Backward bifurcation of predator–prey model with anti-predator behaviors

In this study, we consider a predator–prey model with stage structure and anti-predator behavior such that the adult prey can counterattack their predators. We first investigate the existence and stability of the equilibria. Especially, we verify that there can exist at most one positive equilibrium, which is always stable whenever it exists, if the predator only feed on one age class. We then prove that the system can undergo either a forward bifurcation or a backward bifurcation. Numerical analyses show that anti-predator behavior is beneficial to the growth of prey population, especially helps the equilibrium level of the prey population increase, by enhancing the pressure on the predator. Moreover, anti-predator behavior makes the coexistence of the predator and prey less likely by shrinking the coexistence region with respect to the initial conditions or weakening the existence and stability of the positive equilibrium.