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Stable coexistence states in the Volterra-Lotka competition model with diffusion. (English) Zbl 0562.92012

The authors consider the Lotka-Volterra system \[ u_ t=k_ 1\Delta u+u[a-bu-cv],\quad v_ t=k_ 2\Delta v+v[d-eu-fv] \] in a cylinder \(x\in \Omega\), \(0<t<\infty\), where \(\Omega\) is an open, bounded and smooth domain in \({\mathbb{R}}^ 2\). The functions a, b, c, d, e, f are assumed to be smooth and nonnegative on \({\bar \Omega}\times (0,\infty)\), and the diffusion constants \(k_ 1\) and \(k_ 2\) are positive. These equations are supplemented by von Neumann boundary conditions when the growth rates are periodic, and by Dirichlet boundary conditions when the growth rates are constant.
The existence of periodic solutions when the growth rates are periodic, and existence, uniqueness and stability of time independent solutions when the growth rates are constant are proved.
Reviewer: I.Onciulescu

MSC:

92D25 Population dynamics (general)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs
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