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

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  • 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.

    Mathematics Subject Classification: Primary: 35K57; Secondary: 35B35.

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