Chaos and Control in a Delayed Leslie-Gower Predator-Prey Model via Z-type control strategy

Abstract

This study explores the dynamics and control of a delayed Leslie-Gower predator-prey system incorporating an indirect Z-type control strategy. The model accounts for two biologically motivated delays: one arising from crowding effects in the prey population, and another representing feedback delay in predator growth. It also includes a Holling type-II functional response. Through numerical simulations and bifurcation analysis, we demonstrate that these delays can destabilize the system, leading to sustained oscillations and chaotic behavior, as confirmed by a positive maximum Lyapunov exponent. To mitigate such complex dynamics, we implement an indirect Z-type controller targeting the prey population via predator regulation. This control mechanism dynamically adjusts predator density based on a tracking error between the actual and desired prey levels. Our results show that the Z-type control successfully suppresses both limit cycle and chaotic oscillations, guiding the system toward a stable equilibrium or a biologically feasible periodic solution. These findings highlight the potential of Z-type control as a robust tool for stabilizing ecological systems subject to delay-induced instabilities.