The dynamic behavior of time-delayed Leslie-Gower predator-prey model

In this paper, we examine a delayed Leslie-Gower (LG) predator-prey system with the representation of interactions using a type II response function. It is considered that a prey that is afflicted can transmit the infection to another prey that is susceptible. The infection does not spread instantaneously but with a time delay to account for the necessary incubation period. A delay has the potential to disrupt a stable internal equilibrium. Given that adding harvesting to the prey population capable of mitigating abrupt population fluctuations, when surpassing the threshold level, we can stabilize the system. In this study, we consider both delayed and delay-free LG models. It is shown that the possibility of delay may diminish the stability region and create more complex border structures. New dynamical phenomena such as Hopf and Bautin bifurcations in both delayed and delay-free systems are investigated. Moreover, double-Hopf and resonance bifurcations are explored in delayed systems. By employing the method of multiple scales, we calculate the normal form coefficients for each bifurcation. A comprehensive analysis of the system's dynamic behavior in relation to two parameters, encompassing co-dimension 1, co-dimension 2, and the basin of attraction, is conducted using Lyapunov exponents and Poincaré sections. Numerical simulations validate our analytical findings.