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*all possible*solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done

*without*the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.

$ d\Delta u+g(x)u^{2}(1-u)=0 \ $
in Ω ,

$ 0\leq u\leq 1 $in Ω and $ \frac{\partial u}{\partial\nu}=0 $ on ∂Ω,

where $\Delta$ is the Laplace operator, $\Omega$ is a bounded
smooth domain in $\mathbb{R}^{N}$ with $\nu$ as its unit outward
normal on the boundary $\partial\Omega$, and $g$ changes sign in $\Omega$. This equation models the "complete dominance" case in population genetics of two alleles. We show that the
diffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ play
important roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines the
limiting profile of solutions as $d$ tends to $0$. In particular, a conjecture of Nagylaki and Lou has been largely resolved.

Our results and methods cover a much wider class of nonlinearities than $u^{2}(1-u)$, and similar results have been
obtained for Dirichlet and Robin boundary value problems as well.

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