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On a system of reaction–diffusion equations describing a population with two age groups. (English) Zbl 1031.35057

In this paper the authors study the problem \[ \begin{alignedat}{2} - d_1\Delta u &= a(x)v- c(x)u- eu[u+ v]\quad &&\text{for }x\in\Omega,\\ - d^2\Delta v &= b(x)u- d(x)v- fv[u+ v]\quad &&\text{for }x\in\Omega,\\ {\partial u\over\partial n}(x) &= {\partial w\over\partial n} (x)= 0\quad &&\text{for }x\in \partial\Omega,\end{alignedat} \] where \(\Omega\) is a bounded region with smooth boundary in \(\mathbb{R}^N\), \(a\), \(b\), \(c\), \(d\) are smooth functions positive on \(\Omega\) and \(d_1\), \(d_2\), \(e\), \(f\) are positive constants.
This problem models the steady-state solution of a population which is subdivided into two sub-population: adults and juveniles.
Necessary and sufficient conditions for the existence of a classical positive solution and stability results for nonnegative solutions in terms of the principal eigenvalue of the corresponding linearized system are given. It is used the fact that the corresponding linearized system is a cooperative system (there is defined a notion of semilinear elliptic cooperative system).

MSC:

35K57 Reaction-diffusion equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B35 Stability in context of PDEs
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References:

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