Robust Permanence for Ecological Maps | SIAM Journal on Mathematical Analysis | Vol. 49, No. 5 | Society for Industrial and Applied Mathematics

We consider ecological difference equations of the form xt+1 = x i tAi(xt), where x i t is a vector of densities corresponding to the subpopulations of species i (e.g., subpopulations of different ages or living in different patches), xt = (xt , x 2 t , . . . , x m t ) is the state of the entire community, and Ai(xt) are matrices determining the update rule for species i. These equations are permanent if they are dissipative and the extinction set {x : ∏i ‖xi‖ = 0} is repelling. If permanence persists under perturbations of the matrices Ai(x), the equations are robustly permanent. We provide sufficient and necessary conditions for robust permanence in terms of Lyapunov exponents for invariant measures supported by the extinction set. Applications to ecological and epidemiological models are given.