Global dynamics of a Lotka-Volterra competition patch model

TL;DR

This paper analyzes the long-term outcomes of a two-species competition model across multiple patches, revealing conditions for species coexistence or exclusion based on competition strength and dispersal rates.

Contribution

It provides a comprehensive classification of global dynamics for the Lotka-Volterra competition patch model with asymmetric dispersal, using monotone dynamical systems and graph theory.

Findings

Either one species goes extinct or both coexist in equilibrium.

Outcome depends on inter-specific competition and dispersal rates.

Conditions for coexistence or exclusion are explicitly characterized.

Abstract

The global dynamics of the two-species Lotka-Volterra competition patch model with asymmetric dispersal is classified under the assumptions of weak competition and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. It is shown that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs follow the theory of monotone dynamical system and a graph-theoretic approach based on the Tree-Cycle identity.