Neimark‐Sacker bifurcation and hybrid control in a discrete‐time Lotka‐Volterra model

Abstract

We explore the local dynamics, N‐S bifurcation, and hybrid control in a discrete‐time Lotka‐Volterra predator‐prey model in . It is shown that parametric values, model has two boundary equilibria: and , and a unique positive equilibrium point: if . We explored the local dynamics along with different topological classifications about equilibria: , , and of the model. It is proved that model cannot undergo any bifurcation about and but it undergoes an N‐S bifurcation when parameters vary in a small neighborhood of by using a center manifold theorem and bifurcation theory and meanwhile, invariant close curves appears. The appearance of these curves implies that there exist a periodic or quasiperiodic oscillations between predator and prey populations. Further, theoretical results are verified numerically. Finally, the hybrid control strategy is applied to control N‐S bifurcation in the discrete‐time model.