Stability analysis on a predator-prey system with distributed delays

This paper concerns the local and global dynamical properties of the nonnegative and positive equilibria of a Lotka-Volterra predator-prey system with distributed delays. It is shown that, while the positive equilibrium does not exist, the nonnegative equilibrium is globally asymptotically stable or globally attractive as long as the delays are small enough. If the positive equilibrium exists, it is shown that it is locally asymptotically stable when the delays are suitably small. Furthermore, an explicit asymptotic stability region for the positive equilibrium is also obtained based on a Liapunov functional.