Symmetries and Pattern Formation in Hyperbolic Versus Parabolic Models of Self-organised Aggregation

The study of self-organised collective animal behaviour, such as swarms of insects or schools of fish, has become over the last decade a very active research area in mathematical biology. Parabolic and hyperbolic models have been used intensively to describe the formation and movement of various aggregative behaviours. While both types of models can exhibit aggregation-type patterns, studies on hyperbolic models suggest that these models can display a larger variety of spatial and spatio-temporal patterns compared to their parabolic counterparts. Here we use stability, symmetry and bifurcation theory to investigate this observation more rigorously, an approach not attempted before to compare and contrast aggregation patterns in models for collective animal behaviors. To this end, we consider a class of nonlocal hyperbolic models for self-organised aggregations that incorporate various inter-individual communication mechanisms, and take the formal parabolic limit to transform them into nonlocal parabolic models. We then discuss the symmetry of these nonlocal hyperbolic and parabolic models, and the types of bifurcations present or lost when taking the parabolic limit. We show that the parabolic limit leads to a homogenisation of the inter-individual communication, and to a loss of bifurcation dynamics (in particular loss of Hopf bifurcations). This explains the less rich patterns exhibited by the nonlocal parabolic models. However, for multiple interacting populations, by breaking the population interchange symmetry of the model, one can preserve the Hopf bifurcations that lead to the formation of complex spatio-temporal patterns that describe moving aggregations.