Global Dynamics of a Discrete Two-species Lottery-Ricker Competition Model

TL;DR

This paper analyzes the global behavior of a discrete two-species competition model, establishing conditions for species persistence, coexistence, and the structure of attractors, with implications for ecological modeling.

Contribution

It introduces the concept of relative permanence and characterizes the global dynamics and attractors of the model, extending understanding of coexistence in ecological systems.

Findings

Existence of a stable interior period-2 orbit under certain conditions

Presence of a compact interior attractor attracting almost all interior points

The basin of attraction may be an infinite union of connected components

Abstract

In this article, we study the global dynamics of a discrete two dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except a Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.