Selection and Impact of Anti-Predation Strategies in a Predator-Prey Model with Prey Refuge and Group Defense

In population ecology, the interactions between predators and prey encompass more than just direct predation and killing. Various indirect effects also significantly influence the system's dynamics. In this paper, we consider a refuge proportional to the prey population and assume that only unsuccessful refuge prey engage in group defense mechanisms. We introduce an exponential response function to model this behavior, taking into account the negative impact of defense measures on the prey growth rate. Subsequently, we analyze the non-negativity, boundedness, and persistence of the system. Furthermore, we study the existence and stability of different types of equilibrium points, and provide theoretical proofs for bifurcations in the system, such as transcritical, pitchfork, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Finally, we verify our theoretical results through numerical simulations. Our study reveals that variations in refuge size and the intensity of group defense can alter the number and stability of equilibrium points, leading to phenomena such as bubbling and periodic oscillations. Additionally, group defense can induce bistability. Both prey refuge behavior and excessive group defense can lead to a reduction in predator numbers and eventually to their extinction. Moreover, compared to prey refuge, group defense as an anti-predator behavior has a greater effect on predators.