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February  2014, 34(2): 821-841. doi: 10.3934/dcds.2014.34.821

A nonlinear diffusion problem arising in population genetics

 1 Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China, China 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  September 2012 Revised  April 2013 Published  August 2013

In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
Citation: Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821
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