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Analysis of the periodically fragmented environment model. I: Species persistence. (English) Zbl 1066.92047
Summary: This paper is concerned with the study of stationary solutions of the equation \[ u_t - \nabla \cdot (A(x)\nabla u) = f(x,u), \quad x \in \mathbb R^N, \] where the diffusion matrix A and the reaction term \(f\) are periodic in \(x\). We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity conditions. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environments. The effects of various aspects of heterogeneities, such as environmental fragmentation, are also discussed.

92D40 Ecology
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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