Dynamic complexity in a discrete-time predator-prey system with Holling type I functional response: Gompertz growth of prey population

Abstract

We consider a discrete-time predator-prey system with Holling type I functional response and Gompertz growth of prey population to study its dynamic behaviors. We algebraically show that the predator-prey system undergoes a flip bifurcation (FB) and Neimark-Sacker bifurcation (NSB) in the interior of R2+ when one of the model parameter crosses its threshold value. We determine the existence conditions and direction of bifurcations by using the center manifold theorem and bifurcation theorems. We present numerical simulations to illustrate theoretical results which include the bifurcation diagrams, phase portraits, appearing or disappearing closed curves, periodic orbits, and attracting chaotic sets. In order to justify the existence of chaos in the system, maximum Lyapunov exponents (MLEs) and fractal dimension (FD) are computed numerically. Finally, chaotic trajectories have been controlled by applying feedback control method.

Keywords

• predator-prey system
• gompertz growth
• bifurcations
• lyapunov exponents
• feedback control