Balancing Fear and Survival: Stability and Chaos in a Three-Species Web

This study investigates a discrete-time ecological system with one prey and two predators, incorporating modified Leslie-Gower (LG) type growth for the top predator. The system explores the intricate dynamics arising from the unique ecological interactions within the predator-prey framework. Notably, one predator is entirely reliant on the prey, while the top predator contends with a pronounced scarcity of its primary food source. This scarcity imposes constraints on the top predator's growth, leading to a complex interplay of population dynamics. Additionally, the growth of the middle predator is influenced by a fear effect induced by the top predator. The present study systematically explores various fixed points, delineating their stability dynamics and establishing threshold values for key parameters. Feasibility, local stability conditions of equilibria, and a comprehensive set of sufficient conditions for the global stability of the interior equilibrium point are identified. Analytical investigations uncover the existence of periodic points and one-parameter Hopf bifurcations. Furthermore, the research delves into the parameter ranges associated with different bifurcations, including a comprehensive examination of a two-parameter bifurcation scenario. The simultaneous impact of fear effect and intraspecific competition is explored, revealing a novel pattern in three-dimensional space. Two-parameter stability regions shed light on the system's complexity, highlighting the top predator's pivotal role in determining long-term species densities, contrary to the traditional emphasis on prey dependence. The study concludes by discussing the presence of chaos and its hybrid control within this ecological context.