May  2010, 27(2): 643-655. doi: 10.3934/dcds.2010.27.643

An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity

1. 

Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210

2. 

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

3. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Received  October 2009 Revised  February 2010 Published  February 2010

We study a genetic model with two alleles $A_{1}$ and $A_{2}$ in a bounded smooth habitat $\Omega$. The frequency $u$ of the allele $A_{1}$, under the combined influence of migration and selection, obeys a parabolic equation of the type

$ u_{t}=d\Delta u+g(x)f(u),~0\leq u\leq 1 $ in Ω × (0, ∞),
$ \frac{\partial u}{\partial\nu}=0 $ on ∂ Ω × (0, ∞),

where $\Delta$ denotes the Laplace operator, $g$ may change sign in $\Omega$, and $f(0)=f(1)=0$, $f(s)>0$ for $s\in(0,1)$. Our main results include stability/instability of the trivial steady states $u\equiv 0$ and $u\equiv 1$, and the multiplicity of nontrivial steady states. This is a continuation of our work [12]. In particular, the conjecture of Nagylaki and Lou [11, p. 152] has been largely resolved. Similar results are obtained for Dirichlet and Robin boundary value problems as well.

Citation: Yuan Lou, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 643-655. doi: 10.3934/dcds.2010.27.643
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