A nonlinear diffusion problem arising in population genetics

Peng Zhou; Jiang Yu; Dongmei Xiao

doi:10.3934/dcds.2014.34.821

Article Contents

2014, Volume 34, Issue 2: 821-841. Doi: 10.3934/dcds.2014.34.821

This issue Previous Article Local Well-posedness and Persistence Property for the Generalized Novikov Equation Next Article Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

A nonlinear diffusion problem arising in population genetics

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

Received: August 31, 2012

Revised: March 31, 2013

Published: January 2014

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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].

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Mathematics Subject Classification: Primary: 35K61, 35B40; Secondary: 92D10.

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References

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