Determine dynamical behaviors by the Lyapunov function in competitive Lotka-Volterra systems

TL;DR

This paper constructs explicit Lyapunov functions for 2D, 3D, and May-Leonard competitive Lotka-Volterra systems, providing insights into their global dynamics, including bistability and cycles.

Contribution

It introduces a method to explicitly construct Lyapunov functions for complex 3D competitive Lotka-Volterra systems, enhancing understanding of their global behaviors.

Findings

Explicit Lyapunov functions for 2D, 3D, and May-Leonard models

Analysis of bistable and cyclical dynamics

Method aids in understanding limit cycles in 3D systems

Abstract

Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while the model of May-Leonard is not. Our method is helpful to understand the limit cycle problems in general 3-dimensional case.