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Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions. (English) Zbl 0702.35123
Author’s abstract: “This paper concerns a system of reaction-diffusion equations that describes the evolution of population densities of a prey species u and a predator species v inhabiting the same bounded domain. Under homogeneous Dirichlet boundary conditions, asymptotic stability properties of nonnegative steady states are discussed. The corresponding steady-state problem has nonnegative solutions of three different types; the trivial solution (0,0), the semitrivial solutions (u,0), (0,v) with u,v positive, and a positive solution (u,v) with both components positive. Stability properties of the trivial and semitrivial solutions are determined completely. The stability and uniqueness of positive solutions are also studied. This method is based on spectral analysis, comparison principle, and bifurcation theory.”
Reviewer: G.Gussi

35K55 Nonlinear parabolic equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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