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Existence and uniqueness of coexistence states for a predator-prey model with diffusion. (English) Zbl 0823.35050
The authors investigate the problem $-\Delta u= \lambda u- u^ 2- b{\textstyle {{uv} \over {\gamma+u}}},\;\;-\Delta v= \mu v- v^ 2+ c{\textstyle {{uv} \over {\gamma+u}}} \text{ in } \Omega \subset \mathbb{R}^ n, \quad u,v=0 \text{ on } \partial \Omega.$ $$\lambda$$, $$\mu$$, $$b$$ and $$c$$ are nonnegative constants and $$\gamma$$ is a positive constant. This is a predator-prey problem modelling the interaction between a predator, with population density $$v(x)$$ and a prey, with population density $$u(x)$$ living in the region $$\Omega$$. One is interested in positive solutions $$(u,v)$$. First the authors present numerical results for coexistence states for sets of parameter values. Then, some necessary and some sufficient conditions for coexistence are given. The problem of uniqueness and of stability is also discussed. Simple conditions insuring coexistence in terms of the principal eigenvalues of some semilinear elliptic equation are given.
Reviewer: G.Porru (Cagliari)

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 47J05 Equations involving nonlinear operators (general)
##### Keywords:
uniqueness; stability; principal eigenvalues