# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2010, 27 (2) : 643-655. doi: 10.3934/dcds.2010.27.643
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