Direction and stability of Hopf bifurcation in an eco-epidemic model with disease in prey and predator gestation delay using Crowley-Martin functional response

In this work, we have studied an eco-epidemic model using the Crowley-Martin functional response that includes disease in prey and gestation delay in the predator population. The model possesses three equilibria, namely the disease-free, Predator-free, and the interior equilibrium point. In addition, we examined the stability of the equilibrium points varying the infection rate and time delay parameter. Detailed analysis of Hopf bifurcation of the interior equilibrium point contains two situations: with delay and without delay. Moreover, we have studied the direction of the Hopf bifurcation and the stability of periodic solutions utilizing normal form theory and the center manifold theorem. It is emphasized that Hopf bifurcation occurs when the time delay exceeds the critical value and that the critical value of the delay is strongly impacted by the infection rate in prey. A detailed numerical simulation is provided to verify the analytical results.