Bifurcation and chaotic behaviour of a discrete-time variable-territory predator-prey model

Abstract

The dynamics of a discrete-time variable-territory predator–prey model is investigated in the closed first quadrant R2+. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of R2+ by using centre manifold and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis but also to exhibit the complex dynamical behaviours, such as period-9, -10, -17, -18, -35 orbits, cascades of period-doubling bifurcation in period-2, -4, -8, -16, -6, -12 orbits, quasi-periodic orbits and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviours.