Complex Dynamics of an Eco-epidemiological System with Fear and Allee Effect

In this paper, we propose a predator-prey model with infection in prey population only. We incorporate the fear effect in the growth rate of susceptible prey and the Allee effect in the predator population. Both are the most crucial factors and have a great impact in regulating population size, community structure, and system dynamics. The level of fear responsible for the reduction of the birth rate of susceptible prey causes instability in the system. Different biologically feasible equilibria and their stability conditions are discussed. The existence of Hopf bifurcation is carried out by taking Allee as a bifurcation parameter. Also, we derive the conditions for the direction of stability of Hopf bifurcation by using the normal form theory. Our numerical results suggest that both the fear and Allee stabilize the system. Also, we noted that if the Allee parameter is increased, then the system settles to stable focus from limit cycle oscillations; further increase in the Allee parameter leads to the extinction of predator species. Also, we observed that the combined impact of fear and Allee can produce the occurrence of bistability and other bifurcation points such as generalized Hopf (GH), Bogdanov-Takens (BT), zero Hopf (ZH), etc.