A nonlocal kinetic model for predator-prey interactions in two dimensions

Eulerian models based on integro-differential equations may be used to model collective behaviour, by treating the group of individuals as a population density. In comparison with Lagrangian models, where one tracks distinct individuals, Eulerian models are formulated as evolution equations for the density field, and hence permit rigorous analysis to be performed. The population densities are influenced by the social interactions of attraction, repulsion and alignment. We introduce a new model for predator-prey dynamics that generalizes a previous integro-differential equation model by introducing the predator dynamics and a blind zone for the prey. Extensive simulations were performed to showcase the realism of the model, and these simulations are presented in four stages. First, the prey reacts solely due to interactions with itself. Second, a stationary predator distribution is introduced. Third, the predator’s distribution remains fixed but moves in a predetermined fashion. Finally, the predator dynamics are governed by equations analogous to those of the prey. Variations in the size of the blind zone for the prey are explored that can determine whether a prey cluster stays together or splits apart. The prey and predator demonstrate realistic behaviours that are seen in nature.