The influence of fear effect and phase of chaos control of discrete predator–prey system with mixed Holling types

In this paper, a discrete-time predator–prey system with mixed Holling types I and III functional responses and fear effect is discussed. Three fixed points are determined and their stability is established. Analytical criteria for Neimark Sacker bifurcation (NSB) and flip bifurcation (FB) are proposed. Applying the technique of state feedback control, we introduce the corresponding controlled system. At an unstable fixed point the chaotic orbits can be stabilized. We show that the fear effect of prey species has a negative effect on the density of predator species in our model. Furthermore, fear effect may increase the possibility that prey species will go extinct, which implies that Allee effect may occur in this case. These analytical studies are illustrated with some numerical simulations.