Stability Analysis of a Four-Species Periodic Diffusive Predator–Prey System with Delay and Feedback Control

In this work, we present a novel four-species periodic diffusive predator-prey model, which incorporates delay and feedback control mechanisms, marking substantial progress in ecological modeling. This model offers a more realistic and detailed portrayal of the intricate dynamics of predator-prey interactions. Our primary objective is to establish the existence of a periodic solution for this new model, which depends only on time variables and is independent of spatial variables (we refer to it as a spatially homogeneous periodic solution). By employing the comparison theorem and the fixed point theorem tailored for delay differential equations, we derive a set of sufficient conditions that guarantee the emergence of such a solution. This analytical framework lays a solid mathematical foundation for understanding the periodic behaviors exhibited by predator-prey systems with delayed and feedback-regulated interactions. Moreover, we explore the global asymptotic stability of the aforementioned periodic solution. We organically combine Lyapunov stability theory, upper and lower solution techniques for partial differential equations with delay, and the squeezing theorem for limits to formulate additional sufficient conditions that ensure the stability of the periodic solution. This stability analysis is vital for forecasting the long-term outcomes of predator-prey interactions and evaluating the model's resilience against disturbances. To validate our theoretical findings, we undertake a series of numerical simulations. These simulations not only corroborate our analytical results but also further elucidate the dynamic behaviors of the four-species predator-prey model. Our research enhances our understanding of the complex interactions within ecological systems and carries significant implications for the conservation and management of biological populations.