Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.