Hopf bifurcation and Bautin bifurcation in a prey–predator model with prey’s fear cost and variable predator search speed

This paper considers an extended predator–prey model, in which the fear of predators reduces prey reproduction and the search speed of predators depends on prey density. First, our mathematical analysis shows that high levels of fear can stabilize the coexistence steady state, while low levels would result in periodic oscillation. Comparing to the model ignoring fear where supercritical bifurcation occurs, Hopf bifurcation in our model can be both supercritical and subcritical, which leads to bi-stability and two limit cycles. Second, our analysis demonstrates that a relatively small search speed of predators can promote stability of the coexistence steady state, while a large speed would lead to periodic oscillation. Comparing to the model with invariant search speed where Hopf bifurcation takes place, Bautin bifurcation occurs in our model, which results in tri-stability and three limit cycles. While the paradox of enrichment always takes place in the Holling-type II predation model, it does not occur here when the search speed is small. Even when the speed is large, the prey species can adapt by enhancing their fear level and stabilize the system effectively. Third, our analysis shows that enhancing prey’s sensitivity to predation risk or slowing the predator search speed, can stabilize the coexistence steady state, while a low sensitivity and a high speed will lead to periodic oscillation. Numerical simulations confirm and extend our results.