Population oscillations in a three-species food chain model and their possible control: a Z-type control approach

Abstract

In our experiment, we investigate a model of a three-species food chain that includes the predation-driven Allee effect. Within this framework, we propose and analyze an application of a Z-type control mechanism. Our findings reveal that the predation pressure exerted by the intermediate predator on the prey population induces chaos through period-doubling bifurcation. To confirm the presence of chaos, we compute the Lyapunov exponents. Subsequently, we introduce an indirect Z-control mechanism across different trophic levels of the food chain model. Specifically, we apply the Z-control method to the top predator to regulate the population fluctuations of the intermediate predator. Furthermore, we employ the Z-type controller on the intermediate predator population to govern the oscillations in the prey population, leading to the eradication of chaos from the system. Additionally, we observe that gestation delay in the top predator can trigger population oscillations within the system. We also find that implementing the indirect Z-controller to the top predator population can suppress population oscillations in the delayed system. Our research establishes the fundamental concept that the Z-type control method serves as the most efficient controller across various dynamical regimes of the uncontrolled system. It facilitates the regulation of all kinds of oscillations within the system and can replace population oscillation with a predefined stable steady-state, which is vital for promoting ecosystem stability. Our experiment emphasizes the importance of control mechanisms in managing population fluctuations and highlights the potential of Z-type control mechanisms in predator–prey interaction strategies.