Elliptic Systems in Population Dynamics: A Study with Nonlocal Diffusion Coefficients

• Autores: Marcos Antonio Viana Costa
• Directores de la Tesis: Antonio Suárez Fernández (dir. tes.) , Marcos Tadeu de Oliveria Pimenta (dir. tes.)
• Lectura: En la Universidad de Sevilla ( España ) en 2025
• Idioma: inglés
• Número de páginas: 155
• Enlaces
  • Tesis en acceso abierto en: Idus
• Resumen

  • This work investigates the existence and uniqueness of coexistence states in nonlinear elliptic systems with non-local diffusion, used to model the interaction between two biological species. In these models, the diffusion rate of each species depends either on the total population of that same species or on the population of the other species. The system is defined in a bounded, regular domain in Euclidean space, where the diffusion functions are continuous and nonlinear, and the growth functions describing the interaction dynamics are also continuous.

    In the first model analyzed, we study the interaction between a bacterium and a nutrient, incorporating nonlinear and population-dependent diffusion. To determine conditions that ensure the existence and uniqueness of positive solutions, we apply analytical techniques such as Local and Global Bifurcation Theory and the Implicit Function Theorem.

    The second model explores Lotka–Volterra-type systems with non-local cross-diffusion, representing different ecological relationships such as competition, predator-prey dynamics, and symbiosis. We analyze the stability of solutions where only one species survives (semi-trivial solutions) and establish conditions for the existence of coexistence states, where both species persist over time.

    Finally, in the third model, we focus on a Lotka–Volterra competition framework with non-local diffusion. In this case, the movement of each species depends on its own spatially averaged population. We demonstrate the existence of coexistence states by employing the competitive exclusion principle.

    Overall, the results emphasize how non-local diffusion plays a significant role in capturing realistic features of species interactions and in shaping the conditions under which coexistence can occur in ecological systems.