Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system

Abstract

In this study, complex dynamics of a classical discrete-time predator–prey system are investigated. Rigorous results on the existence and stability of fixed points of this system are derived. It can also be shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation and codimension-two bifurcation associated with 1:2 resonance using the ideas of center manifold theorem, bifurcation theory and the normal form method. Specially, we give the explicit approximate expression of the invariant curve which is caused by the Neimark–Sacker bifurcation. At the same time, bifurcation phenomena and chaotic features are justified numerically via computing Lyapunov exponent spectrum. Results of numerical simulation verify our theoretical analysis. Finally, we extend the hybrid control strategy (state feed back and parameter perturbation) to control flip bifurcation and Neimark–Sacker bifurcation in two-dimensional discrete system.