Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems

作者：Tang, Ying^*; Yuan, Ruoshi; Ma, Yian

来源：Physical Review E, 2013, 87(1): 012708.

DOI：10.1103/PhysRevE.87.012708

• Basins of attraction
• Boltzmann-Gibbs distribution
• Competitive Lotka-Volterra systems
• Dynamical behaviors
• Generalized gradients
• Multiplicative noise
• Steady state

Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added. DOI: 10.1103/PhysRevE.87.012708

• 出版日期2013-1-16
• 单位上海交通大学