Torus doubling route to chaos and chaos eradication in delayed discrete-time predator–prey models

Abstract

This paper analyses a class of discrete-time delayed predator–prey models where time delay is incorporated into the prey's density-dependent growth term. We explore how time delay, carrying capacity, and harvesting can generate or suppress chaotic motion. Analytical conditions for local stability of the fixed point are derived by using the Routh–Hurwitz criterion. Increasing the time delay has the great potential to change the coexisting stable fixed point to unstable via a Neimark–Sacker bifurcation. Thus, the systems may experience quasi-periodic and chaotic motion when the coexisting fixed point is unstable. The existence of a quasi-periodic loop, which generates a 2-quasi-periodic loop, and then 4, 8, and 16 quasi-periodic loops are successively detected as the carrying capacity increases. Further increase of carrying capacity induces chaos. Thus, a torus doubling route to chaos is explored in our article, which has been relatively less reported in the existing literature. Nevertheless, we show that harvesting either prey or predator is more likely to suppress chaos and stabilize the coexisting steady state. We draw all the bifurcation diagrams when delay, carrying capacity, and harvesting strength are varied. The qualitative nature of the phase portraits is determined by computing all the Lyapunov exponents. The analysis of these delayed models in a discrete-time framework might attract a wide range of researchers to work in this direction. In the conclusion section, we also disclose some remaining questions and future perspectives. © 2022 John Wiley & Sons, Ltd.