Complex Dynamics of a Three-Species Food Chain Model with Fear and Allee Effect

In ecology, predator-prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings-Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.