Permanence of a general discrete-time two-species-interaction model with non-monotonic per capita growth rates

TL;DR

This paper investigates the long-term persistence of a two-species interaction model with non-monotonic growth rates, introducing new conditions based on Lyapunov functions and extended relative nonlinearity.

Contribution

It provides the first analysis of permanence in discrete-time two-species models with non-monotonic growth rates using novel mathematical techniques.

Findings

Derived a simple sufficient condition for species permanence.

Extended the concept of relative nonlinearity to characterize nonlinear effects.

Applied results to competition and predator-prey models with non-monotonic growth.

Abstract

Combined with all density-dependent factors, the per capita growth rate of a species may be non-monotonic. One important consequence is that species may suffer from weak Allee effects or strong Allee effects. In this paper, we study the permanence of a discrete-time two-species-interaction model with non-monotonic per capita growth rates for the first time. By using the average Lyapunov functions and extending the ecological concept of the relative nonlinearity, we find a simple sufficient condition for guaranteeing the permanence of systems that can model complicated two-species interactions. The extended relative nonlinearity allows us to fully characterize the effects of nonlinearities in the per capita growth functions with non-monotonicity. These results are illustrated with specific two species competition and predator-prey models of generic forms with non-monotone per capita growth rates.