Bifurcations of a prey-predator system with fear, refuge and additional food

In the predator-prey system, predators can affect the prey population by direct killing and inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. Therefore, it proposes a predator-prey model with anti-predation sensitivity induced by fear and Holling-Ⅱ functional response in the present paper. Through investigating the system dynamics of the model, we are interested in finding how the refuge and additional food supplement impact the system stability. With the changes of the anti-predation sensitivity (the refuge and additional food), the main result shows that the stability of the system will change accordingly, and it has accompanied with periodic fluctuations. Intuitively the bubble, bistability phenomena and bifurcations are found through numerical simulations. The bifurcation thresholds of crucial parameters are also established by the Matcont software. Finally, we analyze the positive and negative impacts of these control strategies on the system stability and give some suggestions to the maintaining of ecological balance, we perform extensive numerical simulations to illustrate our analytical findings.