The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small 5]. It has been conjectured by Nagylaki and Lou 2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in 6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.