UNIVERSIDADE ESTADUAL PAULISTA – UNESP Instituto de Biociências, Letras e Ciências Exatas - Campus de São José do Rio Preto MARCOS ANTONIO VIANA COSTA ELLIPTIC SYSTEMS IN POPULATION DYNAMICS: A STUDY WITH NONLOCAL DIFFUSION COEFFICIENTS São José do Rio Preto 2025 MARCOS ANTONIO VIANA COSTA ELLIPTIC SYSTEMS IN POPULATION DYNAMICS: A STUDY WITH NONLOCAL DIFFUSION COEFFICIENTS Tese apresentada, em regime de co- tutela, à Universidade Estadual Paulista (UNESP), Instituto de Biociências, Letras e Ciências Exatas, São José do Rio Preto e à Universidad de Sevilla (US), para obtenção do título de Doutorado em Matemática (UNESP) e em Doctorado en Matemáticas (US). Área de Concentração: Análise Aplicada Orientador: Prof. Dr. Marcos Tadeu de Oliveira Pimenta (UNESP) Coorientador: Prof. Dr. Antonio Suárez Fernández (US) São José do Rio Preto - Brasil Sevilla - España 2025 C837e Costa, Marcos Antonio Viana Elliptic systems in population dynamics : a study with nonlocal diffusion coefficients / Marcos Antonio Viana Costa. -- São José do Rio Preto, 2025 161 p. Tese (doutorado) - Universidade Estadual Paulista (UNESP), Instituto de Biociências Letras e Ciências Exatas, São José do Rio Preto Orientadora: Marcos Tadeu de Oliveira Pimenta Coorientadora: Antonio Suárez Fernández 1. Dinâmica de Populações. 2. Sistemas Elípticos Não-Locais. 3. Bifurcação Local e Global. 4. Índice de Ponto Fixo. 5. Método de Sub-Supersolução. I. Título. Sistema de geração automática de fichas catalográficas da Unesp. Dados fornecidos pelo autor(a). MARCOS ANTONIO VIANA COSTA ELLIPTIC SYSTEMS IN POPULATION DYNAMICS: A STUDY WITH NONLOCAL DIFFUSION COEFFICIENTS Tese apresentada à Universidade Estadual Paulista (UNESP), Instituto de Biociências, Letras e Ciências Exatas, São José do Rio Preto, para obtenção do título de Doutor em Matemática. Área de Concentração: Análise Aplicada. Data da defesa: 26/09/2025 Banca Examinadora: ______________________________________ Prof. Dr. Marcos Tadeu de Oliveira Pimenta Orientador ______________________________________ Prof. Dr. Antonio Suárez Fernández Co-orientador ______________________________________ Prof.ª Dr.ª Mónica Molina Becerra Universidad de Sevilla (US) ______________________________________ Prof.ª Dr.ª Vanessa Avansini Botta Pirani Universidade Estadual Paulista (UNESP) ______________________________________ Prof. Dr. Willian Cintra da Silva Universidade de Brasília (UnB) Aos meus pais, Antonio e Regina. À minha tia, Leila (in memoriam). AGRADECIMENTOS Agradeço, primeiramente, à Santíssima Trindade, que me concedeu sabedoria e discernimento para tomar decisões acertadas e alcançar esta conquista. Sou grato também pelas pessoas certas colocadas ao meu redor ao longo da caminhada, que me apoiaram e orientaram nos momentos decisivos. À minha família, pelo apoio, sustento e incentivo que foram essenciais e determinantes para essa realização. Cada palavra de encorajamento, cada gesto de cuidado, cada renúncia silenciosa para que eu pudesse estudar e seguir meus sonhos foram fundamentais ao longo dessa jornada. Em especial, à minha mãe, Regina, que desde cedo me mostrou o valor da educação na construção de uma vida justa, íntegra e de sucesso. Aos amigos que, de alguma forma, contribuíram para a concretização deste trabalho, em especial Alex (Japa), Beatriz (Be), Bruno Belorte, Bruno GoisToso, Enrico, Ezequiel, Felipe Cruz, Giovana, Gustavo (Bombinha), Ismael, Izabella, José Vanterler (Animal), Karina, Khetlen, Maria, Mayanna (May), Mayk, Miguel (Miguelzinho), Milena (Mi), Nathalia, Patrik GoisToso, Paulo (Paulão), Ricardo, Rodiak, Roberto Morales, Sorrana, Vinicius (Koba), Wendy e Yino, levarei cada um em meu coração por toda a vida. Aos professores do Departamento de Ecuaciones Diferenciales y Análisis Numérico da Universidad de Sevilla, pelo acolhimento caloroso durante os dois períodos de doutorado sanduíche que realizei nessa instituição. Em especial, ao Professor Cristian Morales-Rodrigo, pelas inúmeras colaborações presentes nesta tese e por toda a ajuda prestada em diversos momentos. Ao Professor Pedro Rubio, pela generosidade ao compartilhar, sempre com muita alegria, as histórias e curiosidades sobre a cidade de Sevilla. Ao meu orientador no Brasil, Professor Marcos Pimenta, pela excelente orientação desde a graduação, passando pelo mestrado até o doutorado. Agradeço por toda a disposição, paciência e valiosas sugestões que contribuíram imensamente para minha formação acadêmica e para a elaboração desta tese. Sua generosidade intelectual, sua clareza nas explicações e sua constante busca pela excelência foram exemplos que levarei para a vida. Sou também profundamente grato pelos inúmeros conselhos pessoais que me deu ao longo dos anos — conselhos que ultrapassaram os muros da academia e me ajudaram a crescer como ser humano, a tomar decisões mais conscientes e a enfrentar com mais sabedoria os desafios da vida. Por fim, agradeço pelo apoio constante, que tornou possível a realização de dois períodos de doutorado sanduíche na Universidad de Sevilla, viabilizando a presente tese em regime de cotutela. Ao meu orientador na Espanha, Professor Antonio Suárez, por todo o conhecimento transmitido durante minhas experiências em Sevilla. Agradeço, especialmente, pela paciência e dedicação com as quais me guiou, mesmo diante das minhas dificuldades, permitindo que juntos alcançássemos importantes resultados aqui apresentados. Sou também grato por ter me apresentado parte da cultura espanhola, em especial o amor que a população de Sevilla tem por suas tradições e pelo futebol. Por fim, agradeço à Universidade Estadual Paulista (UNESP), instituição pública e de excelência, por ser a base da minha formação acadêmica e por despertar em mim o compromisso com a educação de qualidade. À Universidad de Sevilla (US), pela parceria e apoio durante minha formação internacional, que ampliaram minha visão científica e cultural. Ambas as universidades foram fundamentais para a construção desta tese. O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001, à qual agradeço. AGRADECIMIENTOS Agradezco, en primer lugar, a la Santísima Trinidad, que me concedió sabiduría y discernimiento para tomar decisiones acertadas y alcanzar esta conquista. También estoy agradecido por las personas adecuadas que fueron puestas a mi alrededor a lo largo del camino, que me apoyaron y orientaron en los momentos decisivos. A mi familia, por el apoyo, el sustento y el aliento que fueron esenciales y determinantes para esta realización. Cada palabra de ánimo, cada gesto de cuidado, cada renuncia silenciosa para que yo pudiera estudiar y seguir mis sueños fueron fundamentales a lo largo de este recorrido. En especial, a mi madre, Regina, quien desde temprano me mostró el valor de la educación en la construcción de una vida justa, íntegra y exitosa. A los amigos que, de alguna manera, contribuyeron a la concreción de este trabajo, en especial Alex (Japa), Beatriz (Be), Bruno Belorte, Bruno GoisToso, Enrico, Ezequiel, Felipe Cruz, Giovana, Gustavo (Bombinha), Ismael, Izabella, José Vanterler (Animal), Karina, Khetlen, Maria, Mayanna (May), Mayk, Miguel (Miguelzinho), Milena (Mi), Nathalia, Patrik GoisToso, Paulo (Paulão), Ricardo, Rodiak, Roberto Morales, Sorrana, Vinicius (Koba), Wendy y Yino. Os llevaré a todos en mi corazón por el resto de mi vida. A los profesores del Departamento de Ecuaciones Diferenciales y Análisis Numérico de la Universidad de Sevilla, por la cálida acogida durante los dos períodos de doctorado en modalidad de cotutela que realicé en esta institución. En especial, al Profesor Cristian Morales-Rodrigo, por las numerosas colaboraciones presentes en esta tesis y por toda la ayuda brindada en diversos momentos. Al Profesor Pedro Rubio, por su generosidad al compartir, siempre con mucha alegría, historias y curiosidades sobre la ciudad de Sevilla. A mi director de tesis en Brasil, el Profesor Marcos Pimenta, por su excelente orientación desde la licenciatura, pasando por el máster y hasta el doctorado. Le agradezco por su disposición, paciencia y valiosas sugerencias que contribuyeron inmensamente a mi formación académica y a la elaboración de esta tesis. Su generosidad intelectual, su claridad en las explicaciones y su constante búsqueda de la excelencia han sido ejemplos que llevaré para toda la vida. También estoy profundamente agradecido por los numerosos consejos personales que me dio a lo largo de los años —consejos que trascendieron los muros de la academia y me ayudaron a crecer como ser humano, a tomar decisiones más conscientes y a enfrentar los desafíos de la vida con más sabiduría. Por último, le agradezco por su apoyo constante, que hizo posible la realización de dos estancias de investigación en la Universidad de Sevilla, lo que permitió el desarrollo de esta tesis en régimen de cotutela. A mi director de tesis en España, el Profesor Antonio Suárez, por todo el conocimiento transmitido durante mis experiencias en Sevilla. Agradezco especialmente la paciencia y dedicación con las que me guió, incluso ante mis dificultades, permitiendo que juntos alcanzáramos importantes resultados aquí presentados. También le agradezco por haberme presentado parte de la cultura española, especialmente el amor que el pueblo sevillano tiene por sus tradiciones y por el fútbol. Por último, agradezco a la Universidade Estadual Paulista (UNESP), institución pública y de excelencia, por haber sido la base de mi formación académica y por despertar en mí el compromiso con una educación de calidad. A la Universidad de Sevilla (US), por la colaboración y el apoyo durante mi formación internacional, que ampliaron mi visión científica y cultural. Ambas universidades fueron fundamentales en la construcción de esta tesis. El presente trabajo fue realizado con el apoyo de la Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiación 001, a la cual agradezco. ACKNOWLEDGEMENT First and foremost, I thank the Holy Trinity, who granted me wisdom and discernment to make the right decisions and achieve this milestone. I am also grateful for the right people who were placed around me along this journey—those who supported and guided me in decisive moments. To my family, for the support, care, and encouragement that were essential and decisive in making this accomplishment possible. Every word of encouragement, every act of care, every silent sacrifice so I could study and pursue my dreams played a fundamental role throughout this path. In particular, to my mother, Regina, who from an early age taught me the value of education in building a life that is just, honest, and successful. To the friends who, in one way or another, contributed to the completion of this work, especially Alex (Japa), Beatriz (Be), Bruno Belorte, Bruno GoisToso, Enrico, Ezequiel, Felipe Cruz, Giovana, Gustavo (Bombinha), Ismael, Izabella, José Vanterler (Animal), Karina, Khetlen, Maria, Mayanna (May), Mayk, Miguel (Miguelzinho), Milena (Mi), Nathalia, Patrik GoisToso, Paulo (Paulão), Ricardo, Rodiak, Roberto Morales, Sorrana, Vinicius (Koba), Wendy, and Yino. I will carry each of you in my heart for the rest of my life. To the professors of the Departamento de Ecuaciones Diferenciales y Análisis Numérico at the Universidad de Sevilla, for their warm welcome during the two research periods I spent at the institution as part of my cotutelle PhD. In particular, to Professor Cristian Morales-Rodrigo, for the many collaborations included in this thesis and for all the help offered at various moments. To Professor Pedro Rubio, for his generosity in joyfully sharing stories and curiosities about the city of Sevilla. To my advisor in Brazil, Professor Marcos Pimenta, for his excellent guidance since my undergraduate studies, through my master’s and into my PhD. I am deeply thankful for his availability, patience, and valuable suggestions, which contributed immensely to my academic growth and to the development of this thesis. His intellectual generosity, clarity of explanation, and constant pursuit of excellence have been examples I will carry throughout my life. I am also sincerely grateful for the many personal pieces of advice he has given me over the years—advice that went beyond the academic realm and helped me grow as a person, make more conscious decisions, and face life’s challenges with greater wisdom. Finally, I thank him for the unwavering support that made it possible for me to carry out two research periods at the University of Sevilla, which were essential for the development of this cotutelle thesis. To my advisor in Spain, Professor Antonio Suárez, for all the knowledge shared during my time in Sevilla. I am especially thankful for the patience and dedication with which he guided me, even when I faced difficulties, allowing us to achieve important results presented here. I am also grateful for introducing me to aspects of Spanish culture, especially the deep love the people of Sevilla have for their traditions and for football. Lastly, I thank Universidade Estadual Paulista (UNESP), a public and excellent institution, for being the foundation of my academic formation and for awakening in me a strong commitment to quality education. And to the Universidad de Sevilla (US), for their partnership and support during my international academic training, which broadened my scientific and cultural perspective. Both institutions were fundamental to the development of this thesis. This work was carried out with the support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Financing Code 001, to which I thank. “Every second is time to change everything forever.” Charlie Chaplin RESUMO Este trabalho investiga a existência e unicidade de estados de coexistência em sistemas elípticos não-locais que modelam interações entre duas espécies, com difusão dependente da população de outra espécie ou da própria. O problema geral analisado é representado por: −m (∫ Ω u, ∫ Ω v ) ∆u = f(x, u, v) em Ω, −n (∫ Ω u, ∫ Ω v ) ∆v = g(x, u, v) em Ω, u = v = 0 sobre ∂Ω, (P) onde Ω é um domínio regular limitado em RN , com N ≥ 1, m,n : R2 + → [0,+∞) são funções contínuas não-lineares e f, g : Ω × R2 → R funções contínuas. No primeiro modelo, analisamos a interação entre uma bactéria e um nutriente, com difusão não-linear. Usamos Bifurcação Local e Global e o Teorema da Função Implícita para determinar condições de existência e unicidade de soluções positivas. No segundo modelo, estudamos sistemas de Lotka-Volterra com difusão cruzada não- local, modelando interações de competição, predador-presa e simbiose. Investigamos a estabilidade de soluções semi-triviais e a coexistência. No terceiro modelo, abordamos a competição de Lotka-Volterra com difusão não-local, onde a difusão depende da população de cada espécie. Garantimos a coexistência com o princípio da exclusão competitiva. Os resultados destacam a importância da difusão não-local na modelagem de interações biológicas e na dinâmica de coexistência. PALAVRAS-CHAVE: dinâmica de populações; sistemas elípticos não-locais; bifurcação local e global; índice de ponto fixo; método de sub-supersolução. RESUMEN Este trabajo investiga la existencia y unicidad de estados de coexistencia en sistemas elípticos no locales que modelan interacciones entre dos especies, con difusión dependiente de la población de otra especie o de la misma. El problema general estudiado está representado por: −m (∫ Ω u, ∫ Ω v ) ∆u = f(x, u, v) en Ω, −n (∫ Ω u, ∫ Ω v ) ∆v = g(x, u, v) en Ω, u = v = 0 sobre ∂Ω, (P) donde Ω es un dominio regular acotado en RN , con N ≥ 1, m,n : R2 + → [0,+∞) son funciones no lineales continuas, y f, g : Ω × R2 → R son funciones continuas. En el primer modelo, analizamos la interacción entre una bacteria y un nutriente con difusión no lineal. Utilizamos bifurcación local y global y el teorema de la función implícita para determinar las condiciones de existencia y unicidad de soluciones positivas. En el segundo modelo, estudiamos sistemas de Lotka-Volterra con difusión cruzada no local, modelando interacciones de competencia, depredador-presa y simbiosis. Investigamos la estabilidad de soluciones semi-triviales y la coexistencia. En el tercer modelo, abordamos la competencia de Lotka-Volterra con difusión no local, donde la difusión depende de la población de cada especie. Garantizamos la coexistencia con el principio de exclusión competitiva. Los resultados destacan la importancia de la difusión no local en la modelización de interacciones biológicas y en la dinámica de la coexistencia. PALABRAS CLAVE: dinámica de poblaciones; sistemas elípticos no-locales; bifurcación local y global; índice de punto fijo; método de sub-supersoluciones. ABSTRACT This work investigates the existence and uniqueness of coexistence states in non-local elliptic systems that model interactions between two species, with diffusion dependent on the population of another species or on the same species. The general problem studied is represented by: −m (∫ Ω u, ∫ Ω v ) ∆u = f(x, u, v) in Ω, −n (∫ Ω u, ∫ Ω v ) ∆v = g(x, u, v) in Ω, u = v = 0 on ∂Ω, (P) where Ω is a regular bounded domain in RN , with N ≥ 1, m,n : R2 + → [0,+∞) are continuous nonlinear functions, and f, g : Ω × R2 → R are continuous functions. In the first model, we analyze the interaction between a bacterium and a nutrient with nonlinear diffusion. We use Local and Global Bifurcation and the Implicit Function Theorem to determine conditions for the existence and uniqueness of positive solutions. In the second model, we study Lotka-Volterra systems with non-local cross-diffusion, modeling competition, predator-prey and symbiosis interactions. We investigate the stability of semi-trivial solutions and coexistence. In the third model, we address Lotka-Volterra competition with non-local diffusion, where diffusion depends on the population of each species. We guarantee coexistence using the competitive exclusion principle. The results highlight the importance of non-local diffusion in modeling biological interactions and coexistence dynamics. KEYWORDS: population dynamics; non-local elliptic systems; local y global bifurcation; fixed-point index; sub-supersolution method. List of Figures 1 BEHAVIOR OF THE CONTINUUM OF SEMI-TRIVIAL SOLUTION C IN THE COMPETITION CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 BEHAVIOR OF THE CONTINUUM OF SEMI-TRIVIAL SOLUTION C IN THE PREDATOR-PREY CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 BEHAVIOR OF THE CONTINUUM OF SEMI-TRIVIAL SOLUTION C IN THE SYMBIOTIC CASES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 BIFURCATION DIAGRAMS OF PROBLEM (10) . . . . . . . . . . . . . . . . . . . . . 28 5 THE COEXISTENCE REGION OF PROBLEM (12) IN THE COMPETITION CASE. IN THIS CASE, WE ASSUME THAT m AND n VERIFY THE ASSERTION (A) OF THEOREM 6, WHICH ENSURES CERTAIN PROPERTIES OF THE SEMI-TRIVIAL SOLUTIONS. ADDITIONALLY, WE ASSUME THAT n′(0) < 0, IMPLYING THAT F ′(m(0)λ1) < 0. UNDER THESE CONDITIONS, IT FOLLOWS THAT A COEXISTENCE STATE MAY OCCUR EVEN FOR VALUES OF µ SATISFYING µ < n(0)λ1. 28 6 COEXISTENCE REGION OF PROBLEM (12) IN THE PREY- PREDATOR CASE, CORRESPONDING TO THE SCENARIO ANALYZED UNDER SPECIFIC ASSUMPTIONS FOR THE INTERACTION COEFFICIENTS. . . . . . . . . . . . . . . . 28 7 COEXISTENCE REGION OF PROBLEM (12) IN THE SYMBIOSIS CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 8 THE COEXISTENCE REGION OF PROBLEM (14) DEFINED BY CONDITION (16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 9 BEHAVIOR OF THE CONTINUUM OF SEMI-TRIVIAL SOLUTION C(µ,θ,0) WHEN m IS INCREASING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 10 BEHAVIOR OF THE CONTINUUM OF SEMI-TRIVIAL SOLUTION C(µ,θ,0) WHEN m IS INCREASING, λ ≈ m(0)λ1 AND n′(0) < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 11 BIFURCATION DIAGRAM WHEN m AND n BOTH INCREASE. IN THIS CASE, THERE EXISTS A UNIQUE SEMI-TRIVIAL SOLUTION (uλ, 0) AND (0, vµ). . . . 28 12 THE COEXISTENCE REGION OF PROBLEM (14) DEFINED BY CONDITIONS (16) AND (19). IN THIS CASE, THE REGION DEFINED BY CONDITION (19) IS LARGER THAN THE ONE DEFINED BY (16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 1.1 BIFURCATION DIAGRAM. IN THIS CASE, γ0 BIFURCATES FROM THE TRIVIAL SOLUTION, WHEREAS γ1 AND γ2 BIFURCATE FROM NON-TRIVIAL SOLUTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2 GEOMETRIC INTERPRETATION OF THE DEFINITION 1.43. . . . . . .28 1.3 GEOMETRIC INTERPRETATION OF THE THEOREM 1.46. . . . . . . . . 28 1.4 GEOMETRIC INTERPRETATION OF THE THEOREM 1.49. . . . . . . . . 28 2.1 BIFURCATION DIAGRAM OF PROBLEM (P1). . . . . . . . . . . . . . . . . . . . . . 28 3.1 THE COEXISTENCE REGION OF PROBLEM (P2) IN THE COMPETITION CASE. IN THIS CASE, WE ASSUME THAT m AND n VERIFIES THE CONDITION (H+). ADDITIONALLY, IT IS ASSUMED THAT n′(0) < 0 SUCH THAT F ′(m(0)λ1) < 0. UNDER THESE CONDITIONS, IT IS NOTABLE THAT A COEXISTENCE STATE EXISTS FOR µ < n(0)λ1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 COEXISTENCE REGION OF PROBLEM (P2) IN THE PREY- PREDATOR CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 COEXISTENCE REGION OF PROBLEM (P2) IN THE SYMBIOSIS CASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 THE COEXISTENCE REGION OF (3.26). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 REGION DEFINED BY CONDITION (4.14). IF (λ, µ) ∈ R, THEN PROBLEM (P3) POSSESSES AT LEAST A COEXISTENCE STATE. . . . . . . . . . . . . . . . . . . . . 28 4.2 POSSIBLE BIFURCATION DIAGRAMS IN THE CASE OF TWO SEMITRIVIAL SOLUTIONS (u, 0) AND ONE SEMTRIVIAL SOLUTION (0, v). . . . . . . . . . 29 4.3 BEHAVIOUR OF THE GLOBAL CONTINUUM C(µ,θ,0). . . . . . . . . . . . . . . . 29 4.4 A POSSIBLE COEXISTENCE REGION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 BIFURCATION DIAGRAM FOR λ ≈ m(0)λ1, cd < 1 AND n′(0) < 0. IN THIS CASE, THE BIFURCATION IS SUBCRITICAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6 BEHAVIOR OF THE CONTINUUM FOR m AND n INCREASE. . . . . . 29 4.7 COMPARISON OF THE COEXISTENCE REGIONS DEFINED BY (4.14) AND (4.36). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Contents INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1 BASIC CONCEPTS AND DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . .25 1.2 THE MAXIMUM PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3 EIGENVALUE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 SUB-SUPERSOLUTION METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 BIFURCATIONS METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 1.6 FIXED POINT INDEX THEORY WITH RESPECT TO THE POSITIVE CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 ANALYSIS OF A LOGISTIC PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . .25 2 POPULATION DYNAMICS BETWEEN BACTERIA AND A LIVING NUTRIENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 A PRIORI BOUNDS AND NON-EXISTENCE RESULTS OF COEXISTENCE STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 LOCAL BIFURCATION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 GLOBAL BIFURCATION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 UNIQUENESS OF COEXISTENCE STATE . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 COEXISTENCE STATE FOR THE CASE m(0) = 0 . . . . . . . . . . . . . . . .25 3 MODELING POPULATION DYNAMICS IN LOTKA-VOLTERRA SYSTEMS WITH NONLOCAL CROSS-DIFFUSIVITY TERMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 A PRIORI BOUNDS AND NON-EXISTENCE RESULTS OF COEXISTENCE STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 CURVES OF CHANGE OF STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 FIXED POINT INDEX ANALYSIS WITH RESPECT TO THE POSITIVE CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 STUDY OF THE COEXISTENCE STATES REGION . . . . . . . . . . . . . . 25 3.5 COMPARISON OF THE PROPOSED MODEL AND THE LOCAL LOTKA-VOLTERRA SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 MODELING POPULATION DYNAMICS IN LOTKA- VOLTERRA SYSTEMS WITH NONLOCAL COEFFICIENT DIFFUSION . . . 25 4.1 A PRIORI BOUNDS AND NON-EXISTENCE RESULTS OF COEXISTENCE STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 SUB-SUPERSOLUTION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 LOCAL BIFURCATION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 GLOBAL BIFURCATION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 CONCLUSIONS / CONCLUSIONES / CONCLUSÕES . . . . . . 25 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 etoolbox Introduction In the study of Population Dynamics, the focus lies on understanding the complex interactions among one or more populations of living organisms over time and across domains. This field combines mathematical modeling, theoretical analysis, and empirical observations to investigate how populations change in size, structure, and distribution. Such changes are driven by various biological, environmental, and ecological factors, including birth and death rates, migration patterns, competition for resources, and predation. For example, in marine ecosystems, elevated mortality rates among orcas, a predator, can disrupt the balance of the food chain, leading to an overpopulation of their prey, such as fish and other marine species. For this study, it is essential to understand some fundamental concepts, starting with population density, which is defined as the number of living organisms within a population per unit of area. In other words, it represents the size of the population in relation to a specific spatial domain. In addition to population density, several other factors play a pivotal role in shaping population dynamics. These include birth and death rates, which dictate the growth or decline of populations; migration patterns, encompassing both immigration and emigration, which influence the redistribution of populations across regions; and interactions between species, such as competition, predation, and cooperation, which drive ecological balance. External influences like environmental changes, availability of resources, and human interventions further add layers of complexity to these studies. In general, birth and immigration rates contribute to an increase in population density by adding new individuals to the population within a given area. Conversely, death and emigration rates lead to a decrease in population density by reducing the number of individuals present. These opposing processes form the foundation of population dynamics, creating a delicate balance that determines whether a population grows, shrinks, or remains stable over time. In this context, Mathematical Modeling aims to describe and predict the behavior of one or more populations of living organisms, chemical substances, or viruses within a specific location. By employing mathematical tools and frameworks, these models seek to explain the growth or decline of population density through a detailed analysis of the factors involved. These factors can be classified as internal, such as interactions between species, including competition, predation, and cooperation, or external, such as environmental changes, resource availability, and anthropogenic influences. Mathematical models provide a systematic way to capture the complexity of population dynamics, enabling researchers to simulate various scenarios and identify critical thresholds or tipping points. This approach not only enhances our understanding 26 Introduction 27 of population behavior but also supports practical applications, such as conservation strategies, resource management, and policy-making in the context of ecological sustainability. In 2003, J. D. Murray (see [54]) published a book with an extensive study of biological problems modeled mathematically, among which we highlight two. The first, proposed by Malthus in 1798, is a classical formulation of continuous population growth used to describe the dynamics of a single population. This model, which assumes that population growth, is described by the following differential equation: dN dt = bN − dN = (b− d)N, (1) where N(t) represents the population density as a function of time, b is the birth rate, and d is the death rate. The solution to this equation, given by: N(t) = N0e (b−d)t, where N0 is the initial population, indicates exponential growth or decline of the population depending on b and d, as follows: • The population grows exponentially when b > d; • The population declines exponentially when b < d; and • The population remains constant over time when b = d. This model is considered simplistic for describing real populations since it does not account for environmental limitations or external interactions, such as resource competition or predation. However, its importance lies in introducing the fundamental principles of population dynamics and inspiring the development of more realistic models. The second, proposed by P. F. Verhulst in 1836, is a more realistic generalization of population growth compared to Malthus exponential model. It incorporates environmental resource limitations and intraspecific competition, being described by the following differential equation: dN dt = rN ( 1 − N K ) , (2) whereN(t) represents the population density as a function of time, r is the intrinsic growth rate of the population, and K is the carrying capacity of the environment, representing the maximum number of individuals that the environment can sustain. This model predicts that population growth is initially rapid but slows as the population approaches the carrying capacity K. The explicit solution to this equation is given by: N(t) = N0Ke rt K +N0(ert − 1) . Also known as the Logistic Model, it is widely used to describe natural populations subject to environmental constraints and to study density-dependent regulatory mechanisms. Despite its limitations, the model provides an essential foundation for Introduction 28 understanding more complex biological systems and is frequently used as a starting point for incorporating additional factors, such as predation, migration, and interspecies interactions. In 2004, R. S. Cantrell and C. Cosner (see [12]) published a comprehensive book that presents a unified perspective of spatial ecology through Reaction-Diffusion Model, analyzing how biological interactions and movement processes manifest in heterogeneous environments. The work spans from classical nonspatial models—such as the Lotka- Volterra systems that describe basic population dynamics—to more sophisticated mathematical formulations of spatially explicit models. This theoretical framework not only bridges ecological scales but also provides powerful tools to study patterns of species coexistence and segregation. Among these problems, we highlight the Logistic Problem with a local diffusion term, given by:  −α∆w = γw − w2 in Ω, w = 0 on ∂Ω, (3) where Ω is a bounded regular domain in RN , N ≥ 1, γ ∈ R, α > 0 and ∆w := N∑ k=1 ∂2w ∂x2 k denotes the Laplacian Operator. This problem arises from the analysis of the steady-state behavior of the following parabolic problem:  ∂w ∂t − α∆w = γw − w2 in Ω × (0,+∞), w = 0 on ∂Ω × (0,+∞), w(x, 0) = w0(x) in Ω, where w = w(x, t) represents, for example, the density of a population over time, and w0(x) is the initial condition, typically assumed to be nonnegative and nontrivial. For a detailed study of this model and its dynamical properties, see, for instance, [12]. Problem (3), which is a generalization of the Equation (2), is a semilinear elliptic problem known for combining a diffusion term, represented by the Laplacian operator −α∆w, and the reaction term with logistic growth, γw −w2. The parameter α serves as a diffusion coefficient, directly influencing the spatial dynamics of the solutions. Higher values of α enhance the diffusive effect, leading to a more uniform spatial distribution of the variable w throughout the domain. Moreover, it is frequently used to model biological phenomena, such as population dynamics with logistic growth, subject to limitations imposed by finite resources. The domain Ω represents the region where the solution is studied, and the boundary condition ∂Ω imposes that the population density is zero at the boundaries of the domain, simulating a confined environment. Moreover, it is important as it allows for detailed analysis regarding the existence, uniqueness, and stability of solutions, as well as critical behaviors such as bifurcations and transitions between steady states. It serves as a basis for more complex investigations involving nonlocal terms, multi-species interactions, or heterogeneous environments. Introduction 29 In 1989, J. Furter and M. Grinfeld (see [40]) examined several biological models involving a single species, in which they included and deepened the importance of nonlocal interactions, such as competition for shared resources. The nonlocal term, addressed in the text, plays a central role in extending the traditional modeling based on reaction-diffusion equations, which typically consider only local interactions. Its introduction makes it possible to represent situations where interactions between populations are not restricted to immediate proximity, such as in resource competition. This approach adds realism to the models, making it possible to explore more complex spatial patterns. Moreover, the nonlocal term enables the study of novel dynamics, such as the formation of stable patterns in contexts where this would be unlikely without these effects. These characteristics make the models more robust and relevant for practical applications, particularly in ecology and biology, where large-scale phenomena are essential to describe the organization and evolution of populations. The text further emphasizes the importance of studying the conditions that ensure the stability of these patterns, highlighting the critical role of nonlocal interactions in understanding more complex natural systems. The inclusion of a nonlocal term in the diffusion coefficient deserves special attention. In general, the diffusion rate is modeled by: v⃗ = −α∇w, and this modeling choice results in a partial differential equation that takes the following general form:  −div (α∇w) = f(x,w) in Ω, w = 0 on ∂Ω. The term f(x,w) plays a fundamental role in describing the local dynamics of the quantity w. Depending on the context, this term can model various biological, chemical, or physical processes. For instance, in population dynamics, f(x,w) may describe the growth and interaction of a population at a given location x ∈ Ω, incorporating effects such as spatially varying birth and death rates, as well as intraspecific and interspecific competitive interactions. Moreover, depending on the nature of the diffusion coefficient α, we can identify three distinct modeling frameworks, each capturing different ecological assumptions and leading to qualitatively different mathematical structures: (a) When α is a positive constant, we obtain the classical Laplace operator: −div (α∇w) = −α∆w. In this case, α represents the diffusion rate, which remains constant throughout the entire habitat Ω of the species w. (b) When α depends on the population density w, a nonlinearity arises in the diffusive term: −div (α∇w) = −div (α(w)∇w) . In this case, the diffusion coefficient α(w) varies with the population density w. This model captures situations where the dispersal rate of individuals changes according to local density: Introduction 30 – If α is increasing: It indicates that diffusion is more intense in high-density regions, suggesting that individuals tend to spread out more when competition for resources is stronger. – If α is decreasing: It implies reduced diffusion in densely populated areas, possibly due to social behavior or the benefits of aggregation. (c) In contexts where pointwise measurements are impractical, an alternative approach is to consider nonlocal dependencies. For example, replacing α by a function of the spatial average of the population over a region B(x, r), that is: α (∫ B(x,r) w(y) ) . This leads to equations with nonlocal diffusion, such as: −div (α∇w) = −div [ α (∫ Ω K(x, y)w(y) ) ∇w ] . In this case, the diffusion coefficient is influenced by a weighted average of the population density around the point x, determined by the Kernel Function K(x, y). The kernel K : Ω × Ω → R is a nonnegative, measurable function that describes the influence of the point y on the location x. Typically, K(x, y) is chosen to decay as the distance between x and y increases, reflecting the idea that individuals at nearby locations have a stronger impact on the movement at point x than those farther away. This nonlocal term captures long-range interactions, where the movement at location x is affected by the population density at surrounding points y, thus extending the classical diffusion model to incorporate spatial memory or nonlocal perception effects. Such formulations have been studied, for example, in [55], where diffusion depends on the range of interactions, and in [3, 29], which explore intermediate local-nonlocal elliptic problems. Later, M. Chipot and collaborators investigated the role of nonlocal terms in elliptic problems arising in population dynamics. Among the nonlocal terms considered, one particularly notable example involves dependence on the total population within a subdomain, that is: α = α (∫ Ω′ w(x) dx ) , where Ω′ ⊆ Ω denotes a subdomain, which may represent, for example, a specific area of interest where the population density is being monitored or where the environmental feedback is concentrated. This formulation is especially relevant for modeling ecological phenomena such as: • Aggregation-driven dispersal: If α decreases with population density, it describes species that tend to avoid overcrowded areas. • Population attraction: If α increases with density, it models species that are drawn to regions of high population concentration. In 1992, M. Chipot and J. F. Rodrigues (see [17]) studied a class of elliptic problems with nonlocal terms, addressing both theoretical and applied aspects. These problems Introduction 31 are characterized by the global dependence of the solution, which distinguishes them from traditional local problems. More specifically, the authors considered the following problem:  −a (∫ Ω u ) ∆u+ λu = f in Ω, ∂nu+ γ (∫ Ω′ u ) = 0 on ∂Ω, (4) where λ > 0, f represents the supply of beings by external sources, a a positive factor that depends on the total population, γ a positive factor that represents the influence of the population in Ω′, and ∂nu denotes the normal derivative of u on the boundary ∂Ω, that is, the derivative of u in the direction of the outward unit normal vector to the boundary. Problem (4) describes the behavior of a bacteria with density u within a container Ω, considering population dispersion, an external source f , and a mortality rate λ, which is proportional to the density u. The population dispersion is represented by a term that depends on the gradient of the density ∇u, with a factor a that varies according to the total population within Ω, that is, a (∫ Ω u ) . Additionally, the population is expelled from the container through a total flux at the boundary ∂Ω, which depends both on the total population within Ω and on a dominant group within a subdomain Ω′. The complete problem is then described by two equations, where the first equation describes the population dynamics inside the container, and the second specifies the boundary flux conditions. The nonlocal term included in this problem is essential for modeling the global interaction within the system, where the population dynamics at a given point depend not only on local conditions but also on global characteristics, such as the total population within the container and the presence of a dominant group in a subdomain. In 1999, M. Chipot and B. Lovat (see [16]) present a problem to illustrate the study of nonlinear diffusion in the nonlocal case. This problem aims to model population density, where the diffusion rate depends on the integral over a certain region of the domain. More specifically, the authors consider the following problem: ut − a (l(u(·, t))) ∆u = f in Ω × (0, T ), u(·, t) ∈ V for t ∈ (0, T ), u(x, 0) = u0(x) in Ω, (5) where V a subspace of H1 (Ω) that accounts for the boundary conditions of problem, u0 the initial condition, T > 0 some fixed time, and f = f(x) a source term, which may represent, for example, the population growth rate. Moreover, the continuous linear form l is defined as the integral of the product of L2 with a function g ∈ L2 (Ω), that is: l(u) = lg(u) = ∫ Ω′ g(x)u(x), where Ω′ is some subdomain of Ω. Problem (5) models population dynamics through a nonlocal diffusion process. The term ∆u describes standard dispersal from high to low density regions, while f accounts for births, deaths, and migration. The key feature is the density-dependent coefficient a(l(u)) which represents a weighted average population in subregion Ω′. This formulation Introduction 32 captures faster diffusion in crowded areas (when a increasing), and attraction to populated zones (when a decreasing). Biologically, this model can be applied to the study of bacteria moving in search of nutrients in a homogeneous medium, animals migrating between habitats based on global perception of density, or even human populations responding to social and economic stimuli. Thus, the model not only incorporates the local and global interactions of the population but also allows us to explore how these processes influence the spatial and temporal distribution of population density, with significant implications for ecology, population biology, and natural resource management. In the course of this work, three methods will be crucial: the Sub-Supersolution Method, Bifurcation Theory, and Fixed Point Index Theory. We will present a brief overview of these techniques, highlighting key results from the literature and emphasizing their applicability. These methods will be systematically employed throughout the work to establish existence and uniqueness for the nonlocal elliptic systems under study. The first method, the Sub-SuperSolution technique, is used to prove the existence of solutions to differential equations, especially when these solutions are difficult to obtain directly due to nonlinearity or the complexity of boundary conditions. This method is particularly useful for studying elliptic equations involving nonlinear terms, such as equations with reaction terms, diffusion terms, and other nonlinear interactions. The central idea of this method is to use auxiliary functions, called subsolution and supersolution, to establish lower and upper bounds for the solutions of problem. These bounds help identify a solution that lies between these two values and, in many cases, can be used to guarantee the existence of a solution. One of the first authors to work with this method was G. Scorza Dragoni in 1931 (see [31]), who studied the existence of an ordered pair of solutions to a differential inequality in order to determine the existence of a solution to a boundary value problem for a nonlinear second-order differential equation. Later, many authors refined this method in the context of elliptic problems (see [43]) and presented an extensive study of the method in the context of elliptic and parabolic equations (see [56]). In 2016, Y. Baoqiang and M. Tianfu (see [7]) investigate existence and multiplicity results for positive solutions to an important class of nonlocal problems given by: −a (∫ Ω |u|γ dx ) ∆u = fλ(x, u) in Ω, u > 0 on ∂Ω, u = 0 on ∂Ω, (6) where γ ∈ (0,+∞), a : [0,+∞) → (0,+∞) is a continuous function with inf t∈[0,+∞) a(t) > 0 , and fλ : Ω × R → R is a nonlinearity that may depend both on the spatial position x ∈ Ω and on the solution u, possibly involving a control parameter λ. The authors emphasize that such problems naturally arise in various contexts within Applied Mathematics, particularly in ecological models describing the spatial distribution of populations subject to non-standard dispersal mechanisms. A notable special case occurs when γ = 2, in which the problem reduces to a generalized version of the well- known Carrier Equation, originally studied in the context of nonlinear vibration problems in solid mechanics. Introduction 33 The most prominent mathematical feature of this problem lies in the nonlocal nature of the main differential operator, represented by a (∫ Ω |u|γ ) ∆u. This nonlocality arises from the dependence of the diffusion coefficient on the Lγ-norm of the solution over the entire domain Ω. In particular, for γ > 1, densely populated regions exert greater influence on the diffusion process. Conversely, for 0 < γ < 1, or for γ sufficiently large or small, the influence of sparsely populated areas is amplified, enhancing the role of low-density regions in shaping the dynamics. To demonstrate the existence of at least one classical solution for Problem (6), the authors employed the Sub-Supersolution Method and ensured this existence when, for α, β ∈ C1(Ω) ∩ C2 (Ω) such that: (a) α is a subsolution and β is a supersolution of Problem (6), in the sense that they satisfy the differential inequality associated with the problem, with α lying below and β above the nonlinearity, and both vanish on the boundary (see the definition in Section 1.4); (b) α(x) ≤ β(x) for all x ∈ Ω; and (c) |fλ(x, u)| ≤ h(x) for all α(x) ≤ u ≤ β(x), with h ∈ Lp (Ω) and p > N . In the final part of the study, the authors showcase the versatility of this result by applying it to several classes of nonlinearities fλ(x, u). Sub-Supersolution Method faces significant limitations when applied to problems involving nonlocal terms. This is because, in nonlocal problems, the operators depend on integrals or global averages of the solution, which hinders the use of the method traditionally employed to construct solutions via monotone iteration. Furthermore, the lack of properties such as the classical maximum principle prevents the direct application of this method. The second method, which will be the most used throughout this work, is Bifurcation Theory. This method can be divided into two cases: local and global. Broadly speaking, this theory deals with the transition between different types of solutions as one or more parameters of the problem are varied. This is particularly important in nonlinear problems, where parameter changes can lead to the emergence of new solutions, changes in the number of existing solutions, or even the loss of stable solutions. In the context of biological models, bifurcation theory is especially relevant for studying positive solutions, which correspond to meaningful population densities or concentrations. Understanding how positive solutions arise and evolve as parameters vary allows us to analyze critical biological phenomena such as species coexistence, extinction thresholds, and pattern formation in ecosystems. In 1971, M. G. Crandall and P. H. Rabinowitz (see [23]) introduced a detailed investigation into Bifurcation Theory, extending it to a general context with the primary goal of determining the structure of the zero set of an equation G(w) = 0 in the neighborhood of a point on a known curve of zeros. Focusing on bifurcations at simple eigenvalues, a topic widely studied in the literature, the authors consolidated existing results and expanded the applicability of the theory. The main result of the paper, Theorem 1, is frequently employed in the context of Local Bifurcation and is widely known as the Crandall-Rabinowitz Theorem. This result establishes precise conditions for classifying a point on a zero curve as a bifurcation Introduction 34 point. Based on the properties of the derivatives of G, the theorem ensures that, in a neighborhood of this point, the set of solutions forms two continuous curves that intersect uniquely at the bifurcation point. Although there were earlier results, the main result of this paper has become the most applicable in the context of Local Bifurcation due to its generality. Also in 1971, P. H. Rabinowitz (see [58]) extended the results previously published with M. G. Crandall by addressing the global behavior of the solution curve obtained through the Crandall-Rabinowitz Theorem. Focusing on the existence of a continuum, that is, a closed and connected sets of solutions, the author demonstrated that this continuum extends globally from an eigenvalue of odd multiplicity. This result indicates that bifurcation from eigenvalues with odd multiplicity is not merely a local phenomenon but has a global nature. The main result of this article, Theorem 1.3, is frequently employed in the context of Global Bifurcation and is widely known as the Rabinowitz Theorem. This result established the existence of a global continuum of solutions for nonlinear eigenvalue problems in Banach spaces, originating from an eigenvalue of odd multiplicity. Specifically, it asserts that if an eigenvalue µ has odd multiplicity, there exists a maximal subcontinuum, denoted by Cµ, within the solution set, such that (µ, 0) ∈ Cµ. Moreover, this continuum satisfies one of the following two alternatives: • Cµ is an unbounded continuum in R×E , meaning it is not contained in any bounded subset of R × E ; or • Cµ intersects (λ, 0), where λ is another eigenvalue distinct from µ. This result, as previously mentioned, extends the theory of Local Bifurcation by demonstrating that bifurcation from eigenvalues of odd multiplicity is a global phenomenon. The proof of Theorem 1.3 employs the Leray-Schauder degree and the topological analysis of the solution set. In 2018, T. S. Figueiredo-Sousa, C. Rodrigo-Morales and A. Suárez (see [39]) studied a logistic equation with a nonlocal diffusion coefficient, modeling the dynamics of a population within a bounded domain. Through bifurcation methods (local and global) and fixed-point arguments, they determined conditions for the existence of positive solutions and analyzed the global behavior of the solutions depending on the nonlocal diffusion function and parameters λ. The model proposed by the authors is as follows: −a (∫ Ω q(x)up ) ∆u = λu− b(x)u2 in Ω, u = 0 on ∂Ω, (7) where p > 0, λ ∈ R, a ∈ C(R) a positive functions, b ∈ C1(Ω) a nonnegative and nontrivial function, and q(x) a bounded, nonnegative and nontrivial function in Ω. In Problem (7), the Local Bifurcation Method was employed to investigate the existence and behavior of positive solutions. The central idea is to explore the bifurcation of positive solutions from the trivial solution u ≡ 0 when the parameter λ reaches certain critical values. Initially, the nonlocal logistic equation is reformulated into a suitable form for bifurcation analysis, using compact and well-defined operators within the relevant Banach space. This Problem is expressed as an equation of the form: F(λ, u) = a (∫ Ω q(x)up ) ∆u+ λu− b(x)u2. Introduction 35 The analysis of this operator reveals that bifurcation occurs at (a(0)λ1, 0). Here, λ1 denotes the principal eigenvalue of the Dirichlet Laplacian, that is, the unique real number λ for which there exists a nontrivial function φ ∈ H1 0 (Ω), φ > 0 in Ω, satisfying: −∆u = λφ in Ω, φ = 0 on ∂Ω. To ensure the existence of a curve of nontrivial solutions emanating from these critical values, the authors apply the Crandall-Rabinowitz Theorem. This requires verifying the theorem conditions: that the linearized operator has a one-dimensional kernel and satisfies the transversality condition. Furthermore, the authors investigated the direction of bifurcation (supercritical or subcritical) by analyzing the sign of certain terms in the Taylor expansion of the nonlinear operator. They concluded that the bifurcation direction depends on the parameter p and the derivative of the diffusion coefficient a′(0). There is a wide range of problems where Bifurcation Theory can be employed to prove the existence of solutions. For further applications, we recommend consulting articles [5, 34, 37, 38] and their references. The last method we will use in this work is the Fixed Point Index Theory, which is widely employed to study the existence and multiplicity of solutions to nonlinear differential equations, as well as to identify the behavior of solutions as system parameters vary. An important aspect of this method is its connection to Bifurcation Theory: the change in the fixed point index can indicate the presence of a bifurcation. Specifically, the change in the index of an operator around a solution can sign the transition from a trivial solution to a nontrivial solution, or even the existence of multiple solutions from a single parameter value. Although this method had been previously used, for instance in [58], we will focus on two works. The first, published in 1976 by H. Amann (see [4]). In his research, he mainly studied results in cones with nonempty interior and demonstrated the existence of solutions for nonlinear equations under certain conditions. However, Amann conditions required that the cone have a nonempty interior, limiting the applicability of his results to specific domains. Later, in 1983, E. N. Dancer (see [25]), generalized this method and applied it to more general those ones, including cones with empty interior. This was a significant breakthrough because many domains used in differential equations and bifurcation problems have cones with empty interior. By applying fixed point index theory to positive cones, Dancer was able to extend Amann’s results to these more general situations, allowing for the analysis of bifurcations in broader contexts, such as elliptic problems in irregular domains and with mixed boundary conditions. In 2020, B. Yan and C. An (see [61]) investigate the existence of sign-changing solutions for a class of nonlocal elliptic problems posed on an annulus, specifically addressing the following problem:  −a (∫ Ω |u|γ ) ∆u = f(u) in Ω, u = 0 on ∂Ω, (8) where Ω = {x ∈ RN ; 0 < r1 < |x| < r2} is an annulus, γ ∈ (0,+∞), a ∈ C ([0,+∞), (0,+∞)), and f ∈ C ((0,+∞), (0,+∞)). Introduction 36 To establish the existence of sign-changing solutions for Problem A, the authors employ Fixed Point Index Theory. To this end, they transform the original problem on an annulus into a nonlocal boundary value problem for ordinary differential equations, in such a way that the solutions of Problem A correspond to the fixed points of a certain operator. Thus, finding solutions to the original problem reduces to identifying fixed points of the operator T . An important mathematical model in the context of population dynamics is the Lotka- Volterra system. Developed by A. J. Lotka in 1925 (see [51]) and V. Volterra in 1926 (see [60]), these systems consist of differential equations modeling the evolution of two or more interacting species, such as predators and prey, competitors, or cooperative populations. The Lotka-Volterra system generalizes the Logistic Equation, which models the growth of a single species considering environmental resource limitations. While the logistic equation includes a growth term limited by the environment’s carrying capacity, Lotka- Volterra systems extend this by incorporating interactions between multiple populations, such as predation, competition, and cooperation. This generalization enables the study of more complex and realistic scenarios in ecology and other fields. There are various ways to express the Lotka-Volterra system; however, for the purposes of this work, we will focus on the following stationary system: −∆u = λu− u2 − cuv in Ω, −∆v = µv − v2 − duv in Ω, u = v = 0 on ∂Ω, (9) where c, d, λ, µ ∈ R. From the population dynamics perspective, Problem (9) models the behavior of two species, u and v, inhabiting the habitat Ω. Since we are considering homogeneous Dirichlet boundary conditions, the habitat is entirely surrounded by inhospitable region, ∂Ω. The terms −∆u and −∆v describe the spatial movement of the species. In the reactions functions, λ and µ stand for the intrinsic growth rate of each species, and c and d describe the growth limitations on the other population. This can model competition, prey-predator or cooperation interactions depending on the signs of the constants: • Competition when c and d are positive; • Prey-Predator when c is positive and d negative; and • Cooperation when c and d are negative. The first results on local and global bifurcation for Lotka-Volterra systems date back to the 1970s and 1980s. Among these works, we highlight the one by E. N. Dancer (see [24]), which demonstrates the existence of a global continuum for semilinear elliptic boundary value problems emanating from the trivial solution. Classical references on Local and Global Bifurcation in Lotka-Volterra systems include, in addition to those already mentioned, the works of J. Blat and K. J. Brown (see [8, 9]), which address the existence of global branches of coexistence solutions in Competition and Predator-Prey cases. Also noteworthy are the contributions of R. S. Cantrell and C. Cosner [11], who analyze the steady-state problem with diffusion, and P. Korman and A. Leung [46], who investigate the existence and uniqueness of positive steady states in the Volterra-Lotka ecological model with diffusion. Introduction 37 In 1994, J. López-Gómez (see [47]) applied a Global Bifurcation theorem, originally developed by the authors themselves, to analyze the existence of coexistence states in Lotka-Volterra reaction-diffusion systems involving two species. Their focus was on proving the existence of global continuum of coexistence solutions for these systems, employing an optimized version of Rabinowitz Theorem. The main result of this article, Theorem 4.1, is frequently employed in the context of Global Bifurcation for systems. It establishes that, starting from a nondegenerate positive solution (θλ, 0) of problem associated with a semi-trivial solution, there exists a continuum C+ of coexistence states for the system. Basically, in Problem (9), the trivial solution (0, 0) always exists, along with the semi-trivial solutions of the form (u, 0) and (0, v). The semi- trivial solution (θλ, 0) exists if, and only if, λ > λ1. Similarly, the semi-trivial solution (0, θµ) exists if, and only if, µ > λ1. The main bifurcation result for this problem states that, fixing λ > λ1 and taking µ ∈ R as the bifurcation parameter, there exists a critical value given by µ = F (λ) := σ1 [−∆ + dθλ] , from which a continuum of positive solutions C+ bifurcates, where σ1 [−d∆ + a], with d > 0, denotes the principal eigenvalue of the weighted eigenvalue problem: −d∆φ+ a(x)φ = λφ in Ω, φ = 0 on ∂Ω, with a(x) representing the corresponding weight function depending on the context (see [48]). Additionally, the result asserts that this continuum satisfies at least one of the following properties: • C+ is unbounded in the space of R×X; with X a suitable Banach space where the solutions live, indicating that the set of solutions can extend indefinitely • C+ intersects the curve of semi-trivial solutions at the point (µ∞, 0, θµ∞), where θµ∞ is a positive solution of another related problem; or • C+ intersects the curve of semi-trivial solutions at the point (λ∞, θλ∞ , 0), where θλ∞ is another positive solution of the original problem; or • C+ connects to the trivial state (0, 0) at another point. In fact, it can be proved that this continuum satisfies one of the following alternatives: (1) C+ is unbounded in R ×X; or (2) C+ intersects the curve of semi-trivial solutions at the point (µ∞, 0, θµ∞), where θµ∞ > 0 is a positive solution and λ = G(µ∞) := σ1 [−∆ + cθµ∞ ] . In the case of Competition (c, d > 0), it can be shown that C+ is bounded. Therefore, only alternative (2) can occur. Moreover, there exists a coexistence state of Problem (9) when µ ∈ (F (λ), µ∞) ∪ (µ∞, F (λ)) Introduction 38 On the other hand, in the cases of Predator-Prey (d < 0 < c) and Symbiosis (c, d < 0 with cd < 1), one can guarantee that alternative (2) does not occur, and therefore C+ is unbounded. In these scenarios, for all µ > F (λ), there exists a positive coexistence solution. Below we illustrate the bifurcation diagrams corresponding to the Competition, Predator-Prey, and Symbiotic cases: Figure 1: Behavior of the Continuum of semi-trivial solution C in the Competition case. Source: Prepared by the author. Figure 2: Behavior of the Continuum of semi-trivial solution C in the Predator-Prey case. Source: Prepared by the author. Introduction 39 Figure 3: Behavior of the Continuum of semi-trivial solution C in the Symbiotic cases. Source: Prepared by the author. In all cases, the region where both species can coexist is determined by the parameters for which positive solutions exist simultaneously. This region is delimited by the region: R1 := {(λ, µ); (µ− F (λ))(λ−G(µ)) > 0} . This coexistence region, defined by the inequality above, is illustrated in Figure 3.4, where we highlight the set of parameters (λ, µ) for which coexistence states are possible. In this work, we study systems of elliptic equations with nonlocal diffusion, which arise as models in Population Dynamics. These systems have garnered significant attention in the literature due to their ability to capture complex interactions between populations and the effects of nonlocal terms on spatial dispersion. Motivated by applications in Ecology, we consider systems where each variable represents the density of a specific population in a heterogeneous environment. Our primary focus is on investigating the existence of simultaneous positive solutions, known as coexistence states, which represent scenarios where both populations can persist stably within the system. Moreover, we aim to analyze the conditions under which these solutions emerge, as well as the impact of ecological and mathematical parameters on the outcomes. Whenever possible, we interpret the obtained results in an ecological context, establishing a direct connection between theoretical aspects and phenomena observed in nature. The main objective of this work is to investigate sufficient conditions for the existence and uniqueness of coexistence states in coupled nonlocal elliptic systems. More precisely, we consider problems with the following general structure: −m (∫ Ω u, ∫ Ω v ) ∆u = f(x, u, v) in Ω, −n (∫ Ω u, ∫ Ω v ) ∆v = g(x, u, v) in Ω, u = v = 0 on ∂Ω, Introduction 40 where m,n : R2 + → [0,+∞) are continuous nonlinear functions and f, g : Ω ×R2 → R are continuous functions. Our approach will explore three distinct applications of this general problem, each treated with a different analytical method, namely: the Sub-Supersolution Method, Local and Global Bifurcation Theory, and Fixed Point Index Theory. In developing the results for these applications, we draw inspiration from techniques already established in the literature for nonlocal problems involving a single equation. In this way, we extend such theories to the more complex setting of nonlocal systems involving two coupled equations. We have organized this work into five chapters, each designed to progressively develop the theoretical framework and main results of the study. The structure is as follows: In Chapter 1, we provide a review of the main mathematical concepts necessary for the reader to properly understand the subsequent chapters. To this end, we cover topics such as the basics of Partial Differential Equations, Functional Analysis, and Nonlinear Analysis. Next, we discuss key results related to the Maximum Principle and review the Eigenvalue Problem, presenting its fundamental properties. We also explore the literature on the Method of Sub-Supersolution, Bifurcation Methods - Local and Global — and Fixed Point Index Theory. Finally, we highlight well-studied results related to the Logistic Problem, establishing a solid foundation for understanding problems addressed in the following chapters. Additionally, we present in this chapter the results concerning the logistic equation with both local and nonlocal diffusion, specifically Theorem 7. In Chapter 2, we will present the results obtained in [13], where we study the existence of coexistence states for the following nonlocal elliptic system: −m (∫ Ω v ) ∆u = λu− u2 + cuv in Ω, −∆v + σv = ρu in Ω, u = v = 0 on ∂Ω, (10) where m : R → [0,+∞) a continuous function, c, λ ∈ R, ρ ≥ 0, and σ > 0. A particular case of Problem (10) was introduced in [14] to model the behavior of a bacterium, with density v, located in a habitat Ω, where u represents the nutrient. Specifically, in [14], the first equation of (10) is: −m (∫ Ω v ) ∆u = f(x), in Ω where f is a constant rate of the nutrient. In our model, the nutrient is represented by another living organism that grows according to a logistic law, λu − u2, where λ is the growth rate. This nutrient interacts with the bacteria at a rate c, which can be either competitive or cooperative, depending on the sign of c: negative in the first case and positive in the second. In the specific case where c = 0, no interaction occurs. Additionally, the bacteria have a constant death rate σ and a source rate ρ, which depends solely on the nutrients. It is important to note that the main innovation in this model is that the nutrient’s diffusion depends nonlocally and nonlinearly on the bacteria population. For related works involving nonlocal diffusivity terms, see also [33], [44], and [59]. Regarding the results of this chapter, we can summarize the findings on coexistence states through the following result: Theorem 1. The following assertions hold: Introduction 41 (a) Assume that c = 0. Problem (10) possesses at least one coexistence state when λ > m(0)λ1. Moreover, the coexistence state is unique when m is increasing. (b) Assume that c < 0. Problem (10) possesses at least one coexistence state when λ > m(0)λ1 and does not have a coexistence state when λ ≤ λ1 min s≥0 m(s). (c) Assume that c > 0. Problem (10) possesses at least one coexistence state when λ > m(0)λ1 and one of the following conditions holds: c is small, ρ is small, σ is large, or lim s→∞ m(s) s = ∞. (11) Moreover, it does not possess a coexistence state when λ ≤ λ0, for some λ0 ∈ R. Although the existence results in all cases are quite similar, their derivations differ. In particular, we will use the bifurcation technique, and the main difference across the cases lies in the method used to obtain the a priori bounds. It is important to highlight that in the cooperative case, we have obtained the a priori bounds in two distinct ways. In the first case, we primarily use arguments based on the maximum principle, while in the second case, we argue by contradiction and make use of the fact that the diffusion coefficient grows very rapidly. In all cases, we prove the existence of an unbounded continuum C ⊂ R×C1 0(Ω)×C1 0(Ω) of positive solutions for Problem (10). Specifically, we prove: Theorem 2. Assume that m(0) > 0. From the trivial solution (u, v) = (0, 0) emanates an unbounded continuum C ⊂ R × C1 0(Ω) × C1 0(Ω) of positive solutions of Problem (10) at λ = m(0)λ1. Moreover, if one of the following conditions holds: c ≤ 0, c ≥ 0 and c is small, c > 0 and ρ is small, c > 0 and σ is large, or c > 0 and m satisfies (11), then (m(0)λ1,∞) ⊂ ProjR(C) ⊂ (λ0,∞), for some λ0 ≤ 0, where ProjR(λ, u, v) = λ for (λ, u, v) ∈ C. As a consequence, there exists at least a positive solution for λ > m(0)λ1. We also study the local bifurcation, including the bifurcation direction. This direction depends on the relative size of the coefficients of (10) and m′(0). Specifically: Theorem 3. Assume that m(0) > 0. Then, the bifurcation direction from the trivial solution (u, v) = (0, 0) at λ = m(0)λ1 is: (a) Supercritical when m′(0) > (cρ− λ1 − σ) ∥φ1∥3 3 λ1ρ ∥φ1∥1 . (b) Subcritical when m′(0) < (cρ− λ1 − σ) ∥φ1∥3 3 λ1ρ ∥φ1∥1 . Introduction 42 In the Figure 2.1 we have illustrated two possible bifurcation diagrams. (a) Supercritical bifurcation (b) Subcritical bifurcation Figure 4: Bifurcation diagrams of Problem (10) Source: Prepared by the author. Moreover, we prove a uniqueness result. We show that when m is increasing, there exists a unique coexistence state of Problem (10) for c ∈ (−c0, c0) for some c0 > 0. Observe that this uniqueness is optimal in the following sense: if m is increasing and c is large, the bifurcation direction is subcritical, and the multiplicity of positive solutions occurs for λ ∈ (m(0)λ1 − δ,m(0)λ1) for some δ > 0 small. Finally, we analyze the case m(0) = 0. In this case, we cannot apply the bifurcation method directly, but we can use a compactness argument to show the existence of a positive solution to Problem (10) for all λ > 0. In Chapter 3, we will present the results obtained in [22], where we study the existence of coexistence states for the following nonlocal system with cross-diffusion: −m (∫ Ω v ) ∆u = λu− u2 − cuv in Ω, −n (∫ Ω u ) ∆v = µv − v2 − duv in Ω, u = v = 0 on ∂Ω, (12) where c, d, λ, µ ∈ R, and m,n : R → [0,∞) are continuous functions. From the perspective of population dynamics, Problem (12) describes the behavior of two species, u and v, inhabiting the habitat Ω. Since we consider homogeneous Dirichlet boundary conditions, the habitat is entirely surrounded by an inhospitable region. In the reaction functions, we have used classical Lotka-Volterra type reaction terms, where λ and µ represent the intrinsic growth rates of each species, and c and d describe the interaction rates between the species. This can model interactions of competition, predator-prey, or cooperation, depending on the signs of the constants c and d. Introduction 43 In population dynamics, the nonlocal term was included in [40] in the reaction term to account for the interaction between species, stating that, in the ecological context of species interactions, there is no real justification for assuming that they occur locally. On the other hand, in recent years, this kind of nonlocal term has also been incorporated into the diffusion term, considering expressions of the form: −a (∫ Ω u ) ∆u to model the fact that the velocity of migration depends on the total population in the domain, see for instance [17]. Here, a is an increasing function if the species tends to leave crowded zones, while if we are dealing with species attracted by the growing population in Ω, we will assume a to decrease. This idea can be extended to systems, see for instance [14, 33, 44] and their references. That is, the diffusion of one species u, respectively v, depends on the other species in the form −m (∫ Ω v ) ∆u, respectively −n (∫ Ω u ) ∆v. Let us explain the first case. In this case, if the total population of the species v increases, then u can act in two different ways: • The species u tends to leave the crowded region when m is increasing; and • The species u is attracted by v when m is decreasing. The second case is analogous. This diffusion behavior is combined with and complemented by the interaction between species. Assume that u and v compete. In this case, if m increases, species u tends to escape its competitor v, while if m decreases, species u does not leave crowded zones; that is, despite the competition, u remains in a densely populated area, even if it is occupied by its competitor. However, if u and v cooperate, when m increases, species u leaves the crowded region. In this case, the cooperation with v is not being fully exploited. Conversely, if m decreases, species u benefits from its cooperation with v. Finally, in the predator-prey scenario where u is a prey and v is a predator, when m increases, the prey u escapes from v. On the other hand, if m decreases, the prey does not leave the area occupied by the predators. One of the main objectives of this chapter is to study the region of existence (as a function of the parameters λ and µ) of positive solutions, depending on the behavior of the functions m and n, as well as the type of interaction between the species. In relation to the results of this chapter, we show that the semi-trivial solutions of Problem (12), denoted by: (u, v) = (θλ, 0) and (u, v) = (0, θµ) which are, respectively, solutions of the following problems: −m (0) ∆u = λu− u2 in Ω, u = 0 on ∂Ω, Introduction 44 and  −n (0) ∆v = µv − v2 in Ω, v = 0 on ∂Ω, exist if, and only if, λ > m(0)λ1 and µ > n(0)λ1, respectively. We define the maps F : [m(0)λ1,∞) → R and G : [n(0)λ1,∞) → R given by: F (λ) := σ1 [ −n (∫ Ω θλ ) ∆ + dθλ ] and G(µ) := σ1 [ −m (∫ Ω θµ ) ∆ + cθµ ] , where σ1 [−d∆ + a], with d > 0. First, we provide some results characterizing the local stability of semi-trivial solutions: Theorem 4. The following assertions hold: (a) Assume that λ > m(0)λ1. The semi-trivial solution (θλ, 0) is linearly asymptotically stable, respectively unstable, when µ < F (λ), respectively µ > F (λ). (b) Assume that µ > n(0)λ1. The semi-trivial solution (0, θµ) is linearly asymptotically stable, respectively unstable, when λ < G(µ), respectively λ > G(µ). Using the Theory of Fixed Point Index with respect to cones in Banach spaces, originally developed by [4] and later extended and refined by [25], we are able to establish important existence results for positive solutions. These foundational tools allow us to rigorously analyze nonlinear operators in ordered function spaces, leading to the following conclusions: Theorem 5. The following assertions hold: (a) Assume that λ > m(0)λ1 and µ > n(0)λ1. Problem (12) possessat least one coexistence state when (λ−G(µ))(µ− F (λ)) > 0. (13) (b) Assume that λ > m(0)λ1 and µ ≤ n(0)λ1. Problem (12) possessat least one coexistence state when µ > F (λ). (c) Assume that λ ≤ m(0)λ1 and µ > n(0)λ1. Problem (12) possessat least one coexistence state when λ > G(µ). Subsequently, we will conduct a detailed study of the coexistence region defined by (13), which plays a central role in understanding the structure of solutions. To this end, we will closely examine the behavior of the functions F (λ) and G(µ), which are fundamental in characterizing the boundaries of this region. Under the assumption that the functions m and n are differentiable, it follows that both F and G inherit differentiability. Our analysis will focus on their properties and asymptotic behavior, particularly near the lower bounds λ = m(0)λ1 and µ = n(0)λ1, as well as in the limit as λ and µ tend to +∞. This investigation will provide insights into the geometry and structure of the coexistence region. Theorem 6. The following assertions hold: Introduction 45 (a) Assume that d > 0. If n(s) ≥ ks−α for all s ≥ s0, for some s0 > 0, where 0 < α < 1 and k > 0, then lim λ→+∞ F (λ) = +∞. (b) Assume that d < 0. If n(s) ≤ csα for all s ≥ s0, for some s0 > 0, where 0 < α < 1 and c > 0, then lim λ→+∞ F (λ) = −∞. (c) Assume that d < 0. If n(s) ≥ csα for all s ≥ s0, for some s0 > 0, where α > 1 and c > 0, then lim λ→+∞ F (λ) = +∞. By symmetry, the same results apply to the function G. Hence, we can construct the coexistence regions of Problem (12) in the λ − µ plane for all cases, depending on the conditions satisfied by m and n. See the figures below. Figure 5: The coexistence region of Problem (12) in the Competition Case. In this case, we assume that m and n verify the Assertion (a) of Theorem 6, which ensures certain properties of the semi-trivial solutions. Additionally, we assume that n′(0) < 0, implying that F ′(m(0)λ1) < 0. Under these conditions, it follows that a coexistence state may occur even for values of µ satisfying µ < n(0)λ1. Source: Prepared by the author. Introduction 46 (a) In this case, we assume that m verifies the Assertion (a) of Theorem 6 and n the Assertion (b), according to the hypotheses established in the framework of the analysis. (b) In this case, we assume that m verify the Assertion (a) of Theorem 6 and n Assertion (c), under the conditions considered throughout the current study. Figure 6: Coexistence region of Problem (12) in the Prey-Predator Case, corresponding to the scenario analyzed under specific assumptions for the interaction coefficients. Source: Prepared by the author. (a) In this case, we assume that m and n verifies the Assertion (b) of Theorem 6. (b) In this case, we assume that m verifies the Assertion (a) of Theorem 6 and n Assertion (c). Figure 7: Coexistence region of Problem (12) in the Symbiosis Case. Source: Prepared by the author. Finally, we compare our model with the linear model and observe the following differences: Introduction 47 • Competition: In the linear diffusion case, there is no coexistence state if λ ≤ λ1 or µ ≤ λ1. In other words, it is a necessary condition for coexistence that both semi- trivial solutions exist, that is, for two competing species to coexist, each must survive in the absence of competition. However, surprisingly in our case, both competing species can coexist even when one semi-trivial solution does not exist. This occurs when the growth rate of species v, µ is small, and the velocity of diffusion of this species is decreasing (n′(0) negative) with respect to the population of the other species u. See Figure 5. • Prey-predator: In the linear diffusion case, for a fixed predator growth rate µ, there exists at least one coexistence state if the prey growth rate λ is large. This significantly changes in our case when n is large. Indeed, for fixed µ, there is no coexistence state for large λ. This happens because as λ increases, the prey u grows sufficiently, and therefore so does n( ∫ Ω u). As a result, the predator leaves the area populated by the prey and does not benefit from this situation. • Cooperation: In the linear diffusion case, for a fixed growth rate µ of species v, both species coexist for large values of the growth rate λ of the other species. This is true even for negative values of µ. However, in our case, when n grows strongly, if λ increases, the species u also increases, and the high diffusion rate of v, n( ∫ Ω u), drives species v to extinction because, again, v does not take advantage of the cooperation. Regarding this comparison, we can conclude the following in relation to the model proposed by Problem (12): • In the competition case, the coexistence region is expanded when the densities of the species are small and neither species leaves the region of competition. In this scenario, it is in their interest to remain in contact with the other species in order to survive. • In the prey-predator case, the region of coexistence decreases when the predator abandons areas of high population density. As the prey population increases, the predator’s strategy of avoiding these areas prevents coexistence. • In the cooperation case, a similar situation arises: if a species leaves high-population zones, it no longer benefits from the cooperation, leading to the eventual extinction of both species. In Chapter 4, we will present the results obtained in [21], where we study the existence of coexistence states for the following system with nonlocal diffusion:  −m (∫ Ω u ) ∆u = λu− u2 − cuv in Ω, −n (∫ Ω v ) ∆v = µv − v2 − duv in Ω, u = v = 0 on ∂Ω, (14) where c, d, λ, µ ∈ R, and m,n : R → [0,∞) are continuous functions. From the perspective of Population Dynamics, (14) models the behavior of two species with population densities u(x) and v(x), inhabiting a habitat Ω. Since we consider Introduction 48 homogeneous Dirichlet boundary conditions, the habitat is entirely surrounded by an inhospitable region. Unlike Problem (12), in this problem, the diffusion terms are not cross-diffusive, that is, the diffusion of the species u, respectively v, depend on their own populations as follows −m (∫ Ω u ) ∆u, respectively −n (∫ Ω v ) ∆v. This type of diffusion was extensively studied in 2018 by T. S. Figueiredo-Sousa, C. Rodrigo-Morales, and A. Suárez (see [39]). Let’s analyze the first expression, the second follows similarly. • If m is an increasing function, the species tends to leave crowded zones. • If m is a decreasing function, the species is attracted to areas with a growing population. As in Chapter 4, our main objective in this chapter is to study the region of existence (as a function of the parameters λ and µ) of positive solutions, depending on the behavior of the functions m and n, as well as the type of interaction between the species. Regarding the results of this chapter, we will first study the semi-trivial solutions of Problem (14), which we will denote by (u, v) = (θ[λ,m(·)], 0) and (u, v) = (0, θ[µ,n(·)]) which are, respectively, solutions of the following problems: −m (∫ Ω u ) ∆u = λu− u2 in Ω, u = 0 on ∂Ω, and  −n (∫ Ω v ) ∆v = µv − v2 in Ω, v = 0 on ∂Ω. Note that, when one species is absent, the other one follows the following logistic equation with a nonlocal diffusivity coefficient: −g (∫ Ω w ) ∆w = γw − w2 in Ω, w = 0 on ∂Ω, (15) where γ ∈ R and g : R −→ (0,+∞) is a continuous function. This problem has been studied in [39] in a more general situation. In this thesis, we complete the results of [39], and we prove the following result: Theorem 7. The following assertions hold: Introduction 49 (a) If γ ≤ λ1gL, where gL := inf s∈R g(s), Problem (15) does not possess positive solutions. (b) If g(0) > 0, there exists at least a positive solution of Problem (15) when γ > g(0)λ1, Moreover, if g is increasing, there exists a unique positive solution of Problem (15) if, and only if, γ > g(0)λ1. (c) Let wγ a positive solution of Problem (15). Then,∫ Ω wγ dx → ∞ and wγ(x) → ∞, as γ → ∞. We define the maps F : [m(0)λ1,∞) → R and G : [n(0)λ1,∞) → R given, respectively, by: F (λ) := σ1 [−n(0)∆ + dθ[λ,m(·)]] and G(µ) := σ1 [−m(0)∆ + cθ[µ,n(·)]] . It is important to emphasize that, despite the notation, F and G are not proper maps in the strict functional sense, since for each value of λ or µ, the corresponding θ[λ,m(·)] or θ[µ,n(·)] is not necessarily unique. This non-uniqueness stems from the fact that the auxiliary problems that define θ[λ,m(·)] and θ[µ,n(·)] may admit more than one positive solution. This subtlety is one of the particular features of the nonlocal structure of the system under consideration and must be kept in mind throughout the analysis. Since they are already widely studied in the literature, we will not address the results of these maps in this chapter, but we emphasize that they are well-defined and continuous. Furthermore, the following result holds: Theorem 8. Assume that m and n are increasing. The following assumptions hold: (a) The semi-trivial solutions θ[λ,m(·)] and θ[µ,n(·)] are unique if, and only if, λ > m(0)λ1 and µ > n(0)λ1, respectively. (b) The maps F and G are increasing. (c) lim λ→+∞ F (λ) = +∞ and lim µ→+∞ G(µ) = +∞. Using the Sub-Supersolution Method, we will prove the following coexistence state existence result: Theorem 9. Assume that cd < 1. If the conditions λ > mMλ1 + cµ and µ > nMλ1 + dλ (16) are satisfied, then Problem (14) possessat least a coexistence state In Figure 8, we represent the coexistence region R, defined by: R := {(λ, µ) ∈ R2; (λ, µ) verifies (16)}; in gray, which is ensured by the previous theorem, that is, if (λ, µ) ∈ R, then Problem (14) possessat least a coexistence state. The region R is Introduction 50 Figure 8: The coexistence region of Problem (14) defined by Condition (16). Source: Prepared by the author. Applying the results of Local Bifurcation, where we will strongly rely on apply Theorem 1.7 of [23], we will present a bifurcation point for one of the semi-trivial solutions of Problem (14). At this point, it is essential to highlight two major difficulties specific to our setting. First, the semi-trivial solutions of Problem (14) are not necessarily unique, which complicates the direct application of classical bifurcation results. Second, even when a semi-trivial solution is available, we cannot guarantee that it is non-degenerate. For this reason, we must explicitly impose condition (17). Theorem 10. Assume that λ > 0. Suppose there exists a positive solution θ[λ,m(·)] of Problem (15), with g ≡ m such that 1 + m′ (∫ Ω θ[λ,m(·)] ) m (∫ Ω θ[λ,m(·)] ) ∫ Ω (λθ[λ,m(·)] − θ2 [λ,m(·)])eλ ̸= 0. (17) Then, (µ, u, v) = (F (λ), θ[λ,m(·)], 0) is a bifurcation point from the semi-trivial solution (µ, θ[λ,m(·)], 0). In the previous result, eλ denotes the unique positive solution of the following auxiliary problem, whose existence is guaranteed by the Maximum Principle: −m (∫ Ω θ[λ,m(·)] ) ∆eλ + (2θ[λ,m(·)] − λ)eλ = 1 in Ω, eλ = 0 on ∂Ω. Introduction 51 Furthermore, we will present a result addressing the bifurcation direction from the semi-trivial solution (θ[λ,m(·)], 0) arising from the previous theorem. Using the bifurcation point determined through Local Bifurcation, we will employ Global Bifurcation to ensure the existence of a continuum of positive solutions for Problem (14). More precisely, we will show that: Theorem 11. Assume that m is increasing. The following assumptions hold: (a) At µ = F (λ) emanates from (θ[λ,m(·)], 0) a continuum C of positive solutions of Problem (14). (b) There exists a sequence (µn, un, vn) ∈ C such that (µn, un, vn) → (µ∗, 0, v∗ µ), where v∗ µ is a positive solution of Problem (15) and µ∗ satisfies the following condition λ = G(µ∗). (18) In this case, there exists a coexistence state of Problem (14) when µ ∈ (min{F (λ), µ∗},max{F (λ), µ∗}). (19) Furthermore, assuming that n is increasing, there exists a unique semi-trivial solution (0, θ[µ,n(·)]) and a unique value µ∗ that verifies the Condition (18). In the following figures, we will represent several cases of the Continuum C based on the behavior of the functions m and n, as well as the semi-trivial solutions. Figure 9: Behavior of the Continuum of semi-trivial solution C(µ,θ,0) when m is increasing. Source: Prepared by the author. Introduction 52 Figure 10: Behavior of the Continuum of semi-trivial solution C(µ,θ,0) when m is increasing, λ ≈ m(0)λ1 and n′(0) < 0. Source: Prepared by the author. A significant result obtained was the proof of the existence of coexistence states under the assumption that the functions m and n are increasing. More precisely, we established the following result: Theorem 12. Assume that m and n are increasing. The following assumptions hold: (a) Problem (14) does not possess coexistence states when λ ≤ m(0)λ1 or µ ≤ n(0)λ1. (b) Problem (14) possessat least a coexistence state when (µ− F (λ)) (λ−G(µ)) > 0. In the following figure, we present the bifurcation diagram in the case where both m and n are increasing. Introduction 53 Figure 11: Bifurcation diagram when m and n both increase. In this case, there exists a unique semi-trivial solution (uλ, 0) and (0, vµ). Source: Prepared by the author. To conclude the chapter, we will demonstrate an important relationship between the existence result obtained through the Sub-Supersolution Method and the Global Bifurcation Theory. More precisely, we will show that: Theorem 13. If the pair (λ, µ) ∈ R2 satisfies Condition (16), then this pair also satisfies Condition (19). In the Figure 12, we represent the curves generated by conditions (16) and (19). These curves define the coexistence regions, which correspond to the areas enclosed by the curves. Figure 12: The coexistence region of Problem (14) defined by conditions (16) and (19). In this case, the region defined by Condition (19) is larger than the one defined by (16). Source: Prepared by the author. Introduction 54 It is pertinent to highlight some relevant observations that complement and reinforce the aspects we will address in this chapter. • It is quite surprising that the competitive exclusion principle holds regardless of the behavior of the functions m and n. Indeed, while it is well known that this principle is satisfied in the case of linear diffusion, that is, when m ≡ n ≡ 1, it remains valid for any choice of the functions m and n. • Condition (16) is obtained using the Sub-Supersolution Method and does not require any monotonicity assumption on either n or m. However, it is necessary that cd < 1. • Unlike the linear diffusion case, the bifurcation method cannot, in general, be applied for two main reasons. First, the semi-trivial solution is not unique. Second, it is not, in general, nondegenerate. However, under a specific condition (see (17)), it is possible to establish the bifurcation of a continuum of positive solutions. In particular, Condition (17) is satisfied if m is increasing. • In the linear diffusion case, the continuum of positive solutions bifurcates from one semi-trivial solution and connects to the other. However, in the nonlocal diffusion case, the behavior of the continuum is not predetermined — multiple scenarios may occur, and the structure of the solution set can be significantly more complex. • When the bifurcation method can be applied, the results obtained are more robust than those achieved through the Sub-Supersolution Method, as they provide a broader description of the set of solutions and, in particular, expand the region of coexistence. • The inclusion of nonlocal terms leads to a more intricate behavior of the system, resulting, for instance, in the multiplicity of positive solutions under conditions where the corresponding system with linear diffusion admits a unique coexistence state. In Chapter 5, we present the general conclusions drawn from the study of the nonlocal elliptic systems analyzed throughout the previous chapters. We discuss the main results concerning the existence, uniqueness, and structure of positive solutions, with particular emphasis on coexistence states in population dynamics models. Furthermore, we highlight the methodological contributions employed, such as Bifurcation Theory, Sub- Supersolution Method, and the use of Fixed Point Index Theory in Positives Cones, as well as the challenges introduced by the presence of nonlocal terms in the diffusion coefficients. This chapter also underscores the differences between nonlocal models and their local counterparts, revealing new and ecologically relevant phenomena. We conclude this introduction by highlighting some open questions throughout this work. Some of them are difficulties that we couldn’t resolve within the scope of the research conducted, while others represent areas of interest that, although not addressed here, will serve as a guide for future investigations. 1. Applicability of the bifurcation method when m(0) = 0 or n(0) = 0: In the case where m(0) = 0 or n(0) = 0, possibly both, the bifurcation method cannot be applied directly, as the associated linearized operator may lose key properties such as coercivity or invertibility. This scenario requires the use of compactness Introduction 55 arguments or alternative variational techniques to prove the existence of solutions. It highlights an inherent limitation of the bifurcation method in degenerate cases and calls for more robust analytical tools. 2. Uniqueness of the coexistence state in systems with nonlocal diffusion: Establishing the uniqueness of coexistence states in systems involving nonlocal diffusion terms is a major open problem. The global nature of these terms introduces strong coupling and nonlinearity, making it difficult to derive sufficient conditions for uniqueness. This issue is particularly relevant in applications, where uniqueness is often associated with predictability and stability of the model. 3. Stability of solutions: Analyzing the stability of the obtained solutions is crucial for understanding the long-term dynamics of the modeled populations. Determining whether a solution is stable or unstable under small perturbations allows for predictions about species persistence or extinction. However, the interplay between the coupling and the nonlocal terms significantly complicates the spectral and dynamical analysis required to assess stability. 4. Behavior in the cooperative case under strong cooperation: In cooperative systems, where species benefit mutually from each other’s presence, it remains unclear how strong cooperation affects the existence, multiplicity, and stability of solutions. Intense cooperation may lead to new bifurcation phenomena or even loss of stability. A better understanding of this scenario is essential to model ecological systems with high levels of mutualistic interaction accurately. 5. Eigenvalue problems in systems with nonlocal terms: The inclusion of nonlocal terms in the diffusion operator adds a layer of complexity to the associated eigenvalue problems. Such operators are typically nonlinear and lack compactness, making it challenging to characterize the spectrum. A thorough spectral analysis is fundamental for understanding bifurcation structure, stability thresholds, and the qualitative behavior of solutions in these systems. Moreover, through this analysis, we guarantee uniqueness, as pointed out in item 2. 1 Preliminaries 1.1 Basic Concepts and Definitions In this section, we will present some concepts and definitions that will be used throughout this work, which are essential for understanding the more complex topics introduced later. For further details on the material covered here, we recommend foundational texts on Linear and Nonlinear Algebra, Functional Analysis, Topology, and Analysis (see [4, 6, 10, 27, 45]). Throughout this work, we will often consider a specific subset of RN that possesses certain specific properties, namely: Definition 1.1. We say that Ω is a bounded regular domain in RN , with N ≥ 1, when Ω is an open and connected subset of RN whose boundary, ∂Ω, is sufficiently regular. The following definition introduces a category of operators in Banach spaces that are fundamental in Functional Analysis, especially in Differential Equations, due to their stability and connection with topological and spectral properties. Definition 1.2. Let E,F be Banach spaces and T : E → F a continuous linear operator. We say that T is a Fredholm operator when: (a) dim[Ker(T )] < ∞; and (b) codim[Rg(T )] < ∞. For operators that satisfy this definition, we can introduce the concept of an index, defined as follows: ind(T ) := dim[Ker(T )] − codim[Rg(T )]. We will denote by Fred0(E,F ) the set of all Fredholm operators with index zero. In the next result, we present an important property of Fredholm operators, which will be utilized in future applications. Theorem 1.3 (Fredholm Alternative). Let E be a Banach space, T : E → E a compact operator and T ∗ the adjoint operator associated to T . The following assumptions hold: (a) Ker[I − T ] and Ker[I − T ∗] have finite dimension. Moreover dim[Ker(I − T )] = dim[Ker(I − T ∗)]. 56 Basic Concepts and Definitions 57 (b) Rg[I − T ] and Rg[I − T ∗] are closed. Moreover Rg[I − T ] = Ker[I − T ∗]⊥ and Rg[I − T ∗] = Ker[I − T ]⊥. (c) Ker[I − T ] = {0} if, and only if, Rg[I − T ] = E. (d) dim[Ker(I − T )] = dim[Ker(I − T ∗)]. Proof. See Theorem 6.6 of [10]. Let E be a Banach space, I : E → E the identity operator in E and K : E → E a compact linear operator. Note that, I − K, called compact perturbation of the identity, is a Fredholm operator of index zero. Indeed, by Assumption (a) of Theorem 1.3, dim[Ker(I −K)] = dim[Ker(I −K∗)] < ∞ and, by Assumption (b), dim[Ker(I −K)] = dim[Rg(I −K)⊥]. Consequently dim[Rg(I −K)⊥] < ∞. Since Rg(I −K) is closed, by Proposition 11.13 of [10], dim[Rg(I −K)⊥] = codim[Rg(I −K)]. Therefore ind(I −K) = 0. To introduce the next concept, which will be used throughout this text, we first need to recall some fundamental ideas from Linear Algebra. Definition 1.4. Let V be a vector real space and ≺ a ordering in V . We say that ≺ is a linear ordering when the following properties are satisfied: (a) Given x, y ∈ V , for all z ∈ V , x ≺ y implies that x+ z ≺ y + z; and (b) Given x, y ∈ V , for all α ∈ R+, x ≺ y implies that αx ≺ αy. We will refer to the space V with a linear ordering as an ordered vector space (OVS) and denote it by (V,≺), or simply V , when the ordering is clear. Definition 1.5. Let V be a ordered vector space. We say that the set PV := {v ∈ V ;