Double Hopf bifurcation induced by spatial memory in a diffusive predator-prey model with Allee effect and maturation delay of predator

In this paper, we delve into double Hopf bifurcation induced by memory -driven directed movement in a spatial predator-prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non -resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode -5 to another with mode -4 and the coexistence of them can be observed.