Complexity and chaos control in a discrete-time Lotka–Volterra predator–prey system

Abstract

ABSTRACT In this paper, we investigate a discrete-time Lotka–Volterra predator–prey model that can undergo flip bifurcations and Hopf bifurcations by using the centre manifold theorem and bifurcation theory. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits verify the theoretical analysis, which exhibits complex dynamical behaviours such as period-7, 13, 14, 21, 27, 32 orbits, a cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. Furthermore, the feedback control method is used to stabilize the chaotic orbits at an unstable fixed point. This result further reveals richer dynamics of the discrete model compared with the continuous model. It is found out that the appropriately carrying capacity can be helpful to stabilize the model. Instead, the higher or lower carrying capacity may destabilize the system producing more complex dynamical behaviours.