Qualitative analysis of a spatiotemporal prey-predator model with multiple Allee effect and schooling behaviour

In this paper, a mathematical model is proposed to study a predator-prey system with multiple Allee effect acting on the growth rate of the prey population. Cosner-type functional response is incorporated to model schooling or conspecific aggregation behaviour of predator population. Cosner-type functional response emphasizes the strong Allee effect among the predator's population. We determine the existence of biological feasible equilibrium points with their stability properties. Besides, bifurcation scenarios such as saddle-node, Hopf and Bogdanov-Takens have been observed that take place in the dynamics during the parameter variation. Allee effect and growth rate of the prey population give rise to the multistable behaviour of the system. We attempt to portrait the dynamical behaviour of the system in the presence of diffusion. It has been observed that the predator adopts different strategies to surround the prey population. These strategies are depending on the prey's school size, density and predator's agility. Moreover, severity of the Allee effect also brings dramatic changes in the predator-prey distribution in the system. It is found that under the strong Allee effect, the predator-prey populations form high-density schools which are completely justified in the biological point of view. Various types of patterns for different time steps and diffusion rates are captured that confer the spatiotemporal complexity of the system.