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Existence and uniqueness for some competition models with diffusion. (English. Abridged French version) Zbl 0741.92018
Let \(D\) be a bounded domain in \(R^ n\) with smooth boundary, and let \(a\), \(b\), \(c\), \(d\) be real constants with \(b\), \(c\) positive. Consider the stationary Lotka-Volterra system for two competing species: \[ -\Delta u=au-u^ 2-buv, \quad -\Delta v=dv-cuv-v^ 2\hbox{ in } D,\tag{*} \] under homogeneous Dirichlet boundary conditions. The authors prove an existence and uniqueness result for positive solutions (coexistence states) to (*) provided the constants \(a,\dots,d\) satisfy certain inequalities involving the positive solutions of the equation \(-\Delta w=kw-w^ 2\), \(k\in \)R. In particular, these inequalities are satisfied if \(b\) and \(c\) are sufficiently small.

MSC:
92D40 Ecology
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
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