Criteria for C Robust Permanence

Let x* i=xi fi (x) (i=1, ..., n) be a C r vector field that generates a dissipative flow , on the positive cone of R. , 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 fi d+ with respect to invariant measures + are derived. The necessary condition requires that inf+ maxi fi 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 [M1 , ..., Mk] such that every Mj satisfies min+ max i fi d+>0 where the minimum is taken over invariant measures with support in Mj . 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.