Bifurcation and chaos in a discrete-time predator–prey system of Holling and Leslie type

Abstract

A discrete-time predator–prey system of Holling and Leslie type with a constant-yield prey harvesting obtained by the forward Euler scheme is studied in detail. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, phase portraits display new and rich nonlinear dynamical behaviors. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period-1,2,11,17,19,22 orbits, attracting invariant cycles, and chaotic attractors of the discrete-time predator–prey system of Holling and Leslie type with a constant-yield prey harvesting. These results demonstrate that the integral step size plays a vital role to the local and global stability of the discrete-time predator–prey system with the Holling and Leslie type after the original continuous-time predator–prey system is discretized.