Stability and Hopf bifurcation of a multiple delayed predator–prey system with fear effect, prey refuge and Crowley–Martin function.

In this paper, we consider a predator–prey model with fear response delay, gestation delay, fear effect, prey refuge, harvesting and Crowley–Martin type functional response. First, we discuss the model without delays and corroborate the positivity and boundedness of the solution. Then, we give some sufficient conditions for the existence and stability of three equilibriums. For the model with delays, we not only analyze the local stability of the positive equilibrium and the occurrence of Hopf bifurcation, but also obtain the crossing curves to study the stability switches of the positive equilibrium on the delays plane. Furthermore, we calculate the normal form of Hopf bifurcation and hence get the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. At last, we support our findings by numerical simulations. • Considering the cost of fear into both reproduction and death term of prey. • Fear that influence birth and death rate of prey have similar impact on populations. • The stability crossing curve method is creatively applied to the model. • The model can exhibit stability switches on the delays plane. [ABSTRACT FROM AUTHOR]