Bifurcation Analysis of a Discrete-Time Two-Species Model

We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant . It is proved that model has two boundary equilibria: , and a unique positive equilibrium under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: . It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.