Complex Dynamics of a Memory-Induced Stage-Structured Diffusive System With Maturation Delay and Strong Allee Effect

In this work, a memory-induced stage-structured prey–predator diffusive system with maturation delay and strong Allee effect is proposed. First, the positivity of solutions and survival of the non-spatial system are studied. The results indicate that strong Allee effect affects the coexistence of two populations to maintain the harmonious development of the ecosystem, and they can coexist if and only if the predator's fertility is greater than its mortality when the prey reaches its peak. The non-spatial system can undergo Hopf bifurcation caused by the maturation delay. Then we obtain complex dynamics for the spatial system with spatial memory. On one hand, spatial memory diffusion and memory delay can bring about not only Hopf bifurcation and Turing bifurcation but also Turing-Hopf bifurcation and Bogdanov-Takens bifurcation with strong Allee effect. On the other hand, spatial memory delay and maturation delay could induce double Hopf bifurcation. Furthermore, we also investigate the global continuation of local periodic solutions for the spatial system without spatial memory. These interesting results may provide new clues for the investigation of the coexistence for the populations and understanding the complex dynamics of prey–predator systems.