A NONLINEAR DIFFUSION PROBLEM ARISING IN POPULATION GENETICS

作者：Zhou, Peng^*; Yu, Jiang; Xiao, Dongmei

来源：Discrete and Continuous Dynamical Systems, 2014, 34(2): 821-841.

DOI：10.3934/dcds.2014.34.821

• Nonlinear diffusion equation
• existence
• stability
• limiting profile
• bifurcation

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 -> 0 or d -> infinity. Our results show that as d -> 0, 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].

• 出版日期2014-2
• 单位上海交通大学