Estimating the boundary of the asymptotic stability region of Lotka–Volterra system by using the trajectory reversing method

Abstract

This paper proposes a topological approach for plotting the boundary of the region of asymptotic stability (RAS) of Lotka–Volterra predator-prey system. First, stability analysis was used to determine the specific saddle point that has eigenvalues with one positive and two negative real parts in a linearized Jacobian matrix. A set of initial states located around the saddle point on the specific eigenplane spanned by the two stable eigenvectors was then selected. Finally, the trajectory reversing method was used and the trajectories that had initial states on the eigenplane delineated the boundary of the asymptotic stability region. The trajectories of the initial states that started from the opposite sides of the RAS exhibited different dynamic behaviour. The numerical simulation are presented to demonstrate the effectiveness of the proposed approach.