Dynamics of additional food provided predator-prey system with habitat fragmentation

Due to human-caused deforestation, global warming, and other environmental factors, habitat fragmentation became widespread. This fragmentation has a variety of detrimental repercussions for many species as well as humans, especially in the agricultural economy. It causes insect outbreaks, the expansion of alien species, and disrupts biological management by rendering the habitat unsuitable for natural enemies in agriculture. Providing natural enemies with additional food is one method to improve the ecosystem and support them. In this study, we assumed that the ecosystem is separated into two patches and that predators can easily migrate from one patch to the next, while prey stays inside its patch's territory. We looked at the impact of offering more food to predators in a patchy environment using dynamical systems theory. The permanence, stability, and various bifurcations that occur in the system are studied using a rigorous mathematical analysis. The study looks at how predator's access to other food sources affects pest management. By adjusting the provided food's characteristic qualities, such as (nutritional) quality and quantity, one can limit and manage the pest in one or both patches, as well as eliminate predators from the ecosystem. This research reveals that providing predators with additional food (of specified quality and quantity) can help in controlling chaotic behavior in the system. The findings of the study are supported by numerical simulations.