Dynamical analysis of a two-dimensional discrete predator–prey model

Abstract

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.