Detection of Hopf bifurcations induced by pregnancy and maturation delays in a spatial predator–prey model via crossing curves method

Abstract

We investigate the stability switches behaviors due to Hopf bifurcations triggered by maturation and pregnancy delays for a diffusive predator–prey model under the joint impacts of fear and Allee effect of predators. We firstly carry out the linear analysis of the proposed model by applying the recently developed crossing curves method to cope with the corresponding characteristic equation incorporating wave number as well as delay-dependent parameters. We then procure the Hopf bifurcation curves through which delays pass, the stability of the associated constant equilibrium can change accordingly. Thereupon, we deduce the normal form regarding Hopf bifurcation and thus clarify its properties. Finally, we present a numerical example and perform some simulations to verify the accuracy of the obtained results. It is shown that both the maturation and pregnancy periods can respectively induce stability switches.