Robust permanence for ecological maps

TL;DR

This paper establishes conditions for the robust permanence of ecological difference equations, ensuring species coexistence remains stable under perturbations, with applications to ecological and epidemiological models.

Contribution

It provides necessary and sufficient conditions for robust permanence based on Lyapunov exponents, advancing understanding of stability in ecological systems.

Findings

Conditions for permanence and robust permanence are characterized.

Lyapunov exponents determine stability of species coexistence.

Applications demonstrate ecological and epidemiological relevance.

Abstract

We consider ecological difference equations of the form $X_{t+1}^i =X_t^i A_i(X_t)$ where $X_t^i$ is a vector of densities corresponding to the subpopulations of species $i$ (e.g. subpopulations of different ages or living in different patches), $X_t=(X_t^1,X_t^2,\dots,X_t^m)$ is state of the entire community, and $A_i(X_t)$ are matrices determining the update rule for species $i$. These equations are permanent if they are dissipative and the extinction set $\{X: \prod_i \|X^i\|=0\}$ is repelling. If permanence persists under perturbations of the matrices $A_i(X)$, 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.