Long-term dynamics of discrete-time predator–prey models: stability of equilibria, cycles and chaos

In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.