Criteria for Cr Robust Permanence

Abstract

Let x* i =x i f i (x) (i=1, ..., n) be a C r vector field that generates a dissipative flow , on the positive cone of R n . , is called permanent if the boundary of the positive cone is repelling. , is called C r robustly permanent if , remains permanent for sufficiently small C r perturbations of the vector field. A necessary condition and a sufficient condition for C r robust permanence involving the average per-capita growth rates f i d+ with respect to invariant measures + are derived. The necessary condition requires that inf + max i f i d+>0, where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone. The sufficient condition requires that the boundary flow admit a Morse decomposition [M 1 , ..., M k ] such that every M j satisfies min + max i f i d+>0 where the minimum is taken over invariant measures with support in M j . As applications, we provide a sufficient condition for C r robust permanence of Lotka Volterra models and a topological characterization of C r robust permanence for food chain models. Academic Press

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