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Predator–prey interactions are the ubiquitous and natural phenomenon in an ecological system. Predators reduce the prey population’s density by direct killing, which is an essential part of any ecological system. Based on the experimental works, for overcoming predation pressure, prey uses a variety of mechanisms. With Holling type-II functional response, we examined a prey–predator system incorporating anti-predator behavior and the cost of fear into prey. Prey anti-predator activity is a counterattacking strategy in which adult prey targets adolescent predators in order to counteract the potential predation pressure. Fear of predation may disrupt the physiological state of prey species and lead to long loss of prey species. In this study, we investigated this aspect to use a dynamical modeling approach. This research finds a plethora of fascinating phenomena. The studied system exhibits a wide range of dynamics and bifurcations, including saddle-node, Hopf, homoclinic, and a Bogdanov–Takens bifurcation in co-dimension two are among the dynamics and bifurcations observed in the analyzed system. We performed some numerical simulations to investigate the effects of anti-predator behavior and fear on prey and found both affect the prey–predator dynamics significantly. Our numerical examples clearly show that as prey carrying capacity increases, so does the prey’s ability to perceive the risk of predation.

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.