Lotka-Volterra competition models on finite graphs

TL;DR

This paper analyzes Lotka-Volterra competition models on finite graphs, revealing conditions for species coexistence or extinction based on competitiveness and initial populations, using upper and lower solutions methods.

Contribution

It introduces a novel application of upper and lower solutions techniques to weakly coupled parabolic systems on finite graphs for the first time.

Findings

Species coexistence depends on competitiveness and initial populations.

Extinction or coexistence outcomes are characterized for different boundary conditions.

Partially answers a question posed by Slavík (2020).

Abstract

In this paper, we study three two competing species Lotka-Volterra competition models on finite connected graphs, with Dirichlet, Neumann or no boundary conditions. We get that when time goes to infinity, either one specie extincts while the other becomes surviving or both competing species coexist, which depend crucially on the strengh of species' competitiveness and the size of the initial population under the Neumann boundary condition and the condition that there is no boundary condition. One of our results partially answer a question posed by Slav\'{\i}k in [SIAM J. Appl. Dyn. Syst., 19 (2020)]. The critical techiniques in the proof of our main results are upper and lower solutions method, which are developed for weakly coupled parabolic systems on finite graphs in this article.