Stability and Hopf bifurcation analysis for a three-species food chain model with fear and two different delays

In this paper, we consider a Hastings–Powell type model, which consists of a three-species food chain model with fear effect. For more realistic formulation, we incorporated two time delays into the model, one for prey density another for the gestation of the middle and top predator populations. By choosing time delays as the bifurcation parameters, the essential mathematical features and their dynamics are studied in terms of local stability and Hopf bifurcation analysis. Linearizing the system at the positive equilibrium point and analyzing the distribution of the roots of the associated characteristic equation, the conditions for the existence of Hopf bifurcation are obtained. It is shown that for the gradual increase of the magnitude of delay, the stability of equilibrium point changes and the system exhibits a Hopf bifurcation as the time delay passes through some critical values. Furthermore, an explicit formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are investigated by using the normal form method and center manifold theory. Finally, several numerical simulations are provided to verify the effectiveness of the derived theoretical results and to examine the analytical findings.