Stability and bifurcation in a predator-prey system with effect of fear and additional food

In the present study, we propose and analyze a three-dimensional prey-predator model. The prey grows logistically in the absence of the predator and their relationship follows the Crowley-Martin type functional response. In this paper, we examine the impact of supply of the additional food to the predators and the influence of fear in the prey population. Since the predator depends partially on the provided other resources, we incorporate a novel parameter, the degree of dependence, which basically demonstrates how dependent the predator is on the prey population. We investigate the steady-state solutions, and their local and global behavior, which contributes to understanding the long-term dynamics of the interaction. We have shown that the degree of dependence and the cost of fear both can cause periodic orbits to appear in the system via a Hopf-bifurcation. Our findings show that with the newly introduced parameter, we can control the oscillations from the system, which helps to balance the ecosystem. The direction and stability have also been investigated using the center manifold theorem and normal form theory. Last, we perform an extensive numerical simulation to validate our theoretical findings. Our main goal of this work is to maintain the ecological balance in the presence of fear effect and additional food for predators.