(x),x∈BR∩E2\˚ E1; T∂C 2(x),x∈BR\BR∩˚ E2. We first deal with T1, which can be expressed as the sum of two mappings T1 1and T2 1. The mapping T1 1:BR∩E1−→ BR x7−→ T1 1(x) :=    δh,x∈BR∩Eδ 1; d(x,F1)h,x∈BR∩E1\˚ Eδ 1 is a 0-set contraction. Indeed, let A⊂BR∩E1, then Ais bounded and T1 1(A) = T1 1A∩BR∩Eδ 1∪T1 1A∩BR∩E1\˚ Eδ 1. By using that T1 1A∩BR∩E1\˚ Eδ 1⊂co({0}∪{δh}), and the properties 1, 2,3,6,7of Proposition 1.1.6, we can conclude α(T1 1(A)) ⩽max{α({δh}),α(co({0}∪{δh}))}=0. 2.3 main results 29 Furthermore, the mapping T2 1:BR∩E1−→ BR x7−→ T2 1(x) :=    0,x=0; 1 β1(x)T(β1(x)x),x6=0 is a k2 1-set contraction with k2 1< 1, because it is the restriction to BR∩E1of ˜ Tin Lemma 2.3.1with E=E1and F=F1. Finally, by using statement 2of Proposition 1.1.9, we can assert that T1is a k1=0+k2 1-set contraction with k1< 1. Now, we show that T2is a k2-set contraction with k2=k, because it can be written as the composition T◦S, where S:BR\BR∩˚ E1−→ BR∩E2\˚ E1, which is given by S(x) :=    x,x∈BR∩E2\˚ E1; ∂C 2(x) = β2d(0,F2) ||x|| xd(0,F2) ||x|| x,x∈BR\BR∩˚ E2; is a 1-set contraction. Therefore, statement 1of Proposition 1.1.9is satisfied and, as a consequence, T2is a k-set contraction. We now prove that Sis a 1-set contraction. For this, let us consider λ:BR\BR∩˚ E1−→ R+given by λ(x) :=    1,x∈BR∩E2\˚ E1; β2d(0,F2) ||x|| xd(0,F2) ||x|| ,x∈BR\BR∩˚ E2; which is a continuous function and satisfies supλ(x) : x∈BR∩BR\˚ E1⩽1. Hence, by using statement 3in Proposition 1.1.9, we can conclude that Sis a 1-set contraction, since the identity map also fulfills this property. Finally, applying Corollary 1.1.10 to T1and T2, we get that T|BRis a k-set contraction with k=max{k1,k}< 1. Therefore, the hypotheses of Theorem 1.2.6are satisfied and T|BRhas at least one fixed point x∈BR. The fixed point is in the conical domain C∩(E2\˚ E1) We finish the proof by showing that x∈C∩(E2\˚ E1). To that purpose, we assume that the fixed point belongs to one of the other four sets involved in the definition of T: • Suppose that x=0. Since T(0) = 0, then δkhk=0and this is not possible because δ,khk> 0. 30 krasnosel’skii type results in star convex sets • Assume that x∈BR∩(Eδ 1\{0}). Consequently, T(x) = 1 β1(x)T(β1(x)x) + δh =x, so khk⩽1 δkxk+1 δβ1(x)kT(β1(x)x)k and it is a contradiction with the selection of hin the definition of the mapping T:C−→ C. • Let x∈BR∩E1\Eδ 1. Since xis a fixed point, then T(x) = 1 β1(x)T(β1(x)x) + d(x,F1)h=x, so x−1 β1(x)T(β1(x)x) = d(x,F1)h∈C, due to d(x,F1)⩾0and h∈C. Moreover, β1(x)∈[1,+∞), thus β1(x)x−T(β1(x)x)∈C, where β1(x)x∈C∩F1, which contradicts the hypothesis (C1)for Tbeing a compression of the cone C. • Suppose that x∈BR\(BR∩E2). Let us define yx=d(0,F2) ||x|| x, then T(x) = T∂C 2(x)=T(∂2(yx)) =T(β2(yx)yx)=Td(0,∂2(yx)) kyxk d(0,F2) kxkx. As kyxk=d(0,F2), then T(x) = Td(0,∂2(yx)) kxkx. Take ε=kxk d(0,∂2(yx)) −1, we have that ε>0since kxk d(0,∂2(yx)) > 1. Moreover, we can express xas (1+ε)d(0,∂2(yx)) kxkxand T(x) = Td(0,∂2(yx)) kxkx= (1+ε)d(0,∂2(yx)) kxkx. By using that d(0,∂2(yx)) kxkx∈C∩F2and Td(0,∂2(yx)) kxkx− (1+ε)d(0,∂2(yx)) kxkx=0∈C, we arrive to a contradiction with the hypothesis (C2)of Tbeing a compression of the cone C. Thus, we have shown that the fixed point of Tbelongs to C∩E2\˚ E1. Since Tand Tcoincide on this set, then we conclude that Thas a fixed point in the mentioned set. 2.3 main results 31 2.3.2Expansive case We begin this section by improving the expansion type result in Theorem 1.2.8. The proof developed by Các and Gatica (1979) followed the ideas used by Krasnosel’skii. Nevertheless, by using the idea in (Precup, 2006) to reduce the expansive case to the compressive one, we can provide a better localization of the fixed point in a conical annular set. Finally, the same idea is used to extend the result to star convex sets. Theorem 2.3.2.Let (X,k·k)be a Banach space, Cbe a cone in X,r,R∈R,0 < r < R, and T:Cr,R−→ Cbe a k-set contraction satisfying the expansion condition (5). Then, Thas a fixed point in Cr,R. Proof. We consider an auxiliary mapping ˜ T:Cr,R−→ Cgiven by ˜ T(x) := 1 θ(x)T(θ(x)x), where θ(x) = (r+R)/kxk−1, for every x∈Cr,R. Then, we show that it satisfies the hypotheses of Theorem 1.2.7due to Potter, that is, we prove separately that the mapping ˜ Tis well-defined, it satisfies the compression conditions given in (4) and it is a set contraction. ˜ Tis well-defined We need to show that θ(x)x∈Cr,Rfor every x∈Cr,R. Since θ(x)> 0 and x∈C, we can assert that θ(x)x∈C. Therefore, it remains to prove that r⩽kθ(x)xk⩽R. If x∈Cr,R, then r⩽kxk⩽R, whence r⩽r+R−kxk⩽R, that is, r⩽θ(x)kxk⩽R. Due to this, we can assert that θ(x)x∈Cr,Rand, finally, ˜ T is well-defined. ˜ Tsatisfies the compression condition (4) If kxk=r, then kθ(x)xk=Rand, if kxk=R, then kθ(x)xk=r. As a consequence, Tsatisfying (5) satisfies that ˜ Tverifies (4). ˜ Tis a set contraction It is clear that ˜ Tis continuous since T,θare continuous and θ > 0. Therefore, it remains to prove that, for every A⊂Cr,R(Ais bounded), we have α(˜ T(A)) ⩽˜ kα (A), (12) for some constant 0⩽˜ k < 1, independent of A. For that purpose, we proceed similarly the proof of Lemma 2.3.1or (Potter, 1974, Lemma 3.1). We begin by distinguishing two cases. If α(A) = 0, then Ais compact. As ˜ Tis continuous, then ˜ T(A)is compact and, therefore, α(˜ T(A)) ⩽α(˜ T(A)) = 0=˜ k α(A), for any ˜ k⩾0. 32 krasnosel’skii type results in star convex sets If α(A)6=0, let us assume that k6=0, but, if it is not the case, we consider 0 < ˆ k < 1 as close to kas we wish. We proceed as follows: Step 1:We cover Aby a finite number of subsets such that the restrictions of ˜ T to each of them are α-Lipschitz with a suitable constant. As 0<r<R, then δr,R=R r−r R> 0 and, for each n∈N, we can consider εn r,R:= δr,R/n. Let n∈N,n⩾1, be arbitrarily fixed, for each integer number m⩾0, we define the following sets: An m:= x∈A:θ(x)∈hr R+m εn r,R,r R+ (m+1)εn r,Ri. Since A⊂Cr,R, we can assert that r R⩽θ(x)⩽R r. Moreover, noticing that r R+0 εn r,R=r Rand r R+ [(n−1) + 1]εn r,R=R r, we get A⊂ n−1 [ m=0 An m. From this, by using some properties of the measure of noncompactness in Proposition 1.1.6, it follows that α˜ T(A)⩽α n−1 [ m=0 ˜ T(An m)!=max m∈{0,...,n−1}α˜ T(An m). Step 2:For each m∈{0,...,n−1}, we show that ˜ T|An mis α-Lipschitz with constant kr R+ (m+1)εn r,R/r R+m εn r,R. We study some properties of the following auxiliary mappings: 1 θ|An m :An m−→ C,x7−→ 1 θ(x); Sn m:An m−→ C,x7−→ Sn m(x) := θ(x)x. On the one hand, for each x∈An m, it is satisfied that 1/θ(x)⩽1/(r R+m εn r,R), then sup1 θ(x):x∈An m⩽1 r R+m εn r,R . On the other hand, for every B⊂An m, we have Sn m(B) = {θ(x)x:x∈B} ⊂hλr R+m εn r,R+ (1−λ)r R+ (m+1)εn r,Rix:λ∈[0,1],x∈B =cohr R+m εn r,RiB∪hr R+ (m+1)εn r,RiB. 2.3 main results 33 Now, by using the properties of the measure of noncompactness, we can assert α(Sn m(B))⩽αhr R+m εn r,RiB∪hr R+ (m+1)εn r,RiB =r R+ (m+1)εn r,Rα(B). Consequently, Sn mis α-Lipschitz with constant r/R + (m+1)εn r,R. As Tis a k-set contraction and ˜ T|An m= (T◦Sn m)/θ|An m, we finally get that ˜ T|An m is α-Lipschitz with constant kr R+ (m+1)εn r,R/r R+m εn r,R. Step 3:˜ Tis a set contraction. Indeed, for each m∈{1,...,n−1}, it follows that r R+ (m+1)εn r,R r R+m εn r,R < r R+εn r,R r R . Thus, taking into account the different statements which have been proved, we have α˜ T(A)⩽max m∈{0,...,n−1}α˜ T(An m) ⩽max m∈{0,...,n−1}r R+ (m+1)εn r,R r R+m εn r,R α(An m) ⩽ r R+εn r,R r R k α(A). Since εn r,R→0as n→∞, one has that, for nlarge enough, the number ˜ k:= r R+εn r,R r R k is as close to kas we wish. Therefore, we can guarantee that ˜ k∈(0,1). Existence of a fixed point of Tin Cr,R To finish the proof, we apply Theorem 1.2.7to the mapping ˜ T. Hence, ˜ Thas a fixed point ˜x∈Cr,R, that is, ˜x=˜ T(˜x) = 1 θ(˜x)T(θ(˜x)˜x), or, equivalently, θ(˜x)˜x=T(θ(˜x)˜x). This shows that the point ˆx:= θ(˜x)˜xis a fixed point of Tin Cr,R. The next result extends Theorem 2.3.2to star convex sets. We notice that the proof is quite similar, the main difference appears in the definition of the map θ, which transforms the expansion conditions in compressive ones. 34 krasnosel’skii type results in star convex sets Theorem 2.3.3.Let (X,k·k)be a Banach space, Cbe a cone in X, and E1,E2be star convex sets fulfilling Condition 1. If T:C∩E2\˚ E1−→ Cis a k-set contraction and an expansion of the set C∩E2\˚ E1, then Thas a fixed point in C∩E2\˚ E1. Proof. We consider the auxiliary mapping ˜ T:C∩E2\˚ E1−→ Cgiven by ˜ T(x) := 1 θ(x)T(θ(x)x), where θ(x) = βC 1(x) + β2(x) − 1, for x∈C∩E2\˚ E1. Then, we show that it satisfies the hypotheses of Theorem 2.3.1, that is, we prove separately that ˜ Tis well-defined, it is a compression of the set C∩E2\˚ E1and it is a set contraction. ˜ Tis well-defined We need to show that θ(x)x∈C∩E2\˚ E1, for every x∈C∩E2\˚ E1. Since θ(x)> 0 and x∈C, we clearly have θ(x)x∈C. To prove that θ(x)x∈E2\˚ E1, we note the equivalence between λx ∈E2\˚ E1and the inequality βC 1(x)⩽λ⩽β2(x). Now, let us consider x∈E2\˚ E1, then βC 1(x)⩽1⩽β2(x), and we can assert that βC 1(x)⩽βC 1(x) + β2(x) − 1⩽β2(x), that is, βC 1(x)⩽θ(x)⩽β2(x), which shows that θ(x)x∈E2\˚ E1. Therefore, we conclude that ˜ Tis well-defined. ˜ Tis a compression of the set C∩E2\˚ E1 For that purpose, we prove that, if x∈F1, then θ(x)x∈F2, and, if x∈F2, then θ(x)x∈F1. Consequently, if Tfulfills (E1)then ˜ Tfulfills (C2); and if Tfulfills (E2)then ˜ Tfulfills (C1). This way, ˜ Tis a compression of the set C∩E2\˚ E1. Indeed, if x∈F1, then βC 1(x) = 1and so θ(x) = β2(x). Hence, according to Proposition 2.2.4,θ(x)x∈F2. Similarly, if x∈F2, then β2(x) = 1and so θ(x) = βC 1(x). Thus, by using the definition of βC 1,θ(x)x∈F1. ˜ Tis a set contraction On the one hand, as βC 1,β2and Tare continuous and θ(x)6=0, for all x∈C∩E2\˚ E1, one has that ˜ Tis continuous. On the other hand, for every A⊂C∩E2\˚ E1, we have to prove that α˜ T(A)⩽˜ kα(A), (13) for some constant 0⩽˜ k < 1, independent of A. The proof of (13) is identical to that of formula (12) in the proof of Theorem 2.3.2, once we have shown the existence of two positive numbers rand Rwith r < R such that r R⩽θ(x)⩽R r, for all x∈C∩E2\˚ E1. (14) Indeed, as 0∈E1\F1, there exists r > 0 such that B(0,r)⊂˚ E1. Since E1⊂E2 and E2is bounded, then there exists R > r such that E2⊂B(0,R). Therefore, we 2.4 admissible sets defined by functionals 35 have that C∩E2\˚ E1⊂C∩(B(0,R)\B(0,r)). Hence, for any x∈C∩E2\˚ E1, we obtain x,θ(x)x∈C∩(B(0,R)\B(0,r)), implying r⩽kxk,θ(x)kxk⩽R, which immediately yield (14). Existence of a fixed point of Tin C∩E2\˚ E1 To finish the proof, we apply Theorem 2.3.1to the mapping ˜ T. Thus, ˜ Thas a fixed point ˜x∈C∩E2\˚ E1. Then ˜x=˜ T(˜x) = 1 θ(˜x)T(θ(˜x)˜x), or, equivalently, θ(˜x)˜x=T(θ(˜x)˜x). This shows that the point ˆx:= θ(˜x)˜xis a fixed point of Tin C∩E2\˚ E1. 2.4 admissible sets defined by functionals The fixed point theorems due to Krasnosel’skii, Potter, or Các and Gatica provide the existence of fixed points for a mapping Tin certain subsets of a Banach space (X,|| ·||). An interesting characteristic of the sets involved in these results, either in the compression-expansion conditions required to the mapping Tor in the set where the fixed points are located, is that they are determined by a norm. These sets can be expressed in terms of: Br=C∩{x∈X:||x|| ⩽r},Sr=C∩{x∈X:||x|| =r}, where ris a positive real number. In Section 2.3, we generalized the mentioned results to star convex sets, which in general cannot be expressed by using a norm. Thus, for applications, it is interesting to determine some conditions over a functional ϕ:X−→ [0,+∞) such that, for a real number r > 0, the sets Er:= {x∈X:ϕ(x)⩽r},Fr:= {x∈X:ϕ(x) = r}, satisfy Condition 1. Thus, in this subsection, we first determine some suitable conditions to be imposed the functional ϕ. Then, we derive some corollaries of Theorems 2.3.1and 2.3.3. We notice that, if ϕ=|| ·||, then C∩Er≡Brand C∩Fr≡Sr. However, this approach allows to consider functionals with weaker properties in comparison 36 krasnosel’skii type results in star convex sets with the norm. In order to see that we work with weaker hypotheses, once we know how to choose a suitable functional ϕ, we provide an example of a functional satisfying the new restrictions that is not a norm. Moreover, we will show that we cannot relax the continuity hypothesis, at least by considering an upper/lower semicontinuous functional ϕ. Proposition 2.4.1.Let (X,k·k)be a Banach space, x∈X,r∈R,r>0, and ϕ:X−→ [0,+∞)be a functional satisfying: (F1)if ϕ(x)⩽r, then ϕ(λx)⩽rfor all λ∈[0,1]; (F2)ϕis continuous; (F3)for all x∈Er,ϕ(x) = 0if and only if x=0; (F4)ϕ(λx) = λϕ(x)for all λ∈(0,+∞)and x∈Er; (F5)there exists m∈R,m > 0 such that mkxk⩽ϕ(x)for all x∈Xwith kxk> ϕ(x), or liminf kxk→+∞ ϕ(x)> r. Under these assumptions, Erand Frsatisfy Condition 1. Proof. We proceed step by step, i.e., we prove each of the properties required to Erand Frby using the appropriate hypothesis over ϕ. Indeed: •(F1)is equivalent to Erbeing a star convex set. •(F2)implies that Eris closed and ϕ−1([0,r)) is open in X. Indeed, we can write Er=ϕ−1([0,r]). As [0,r]is a closed subset of ([0,+∞),|·|), where |·|is the absolute value for real numbers, then Eris closed since it is the preimage of a closed set by a continuous function. Besides, we can assert that ϕ−1([0,r)) is open in Xsince [0,r)is an open subset of [0,+∞). •(F3)implies that 0 /∈Fr, because ϕ(x) = r > 0 for all x∈Fr. • By hypothesis (F4), we prove two conditions over the sets Erand Fr. First, we show that Fris the boundary of Er. We have just proved that Er is closed and Er\Fr=ϕ−1([0,r)) is open, so the boundary of Eris a subset of Fr. Let us consider y∈Frarbitrarily fixed, we want to prove that y is a boundary point of Er. As y∈Fr⊂Er, we must prove that, for all ε > 0,B(y,ε)∩(X\Er)6=∅. We take z=1+ε 2kyky∈B(y,ε), then it is satisfied that ϕ(z) = 1+ε 2kykϕ(y), and, as 1+ε 2kyk> 1, we can conclude that ϕ(z)> ϕ(y) = r. Therefore, z∈X\Erand yis a point in 2.4 admissible sets defined by functionals 37 the boundary of Er. Secondly, we prove that the mapping ∂:Er\{0}−→ Fr,x7−→ ∂(x) := r ϕ(x)x satisfies the desired conditions. By using hypotheses (F3)and (F4), we can assert that ∂is well-defined. Let x∈Er\{0},ϕ(∂(x)) = ϕr ϕ(x)x=r, so ∂(x)∈Fr. Then, (F2)implies that ∂is continuous. Now, if x∈Fr, then ϕ(x) = rand, therefore, ∂(x) = x. By using again the hypothesis (F4), we can assert that ∂(λx) = ∂(x)for all x∈Erand λ∈(0,1]. • Finally, (F5)implies that Eris a bounded set. Example 2.4.2.Let (C([0,1],R),k·k∞)be the Banach space in Example 1.2.3for n=1. We consider the functional ϕ:C([0,1],R)−→ [0,∞)given by ϕ(x) := amin t∈[0,1]|x(t)|+bkxk∞, for all x∈C([0,1],R), (15) where a,bare positive real numbers. It is easy to see that ϕsatisfies the hypotheses (F1)-(F5)in Proposition 2.4.1. However, this functional does not fulfill the triangular inequality, then it is not a norm. Indeed, let us consider the functions x(t) = tfor all t∈[0,2];y(t) = 1−tif t∈[0,1],y(t) = t−1if t∈[1,2], then we have ϕ(x+y) = a+3b > ϕ(x) + ϕ(y) = 3b. Next, we make the following question: is it possible to extend this study to upper or lower semicontinuous functionals? In the following, we prove that the answer is negative, since this assumption is not enough. Assume that ϕis upper semicontinuous. We show that Er\Fris open, but Er is not necessarily closed. Indeed: • Let r∈R,r > 0, then ϕ−1([0,r)) is open. We take y∈ϕ−1([0,r)) arbitrarily fixed and show that it is an interior point, i.e., there exists δ∈R,δ > 0 such that B(y,δ)⊂ϕ−1([0,r)). (16) As ϕ(y)< r, we can take ε > 0 such that ϕ(y) + ε < r. Besides, by using that ϕis upper semicontinuous, there exists δy εsuch that, for all x∈B(y,δy ε),0⩽ϕ(x)< ϕ(y) + ε < r. Thus, B(y,δy ε)⊂ϕ−1([0,r)), and it proves that ϕ−1([0,r)) is an open set. 44 krasnosel’skii type results in star convex sets Next, we show that (H4,c1)guarantees that condition (C1)is fulfilled. Indeed, if we assume the contrary, then there exists y∈Cwith  yj ∞⩽rj, for all j∈{1,...,n}, and kykk∞=rkfor some k∈{1,...,n}, such that y(t)⩾T(y)(t), for all t∈[0,1]. Let t0∈[0,1]be such that yk(t0) = kykk∞=rk. From the previous inequality, by using the definition of T, we obtain rk=yk(t0)⩾Tk(y)(t0) = fkt0,Zt0 0 y(s)ds+gk(t0,y(t0)) ⩾fk+gk, which contradicts (H4,c1). Hence, (C1)holds. We conclude by proving that (C2)is also satisfied as a consequence of hypothesis (H4,c2). If we assume the contrary, then there exist ε>0and y∈C with ϕj(yj)⩽Rjfor all j∈{1,...,n}, and ϕk(yk) = Rkfor some k∈{1,...,n}, such that T(y)(t)⩾(1+ε)y(t), for all t∈[0,1]. Let t0∈[0,1]be such that yk(t0) = kykk∞. Then, using the last inequality, the expression of T, and (23), we obtain fk+gk⩾fkt0,Zt0 0 y(s)ds+gk(t0,y(t0))> yk(t0)⩾Rk ak+bk , which contradicts (H4,c2). Hence, (C2)holds. Therefore, since all the assumptions of Theorem 2.3.1are fulfilled, we have the following existence and localization result. Theorem 2.5.1.Under conditions (H1)-(H3)and (H4,c1)-(H4,c2), problem (20)has a non-negative and increasing solution x∈C1([0,1],Rn)such that rk⩽ x0 k ∞,for at least one k∈{1,...,n},and ϕi(x0 i)⩽Ri,for all i∈{1,...,n}.(24) Particular case n=1 In particular, if we assume that n=1, and the following monotonicity condition on fand g: (H5)For each t∈[0,1], the functions f(t,·)and g(t,·)are increasing in R+; then conditions (H4,c1)and (H4,c2), with r1,R1,a1,b1simply denoted by r,R, a,b, turn into (H5,c1)f(t,0) + g(t,r)> r, for all t∈[0,1]; (H5,c2)ft,R b+gt,R b⩽R a+b, for all t∈[0,1]. Let us present two examples. The first one, which is in fact an explicitly solvable equation, is given to test the conditions (H1)-(H3),(H5,c1)-(H5,c2). 2.5 application to a first order implicit differential system 45 Example 2.5.2.Let us consider the equation x0(t) = λx(t) + αx0(t) + β,t∈[0,1]. In this case f(t,s) = λs and g(t,s) = αs +β, for all s∈R,t∈[0,1], where we assume that λ,α⩾0,β>0and λ+α < b a+b. If r < β 1−αand R1 a+b−λ+α b⩾β, (25) then the assumptions of Theorem 2.5.1are fulfilled. One we have determined some sufficient conditions on the parameters, we deal with some particular values and show that the particular solution of problem (20) satisfies condition (24) in Theorem 2.5.1. First, we have that condition (25) is satisfied for a=b=1,r=1,R=6,β=1and λ=α=1/6. In this case, the exact solution of the problem with x(0) = 0is x(t) = 6et 5−1,t∈[0,1], whose derivative is x0(t) = 6 5et 5,t∈[0,1]. Then kx0k∞=6 5e1 5and ϕ(x0) = 6 51+e1 5, and it is easy to see that condition (24) holds. The second example deals with an equation that cannot be explicitly solved. However, for some particular parameter values, this equation reduces to the solvable implicit problem given in Example 2.5.2. Thus, in this particular case, we can see that the sufficient conditions required to the parameters correspond to those in Example 2.5.2. Example 2.5.3.Consider the equation x0(t) = λx(t) + αx0(t) + β+γsin(x0(t)),t∈[0,1]. (26) In this case, f(s) = λs and g(s) = αs +β+γsins(s∈R), where we assume that α,β,γand λare non-negative. Now, we explain how to fulfill the conditions (H1)-(H3),(H5,c1)-(H5,c2)in Theorem 2.5.1. Clearly, condition (H1)holds. Next, if α < 1 −γ, then |g0(s)|=|α+γcoss|⩽α+γ < 1. Therefore, (H2)is satisfied for k=α+γ < 1. To guarantee condition (H3), we need g(R+)⊂R+, which takes place if β⩾γ. 46 krasnosel’skii type results in star convex sets Furthermore, condition (H5)is fulfilled if gis increasing in R+, and this happens if α⩾γ. This condition, together with α < 1 −γ, gives γ⩽α < 1 −γ. Then, obviously, γhas to satisfy 0⩽γ < 1 2. Finally, we have to check conditions (H5,c1)and (H5,c2). For the first, we need r > 0 such that g(r)> r, that is, αr +β+γsinr > r. This clearly happens if αr +β−γ>r, or, equivalently, r < β−γ 1−α, which requires β > γ since rhas to be positive. Condition (H5,c2)reads as λR b+αR b+β+γsin R b⩽R a+b. (27) We show that there exists Rlarge enough that satisfies this inequality. Indeed, if we divide by R b, we obtain λ+α+βb R+γsinR b R b ⩽b a+b. The limit of the left hand side, when Rtends to ∞, being λ+αguarantees the existence of Rprovided that λ+α < b a+bor, equivalently, λ < b a+b−α. In view of λ⩾0, it requires that α < b a+b. Therefore, the conditions of Theorem 2.5.1are fulfilled if the non-negative parameters α,β,γand λsatisfy: γ⩽α < 1 −γ,λ+α < b a+b, and γ < min1 2,β. Under these conditions, for every r < β−γ 1−α, there exists a solution x∈C1([0,1],R)of equation (26) with x(0) = 0that is non-negative, increasing and with kx0k∞⩾r. If, in addition, a number Ris chosen such that inequality (27) holds, then the solution xalso satisfies amin t∈[0,1]x0(t) + bmax t∈[0,1]x0(t)⩽R. We finally show that a similar approach does not work for expansion type conditions, that is, Theorem 2.3.3does not apply for the initial value problem (20), at least if we use arguments like those developed for the compressive case. Indeed, if we take E1={y∈C([0,1],R) : ϕ(y)⩽r},E2={y∈C([0,1],R) : kyk∞⩽R}, 2.5 application to a first order implicit differential system 47 where ϕis the functional defined in Example 2.4.2and r,Rare positive numbers with r < bR; and we proceed similarly to the compression case, we arrive to the following sufficient expansion type conditions: (H4,E1)max t∈[0,1],y∈[0,r b]f(t,y) + max t∈[0,1],y∈[r a+b,r b]g(t,y)⩽r a+b; (H4,E2)min t∈[0,1],y∈[0,R]f(t,y) + min t∈[0,1]g(t,R)> R. These conditions ensure that E1and E2are star convex sets satisfying Condition 1,0∈E1⊂˚ E2and (E1)-(E2)hold; but, unfortunately, they are not compatible with hypothesis (H2), and thus Theorem 2.3.3can not be applied. We prove this incompatibility in the autonomous case, that is, when fand gdo not depend on t, and conditions (H4,E1)-(H4,E2)read as max y∈[0,r b]f(y) + max y∈[r a+b,r b]g(y)⩽r a+b; min y∈[0,R]f(y) + g(R)> R. Subtracting the two inequalities yields g(R)−max y∈[r a+b,r b]g(y)> R −r a+b+max y∈[0,r b]f(y) − min y∈[0,R]f(y). (28) From r < bR, we have [0,r/b]⊂[0,R], whence min y∈[0,R]f(y)⩽min y∈[0,r b]f(y)⩽max y∈[0,r b]f(y). Hence, the right-hand side in (28) is greater than or equal to R−r/ (a+b), so g(R)−max y∈[r a+b,r b]g(y)> R −r a+b. (29) On the other hand, if ˆy∈[r/ (a+b),r/b]is such that g(ˆy) = max y∈[r/(a+b),r/b]g(y), then, by using (H2), we have g(R)−max y∈[r a+b,r b]g(y) = g(R)−g(ˆy)⩽k(R−ˆy) ⩽kR−r a+b< R −r a+b. This together with condition (29) clearly yields a contradiction. Thus, we did not succeed in applying the expansion type result to (20). However, we claim that it may work for other types of initial or boundary value problems, since the expansion conditions are compatible with many other problems involving compact operators, as illustrated extensively in the literature. 48 krasnosel’skii type results in star convex sets 2.6 discussion In this section, we summarize and discuss the research work we have developed throughout the chapter. We present the contents classified in blocks for a better distribution. Interest of the new extensions of Krasnosel’skii fixed point theorem In Section 2.1, we consider some of the several generalizations of Krasnosel’skii compression-expansion fixed point theorem. Then, in Sections 2.2and 2.3, we extend the results for set contractions to star convex sets. In the following, we justify the usefulness of the obtained results. The initial motivation to work in this particular generalization was the possibility to localize different solutions with the same norm. We notice that there exist other results with this potential, like those owed by Güo and Lakshmikantham (1998) and Precup (2006). Our results generalize the ones in (Precup, 2006) (see Section 2.4for details), but do not work for the general domains determined by two open sets in (Güo and Lakshmikantham, 1998). However, the key point of our results is that they consider set contractions instead of continuous and compact maps, and it allows us to deal with more general problems, such as the system of implicit first order differential equations in Section 2.5. The results in (Precup, 2006) extend the applicability of Krasnosel’skii type fixed point theorems to boundary value problems for partial differential equations, such as semi-linear elliptic problems. The key ingredient of these results is the possibility to work in more general domains determined by two norms. Since our fixed point theorems can consider funtionals more general than a norm, we claim that they can be also useful to deal with this type of problems. It is worth mentioning that, apart from the generalization of the results to conical domains determined by two star convex sets, we also improve the expansion type fixed point theorem for balls owed by Các and Gatica (1979). In the work (Potter, 1974), it is said that Theorem 2.3.2follows from similar arguments to those developed for the compressive case, but the proof is not given. Besides, Các and Gatica (1979) follow Krasnosel’ski ideas in the proof of the expansive case in Theorem 1.2.5, but they needed to impose more restrictive expansion conditions and the balls did not play a role in the localization of fixed points. Thus, we realized that a change of variable which transforms the expansive case in the compressive one helps to relax the expansion hypotheses and to localize the fixed points in a conical shell determined by two balls. Complexity of the hypotheses in our main results At the beginning of Section 2.3, we show that the generalization of the existing results for balls to star convex sets cannot be simply adapted by using the classical idea of composing with a homeomorphic transformation. We provided 2.6 discussion 49 different reasons related with the set contraction property, the domain and range of the mapping and the compression-expansion conditions. Besides, in Subsection 2.3.1, where we adapt Potter’s results to star convex sets, we observe that the difficulty when working with k-set contractions comes from the fact that the geometric transformations can change uncontrollably the constant k. Moreover, the use of star convex sets introduces much more complicated geometric transformations connected to their retro-activity property, which have to be put into accordance with the constant k, as Lemma 2.3.1 shows. Relevance of defining the star convex sets by using functionals We already mentioned that Erbe and Wang (1994), Torres (2003), and Zima (2004) apply generalizations of Theorem 1.2.5in terms of the norm to different second order boundary value problems. These fixed point results enable to localize solutions in the general domain C∩(Ω2\Ω1), where Cis a cone and Ω1,Ω2are open sets. However, for applications, they choose the simplest possibility and determine the open sets by using the norm. In this way, they are missing the good localization qualities of this type of results. In our case, the compression-expansion results work for star convex sets more general than balls, so to take advantage of this general framework in applications, we determine conditions over a functional in order to define admissible star convex sets while being more general than a norm. To conclude, we recall that there exist other research studies dealing with generalizations of Krasnosel’skii fixed point theorem in terms of functionals. We mentioned in the Introduction the work by Anderson, Avery, and Henderson (2010), where the conditions required do not meet hypotheses (F1)-(F5), since they consider convex/concave functionals. 3 APPLICATIONS OF FIXED POINT THEORY TO PERIODIC PREDATOR-PREY DIFFERENTIAL EQUATIONS This chapter includes the contents of the research article (Lois-Prados and Precup, 2020)1, in which we contribute to the application of fixed point results to Lotka-Volterra type models. The chapter is organized as follows: We start with an introductory section, where we first revisit different formulations of predator-prey systems, we provide the formulation of the model into consideration and we also describe our initial motivation to develop the present research. Next, we explain the interest of our study in the framework of fixed point theory, where we contribute to the application of Krasnosel’skii type fixed point theorems. Finally, we briefly review the research developed throughout this chapter. Then, in Section 3.2, we prepare the underlying model for the application of fixed point techniques, so we obtain its integral version and we state some useful notations. The central part of the research is comprised in Section 3.3, which is divided into three subsections. In the first one, we state and prove the main result about the existence and localization of periodic solutions as a consequence of the homotopy version of Krasnosel’skii fixed point theorem; and we discuss the possibility of localization for some particular expressions of the model. In Subsection 3.3.2, we study two interesting properties of the localized solutions. In the last subsection, the existence results are improved for the particular case in which the predator functional response does not depend on time. Finally, we include a Discussion section, where we summarize our main contributions to the study of Lotka-Volterra type systems, by means of fixed point theory techniques, and we compare our approach with other similar research works. We also devote some lines to talk about the qualitative properties of the localized solutions. 1Cristina Lois-Prados (Instituto de Matemáticas, Universidade de Santiago de Compostela, Spain) & Radu Precup (Department of Mathematics, Babe¸s Bolyai University, Romania), “Positive periodic solutions for Lotka–Volterra systems with a general attack rate”, Nonlinear Analysis: Real World Applications (ISSN: 14681218),52,2020. The final authenticated version is available online at: https://doi.org/10.1016/j.nonrwa.2019.103024. JCR 2019 (category; impact factor; relative position; quartile): Applied Mathematics; 2.072; 37/261; Q1. PhD student contributions: The authors have equally contributed to the development of the article contents, they were continuously collaborating during the research stay of C. Lois-Prados at Babe¸s Bolyai University. The initial idea of studying the involved Lotka-Volterra type model was conceived by C. Lois-Prados. 51 52 applications to predator-prey differential equations 3.1 introduction and model description In this section, we include the formulation of the non-autonomous predatorprey population model studied in this chapter, as well as some motivations and interests of our research work. We organize the contents by dividing them in blocks with their corresponding descriptive headline. Review of different Lotka-Volterra type systems and research motivation Lotka-Volterra type systems are commonly used to describe interactions between two species, prey and predator. In the autonomous case, these models have a Kolmogorov structure, being of the form    x0=x F(x,y); y0=y G(x,y); and most of them satisfy the following conditions: Fy(x,y)< 0,Gx(x,y)> 0 and Gy(x,y)⩽0 (see, e.g., Brauer and Castillo Chávez, 2001, Section 5.4). For non-autonomous Kolmogorov type systems, we refer the reader to the paper by Zanolin (1992). The original model proposed by Lotka (1925) and Volterra (1926) is given by    x0=ax −λxy; y0= −by +cλxy.(30) For a historical note on this classical model see (Bacaër, 2011). As suggested by Volterra himself, a more realistic prey growth is the logistic one, which was considered by several authors. For instance, Rosenzweig and MacArthur (1963) proposed the following model:    x0=ax 1−x K−φ(x)y; y0= −by +cφ(x)y. Some generalizations of the Rosenzweig-MacArthur model are given in (Van der Hoff and Fay, 2016), where, in particular, it is considered the logistic growth for both prey and predator populations (see also Buffoni, Groppi, and Soresina, 2011). In this paper, we look for periodic solutions for the following periodic Lotka- Volterra type systems with a general prey growth gand a general functional response of predators ϕ:    x0=a(t)xg(x) − ϕ(t,x,y)xy; y0= −b(t)y+c(t)ϕ(t,x,y)xy;(31) 3.1 introduction and model description 53 where a,b,c∈C(R,R+)are ω-periodic with the same period ω > 0,a,b6≡ 0, mins∈[0,ω]c(s)> 0;ϕ∈C(R×R+×R+,R+)is such that ϕ(·,x,y)is ω-periodic for every (x,y)∈R+×R+; and g∈C(R+,R)is decreasing, with g(0)⩽1. We notice that x,yrepresent the prey and predator populations, respectively. In particular, we use g(x)≡1(linear growth of the prey), or g(x)≡1−x K(logistic growth of the prey) and one of the following expressions for the functional response ϕ, ϕI(t,x,y)≡λ(t) + α(t)y; ϕII(t,x,y)≡λ(t) + α(t)y 1+β(t)(λ(t) + α(t)y)x. Here, we assume that α,β,λ∈C(R,R+)are ω-periodic functions and λ,β6≡ 0. The particular expressions for both ϕIand ϕII involving constant coefficients (λ,β > 0 and α⩾0) are used in the literature to simulate the effects of hunting cooperation between predators (see Berec, 2010; Teixeira Alves and Hilker, 2017). The motivation to study the existence of ω-periodic solutions of system (31) comes from the research carried out by Teixeira Alves and Hilker (2017). They consider constant coefficients and the functional response ϕ(x,y) = λ+α y, where λis the attack rate and αrepresents the cooperation term. This model does not sustain predator-prey oscillations in the absence of hunting cooperation (α=0), so they can assert that the observed oscillations are clearly generated by the cooperative behavior. Fixed point theory and predator-prey models The existence of ω-periodic solutions for Lotka-Volterra type models has been studied by means of different fixed point theory approaches. For instance, Tsvetkov (1996) and Zanolin (1992) use topological arguments, such as index theory or Mawhin’s coincidence degree. However, Lv, Lu, and Yan (2010) and Tang and Zou (2006) consider more complex predator-prey models including periodic time-delays, but they apply directly the norm type generalization of Krasnosel’skii fixed point theorem given in Theorem 1.2.9. To our knowledge, there are not research works available in which Krasnosel’skii type compression or expansion results are applied to simpler models such as the one in Tsvetkov (1996). One possible reason is that the most common versions of Krasnosel’skii fixed point theorem do not apply for this class of Lotka-Volterra equations. In this chapter, we contribute to fill this gap in the application of Krasnosel’skii type results for Lotka-Volterra population models. For the nonautonomous predator-prey system (31), which is a generalization of the model 60 applications to predator-prey differential equations where we have used that λ,x0,y0> 0. On the one hand, for each t∈[0,ω], using condition (G2)over ϕ, one has kxk∞⩾x(t)> N1(x,y)(t) =Zt+ω t H1(t,s)[a(s)x(s)(1−g(x(s))) + ϕ(s,x(s),y(s))x(s)y(s)]ds ⩾Zt+ω t H1(t,s)ϕ(s,x(s),y(s)) x(s)y(s)ds ⩾m1q1q2kxk∞kyk∞Zω 0 ψ(s,x(s) + y(s))ds ⩾m1q1q2kxk∞kyk∞Zω 0 ψ(s,k(x,y)kω)ds, which, after dividing by kxk∞, yields 1 > m1q1q2kyk∞Zω 0 ψ(s,R)ds. (42) On the other hand, in a similar way, from y(t)> N2(x,y)(t), for all t∈[0,ω], we deduce that 1 > m2c q1q2kxk∞Zω 0 ψ(s,R)ds. (43) Now, by adding inequalities (42), (43) and using kxk∞+kyk∞=k(x,y)kω=R, we obtain 2 > m3RZω 0 ψ(s,R)ds, which contradicts our assumption (35). Thus, condition (41) is fulfilled. Application of Theorem 1.2.11 We have proved that all the conditions of Theorem 1.2.11 are satisfied, therefore the non-linear operator Nhas a fixed point (x,y)∈Cwith r⩽k(x,y)k⩽R. This fixed point (x,y)is a continuous ω-periodic solution of the Lotka-Volterra type system (31). 3.3.1.1Applicability problems of Theorems 1.2.5,1.2.9and 1.2.10 In this subsection, we show the appropriateness of the homotopy version of the Theorem of Krasnosel’skii for Lotka-Volterra type systems, as compared to the other more popular versions. We first show that the classical Krasnosel’skii compression-expansion fixed point result considered in Theorem 1.2.5is not applicable for the Banach space Cω(R,R2),k·kω, the cone Cin Example 1.2.4and the integral mapping N associated to the Lotka-Volterra type system (31). 3.3 existence,localization and other properties of solutions 61 One of the compression or expansion hypotheses of the Theorem of Krasnosel’skii requires that, for some real number τ > 0, (x,y) − N(x,y)/∈C, for all (x,y)∈Cwith k(x,y)kω=τ. (44) However, if we choose pairs of the form (0,y)∈C, with k(0,y)kω=kyk∞=τ, then, since N(0,y) = (0,0), one has (0,y) − N(0,y) = (0,y)∈C. Consequently, condition (44) does not hold. Next, we use similar arguments to show that the norm type version provided in Theorem 1.2.9can neither be applied to systems of the form (31). We follow the approach in (Lv, Lu, and Yan, 2010; Tang and Zou, 2006), where they also use the Banach space Cω(R,R2),k·kωand the cone Cin Example 1.2.4. Moreover, for some positive real numbers r1< R1,r2< R2, they consider the open and bounded sets Ω1:= (x,y)∈Cω(R,R2),k·kω:kxk∞< r1,kyk∞< r2; Ω2:= (x,y)∈Cω(R,R2),k·kω:kxk∞< R1,kyk∞< R2. One of the hypotheses in conditions (NC),(NE)requires that, for some i∈{1,2}, kN(x,y)kω⩾k(x,y)kω, for all (x,y)∈C∩∂Ωi, (45) where for τj=rjor τj=Rj(for both j∈{1,2}) the elements of ∂Ωisatisfy kxk∞⩽τ1,kyk∞=τ2or kxk∞=τ1,kyk∞⩽τ2. Thus, if we choose pairs of the form (0,y)∈C∩∂Ωi, with kyk∞=τ2, since N(0,y) = (0,0), one has 0=kN(0,y)kω<k(0,y)kω:= kyk∞=τ2. Consequently, condition (45) does not hold. Finally, we consider the classical Lotka-Volterra system with continuous and ω-periodic coefficients and we show that the vector version in Theorem 1.2.10 still cannot be applied, at least by following the proof of Theorem 3.3.1. It is worth mentioning that the result could work for this type of problems if we used a distinct reasoning, but the difficulty intrinsic to the computations obscures the visualization of this possible application. Let us consider the real numbers 0 < r1< R1,0 < r2< R2, the following cones in the Banach space (C(R,R),k·k∞) C1:= {x∈C(R,R) : x(t) = x(t+ω),x(t)⩾q1kxk∞, for all t∈[0,ω]}; C2:= {y∈C(R,R) : y(t) = y(t+ω),y(t)⩾q2kyk∞, for all t∈[0,ω]}; 62 applications to predator-prey differential equations and the mapping N= (N1,N2) : (C1)r1,R1×(C2)r2,R2−→ C1×C2given by N1(x,y)(t) := Zt+ω t H1(t,s)λ(s)x(s)y(s)ds; N2(x,y)(t) := Zt+ω t H2(t,s)c(s)λ(s)x(s)y(s)ds. The non-applicability of the result is deduced once we show that both conditions (V1)and (V2)cannot be satisfied for one of the index values i∈{1,2}. Let us see, for instance, what happens for i=1and condition (V1). If we proceed like in the proof of Theorem 3.3.1, this condition is satisfied provided that the following inequalities hold M1λR2ω < 1 < m1λr2q1q2ω, but this is not possible since m1⩽M1,λ⩽λ,r2< R2and q1q2∈(0,1). 3.3.1.2Viability of conditions (33)-(35) In this subsection, we study the existence of real numbers 0 < r < R satisfying the required hypotheses in Theorem 3.3.1. For general expressions of the prey growth gand the predators functional response ϕ, we prove the existence of small enough r>0fulfilling conditions (33) and (34). However, we study condition (35) in the particular cases of linear and logistic growth, where we also specify how to choose suitable values of r and R. Under the simplest expressions for gand ϕ, we also discuss whether it is possible or not to localize multiple solutions and compare our results with those in (Tsvetkov, 1996). Remark 4.Let us consider the Lotka-Volterra model (31) with a functional response ϕsatisfying conditions (G1)-(G2). If g(0)> 1 −1 M1a, (46) then there exists a number r > 0 such that conditions (33) and (34) hold. Indeed, by using the continuity of gat 0, if (46) is satisfied, then there exists r0> 0 such that g(r)⩾1−1/ M1a, or, equivalently, condition (33) holds for every r∈(0,r0). From (46), we also have g(0)> 1 −2/ M1a, or, equivalently, (1−g(0))M1a < 2, which, together with the increasing character of ηrequired in condition (G1), guarantees (34) for any small enough r > 0. Notice that condition (46) is trivially satisfied when g(0) = 1, which is the case of both linear and logistic growth of the prey population. 3.3 existence,localization and other properties of solutions 63 We consider now the particular expression of gwhich corresponds to the linear or logistic growth for the prey population, and we show how the conditions over r,R > 0 in Theorem 3.3.1look like. Moreover, we study the existence of such numbers rand R, when ϕ≡ϕIor ϕ≡ϕII. For that purpose, let us start by proving that ϕIand ϕII satisfy all the conditions previously required to a general ϕ. It is clear that both functions belong to C(R×R+×R+,R+)and are ω-periodic in the first variable. Concerning conditions (G1)and (G2), for function ϕI, we can take η=ϕIand ψ≡λ, while for function ϕII, we can set η=ϕIand ψ(t,z) = λ(t) 1+β(t)(λ(t) + α(t)z)z. Additionally, we fix the following notations λ:= min s∈[0,ω]λ(s),λ:= max s∈[0,ω]λ(s),α:= max s∈[0,ω]α(s)and β:= max s∈[0,ω]β(s). Linear growth In this case, the localization result provided in Theorem 3.3.1reads as follows. Corollary 3.3.2.Assume that g≡1and conditions (G1),(G2)over ϕare satisfied. If there exist r,R∈R,0 < r < R, such that rZω 0 η(s,r,r)ds ⩽2 M3 , (47) and (35)hold, then the system    x0=a(t)x−ϕ(t,x,y)xy; y0= −b(t)y+c(t)ϕ(t,x,y)xy (48) has an ω-periodic solution (x,y)∈C, being Cthe cone in Example 1.2.4, such that r⩽k(x,y)kω=kxk∞+kyk∞⩽R. Next, for each particular expression of ϕ, we give sufficient conditions for (47) and (35) to hold. Case I: When ϕ=ϕI, conditions (47), (35) read as rZω 0 λ(s) + α(s)r ds ⩽2 M3 ,R⩾2 m3Rω 0λ(s)ds (49) and are respectively satisfied provided that rλ+αr⩽2 M3ω,R⩾2 m3ω λ, 64 applications to predator-prey differential equations or, equivalently, r⩽−λ+qλ2+4 α 2/(M3ω) 2α ,R⩾2 m3ω λ. (50) Therefore, under condition (50), which is satisfied for small enough rand sufficiently large R, Corollary 3.3.2applies. We notice that the existence of small enough r > 0 was already proved in Remark 4. However, the expressions in (50) tell us how to choose suitable values of rand R. The simplicity of the conditions over r,Rgiven in (50) allows us to easily discuss the possibility to localize multiple solutions, reaching a negative conclusion, as we will justify. Remark 5.Since ω-periodic functions are also nω-periodic, for every natural number n⩾2, it makes sense to ask to what extent the numbers rand R depend on the period. For each natural number n⩾1, we denote by rnand Rn the numbers rand Rsatisfying the conditions (50), when ωis replaced by nω. Making some computations, it can be shown that rn+1< rn< Rn< Rn+1, for all n∈N,n⩾1; lim n→∞ rn=0, lim n→∞ Rn=∞. Therefore, by taking a multiple of the period ω, it may happen to localize the same ω-periodic solution for every n⩾1, so that the best localization region is that for n=1. It is worth mentioning that one should take into account that the terms M3and m3involved in the inequalities depend on the period. Remark 6.In the particular case of ϕ=ϕIand α=0, system (48) reduces to the model    x0=a(t)x−λ(t)xy; y0= −b(t)y+c(t)λ(t)xy;(51) which was studied in (Tsvetkov, 1996) by means of index theory. Even in this particular case, our result based on Theorem 1.2.11 gives a better localization of ω-periodic solutions, namely in the annular conical set Cr,R:= {(x,y)∈C:r⩽kxk∞+kyk∞⩽R}, where r=2 M3Rω 0λ(s)ds,R=2 m3Rω 0λ(s)ds. Case II: When ϕ=ϕII, conditions (47), (35) become rZω 0 λ(s) + α(s)r ds ⩽2 M3 ,RZω 0 λ(s) 1+β(s)(λ(s) + α(s)R)Rds ⩾2 m3 (52) 3.3 existence,localization and other properties of solutions 65 and are respectively satisfied provided that r(λ+α r)⩽2 M3ω,Rλ 1+β(λ+α R)R⩾2 m3ω. (53) The first inequality in (53) is satisfied for r⩽−λ+qλ2+4 α 2/(M3ω)/(2α) as it happens in Case I. Next, we study the existence of Ras required by the second inequality in (53). To this aim, we consider the auxiliary function f(z) := Az 1+Bz +Cz2,z∈R+\ {0}, where A:= m3ω λ,B:= β λ and C:= β α. Let us show that there exists R>0 fulfilling the last inequality in (53) if, and only if, A⩾2(2√C+B). (54) One can easily prove that limz→0f(z) = limz→+∞f(z) = 0and f(z)> 0, for all z>0. Additionally, fhas a unique critical point at 1/√C>0and the previous properties ensure that fattains a maximum at zmax := 1/√C. Therefore, if f(zmax)⩾2, or, equivalently, A⩾2(2√C+B), then it is possible to choose Rclose enough or equal to zmax, such that the required inequality holds. Moreover, under assumption (54), we can precise the interval where we can choose R. It is [z1,z2], where z1,z2are the solutions of the equation f(z) = 2, namely z1=A−2B −p(A−2B)2−42C 4C ,z2=A−2B +p(A−2B)2−42C 4C . (55) 3.3.1.3Logistic growth In this particular case, we can rewrite Theorem 3.3.1as follows. Corollary 3.3.3.Assume that g(x)≡1−x/K and conditions (G1),(G2)over ϕare satisfied. If there exist r,R∈R,0 < r < R, such that r⩽K a M1 , (56) a M1 Kr+M3rZω 0 η(s,r,r)ds ⩽2, (57) and (35)hold, then the system    x0=a(t)x1−x K−ϕ(t,x,y)xy; y0= −b(t)y+c(t)ϕ(t,x,y)xy (58) has an ω-periodic solution (x,y)∈C, being Cthe cone in Example 1.2.4, such that r⩽k(x,y)kω=kxk∞+kyk∞⩽R. 66 applications to predator-prey differential equations It is clear how condition (56) can be satisfied. In addition, it is not necessary to study again the existence of R > 0 fulfilling condition (35), since it does not depend on the expression of g, and therefore we can follow the arguments of Subsection 3.3.1.2. Thus, let us state some sufficient conditions on r > 0, such that (57) holds when ϕ≡ϕIor ϕ≡ϕII. We have already mentioned that we can consider η=ϕIfor both expressions of ϕ. Then, for both cases, condition (57) reads as M1a Kr+M3rZω 0 (λ(s) + α(s)r)ds ⩽2, being trivially satisfied provided that M1a Kr+M3ω r λ+α r⩽2. (59) We know, from Remark 4, that there exists a small enough r > 0 fulfilling condition (57). However, condition (59) tells us how to obtain suitable values of r > 0. 3.3.2Properties of solutions We devote this subsection to study some interesting properties of the ω-periodic solutions of the Lotka-Volterra system (31). First, in Subsection 3.3.2.1, we determine the steady states or constant solutions of the model. Then, we use this information and the localization results in Subsection 3.3.1to find sufficient conditions that ensure the nonconstancy of the solutions. Finally, under uniqueness hypotheses, we show that, if the localized solution is nonconstant, then the prey and predator populations do not get extinct. 3.3.2.1Steady states The steady states of system (31) are points (x0,y0)∈R+×R+such that x0(a(t)g(x0) − ϕ(t,x0,y0)y0)=0; y0(c(t)ϕ(t,x0,y0)x0−b(t)) = 0;(60) for all t∈R. It is clear that (0,0)is a solution of (60). To determine other possible solutions, we distinguish three cases: Case 1:x0=0,y0> 0. Under these conditions, the first equation in (60) is obviously satisfied, while from the second one we have y0b(t) = 0, for all t∈[0,ω], which is not possible for y0> 0 and b6≡ 0. Therefore, there are no steady states of the form (0,y0), with y0> 0. Case 2:x0> 0,y0=0. Now the second equation in (60) trivially holds, while the first one gives a(t)g(x0) = 0, for all t∈[0,ω]. As a6≡ 0, one must have 3.3 existence,localization and other properties of solutions 67 g(x0) = 0. Therefore, a point of the form (x0,0),x0> 0, is a steady state if and only if g(x0) = 0. Case 3:x0> 0,y0> 0. Under these conditions, system (60) is equivalent to ϕ(t,x0,y0) = a(t)g(x0) y0 =b(t) c(t)x0 , for all t∈[0,ω]. We collect these results about the existence of steady states for system (31) in the following proposition. Proposition 3.3.4.A point (x0,y0)∈R+×R+is a steady state of system (31)if, and only if, one of the following conditions holds: 1.x0=0and y0=0; 2.x0> 0,g(x0) = 0and y0=0; 3.x0> 0,y0> 0 and ϕ(t,x0,y0) = a(t)g(x0) y0 =b(t) c(t)x0 ,for all t∈[0,ω]. (61) Linear growth According to this proposition, in case of considering the linear growth of the prey population, statement 2is not possible, and steady states of the form (x0,y0)with x0,y0> 0 exist if and only if ϕ(t,x0,y0) = a(t) y0 =b(t) c(t)x0 , for all t∈[0,ω]. (62) Logistic growth If one considers the logistic growth of the prey population, then from statement 2in Proposition 3.3.4we obtain the steady state (K,0), and steady states of the form (x0,y0)with x0,y0> 0 exist if and only if ϕ(t,x0,y0) = a(t)1−x0 K y0 =b(t) c(t)x0 , for all t∈[0,ω]. (63) General prey growth Coming back to the general system (31), let us note that, if there is not any constant k > 0 such that a(t) = kb(t) c(t), for all t∈[0,ω], (64) then the system has no steady states (x0,y0)with x0,y0> 0. Therefore, under condition (64), the orbits of all ω-periodic solutions (x,y), with x(t),y(t)> 0 for all t∈[0,ω], do not reduce to points. 68 applications to predator-prey differential equations 3.3.2.2Nonconstant solutions Here, we deduce sufficient conditions to ensure that the periodic solution obtained in Theorem 3.3.1does not reduce to a steady state, letting it the possibility to be a limit cycle. We state a general result which is useful for the particular case of linear prey growth, but cannot be applied to prey populations with logistic growth. Therefore, we study the latter case separately, and we provide an example to illustrate the theoretical arguments. Theorem 3.3.5.Assume that condition (61)does not hold and g(x)6=0,for all x > 0. (65) Then any ω-periodic solution (x,y)of system (31)with k(x,y)kω> 0 does not reduce to a steady state. Proof. In view of the inequality k(x,y)kω> 0, the result follows once we have proved that the unique steady state is (0,0). From Proposition 3.3.4, we know that there are no steady states of the form (0,y0)with y0> 0; moreover, if g(x)6=0, for all x > 0, then there are no steady states of the type (x0,0)with x0> 0; and, if condition (61) does not hold, then there are no positive (with both positive components) steady states. Therefore, the unique steady state of the system is (0,0), as wished. Linear growth In case of linear growth of the prey population, condition (65) in Theorem 3.3.5 trivially holds. Consequently, given two real numbers 0 < r < R satisfying the required hypotheses in Corollary 3.3.2, if condition (62) is not fulfilled, then, applying Theorem 3.3.5, we can assert that there exists a nonconstant ω-periodic solution (x,y)∈Cwith r⩽k(x,y)kω⩽R. In particular, when ϕ=ϕI, the hypotheses in Corollary 3.3.2read as (49), which can always be satisfied for small enough rand sufficiently large R. Therefore, if we ensure that condition (62) does not hold, then there is an ω-periodic solution which is nonconstant. For instance, if a,b,c,λare constants, but the cooperation coefficient αis nonconstant, then condition (62) is not fulfilled. Logistic growth However, for the logistic growth of the prey population, Theorem 3.3.5does not apply since g(K) = 0. Nevertheless, if condition (63) does not hold, then the steady states of system (58) are (0,0)and (K,0). Recall that we can always consider small enough r > 0 such that conditions (56)-(57) in Corollary 3.3.3 are fulfilled. Therefore, if the exists a number R > 0 such that K > R > r > 0 and condition (35) holds, then the orbit of the ω-periodic solution given by Corollary 3.3.3does not reduce to a point. 3.3 existence,localization and other properties of solutions 69 For the particular cases of ϕ=ϕIand ϕ=ϕII, it is possible to choose a real number 0 < R < K provided that K > 2 m3Rω 0λ(s)ds or z1< K, respectively, where z1is given in (55). In particular, when ϕ=ϕI, if a,b,c,λare constants and αis nonconstant, then condition (63) does not hold. Therefore, for every K > 2 m3ωλ, the ω-periodic solution of system (58), given by Corollary 3.3.3, is nonconstant. We conclude this subsection with an example where all the coefficients, except c, are nonconstant. Example 3.3.6.Let the coefficients of system (58), with ϕ=ϕI, be a(t) := sin2πt,b(t) := cos2πt,c(t) := c∈(0,1); λ(t) := θa(t) + (1−θ)b(t) (θ∈(0,1)),α(t) := (1−b(t))b(t). We start with a brief interpretation of this particular system, which could be suitable to model the interplay between prey and predator populations in which we take into account the following factors: ecological seasonal effects; the prey growth rate aattains its maximum value when the predator mortality rate breaches its minimum and vice versa; the attack rate of predators λ is a convex combination of the prey growth and the predator mortality rates; and the hunting cooperation coefficient αvanishes when the mortality rate of predators attains its maximum or minimum. For this particular system, condition (63) does not hold. Indeed, we can show that there are no constants k > 0 satisfying condition (64). For t=0and any k > 0, we obtain a(0) = sin20=06=k c=kcos20 c=kb(0) c. So, condition (64) does not hold for t=0and, consequently, condition (63) is not fulfilled. Therefore, if K > 2 m3R1 0λ(s)ds =4e(√e−1) c, then the 1-periodic solution given by Corollary 3.3.3is nonconstant. 76 applications to predator-prey differential equations Contributions to the direct application of fixed point theorems to Lotka-Volterra models In Section 3.1, we observe that the existence of periodic solutions to simple generalizations of Lotka-Volterra type systems has been studied by means of fixed point theory based on topological arguments. In our view, for applications, the use of topological techniques may complexify the details of the research development in comparison with the use of some classical restrictions in fixed point theory, i.e., those given directly in terms of the underlying mapping. Thus, we study the applicability of different versions of Krasnosel’skii fixed point theorem, whose hypotheses are directly expressed in terms of the non-linear operator, to the periodic predator-prey system (31). In Subsection 3.3.1, we use the homotopy version to prove the existence of periodic solutions to this class of models. We follow a similar approach to that in (Lv, Lu, and Yan, 2010; Tang and Zou, 2006), where they show the applicability of the norm-type version to more complex predator-prey models including time-delays. However, we prove that both classical and norm-type versions do not apply for system (31). Moreover, for the particular case of the classical Lotka-Volterra system with periodic coefficients, we also show some problems with the applicability of the vector version, which had been used to deal with similar periodic systems (see Precup, 2007). On the other hand, we improve the localization results in (Tsvetkov, 1996), where system (51) was studied by means of index theory. Besides, our approach holds for a more general formulation of Lotka-Volterra type systems. First steps on the qualitative study of system (31) Our main result allows to localize continuous and periodic solutions to the general system (31), which describes a periodic interplay between prey and predator populations. However, these periodic solutions could be in fact steady states, i.e, solutions that do not vary with time, as we showed for the autonomous case (see Subsection 3.3.3). Moreover, the localization domain does exclude the possibility of extinction for some of the species. Therefore, it is interesting to determine sufficient conditions that ensure the nonconstancy of the periodic solution and the positiveness of both populations. We develop this study in Subsection 3.3.2, where we can observe that the sufficient conditions depend on the expressions of the prey growth gand the predators functional response ϕ. Thus, we analyzed their viability for the linear and logistic prey growth and the expressions of ϕI,ϕII, which consider hunting cooperation between predators. This is a preliminary study of the solutions properties, which lets the possibility to localize a limit cycle. A further step on the qualitative study of this class of models would be to determine the stability properties of the steady states and the localized periodic solution. CONCLUSIONS AND FUTURE PROSPECTS I In this research line, we have contributed to both the generalization and application of Krasnosel’skii type compression-expansion fixed point results. In our view, one of the most relevant aspects of our research are the approaches we used, since one of them allows to continue with the extension of the localization region provided by our fixed point results and the other fills a gap in the application of Krasnosel’skii type fixed point theorems to Lotka-Volterra population models. We devote the following lines to describe the research skills we have gained working on the objectives of this part of the thesis. Then, we give details about some ideas to continue with the investigation, which could potentially increase the interest of our work. A. Conclusions In Chapters 2and 3within Research Line I, we include the research we have developed in the framework of fixed point theory. The most relevant results we have obtained are summarized and compared with other present in the related literature at the end of each chapter. In this section, we complement this discussion by reviewing the learning process in which I got involved when I was working in goals G1 and G2. I begin with Chapter 2, where we achieve objective G1. As we already mentioned, an important part of the research carried out to generalize Krasnosel’skii compression-expansion fixed point theorem to set contractions and star convex sets has been developed during my Master degree studies. For instance, it was during that period when my supervisor Rosana Rodríguez López proposed me the idea to adapt the results for balls owed by Các and Gatica (1979) and Potter (1974) to more general sets, which contain the rays connecting the points on its boundary to the origin. Thus, before the beginning of my PhD studies, I had developed this more general setting and proved the compression type result. In this way, I had acquired the capacity to analyze the proof of a result and to adapt it to a more general framework. We had also proved the expansion type result as a generalization of the work in (Các and Gatica, 1979), but the localization domain was not so precise. Moreover, the boundary value problem we had considered to illustrate the applicability of our results satisfied the hypotheses of some classical results, so we had not justified properly the necessity of our generalizations. It was during the PhD studies when we proved both compression and expansion results under the general hypotheses considered in this document. For the expansive case, professor Radu Precup suggested us to use a change of variable to reduce it to the compressive one. For both proofs, I needed to become familiar with the par- 77 ticular behavior of set contractions. Besides, regarding the applications of the obtained results, I worked for the first time with a noncompact mapping and perceived the relevance of defining the star convex sets by means of functionals. The contents of Chapter 3contribute to reach the goal G2 and were completely developed during the PhD studies. In this case, I reviewed some existing literature and decided to study predator-prey models including cooperation between predators by means of fixed point theory. Our main aim was to apply directly some Krasnosel’skii type results, since, in our view, this methodology is clearer and easier to reproduce than topological approaches that have been used to localize solutions of similar models. The main difficulties were to overcome the problems in the application of some popular versions of Krasnosel’skii fixed point theorem and also to find the appropriate result for our problem of interest. Moreover, I also learned how to discuss the possibility to localize multiple solutions. Finally, it is also interesting to comment that, to obtain some basic qualitative properties of periodic solutions, we combined the localization results with some knowledge acquired during my degree studies about ordinary differential equations. B. Future prospects In this section, we provide some ideas that we would like to develop as a continuation of the research carried out in this part of the thesis. It is worth mentioning that we have already started the research on the future prospect I, which is part of a joint work with my supervisor Rosana Rodríguez López. i.generalization of krasnosel’skii compression-expansion fixed point theorem for set contractions to more general domains Along this part of the thesis, we provided different reasons to consider new generalizations of Krasnosel’skii compression-expansion fixed point theorem, such as the higher applicability and the better localization they can provide. Moreover, in the Introductory section of Chapter 2, we also commented the advantages of using classical fixed point arguments rather than topological techniques, both in applications and to obtain theoretical extensions. In the latter case, we also consider that the classical approach exhibits the key points of the proofs in a clearer way, allowing to easily identify the generalization potential of the results. We recall that we have obtained a generalization of the classical compressionexpansion Theorem of Krasnosel’skii for set contractions which localizes the solutions in conical domains determined by star convex sets. A deeper analysis of the proofs reveals that their key points are the properties of set contractions and the shape of the localization domain. Moreover, we also observe that we are not taking advantage of all the possibilities offered by the set contractions, 78 and we claim that the obstacle to obtain better localization results comes from the peculiarities of the cone. Therefore, to continue with the extension of these results, we decided to preserve the ideas in the proof for both compression and expansion cases, as well as the regularity of the mapping, but to construct a more general framework for the localization domains. We notice that, for applications, we must take care of the extension of the cone, since its characteristics are relevant to determine the properties of the localized solution. Apart from the theoretical interest of this research idea, it would also be interesting to justify its potential in applications. ii.qualitative study of the periodic lotka-volterra system with general prey growth and predators functional response In Section 3.1, we mentioned that the motivation to study the predator-prey system (31) came from the research developed in (Teixeira Alves and Hilker, 2017). However, we do not follow their methodology and, inspired by the work in (Tsvetkov, 1996), we decided to use an operator approach to localize periodic solutions. Once we have proved the existence of periodic solutions, we also started to study some of their properties. We first determined sufficient conditions to ensure that the localized solutions do not reduce to steady states. Then, under uniqueness hypotheses, we also proved that the nonconstant periodic solutions are positive, so none of the species ends in extinction. In this way, we slightly contributed to the qualitative study of the nonautonomous system (31). However, there are still many interesting aspects of the dynamical behavior of the model which deserve to be studied. For instance, in Section 3.4, we proposed to study the stability of steady states and the localized periodic solution, which is interesting to understand the long-term behavior of the populations. To that purpose, as a beginner in this field, the first step would be a careful revision of the existing literature on the topic. Moreover, it can also be useful to consider the approach in (Faria and Oliveira, 2019), where they prove the global attraction of a periodic orbit for a nonautonomous model of hematopoiesis. In the research developed in (Teixeira Alves and Hilker, 2017) for an autonomous model, after the stability study, they continue with an analysis of bifurcations, that is, they observe how the stability behavior of the system changes as a parameter is varied. It would also be interesting to develop a similar study for the more general nonautonomous Lotka-Volterra systems (31). There are plenty of research works devoted to the study of bifurcations in autonomous systems; however, the analogous research for nonautonomous differential equations seems to be under construction and it does not exist a whole theoretical framework in which we can base our research. In spite of that, we can start looking at the works in (Langa, Robinson, and Suárez, 2002; 79 Rasmussen, 2007), where two different generalizations of some bifurcation notions are developed for the nonautonomous case. 80 REFERENCES I Anderson, D. R., R. I. Avery, and J. 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(2004). “Fixed point theorem of Legget-Williams type and its applications.” In: Journal of Mathematical Analysis and Applications 299:1, pp. 254– 260. 83 Part II RESEARCH LINE II 92 background definitions and results ii Definition 4.2.2.A closed invariant set Ris called repelling if there exists a neighborhood Uof Rsuch that every point in U\Ris mapped outside Uin a finite number of iterations. An attracting set A⊂Jis a closed invariant set for which there exists a neighborhood Uof Asuch that h(U)⊂Uand ∩∞ i=0hi(U) = A. An attractor Ais an attracting set with a dense orbit. The basin of attraction of an attractor Ais the set of all points converging to A, that is, the points x∈Jsuch that lim n→∞ hn(x)∈A. According to (Avrutin et al., 2019, Section 1.4), discontinuous one-dimensional maps can have four types of attractors: k-cycles and k-band chaotic attractors, k⩾1; and two other types of attractors associated with quasi-periodic orbits. In general, we use the notation oscillatory attractor for those attractors which are not fixed points of the map h. For a fixed point x∗∈J, defined by h(x∗) = x∗, we use some additional notions. Definition 4.2.3.A repelling fixed point is also called unstable. The fixed point x∗is stable if for every neighborhood Vof x∗, there exists another neighborhood Uof x∗such that hn(x)∈Vfor all x∈U,n⩾1. We say x∗is locally asymptotically stable (LAS) if it is stable and an attractor. If x∗is an attractor with U=J, we say that x∗is a global attractor. If a global attractor is stable, then we refer to it as globally asymptotically stable (GAS). We use the term stabilizing when an unstable fixed point becomes LAS under variation of a parameter; and destabilizing when a stable equilibrium loses its stability as a parameter is changed. If x∗is not an attractor, but limn→∞hn(x) = x∗for all x∈[x∗,x∗+ε)or x∈(x∗−ε,x∗]and some ε > 0, then we say x∗is semi-stable. If x∗is not an attractor but limn→∞hn(x) = x∗for Lebesgue almost every xin a neighborhood Uof x∗, we say that x∗is an essential attractor. If U=J, then we call x∗an essential global attractor. In the context of population dynamics, when the fixed point is x∗=0, then we refer to this phenomena as essential extinction. Throughout this monograph, the first step on the study of asymptotic dynamics is to find parameter restrictions which ensure the existence of positive fixed points x∗of the one-dimensional map h, which is usually an easy task. Then, we study the stability of the equilibrium points which are located at the smooth branches of the map hby means of the value h0(x∗). It is also straightforward, specially in the hyperbolic case, that is, when |h0(x∗)|6=1. For fixed points which are locally asymptotically stable, it is natural to study their global stability. Although the statement “LAS implies GAS” is not generally true, it holds when the map hsatisfies some additional conditions. In the work by Coppel (1955), we find a simple condition which ensures the global stability of an equilibrium for continuous maps defined on bounded intervals. If the interval is bounded or unbounded, we have to require some additional conditions (Franco, Perán, and Segura, 2020): 4.2 stability concepts and results 93 Lemma 4.2.1.If the map hrelated to the difference equation (76)is continuous in J, then a fixed point x∗is GAS if and only if h2(x)6=xfor all x∈J\{x∗}and x∗is stable for the dynamical system associated to hn, for some n∈N. In general, it is difficult to ensure analytically the absence of 2-cycles, but based on numerical simulations we can use the result to conjecture GAS. In the literature we can find several papers devoted to find conditions on the map h that are easier to check, an example is the following simple result which works for continuous maps and is a generalization of (Braverman and Liz, 2012) stated in (Liz and Lois-Prados, 2020a): Lemma 4.2.2.If the map hrelated to the one-dimensional discrete-time difference equation (76)is continuous on J= (a,b)(−∞⩽a < b ⩽∞) and has a unique fixed point x∗such that 1.x < h(x)< x∗for all x∈(a,x∗); 2.a < h(x)< x for all x∈(x∗,b); then x∗is GAS on J. This result ensures the global asymptotic stability of the unique positive fixed point for compensatory models, in particular, we obtain the following result. Corollary 4.2.4.Let us consider the Beverton-Holt model given by (80)and the Ricker model given by (81)with 0 < r ⩽1, then the positive fixed point K(K⩽xc) is GAS. Other results in this line make use of Lemma 4.2.1in their proof, this is the case of the following result by Cull and Chaffee (2000) (see also (Cull, 2007)). Lemma 4.2.3.Let φ: [0,∞)−→ [0,∞)be a decreasing function which is positive on (0,l)and so that φ2(x) = xfor all x∈[0,∞). If the map hrelated to the difference equation (76)is continuous on J= [0,∞), has a positive fixed point x∗and 1.φ(x)> h(x)on (0,x∗); 2.φ(x)< h(x)on (x∗,l); 3.h(x)> x on (0,x∗); 4.h(x)< x on (x∗,∞); 5.h(x)> 0 on (0,∞); then x∗is GAS on (0,∞). This result applies for compensatory and overcompensatory models, in particular for those with f0(x)⩾−1for all x∈[0,∞)and at most one point in [0,∞)such that f0(x)=−1. We illustrate it with the Ricker model given by (81). 94 background definitions and results ii Corollary 4.2.5.Let us consider the Ricker model given by (81). If 0 < r ⩽2, then f0(x)⩾−1for all x > 0. Moreover, f0(x) = −1if and only if r=2and x=K=1. In particular, if 0<r⩽2, then the positive equilibrium K=1is GAS in (0,∞). While, if r > 2, then K=1is unstable. Proof. Let us consider r > 0 and x > 0, since f0(x) = (1−rx)er(1−x)and f00(x) = −r(2−rx)er(1−x), it follows that f00(x)< 0 for all x∈(0,2/r),f00(2/r) = 0and f00(x)> 0 for all x > 2/r. Hence, f0attains its global minimum at 2/r with f0(2/r) = −er−2. Therefore, f0(x)<−1if and only if r > 2 and f0(x)=−1if and only if r=2, x=1. Now, we assume 0 < r ⩽2. The global attraction of the positive fixed point K=1follows from Lemma 4.2.3with φ(x) = 2K −x=2−xand h(x) = f(x)for all x∈(0,∞);l=2and x∗=K=1. Let us prove that the required conditions are fulfilled (see Figure 9). First we have φ2(x) = 2− (2−x) = xand it is well-known that the Ricker map fsatisfies conditions 3-5. It remains to prove that φand ffulfill conditions 1,2. On the one hand, since φ(0) = 2 > f(0) = 0,φ(1) = f(1) = 1,φ0(x) = −1for all x > 0 and f0(x)<−1 for all x>0,x6=1; then f(x)< φ(x)for all x∈(0,K) = (0,1). On the other hand, since φ(1) = f(1) = 1,φ0(x) = −1and f0(x)⩾−1for all x > 0 (f0(x) = −1 if and only if r=2,x=1), then f(x)> φ(x)for all x > K =1. Finally, if r > 2, then we have f0(K) = f0(1) = (1−r)<−1, so the equilibrium is unstable. 01234 0.0 0.5 1.0 1.5 Figure 9: Diagram showing the Ricker map f(x) = xer(1−x),r⩽2(compensatory, r=0.7∈(0,1]in black; overcompensatory, r=1.5∈(1,2]in blue) enveloped by φ(x) = 2−x(red line). We also plot y=x(gray, dashed). Notice that we have obtained sharp global stability for the Ricker map, that is, we proved that the positive equilibrium is GAS for all the values of the parameter r > 0 for which the fixed point is LAS. The next result provides a simple criterion to compare global stability of two continuous maps (see (El-Morshedy and Jiménez López, 2008, Theorem B)). Lemma 4.2.4.Assume that the continuous map hrelated to the difference equation (76)has a GAS equilibrium x∗in J= (0,∞), and let H: (0,∞)−→ (0,∞)be a continuous map satisfying 4.2 stability concepts and results 95 1.x < H(x)⩽max{h(x),x∗}, for all x∈(0,x∗); 2.x > H(x)⩾min{h(x),x∗}, for all x∈(x∗,∞); then x∗is a GAS fixed point of xn+1=H(xn). For smooth maps h∈C3, a typical condition required in the global stability results is the negative sign of the Schwarzian derivative. This term was first formulated in 1869 by Hermann A. Schwarz in his work on conformal mappings, but it was not until the work by Singer (1978) that it was introduced in the study of one-dimensional dynamical systems. The following result follows from Theorem 2.7and Addendum in (Singer, 1978). Lemma 4.2.5.Let the map h: [0,∞)−→ [0,∞)related to the one-dimensional difference equation (76)be S-unimodal, then the positive fixed point x∗=Kis GAS in (0,∞)if and only if h0(K)⩾−1. By using that the Ricker map is S-unimodal and the conditions proved in Corollary 4.2.5for its derivative when 0<r⩽2, we can also apply Lemma 4.2.5to prove the sharp global stability of the Ricker model. A generalization of Lemma 4.2.5is given in (El-Morshedy and Jiménez López, 2008), where the condition on the negative Schwarzian derivative is restricted to a suitable subinterval of J. Lemma 4.2.6.Let the map hrelated to the one-dimensional discrete-time difference equation (76)be continuous in a closed interval Jand have a unique fixed point x∗∈J, such that h(x)> x for all x∈J,x < x∗and h(x)< x for all x∈J,x > x∗. Assume that there exist c,d∈J,c < x∗< d such that: 1.h|(c,d)has at most one critical point; 2.h(x)⩽h(c)for all x∈J,x⩽c; 3.h(x)⩾h(d)for all x∈J,x⩾d. If his decreasing at x∗,−1⩽h0(x∗)< 0 and (Sh)(x)< 0 for all x∈(c,d)except for at most one critical point of h; then x∗is GAS in J. For the remainder of this section, assume that 0 < σ < 1 and let the map hrelated to the one-dimensional discrete-time difference equation (76) be a C3 function on J= [0,∞)given by h(x) = (1−σ)x+f(x). For general S-unimodal maps f, the corresponding map hdoes not necessarily inherit all the properties of f, this is the case of the condition on the Schwarzian derivative (see (Liz and Franco, 2010) for details). Thus, the previous results cannot be directly applied to the system (78). This gap on the literature was filled by Liz and Franco (2010, Theorem 1) and the next result from (Liz and Lois-Prados, 2020a) is a generalization. Lemma 4.2.7.Assume that 0 < σ < 1 and f: [0,∞)−→ [0,∞)satisfy the following conditions: 96 background definitions and results ii 1.fσ(x)=(1/σ)f(x)has a unique positive fixed point x∗> 0,fσ(0) = 0and lim x→0+f0 σ(x)> 1 (it can be ∞); 2.fhas a unique critical point xc; moreover, f0(x)> 0 for all x∈(0,xc)and f0(x)< 0 for all x > xc; 3.f00(x)< 0 for all x∈(0,xc); 4.(Sf)(x)< 0 for all x > xc. Then the unique positive equilibrium x∗of equation (78)is GAS in (0,∞)if and only if h0(x∗) = (1−σ) + f0(x∗)⩾−1. (84) Moreover, if condition (84)does not hold, then x∗is unstable. Proof. Equation (78) can be written in the form of equation (4) in (Liz and Franco, 2010), that is: xn+1=αxn+ (1−α)fσ(xn), with α=1−σand fσ(x) = (1/σ)f(x). It is clear that conditions 2-4hold for fσbecause f0 σ(x) = (1/σ)f0(x),f00 σ(x) = (1/σ)f00 σ(x), and (Sfσ)(x) = (Sf)(x). There are two relevant differences with respect to Theorem 1in (Liz and Franco, 2010). On the one hand, condition 4there required (Sf)(x)< 0 for all x6=xc. However, a simple inspection of the proof shows that the less restrictive condition (Sf)(x)< 0 for all x > xcis enough to get the result. On the other hand, the restriction xc< x∗is required in (Liz and Franco, 2010, Theorem 1). In case xc⩾x∗, the map h(x) = (1−σ)x+f(x)defining the right-hand side of (78) satisfies the hypothesis of Lemma 4.2.2. First, the fixed points of hare exactly those of fσ, therefore hhas a unique positive fixed point. Second, since h(x) = xif and only if x∈{0,x∗}, lim x→0+h0(x) = lim x→0+(1−σ) + σf0 σ(x)> 1 and h0(x)> 0 for all x∈(0,x∗), then x<h(x)< x∗for all x∈(0,x∗). Third, as hsatisfies h(x∗) = x∗,h0(x) = (1−σ) + f(x)> 0 and h00(x) = f00(x)< 0 for all x∈(x∗,xc)and h0(x) = (1−σ) + f(x)<(1−σ)< 1 for all x > xc, then 0 < h(x)< x for all x > x∗. Thus, we can conclude that x∗is GAS in (0,∞). 4.3 bifurcation types We distinguish between two categories of bifurcations: local and global. We use the term local for those occurring in a neighborhood of a fixed point and the term global for all the rest. In Chapters 5and 6, once we have determined the stability of fixed points, we use that information to study first local and then global bifurcations. 4.3 bifurcation types 97 In the category of local bifurcations, there exists a well developed classification theory when, at the bifurcation point, the equilibrium lies in a smooth or continuous branch of the map h. We begin reviewing these types of bifurcations and then we continue with the ones which take place when a fixed point collides with a discontinuity point. We use the term smooth bifurcations (SBs) for those bifurcations that are typical of smooth dynamical systems, which are associated with the multiplier h0(x∗)of a fixed point x∗passing through the values ±1. We recall that, when |h0(x∗)|6=1we say that x∗is hyperbolic. We can find the following classification with illustrations in (Wiggins, 1990, Chapter 3): • A saddle-node or fold SB occurs when two coexisting LAS and unstable fixed points become a unique equilibrium with multiplier 1and then disappear. • A transcritical SB occurs when two coexisting LAS and unstable curves of fixed points become a unique equilibrium with multiplier 1, and then the curves of fixed points interchange their stability. • A pitchfork SB occurs when three branches of fixed points collide and then only one remains. In the supercritical case, two LAS fixed points collide with an unstable fixed point and the remaining fixed point is LAS. In the subcritical case, two unstable fixed points collide with a LAS fixed point, and the remaining fixed point is unstable. At the bifurcation point, there is a unique equilibrium with multiplier 1. • A flip or period-doubling SB occurs when a fixed point x∗becomes hyperbolic with multiplier −1and then the fixed point changes its stability and a 2-cycle {p1,p2}appears, where p1< x∗< p2. In the supercritical case, the LAS fixed point becomes unstable and the 2-cycle is LAS. In the subcritical case, the unstable fixed point becomes LAS and the 2-cycle is unstable. We use the term period-halving SB when the bifurcation takes place in the opposite direction. For smooth maps h∈Cr(r⩾2for fold and transcritical; r⩾3for pitchfork and flip), the book by Wiggins (1990) states a list of sufficient conditions for the map hunder which each of these types of SBs takes place. Similar bifurcations related to the multiplier of a fixed point passing through ±1, but occurring under some degeneracy conditions, are referred as degenerate SBs (see (Avrutin et al., 2019, Section 2.2) for details). Before describing local bifurcations of fixed points for piecewise-smooth (continuous or discontinuous) maps, we need to adopt some special notations introduced in (Bernardo et al., 2008). For that purpose, we consider hland hr 98 background definitions and results ii two smooth maps on J, a point d1∈Int(J)such that hl0(d1)6=hr0(d1), and we assume that the map hassociated to the difference equation (76) is defined by h(x) =          hl(x),x < d1; hl(x)or hr(x),x=d1; hr(x),x > d1; (85) which is not differentiable at d1. The point d1where his not differentiable is called a break point. If a break point is a fixed point of h, then we say it is a boundary fixed point. We recall that the map hin (85) is defined by two smooth maps hland hr, so there may are fixed points of hland hrthat are not fixed points of h, we refer to them as virtual fixed points; while the fixed points of h are called admissible fixed points. The local bifurcations at a boundary fixed point where introduced by Nusse and Yorke (1992) and occur when, under infinitesimal parameter variation, a fixed point collides with a break point and the collision leads to a qualitative change on the dynamics, with the fixed point transitioning from being admissible to being virtual or viceversa. We refer to these types of bifurcations as border-collision bifurcations (BCBs). A classification of BCBs in piecewise-smooth continuous discrete-time difference equations is given in (Bernardo et al., 2008, Chapter 3). In Section 3.4, there is a classification for one-dimensional systems, which is given in terms of the derivatives of hl,hrat the boundary fixed point. We pay special attention to the contents in Subsection 3.1.2, where they describe and illustrate the four basic dynamical scenarios which take place at a BCB: • A fold BCB occurs when two coexisting admissible fixed points collide at a break point and become two virtual fixed points. • A persistence BCB occurs when an admissible and a virtual fixed point collide at a break point and interchange their roles. No other periodic points are created or destroyed at the bifurcation point. • A flip or period-doubling BCB occurs when an admissible fixed point x∗ collides with a break point and a 2-cycle {p1,p2}, with p1< x∗< p2, appears. • A period-multiplying BCB occurs when an admissible fixed point collides with a break point and an m-cycle appears, with m > 2. We observe that new dynamic transitions at the bifurcation points occur due to the lack of differentiability of the map h, such as the persistence or periodmultiplying BCBs. Moreover, there exist more complex bifurcations, where after the border collision of an admissible fixed point, the asymptotic dynamics become chaotic. It is well-known that such a transition in general does not occur in smooth systems. 4.3 bifurcation types 99 The previously mentioned bifurcations can also be observed in the long-term dynamics of a piecewise-smooth discontinuous dynamical system, but the discontinuity character increases again the variety of fixed point local bifurcations. We devote the next lines to describe some additional BCBs taking place at discontinuity points of systems that we will consider in Chapter 5(see (Avrutin et al., 2019, Chapters 2and 3) for details): • An existence BCB occurs when a virtual fixed point of hbecomes admissible after a collision with a break point. No other orbits are created or destroyed at the bifurcation point. • A period-adding BCB occurs when a LAS admissible fixed point collides with a break point; at the collision point, there is a homoclinic orbit (we recall the definition at the end of this subsection); after the collision the fixed point becomes virtual, and an attracting m-cycle becomes admissible, giving rise to a period-adding scenario. The period-adding scenario refers to the order of periodicity regions in the parameter space where between two cycles of periods nand lthere is an (n+l)-cycle (see Granados, Alsedà, and Krupa, 2017, for details). In the category of global bifurcations, we consider two different groups: •Boundary-collision bifurcations, which are caused by the collision of an attractor with an unstable fixed point or m-cycle (m∈N,m⩾2). These bifurcations where introduced as crises by Grebogi, Ott, and Yorke (1982) for the case of a chaotic attractor. They use the term boundary crisis when the unstable orbit is on the boundary of the chaotic attractor and the collision causes termination of the attractor; and interior crisis when the collision occurs within the basin of attraction. They mention that the interior crisis often results in a sudden expansion of the basin. •Basin boundary metamorphoses, which are those related with transformations of the basins of attraction. We will find transitions from a simplyconnected to a multiply-connected basin. The transitions can be more complex, as those in (Grebogi, Ott, and Yorke, 1983) from a regular basin to a fractal one. We have already mentioned that a homoclinic orbit appears at the periodadding BCBs, and we will also observe the presence of this type of orbits in boundary-collision bifurcations and basin boundary metamorphoses. We recall that a homoclinic orbit is formed by a homoclinic point, its preimages and its (finite) forward orbit . A point xis homoclinic to an n-cycle p={p1,...,pn}if (fn)m(x) = pkfor some m∈Nand k∈{1,...,n}, and xbelongs to the unstable manifold of pk. For further details, see (Devaney, 1989, Section 1.16) or (Liz, 2010a, Appendix C). Finally, it is worth mentioning that, in general, we study bifurcations of attractors, but bifurcations of other invariant sets also deserve to be investigated since they can influence the asymptotic dynamics as well. 100 background definitions and results ii 4.41-parameter bifurcation diagrams features The last step in the analysis of asymptotic dynamics is to illustrate the influence of parameters. For that purpose, we first reflect the information we have on stability of fixed points and bifurcations in a 2-parameter BD. Then we plot several 1-parameter BDs to get more insight and complete the global picture of the dynamics. In this section we review some relevant features or phenomena that have been observed in 1-parameter BDs of a wide range of discrete one-dimensional dynamical systems. Long-transients and hysteresis Let us first discuss some consequences of focusing on asymptotic dynamics in ecology. In (Hastings et al., 2018), there is a review summary on the long transient phenomena, defined as a dynamical regime that persists for more than a few and as many of ten generations, but which is not the stable longterm dynamics that would eventually occur. In the particular framework of mathematical analysis of the dynamical systems given by (76), the study of non-asymptotic dynamics has not much interest, since long-transients are just iterations of an initial condition by the map h. However, the presence of longtransients can obscure the decisions on the management of ecological systems, so it becomes interesting to categorize different ways in which transients can arise. With this possibility in mind, we start defining the long-term dynamics phenomenon of hysteresis, which can also have unexpected consequences for the management in biological systems. In (Blackwood, Hastings, and Mumby, 2012), for a system with multiple stable states, the authors say that this feature occurs when the previous history of the system influences the convergence of an initial condition to one of the stable states or the others. They also describe the phenomenon in other words, saying that the critical parameter conditions under which some points converging to one stable state switch and converge to another one are different from the conditions that will allow the convergence of such points to the original state. See Figure 22 (A). Bubbles, bistability and hydra effect We now deal with three features that have been observed in bifurcation diagrams of both semelparous and iteroparous population models. We begin with a phenomenon related to the period-doubling sequence of bifurcations as a universal route to chaos, which can be broken and reversed as shown in (Bier and Bountis, 1984) for simple non-linear discrete dynamical systems involving the variation of two or more parameters. As a consequence, the bifurcation diagrams form closed loop-like structures similar to bubbles 4.41-parameter bifurcation diagrams features 101 and the effect is usually referred to as bubbling. Definition 3in (Liz and Ruiz- Herrera, 2012) gives a formal definition of the concept of bubble. The complexity of the bubble structure depends on the point at which the period-doubling sequence is reversed. The simplest bubble occurs when an equilibrium loses its asymptotic stability through a period-doubling bifurcation and, at the next bifurcation point, a period-halving bifurcation occurs, so that the local asymptotic stability of the fixed point is regained, we refer to it as primary bubble. If the period-doubling sequence of bifurcations is reversed after a region of chaotic dynamics, we use the term chaotic bubble. The bistability feature refers to the coexistence of two attractors, and it occurs due to fold bifurcations (either smooth or border-collisions). Bistability has important consequences in population dynamics because, in this case, the longterm behavior of the solutions strongly depends on the initial condition. Another formal definition stated in (Liz and Ruiz-Herrera, 2012) is the hydra effect. This term was used by Abrams (2009) and the references therein for the phenomenon of a population increase in response to an increase in its mortality rate. This feature has been first recognized by Ricker (1954) for the well-know Ricker model which assumes that mortality precedes reproduction. That is one of the three mechanisms underlying hydra effect which are developed in (Abrams, 2009) for non-overlapping populations models. Liz and Ruiz-Herrera (2012) studied the hydra effect for iteroparous population models, in which a percentage of the adult population survives the reproductive season. Extinction windows and sudden collapses The following features are typical of models with Allee effect, where populations cannot survive in the long-term if its abundance is below a critical size, they are called extinction windows and sudden collapses. The term extinction window refers to the survival-extinction dynamics described by Sinha and Parthasarathy (1996) as an unusual structure with alternating regions of survivals and extinction under variation of parameters, so that the population can persist under very low and fairly high values of the parameter, though it is not able to survive at intermediate parameter values. The sudden collapse phenomenon appears also in (Sinha and Parthasarathy, 1996), but a description of the feature is given in (Schreiber, 2001) where it is said that there exists a critical parameter value above which populations are driven to extinction for all initial densities and below which persistence is possible. By persistence we mean that population remains at densities bounded away from zero. Periodic-windows, star-like intersections and effectively chaotic behaviour We finally describe some typical features of bifurcation diagrams for piecewisesmooth maps with flat branches (see (Sinha, 1994) for further details). 108 combinations of cc and th harvesting strategies harvesting strategies. The consideration of constant quotas instead of fixed mortality rates can reduce the risk of overcapacity when the stock decreases, since PH stimulates investment when stock size is large. The results in (Hjerne and Hansson, 2001) show that long-term yield for the precautionary combination of TH and CC (referred to as quasi constant catch, QCC for short) is 10% less than that of PH at MSY. They consider that it is a small difference and emphasize that the much lower inter annual variability of QCC gives the opportunity of better fishery capacity utilization. We notice that they first considered the TCC control rule, and then QCC as a precautionary modification to reduce fishery closures. To our knowledge, the TCC strategy has been considered so far only by Hjerne and Hansson (2001) and in a slightly different form by AlSharawi and Rhouma (2009). We also notice that Punt (2010) cites the article (Butterworth, 1987) to talk about threshold management strategies in which catch becomes constant when the stock size is greater than a target level. We finally provide a different reason to consider this type of control rules. Steiner, Criddle, and Adkinson (2011) looked for a solution to the revenue decline of Bristol Bay sockeye salmon managed with a fixed escapement control rule. In contrast to other fisheries, the landings had remained high, but the prices fell as a consequence of competition resulting from the increased production of trout and salmon farmed species in Chile. They provide several reasons to consider implementing the PTCC strategy rather than TH or PTH: the fact that PTCC induces lower harvests allows to improve the quality of the fish delivered, thus producing a high exvessel price per pound; the lower variability in harvest provides more efficiency of the management operations because the harvest levels will be known at the beginning of the season. H (A) escapement catch ? No harvest T T +H H (B) escapement catch ? No harvest T H No harvest ? Extinction ? escapem. T H catch (C) Figure 11: Different harvesting strategies that combine constant catches with threshold reference points. We represent the catch (red solid line) as a function of the population biomass. The blue line represents the identity map, Tis the threshold and Hthe maximum allowed quota. (A): Precautionary threshold constant catch harvesting. The other panels show threshold constant catch harvesting with (B): H < T and (C): H > T. 5.1 introduction 109 Contextualization in the framework of piecewise-smooth one-dimensional difference equations As far as we know, we formulate and study the PTCC and TCC rules in the framework of one-dimensional discrete-time dynamical systems by means of an analytical approach for the first time. As these harvesting strategies are based on threshold population sizes, the associated map will be composed of different branches (corresponding to high or low/no harvesting) defined on intervals which are separated at the biomass reference points (also called break points in the mathematical literature). As the maps are not differentiable at the threshold points, they give rise to piecewise-smooth dynamical systems (Avrutin et al., 2019; Bernardo et al., 2008). They can exhibit so-called nonsmooth bifurcations that differ substantially from those that occur in smooth dynamical systems, e.g., border-collision bifurcations (Nusse and Yorke, 1992). Non-smooth bifurcations are related to invariant sets colliding with a break point, which is given by the harvesting threshold. In recent years, a lot of progress has been made in understanding the dynamics of piecewise-smooth maps, e.g., (Banerjee et al., 2000; Radi and Gardini, 2018; Sushko, Gardini, and Matsuyama, 2014). However, even though they emerge quite naturally in the context of threshold-based harvesting, their mathematical analysis in the context of fisheries models is just at the beginning (Bischi, Lamantia, and Tramontana, 2014; Franco and Hilker, 2013,2014; Hilker and Liz, 2019,2020; Liz and Lois-Prados, 2020b; Lois-Prados and Hilker, submitted; Segura, Hilker, and Franco, 2016,2020). Overview of the research development Our study is focused on stability and bifurcations; in this regard, we consider the two relevant harvest parameters Hand Tand obtain 1-parameter and 2- parameter bifurcation diagrams that help to understand how a continuous variation of any of them influences the dynamics, and the interplay between both parameters. We identify regions where global attraction, periodic attractors, multi-stability and complex behavior are likely to occur, paying special attention to non-smooth bifurcations. We first study the long-term behavior of the continuous map associated with PTCC, which will serve as a baseline against which we can compare the effects induced by the discontinuity in TCC. For the PTCC rule, we completely determine the asymptotic dynamics and bifurcations for general compensatory population models; we also understand the long-term behavior in some regions of the parameter plane (H,T)for general overcompensatory maps, for which we consider the Ricker map as a case study to give a global picture of the dynamics. For the TCC rule, we just deal with strictly increasing stock-recruitment maps, while in this case all initial conditions would converge to an equilibrium for the PTCC rule, the discontinuity point of TCC gives rise to highly complex dynamics, including multiple attrac- 110 combinations of cc and th harvesting strategies tors, different periodic cycles, homoclinic orbits and even chaotic oscillations. The bifurcations in which these dynamical patterns emerge and disappear involve border- and boundary-collision bifurcations as well as basin boundary metamorphoses. Hence, the discontinuity in the harvest control rule produces rich dynamics that, to our knowledge, have not been observed in continuous harvesting models applied to population maps that are strictly increasing in the absence of harvesting. The research on PTCC totally focuses on dynamical systems theoretical aspects, we notice that for some particular parameter values the long-term behavior is influenced by the dynamics of constant-catch or threshold harvesting. Nevertheless, some characteristic dynamics of CC or TH are not completely preserved, e.g., the extinction attractor of CC or the convergence of almost all solutions to a periodic orbit containing Tof TH. For the TCC rule, we thoroughly combine the qualitative study of population dynamics with management-oriented interpretations; additionally we study the influence of harvesting parameters in long-term average yield and harvest frequency. We pay special attention to the influence of the threshold point (that is, the effect that TCC has in comparison to CC) in population, average yield and harvest frequency behavior. 5.2 models description In this section, for the sake of completeness, we first state the mathematical formulation and recall some results on asymptotic dynamics for the classical control rules that can be considered as particular cases of TCC or PTCC, that is, constant quota (subsection 5.2.1) and threshold harvesting (subsection 5.2.2). In subsections 5.2.3and 5.2.4, we provide the mathematical expression for PTCC and TCC, respectively. We compare the TCC control rule with the precautionary version given by PTCC and the similar strategy considered by Avrutin et al. (2019). 5.2.1Constant catch rule For later reference, we state some well-known results for the CC harvesting rule. For a general map f: [0,∞)−→ [0,∞)and the constant quota H > 0, the model reads xn+1=FCC(xn) = max{0,f(xn) − H}. (86) Let us consider the map g(x) = f(x) − H,x∈[0,∞), then we can rewrite FCC in the form: FCC(x) = max{0,g(x)}. The following result establishes a critical value H20 for the harvesting quota. Harvesting above this level (H > H20) will drive the population to extinction. Below this level, several long-term dynamics can occur. The simplest dynamics take place for a map fsatisfying conditions (A1)-(A2)(defined in Section 4.1) with xc=∞, for which the population will survive provided the initial con- 5.2 models description 111 dition is large enough. By contrast, the most complex dynamics occur when xc< K and there is bistability between the LAS fixed point 0and other attractor. The proposition uses that there exists a unique ˜x>0such that f0(˜x) = 1. This follows from assumptions (A1)-(A2)and the Mean Value Theorem. Moreover, ˜x∈(0,K). Proposition 5.2.1.Assume that H>0and fsatisfies (A1)-(A2). Denote by ˜xthe unique solution of f0(x) = 1. 1. If H < H20 := f(˜x) − ˜x, then ghas two positive equilibria x∗ −and x∗ +with 0<x∗ −<˜x < x∗ +< K. While x∗ −is always unstable and 0is locally asymptotically stable, the fixed point x∗ +can be either stable or unstable: a) If xc=∞, then x∗ +is LAS with basin of attraction (x∗ −,∞)and [0,x∗ −)is the basin of attraction of 0. b) If K < xc<∞, then x∗ +is LAS with basin of attraction (x∗ −,g−1(x∗ −)) and [0,x∗ −)∪(g−1(x∗ −),∞)is the basin of attraction of 0. c) If xc< K, then x∗ +can be LAS with basin of attraction (x∗ −,g−1(x∗ −)) or unstable. If x∗ +is unstable, the dynamics depend on the position of g2(xc) with respect to x∗ −: If g2(xc)> x∗ −, then I= [g2(xc),g(xc)] is absorbing and the initial conditions in (x∗ −,g−1(x∗ −)) enter Iin finite time; besides, [0,x∗ −)∪(g−1(x∗ −),∞) is the basin of attraction of 0. If g2(xc)< x∗ −, then 0is an essential global attractor. 2. If H=H20, then ˜xis the unique positive fixed point of g. The equilibrium ˜xis semi-stable and 0is locally asymptotically stable. If xc=∞,[˜x,∞)and [0, ˜x) are their respective basins of attraction; while if xc<∞they are [˜x,g−1(˜x)] and [0, ˜x)∪(g−1(˜x),∞), respectively. 3. If H > H20, then ghas no positive fixed points and 0is GAS. We refer the reader to (Schreiber, 2001) for a rigorous proof and analysis of the results in Proposition 5.2.1. 5.2.2Threshold harvesting rule We consider now the TH rule, which can be seen as the antipode to the CC harvesting strategy. For a general map f: [0,∞)−→ [0,∞)and a threshold level T > 0, the dynamics of TH are governed by the difference equation: xn+1=FTH(xn) = min{f(xn),T}=   f(xn),f(xn)⩽T; T,f(xn)> T.(87) In the recent work by Hilker and Liz (2019), we find a rigorous theoretical study on the influence of the threshold Ton the dynamics of (87). Under some 112 combinations of cc and th harvesting strategies general conditions for the map f, they show that threshold harvesting can never have a destabilizing effect on the managed population. The following result summarizes the findings in (Hilker and Liz, 2019, Section 2.1). Proposition 5.2.2.Assume that T > 0 and fsatisfies (A1). 1. If T⩽K, then Tis the unique positive equilibrium of (87)and it is GAS. 2. If T > K, the dynamics of the managed system (87)depend on the dynamics of the unmanaged system xn+1=f(xn): a) If Kis GAS for the unmanaged system, then Kis the unique positive equilibrium of (87)and it is GAS. b) If Kis unstable but the unmanaged system has a finite number of periodic orbits, then decreasing threshold induces a sequence of period-halving bifurcations until the equillibrium becomes stable. c) If Kis unstable and the unmanaged system is chaotic, for S-unimodal maps there is a unique periodic orbit that is an essential global attractor of the managed system (87). Decreasing Tfrom f(xc)to K, there is Li-Yorke chaos (see Definition 3.1, Aulbach and Kieninger, 2001) as long as the period of the attracting cycle is not a power of 2. Once the dynamics become simpler due to smaller threshold values, we have the situation considered in the previous case. 5.2.3Precautionary threshold constant catch rule Applying the PTCC harvesting rule to a semelparous population model given by (77), we obtain the difference equation xn+1=FPTCC(xn) =          f(xn),f(xn)⩽T; T,T < f(xn)⩽T+H; f(xn) − H,f(xn)> T +H. (88) The map g(x) = f(x) − Hallows to write FPTCC in the form FPTCC(x) =          f(x),f(x)⩽T; T,g(x)⩽T < f(x); g(x),g(x)> T. The piecewise smooth continuous map FPTCC can also be written in a line as FPTCC(x) = min{f(x),f(x) − min{H,f(x) − T}}=min {f(x),max{g(x),T}}; and depends on the two harvesting parameters Hand T. Throughout the analysis of the PTCC harvesting strategy (88), we will assume that H > 0 and T > 0. It is worth noticing that we get the unmanaged map for the CC and TH rules as particular or limit cases: 5.2 models description 113 • If H=0or T⩾sup{f(x),x⩾0}, then FPTCC ≡f. • In the limit case T=0, (88) becomes the usual constant catch policy, defined by (86). • If T < sup{f(x),x⩾0}⩽H+T, then the PTCC harvesting strategy becomes the pure threshold harvesting rule (87). The typical shape of FPTCC can be seen in Figure 12. Roughly speaking, the graphs of fand gare joined by flat segments defined by T, which typically results in five intervals of smoothness for FPTCC if fis unimodal, and three if f is strictly increasing. (A) 4 (A) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 (B) T ? 6 H 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 FIG. 2: Illustration of the graph of the piecewise smooth map F(blue solid line). We also plot the line y=x(red, dashed) and the graph of f(black, dashed). (A): Unimodal Ricker map f(x)=xe 2.6(1x), with H=0.5andT=0.7. (B): Monotone Beverton-Holt map f(x)=2x/(1 + x), with H=0.2 and T=0.7. TABLE I: Main notations Symbol/concept Meaning H(maximum) harvesting quota Tthreshold harvesting parameter PTCH precautionary threshold constant-catch harvesting TH (pure) threshold harvesting fproduction map governing (1) gg(x)=f(x)H Fmap defining the PTCH rule (2) Ricker map f(x)=xe r(1x),r>0 xccritical point of f(f0(xc)=0) ˜xpoint such that f0(˜x)=1 x(smallest) point such that f0(x)=1 Kpositive fixed point of f p, q positive fixed points of g(0 <pq<K) break point point at which Fis not di↵erentiable boundary fixed point break point which is a fixed point of F admissible fixed point fixed point of F virtual fixed point fixed point of one map defining Fbut not of F BCB border-collision bifurcation SB smooth bifurcation III. FIXED POINTS: LOCATION, STABILITY, AND BIFURCATIONS In this section, we study the fixed points of Fdepending on the parameter values Tand H. In the first subsection we focus on the number of fixed points and their location, in the second one we study their stability properties, and in the third subsection we describe the local bifurcations of fixed points, that is, we determine the critical values of the parameters for which fixed points are created or destroyed, or stability switches occur. One important consequence of conditions (A1) and (A2) is that there is a unique ˜x>0 such that f0(˜x) = 1. Moreover, ˜x2(0,min{K, xc}). This property is a direct consequence of the Mean Value Theorem and the concavity of fon (0,x c). The point ˜xplays an important role in the study of fixed points. (B) 4 (A) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 (B) T ? 6 H 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 FIG. 2: Illustration of the graph of the piecewise smooth map F(blue solid line). We also plot the line y=x(red, dashed) and the graph of f(black, dashed). (A): Unimodal Ricker map f(x)=xe 2.6(1x), with H=0.5andT=0.7. (B): Monotone Beverton-Holt map f(x)=2x/(1 + x), with H=0.2 and T=0.7. TABLE I: Main notations Symbol/concept Meaning H(maximum) harvesting quota Tthreshold harvesting parameter PTCH precautionary threshold constant-catch harvesting TH (pure) threshold harvesting fproduction map governing (1) gg(x)=f(x)H Fmap defining the PTCH rule (2) Ricker map f(x)=xe r(1x),r>0 xccritical point of f(f0(xc)=0) ˜xpoint such that f0(˜x)=1 x(smallest) point such that f0(x)=1 Kpositive fixed point of f p, q positive fixed points of g(0 <pq<K) break point point at which Fis not di↵erentiable boundary fixed point break point which is a fixed point of F admissible fixed point fixed point of F virtual fixed point fixed point of one map defining Fbut not of F BCB border-collision bifurcation SB smooth bifurcation III. FIXED POINTS: LOCATION, STABILITY, AND BIFURCATIONS In this section, we study the fixed points of Fdepending on the parameter values Tand H. In the first subsection we focus on the number of fixed points and their location, in the second one we study their stability properties, and in the third subsection we describe the local bifurcations of fixed points, that is, we determine the critical values of the parameters for which fixed points are created or destroyed, or stability switches occur. One important consequence of conditions (A1) and (A2) is that there is a unique ˜x>0 such that f0(˜x) = 1. Moreover, ˜x2(0,min{K, xc}). This property is a direct consequence of the Mean Value Theorem and the concavity of fon (0,x c). The point ˜xplays an important role in the study of fixed points. Figure 12: Illustration of the graph of the piecewise smooth map FPT CC (blue solid line). We also plot the line y=x(red, dashed) and the graph of f(black, dashed). (A): Unimodal Ricker map f(x) = x e2.6(1−x), with H=0.5and T=0.7.(B): Strictly increasing Beverton-Holt map f(x) = 2x/(1+x), with H=0.2and T=0.7. 5.2.4Threshold constant catch rule When harvesting a population that is growing according to (77) with the TCC rule, we obtain xn+1=FTCC(xn) :=    f(xn),f(xn)< T; max{0,f(xn) − H},f(xn)⩾T.(89) Let us consider the map g(x) = f(x) − H, then we can rewrite the map FTCC in the form FTCC(x) =          f(x),f(x)< T; 0,T⩽f(x)< H; g(x),f(x)⩾max{T,H}. (90) 114 combinations of cc and th harvesting strategies The piecewise-smooth map FTCC depends on the two harvesting parameters Tand H. For the study of TCC harvesting rule (89), we assume H > 0 and T > 0. As particular or limit cases, we obtain the unmanaged map fand the CC rule, asking the same conditions to the harvesting parameters as in PTCC. By contrast, we cannot get the TH harvesting strategy as a particular case of TCC. The typical shape of FTCC applied to strictly increasing maps is shown in Figure 13. The main differences with respect to FPTCC are the existence of a point of discontinuity at xT=f−1(T)and the absence of a flat segment defined by T. 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 (A) (B) ? 6 H T ? 6 H T Figure 13: Illustration of the graph of the piecewise-smooth discontinuous map FTCC (blue solid line). We also plot the line y=x(red, dashed) and the graph of the Beverton-Holt map f(x) = 3x/(1+x).(A) : T=1,H=0.4.(B) : T=0.3, H=0.4. Related harvesting strategy We point out that AlSharawi and Rhouma (2009, Section 4) proposed a harvesting strategy very similar to TCC. It reads xn+1=   f(xn),xn< xth; f(xn) − h,xn⩾xth;(91) where they considered specifically the Beverton-Holt map for f,xth ∈(0,K)is a threshold population size and h∈(0,f(xth)) is the constant quota. The difference with (89) is that TCC compares the threshold with the population size f(xn)after reproduction, whereas (91) compares the threshold with the population size xnbefore reproduction. However, for the strictly increasing population maps considered in this paper, there is a correspondence between (89) and (91). To see this, we note that fis bijective and that we just have to establish the following relation: T:= f(xth),H:= hor, equivalently, xth := f−1(T),h:= H for T∈(0,K),H∈(0,T). The correspondence allows us to use some results obtained by AlSharawi and Rhouma (2009) on the existence of (stable or unstable) 2-cycles and to describe the bifurcations of F2 TCC. For non-monotone population maps like the Ricker map, there can be two break points in the harvest rule (89), and there is no correspondence between (89) and (91). That is, 5.3 precautionary threshold constant catch (ptcc)115 the two harvest strategies could differ qualitatively in their dynamics. As the harvest rule in (91) refers to a measurement of population size that is further in the past, this introduces a time lag that, for overcompensatory population maps, could lead to delayed density-dependent effects which are known to change dynamics quantitatively and qualitatively (Franco and Hilker, 2014). 5.3 precautionary threshold constant catch (ptcc) We organize the section as follows: Subsection 5.3.1is devoted to the study of fixed points. We start giving the location of positive equilibria depending on the relevant parameters. Then we provide stability results for the positive fixed points: while for compensatory models all solutions converge to an equilibrium, in the overcompensatory case there are stability switches and the global picture is more complicated; we give some general results for global stability and study in more detail the Ricker map, which is a prototype for discrete population models, especially in the context of fisheries (Quinn and Deriso, 1999; Ricker, 1954). Finally, we describe all posible bifurcations of fixed points (smooth and border-collision bifurcations). In Subsection 5.3.2, we focus on a particular region of the parameter plane for which chaos and essential attraction can occur. We recall that the latter means that an equilibrium is not globally attracting, but solutions converge to it with probability one. The obtained results allow us to determine some boundary-collision bifurcations. In Subsection 5.3.3, we address two case studies: a simple compensatory model, where only bifurcations of fixed points appear, and an overcompensatory model that exhibits richer dynamics; in both cases numerical bifurcation diagrams help to understand the influence of the parameters. In the overcompensatory case we pay special attention to bifurcations of 2-cycles, bistability regions and the influence of flat branches on the dynamics. 5.3.1Fixed points: location, stability and local bifurcations In this subection, we study the fixed points of FPTCC depending on the parameter values Hand T. We first focus on the number of positive fixed points and their location; secondly we study their stability properties; and thirdly we describe the local bifurcations of fixed points, that is, we determine the critical values of the parameters for which fixed points are created or destroyed, or stability switches can occur. We notice that existence and localization results hold for both compensatory and overcompensatory models, while the stability ones depend on the type of stock-recruitment relationship. We recall that a consequence of conditions (A1)-(A2), the Mean Value Theorem and the concavity of fin (0,xc)is the existence of a unique point ˜xin (0,min{K,xc})such that f0(˜x) = 1. The point ˜xplays an important role in the study of fixed points. 116 combinations of cc and th harvesting strategies 5.3.1.1Existence and localization of positive fixed points The following result provides the number of positive fixed points of the map FPTCC defined in (88). We assume that 0 < T < sup{f(x) : x > 0}and H > 0. Proposition 5.3.1.Assume that (A1)-(A2)hold. Denote by ˜xthe unique solution of f0(x) = 1and by x∗ −,x∗ +(0 < x∗ −⩽˜x⩽x∗ +< K) the positive fixed points of g(x) = f(x) − H, when they exist. The following assertions hold: 1. If T⩾K, then Kis the unique positive equilibrium of (88). 2. If ˜x⩽T < K, then FPTCC has a unique positive fixed point, which is Tif H⩾f(T) − T; or x∗ +∈(T,K)if H < f(T) − T. 3. If 0 < T < ˜x, then: a) FPTCC has a unique positive fixed point x∗ +∈(˜x,K)if H < f(T) − T. b) FPTCC has three positive fixed points (Tand the two fixed points of g) if f(T) − T < H < f(˜x) − ˜x. c) Tis the unique positive fixed point of FPTCC if H > f(˜x) − ˜x. d) FPTCC has two positive fixed points if either H=f(T) − T(Tand x∗ +) or H=f(˜x) − ˜x(Tand ˜x). Proof. In view of (88), Kis a fixed point of FPTCC if and only if K=f(K)⩽T. Moreover, if K⩽T, then Kis the only positive fixed point of FPTCC because FPTCC(x)⩾min{f(x),T}> x if x < K, and FPTCC(x)⩽f(x)< x if x > K. If T < K, then there are two possibilities for the positive fixed points of FPTCC: the threshold Tand the positive equilibria of g. It is obvious from the definition of FPTCC that Tis a fixed point if and only if f(T)⩽T+H, that is, H⩾f(T) − T. The condition T < f(T)holds since T < K. Since g0(x) = f0(x)and g00(x) = f00(x),gcan have at most two positive fixed points x∗ −⩽x∗ +. Then, by the Mean Value Theorem, x∗ −⩽˜x⩽x∗ +, where ˜xis the only point for which f0(˜x) = 1. It also follows that T < x∗ −⩽x∗ +< K, because x∗ −=FPTCC(x∗ −) = g(x∗ −) = f(x∗ −) − H > T, and g(x) = f(x) − H⩽x−H < x for all x⩾K. Now, statements 2and 3follow easily. We include the proof of 2and omit the details of the other assertion since it is analogous, so assume that ˜x⩽T < K. If H⩾f(T) − T, then the threshold Tis the unique equilibrium of FPTCC because g(T) = f(T) − H⩽Tand, for T⩾xc, it follows from g0(x)< 1 for all x > T; while for T < xc, it follows from g0(x)> 0,x < xcand g0(x)< 0,x > xc. See Figure 14 as an illustration of the proof. If H<f(T) − T, then FPTCC has a positive fixed point x∗ +∈(T,K)because g(T) = f(T) − H > T and g(K) = f(K) − H=K−H < K. This fixed point is unique because, if there were two fixed points x∗ −< x∗ +, then T < x∗ −and g(x)< x for all x < x∗ −would imply that g(T)< T, a contradiction. 5.3 precautionary threshold constant catch (ptcc)117 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (A) (B) T T Figure 14: Illustration of the piecewise-smooth continuous map FPTCC (blue solid line). We also plot the line y=x(red, dashed) and the Ricker map f(x) = rx/(1+x)(black, dashed). (A) : r=2.6,T=0.7∈(xc,K)≈ (0.3846,1),H=1 > f(T) − T≈0.827.(B) : r=1.5,T=0.4∈(˜x,xc)≈ (0.3969,0.6667),H=0.6>f(T) − T≈0.5838. Figure 15 illustrates the number of fixed points in the parameter plane (H,T). 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Harvesting quota, H Threshold, T 1fixed point K 1fixed point T 1fixed point x∗ +∈(T,K) 3equilibria ˜x Figure 15: Number of positive equilibria of FPTCC with f(x) = xe2.6(1−x). There are two positive fixed points for the parameter values in the boundaries colored in red: H=f(T) − T,0 < T < ˜x; and H=f(˜x) − ˜x,0 < T < ˜x. There is only one positive fixed point for parameters at the boundaries colored in blue: H=f(T) − T, ˜x < T < 1; and T=1. 5.3.1.2Stability of positive fixed points This section is devoted to study the stability properties of the positive fixed points of FPTCC. It is worth mentioning that, for a map fsatisfying conditions (A1)-(A2),0is a fixed point of FPTCC, and it is always unstable. We first consider the case T⩾K, for which we obtain a global stability result if fis a general map satisfying condition (A1). The proof follows the one of the analogous result for proportional threshold harvesting (Hilker and Liz, 2019, Proposition A.3).