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Effects of a degeneracy in the competition model. II: Perturbation and dynamical behaviour. (English) Zbl 1042.35017
This is the second part of the author in the study of the competition model $$\cases \frac{\partial u}{\partial t}-\Delta u=\lambda u-b(x)u^2-cuv\\ \frac{\partial v}{\partial t}-\Delta v=\mu v-v^2-duv,\endcases\tag 1$$ where $x\in\Omega$ and $t\ge 0$, $\Omega$ denotes a smooth bounded in $\bbfR^N$ $(N\ge 2)$, $b(x)\ge 0$, $\lambda,\mu,c$ and $d$ are positive constants. Moreover, the author supposes that $u$ and $v$ satisfy homogeneous Dirichlet boundary conditions on $\partial \Omega$. The aim of the author is to understand the effects of the degeneracy of $b(x)$ on (1). Here, based on results obtained in Part I [ibid. 181, 92--132 (2002; Zbl 1042.35016)] the author shows that both the classical and generalized steady-states of (1) occur naturally as the limits, when $\varepsilon\to 0$, of the positive classical solutions of the perturbed system $$\cases -\Delta u=\lambda u-\bigl[b(x)+ \varepsilon \bigr]u^2-cuv\\ -\Delta v=\mu v-v^2-duv\\ u\vert_{\partial\Omega}=0,\ v\vert_{\partial\Omega}=0,\endcases\tag 2$$ where $\varepsilon>0$ is a constant.

35K50Systems of parabolic equations, boundary value problems (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
92D25Population dynamics (general)
Full Text: DOI
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