Stability analysis on a predator-prey system with distributed delays (Q1381711)

A Lotka-Volterra predator-prey system with distributed delays is considered and local and global dynamical properties of two possible equilibria \(P_1= (x_0,0)\) and \(P_2= (x^*, y^*)\) are discussed. It is shown that when the delays are sufficiently small, if \(P_2\) does not exist, then \(P_1\) is globally asymptotically stable or globally attractive; otherwise, \(P_2\) is locally asymptotically stable. Furthermore, a region of explicit asymptotic stability is obtained for \(P_2\) based on a Lyapunov functional.