The Dynamic among the Producer, Consumer, and Predator in the Presence of Fear, Hunting Cooperation, and Anti-Predator Behavior

In general, the dynamic interactions that form the relationships between predators and prey within a food chain depend heavily on fear, hunting cooperation, and anti-predator defenses. These elements support species’ survival in their particular habitats, balance the ecosystem, and control the numbers of predators and prey. Therefore, a three-species food chain system consisting of producer-consumer-predator, which plays a crucial role in ecological dynamics, has been formulated mathematically. The objective is to investigate the influence of fear, hunting cooperation, and anti-predators on the dynamic behavior of the food chain system. Two types of functional responses have been utilized. The characteristics of the solution have been investigated. The existence and local stability conditions of the possible equilibrium have been established. The Lyapunov function is used to reach global behavior whenever possible. The persistence requirements of the food chain are determined. Local bifurcations, including Hopf bifurcations, are investigated. A numerical simulation is finally performed to validate the theoretical result and identify the control set of parameters that regulate the food chain system. It is observed that the system is rich in its dynamics, including stable point, periodic, and bi-stable behavior.