## 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 $\begin{cases} \frac{\partial u}{\partial t}-\Delta u=\lambda u-b(x)u^2-cuv\\ \frac{\partial v}{\partial t}-\Delta v=\mu v-v^2-duv,\end{cases}\tag{1}$ where $$x\in\Omega$$ and $$t\geq 0$$, $$\Omega$$ denotes a smooth bounded in $$\mathbb{R}^N$$ $$(N\geq 2)$$, $$b(x)\geq 0$$, $$\lambda,\mu,c$$ 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 $\begin{cases} -\Delta u=\lambda u-\bigl[b(x)+ \varepsilon \bigr]u^2-cuv\\ -\Delta v=\mu v-v^2-duv\\ u|_{\partial\Omega}=0,\;v|_{\partial\Omega}=0,\end{cases}\tag{2}$ where $$\varepsilon>0$$ is a constant.

### MSC:

 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)

Zbl 1042.35016
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### References:

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