# American Institute of Mathematical Sciences

September  2015, 14(5): 2095-2115. doi: 10.3934/cpaa.2015.14.2095

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

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

Received  February 2014 Revised  August 2014 Published  June 2015

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.
Citation: Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
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