Global stability for a Lotka-Volterra competition system with symmetric diffusion matrices.

Abstract

The global stability of a two-species Lotka-Volterra competition n-patch system with symmetric diffusion matrices is a conjecture proposed by Hofbauer, So, and Takeuchi. The cases n = 3, 4, 5 have been demonstrated to hold validity. In this article, it is shown that the Hofbauer-So-Takeuchi conjecture holds true for general n. A concrete example is given to display the global stability for one of the equilibria of the system depending on the magnitude of the parameters. Furthermore, an algebraic curve problem in the system's parameter space for determining the stability regions is proposed.