A nonlinear diffusion problem arising in population genetics

Abstract

In this paper we investigate a nonlinear diffusion equation with theNeumann boundary condition, which was proposed by Nagylaki in[19] to describe the evolution of two types of genes inpopulation genetics. For such a model, we obtain the existence ofnontrivial solutions and the limiting profile of such solutions asthe diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Ourresults show that as $d\rightarrow0$, the location of nontrivialsolutions relative to trivial solutions plays a very important rolefor the existence and shape of limiting profile. In particular, anexample is given to illustrate that the limiting profile does notexist for some nontrivial solutions. Moreover, to better understandthe dynamics of this model, we analyze the stability and bifurcationof solutions. These conclusions provide a different angle tounderstand that obtained in [17,21].