Influence of fear effect and predator-taxis sensitivity on dynamical behavior of a predator-prey model

Experiments show that the fear of predators reduces the birth rate of the prey population, but it does not cause the extinction of the prey population. Even if the fear is sufficiently large, the prey can survive under the saturated fear cost. Moreover, the sensitivity of prey to predator will affect the population density of prey and predator. It is feasible to introduce the saturated fear cost and predator-taxis sensitivity into the predator-prey interactions model. In this paper, we obtain the threshold condition of the persistence for the proposed model and discuss all ecologically feasible equilibrium points and their stability in terms of the model parameters. Furthermore, when choose the fear level as bifurcation parameter, the model will arise single Hopf bifurcation point. However, when choose the predator-taxis sensitivity as bifurcation parameter, the model will arise two Hopf bifurcation points. In order to determine the stability of the limit cycle caused by Hopf bifurcation, the first Lyapunov number is calculated in detail. In addition, by the sensitivity analysis and the elasticity analysis, the saturated fear cost takes on the strong impact on the sensitivity for the model, and the predator death rate has a greater impact on the persistence of the model than the prey death rate. Our numerical illustration also shows that the predator-taxis sensitivity determines the success or failure of the predator invasion under appropriate fear level.