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

We study a genetic model with two alleles A₁and A₂in a bounded smooth habitatω. The frequency u of the allele A₁, under the combined influence of migration and selection, obeys a parabolic equation of the type Math equiten presented where δdenotes the Laplace operator, g may change sign in ω, and f(0) = f(1) = 0,f(s) >0 for sε(0,1). Our main results include stability/instability of the trivial steady states uξO and uξ1, 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 Dirichiet and Robin boundary value problems as well.