Multistability, chaos and mean population density in a discrete-time predator–prey system

Abstract

We investigate a discrete-time system derived from the continuous-time Rosenzweig–MacArthur (RM) model using the forward Euler scheme with unit integral step size. First, we analyze the system by varying carrying capacity of the prey species. The system undergoes a Neimark–Sacker bifurcation leading to complex behaviors including quasiperiodicity, periodic windows, period-bubbling, and chaos. We use bifurcation theory along with numerical examples to show the existence of Neimark–Sacker bifurcation in the system. The transversality condition for the Neimark–Sacker bifurcation at the bifurcation point is derived using a different approach compared to the existing ones. Multistability of different kinds: periodic–periodic and periodic–chaotic are also revealed. The basins of attraction for these multistabilities show complicated structures. The sufficient increase in the nutrient supply to the prey species may have negative effect in form of decrease in mean predator stock which leads to extinction of predator. Therefore, the paradox of enrichment is prominent in our discrete-time system. Further, we introduce prey and predator harvesting to the system. When the system is subjected to prey (or predator) harvesting, it stabilizes into equilibrium state. The system also exhibits complicated dynamics including multistability and Neimark–Sacker bifurcation when prey (or predator) harvesting rate is varied. With prey harvesting, the mean predator density increases when the system exhibits nonequilibrium dynamics. However, we have identified a situation for which the unstable equilibrium predator biomass decreases while mean predator density increases under predator mortality. Thus, this counter-intuitive phenomenon (positive effect on predator biomass) referred to as hydra effect is detected in our discrete-time system.