Robust permanence for interacting structured populations

The dynamics of interacting structured populations can be modeled by where xi Rni, x=(x1,…,xk), and Ai(x) are matrices with non-negative off-diagonal entries. These models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of Ai(x). Necessary and sufficient conditions for robust permanence are derived using dominant Lyapunov exponents λi(μ) of the Ai(x) with respect to invariant measures μ. The necessary condition requires maxiλi(μ)>0 for all ergodic measures with support in the boundary of the non-negative cone. The sufficient condition requires that the boundary admits a Morse decomposition such that maxiλi(μ)>0 for all invariant measures μ supported by a component of the Morse decomposition. When the Morse components are Axiom A, uniquely ergodic, or support all but one population, the necessary and sufficient conditions are equivalent. Applications to spatial ecology, epidemiology, and gene networks are given.