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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 \[ \begin{cases} \frac{\partial u}{\partial t}-\Delta u=\lambda u-b(x)u^2-cuv\\ \frac{\partial v}{\partial t}-\Delta v=\mu v-v^2-duv,\end{cases}\tag{1} \] 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)

Citations:

Zbl 1042.35016
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References:

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