A systematic study of autonomous and nonautonomous predator–prey models with combined effects of fear, migration and switching

In this paper, we study the dynamics of a predator–prey system under the combined effects of fear, migration and switching phenomena. This study is new in the sense that the dynamics of such system was studied either through fear or migration, or switching, but the combined effects of such three factors are yet to be explored. We observe oscillatory behavior of the system in the absence of fear, migration and switching, whereas the system shows stable dynamics if anyone of these three factors is introduced. After analyzing the behavior of system with fear, migration and switching, we find that the system does not possess periodic solution whenever the predator population experiences intraspecies competition, but in the absence of intraspecies competition among predator population, fear of predators destabilizes the system, whereas on increasing the migration rates, the system first undergoes subcritical Hopf-bifurcation and then supercritical Hopf-bifurcation settling the system to stable coexistence. Existence of multiple limit cycles is also observed. Our results show that switching behavior of predator population supports the survival of prey and predator populations. We extend our model by assuming the fear parameters as time dependent. We find that the nonautonomous system exhibits periodic solutions, whereas the corresponding autonomous system shows stable focus. Moreover, we observe that if the autonomous system undergoes a Hopf-bifurcation through limit cycle oscillations, the corresponding nonautonomous system shows higher periodic solutions. Almost periodic behavior of the system is also observed by setting the fear parameters as almost periodic functions of time.