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Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions. (English) Zbl 1042.35016
This paper is devoted to how heterogeneous environmental effects can greatly change the dynamical behaviour of a population model. The author considers the competition model given by $$\cases \frac{\partial u}{\partial t}-d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv\\ \frac{\partial v}{\partial t}-d_2(x)\Delta v=\mu a_2(x)v-e(x)v^2-d(x)uv\endcases\tag 1$$ where $x\in\Omega$ and $t\ge 0$, $\Omega$ denotes a smooth bounded domain in $\bbfR^N$ $(N\ge 2)$, $d_1,d_2,a_1,a_2,b,c,d,e$ are nonnegative functions over $\Omega$, and $\lambda,\mu$ are positive constants. Throughout this paper the author supposes that $u$ and $v$ satisfy homogeneous Dirichlet boundary conditions. The author shows that there exists a critical number $\lambda_*$ such that (1) behaves as if $b(x)$ is a positive constant if $\lambda<\lambda_*$, but if $\lambda>\lambda_*$, then interesting new phenomena occur. This shows that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well.

MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B40 Asymptotic behavior of solutions of PDE 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)
Full Text:
References:
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