Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior

In this paper, we investigated a diffusive predator-prey model with multiple Allee effect, herd behavior. Moreover, we consider the quadratic mortality on predator species. We found that the dynamics of system near the positive equilibria is quite rich. We are more concerned with the spatial dynamics near the positive equilibrium. The necessary conditions for Turing instability occurring are obtained. And the stability and direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that Allee effect does not alter the local stability of the boundary equilibrium and the transcritical bifurcation occurs at the highest intensity of Allee effect. The biomass conversion rate does affect the stability of the system and the occurrence of Turing instability. This indicates that the biomass conversion rate is essentially significant for the predator-prey system, and we can control the biological conversion rate to achieve the coexistence of the predator and prey. Finally, we summarize our findings in the conclusion.