Dynamics of a patchy predator-prey model with non-local predator maturation

This paper presents a comprehensive analysis of a time-delayed predator-prey model with  dispersal among multiple patches, considering the critical threshold parameter, [Formula: see text], as the net reproduction number. Our research highlights the significance of  [Formula: see text] in determining the global behavior of the system. We demonstrate that when  [Formula: see text], the predator-free equilibrium is proven to be globally attractive, while for  [Formula: see text], this equilibrium becomes unstable. Furthermore, we reveal  that, under specific additional conditions, the predator-free equilibrium can exhibit  local asymptotic stability when [Formula: see text]. In cases where [Formula: see text], the model exhibits uniform persistence and supports the existence of at least one coexistence equilibrium.  Our practical application in a two-patch environment with linear release rates of natural enemies yields several noteworthy insights. Firstly, we observe that   [Formula: see text] exhibits a gradual decrease with an increase in maturation delay, indicating the impact of this delay on the system’s stability.  Secondly, we highlight that for cases where  [Formula: see text],  an increase in maturation delay can lead to system destabilization, posing challenges for effective pest control. Finally, we emphasize that alterations in the dispersal rates of both prey and predator species can have a profound influence on the survival or extinction of the predator population.Notably, our findings indicate that increasing the release rates of natural enemies may not always be the optimal strategy for pest control due to the complex interplay of dispersal effects. Our research provides valuable insights into the management of predator-prey interactions in multi-patch environments, offering guidance for more effective ecological pest control strategies.