The impact of delays on prey-predator dynamics with predation-induced fear

This study explores a prey-predator model with a Holling type II functional response, focusing on how predation-induced fear affects prey dynamics. Assuming a decline in prey population growth rate attributed to predator-induced fear, the model incorporates a fear response delay representing prey detection time, along with gestation delay. The system’s positivity, boundedness, and permanence are proved under certain parametric conditions. The local stability is discussed at trivial, semi-trivial, and positive equilibria. The system exhibits Hopf bifurcation with respect to both delays. Hopf bifurcation analysis is done for different combinations of delays. Furthermore, the properties of periodic solutions in the delayed system are also determined. An extensive numerical simulation has been performed to validate analytical findings. The occurrence of Hopf bifurcation is shown for different combinations of delays by plotting eigenvalues.