A nonlocal diffusion population model with age structureand Dirichlet boundary condition

Yueding  Yuan; Zhiming Guo; Moxun  Tang

doi:10.3934/cpaa.2015.14.2095

Article Contents

2015, Volume 14, Issue 5: 2095-2115. Doi: 10.3934/cpaa.2015.14.2095

This issue Previous Article Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions Next Article

A nonlocal diffusion population model with age structure and Dirichlet boundary condition

1.
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006
2.
Department of Mathematics, Michigan State University, East Lansing, MI 48824

Received: January 31, 2014

Revised: July 31, 2014

Published: August 2015

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Abstract

In this paper, we study the global dynamics of a population model with age structure. The model is given by a nonlocal reaction-diffusion equation carrying a maturation time delay, together with the homogeneous Dirichlet boundary condition. The non-locality arises from spatial movements of the immature individuals. We are mainly concerned with the case when the birth rate decays as the mature population size becomes large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of the positive steady states of the model. By establishing an appropriate comparison principle and applying the theory of dissipative systems, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the unique positive steady state.

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Mathematics Subject Classification: Primary: 34B18.

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