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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

A nonlinear diffusion problem arising in population genetics

Pages: 821 - 841, Volume 34, Issue 2, February 2014      doi:10.3934/dcds.2014.34.821

 
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Peng Zhou - Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China (email)
Jiang Yu - Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China (email)
Dongmei Xiao - Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China (email)

Abstract: 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].

Keywords:  Nonlinear diffusion equation, existence, stability, limiting profile, bifurcation.
Mathematics Subject Classification:  Primary: 35K61, 35B40; Secondary: 92D10.

Received: September 2012;      Revised: April 2013;      Available Online: August 2013.

 References