Analysis of the periodically fragmented environment model : I - Influence of periodic heterogeneous environment on species persistence.
This paper is concerned with the study of the stationary solutions of the equation $$u_t-\nabla\cdot(A(x)\nabla u)=f(x,u),\ \ x\in{\mathbb{R}}^N,$$ 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 condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.