Complex Dynamical Behaviors of a Leslie-Gower Predator-Prey Model with Herd Behavior

Abstract

In this paper, we consider a Leslie-Gower predator-prey model with a square root functional response while prey forms a herd as a form of group defense. We show that the solution of the system is non-negative and bounded. By applying the blow-up technique, it can be deduced that the origin displays instability. Moreover, employing the proof-by-contradiction approach, we demonstrate that the unique equilibrium point can be globally asymptotically stable under certain conditions. The sufficient conditions for the occurrence, stability, and direction of Hopf bifurcation are obtained. We further explore the conditions for the existence and uniqueness of the limit cycle. Theoretical results are validated through numerical simulations. Thus, our findings reveal that herd behavior has an important impact on the Leslie-Gower prey-predator system.