The spatiotemporal dynamics of a diffusive predator-prey model with double Allee effect

We introduce a diffusive predator-prey system with the double Allee effect, focusing on the stability and sufficient conditions for the coexistence of prey and predator. Subsequently, we derived the amplitude equation and explore secondary-order dynamic properties using methods such as Taylor series expansion and multiscaling. The novel approach outlined above provides a precise means to thoroughly analyze the predator-prey model. Through this analysis, we demonstrated that the inclusion of the Allee effect and diffusion leads to the system exhibiting more intricate dynamic behaviors compared to systems lacking these factors. On one hand, in the diffusive system without the Allee effect, the pattern formation regarding the distribution of species was relatively scattered, whereas in the diffusive system with the Allee effect, it is more intensive. On the other hand, the system with the Allee effect transitioned from unstable to stable when the diffusion parameter in prey increased, and the aggregation degree of pattern formation in the system with the Allee effect was higher than in the system without it. These findings highlight the significant roles played by the Allee effect and diffusion in determining the dynamic behaviors of prey and predator within the system.