Bifurcation for the Lotka–Volterra competition model

Abstract

We analyze the bifurcation phenomenon for the following two-component competition system:

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 bifurcation locations and determining 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 $\beta \to +\infty$.