Bifurcation for the Lotka-Volterra competition model

TL;DR

This paper investigates the bifurcation phenomena in a two-species competition model described by PDEs, analyzing local and global bifurcations, stability, and limiting behaviors as competition intensifies.

Contribution

It provides a detailed bifurcation analysis of the Lotka-Volterra competition system, including local and global bifurcations, stability results, and asymptotic behavior as competition rate increases.

Findings

Identification of bifurcation points and directions.

Instability of constant solutions under certain conditions.

Asymptotic profiles as competition rate tends to infinity.

Abstract

We analyze the bifurcation phenomenon for the following two-component competition system: \begin{equation*} \begin{cases} -\Delta u_1=\mu u_1(1-u_1)-\beta \alpha u_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, -\Delta u_2=\sigma u_2(1-u_2)-\beta \gamma u_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, \frac{\partial u_1}{\partial n}= \frac{\partial u_2}{\partial n} =0,&\text{on}\ \partial B_1, \end{cases} \end{equation*} where $N\ge 2$, $\alpha>\gamma>0$, $\sigma\ge\mu>0$ and $\beta>\frac{\sigma}{\gamma}$. More precisely, treating $\beta$ as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis. Lastly, under the condition $\sigma=\mu$, we demonstrate the limiting configuration on each bifurcation branch as the competition rate $\beta\rightarrow+\infty$.