Publication: Existencia global de soluciones y estabilidad en sistemas de dos especies depredador-presa con difusión y quimiotaxis
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2019-07
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Abstract
En este trabajo se estudia un sistema no lineal de reacción-difusión de ecuaciones en derivadas parciales que describe la evolución de un sistema biológico depredador-presa con quimiotaxis. El sistema está compuesto por tres ecuaciones, dos de ellas parabólicas correspondientes a los depredadores activos y a las presas, y una ordinaria correspondiente a depredadores inactivos. La quimiotaxis en este contexto afecta a los depredadores activos de modo que dirigen su movimiento hacia las zonas en las que la densidad de depredadores inactivos es mayor. Para llevar a cabo este estudio primero se realiza una revisión bibliográfica sobre algunos sistemas depredador-presa con quimiotaxis y con término de depredadores inactivos. Posteriormente se utilizan métodos clásicos para ecuaciones parabólicas-parabólicas-ordinarias para demostrar la existencia local de soluciones en nuestro sistema, y el método iterativo de Moser-Alikakos para construir acotaciones uniformes. Finalmente se realiza una breve aproximación numérica para obtener una mejor comprensión del comportamiento del sistema biológico.
In the present work a nonlinear system of reaction-diffusion partial differential equations describing the evolution of a prey-predator biological system with chemotaxis is studied. The system consists of three equations, two parabolic equations corresponding to the active predators and preys, and an ordinary equation corresponding to a special term related to the dormant predators. Chemotaxis in this context affects to the active predators such that they will move towards the regions in which the density of resting eggs (dormant predators) is larger. To properly accomplish this study first a literature review on some prey-predator chemotaxis systems and with dormant predators term is conducted. Then we use classical methods for parabolic-parabolic-ordinary equations to prove the local existence of solutions and the Moser-Alikakos iterative method to find bounds in L ∞, hence we obtain global existence of solutions in our system. Finally a brief numerical approximation is achieved in order to get a better understanding of the behavior of our system.
In the present work a nonlinear system of reaction-diffusion partial differential equations describing the evolution of a prey-predator biological system with chemotaxis is studied. The system consists of three equations, two parabolic equations corresponding to the active predators and preys, and an ordinary equation corresponding to a special term related to the dormant predators. Chemotaxis in this context affects to the active predators such that they will move towards the regions in which the density of resting eggs (dormant predators) is larger. To properly accomplish this study first a literature review on some prey-predator chemotaxis systems and with dormant predators term is conducted. Then we use classical methods for parabolic-parabolic-ordinary equations to prove the local existence of solutions and the Moser-Alikakos iterative method to find bounds in L ∞, hence we obtain global existence of solutions in our system. Finally a brief numerical approximation is achieved in order to get a better understanding of the behavior of our system.
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