Bifurcation analysis of a discrete Leslie–Gower predator–prey model with slow–fast effect on predator

Abstract

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter (Formula presented.), which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of (Formula presented.) because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of (Formula presented.) cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period-doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples.