Criteria for Cr Robust Permanence

Let xi=xifi(x) (i=1, …, n) be a Cr vector field that generates a dissipative flow φ on the positive cone of Rn. φ is called permanent if the boundary of the positive cone is repelling. φ is called Crrobustly permanent if φ remains permanent for sufficiently small Cr perturbations of the vector field. A necessary condition and a sufficient condition for Cr robust permanence involving the average per-capita growth rates ∫ fi dμ with respect to invariant measures μ are derived. The necessary condition requires that infμ maxi ∫ fi dμ>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 minμ maxi ∫ fi dμ>0 where the minimum is taken over invariant measures with support in Mj. As applications, we provide a sufficient condition for Cr robust permanence of Lotka–Volterra models and a topological characterization of Cr robust permanence for food chain models.