Abstract / Summary
Abstract Three-dimensional aggregation – idealised here as a cubic herd – is a recurring collective phenomenon among marine and aerial animals (such as schooling fish, krill swarms, and bird flocks), serving as an effective anti-predator mechanism. However, the mechanisms by which such a geometric group structure provides protection, enabling predator-prey coexistence, remain uncertain. In this paper, we propose and analyze a prey-predator model with a Beddington-DeAngelis functional response, in which the prey form a cubic herd, while the predators have additional food sources. We discuss the positivity, uniform boundedness, and local stability of the equilibrium points, and perform a bifurcation analysis with respect to the additional food parameter A. The numerical results support the analytical findings. Moreover, the conditions for Turing instability are derived and verified both analytically and numerically. It has been demonstrated that increasing the amount of additional food can induce bistability, thereby allowing for the coexistence of two equilibria and the formation of spatial spot patterns. These results suggest that additional food is crucial for supporting predator persistence and maintaining ecological balance in prey-predator systems exhibiting cubic herd behavior.