An investigation of a nonlocal hyperbolic model for self-organization of biological groups

Abstract

In this article, we introduce and study a new nonlocal hyperbolic model for the formation and movement of animal aggregations. We assume that the nonlocal attractive, repulsive, and alignment interactions between individuals can influence both the speed and the turning rates of group members. We use analytical and numerical techniques to investigate the effect of these nonlocal interactions on the long-time behavior of the patterns exhibited by the model. We establish the local existence and uniqueness and show that the nonlinear hyperbolic system does not develop shock solutions (gradient blow-up). Depending on the relative magnitudes of attraction and repulsion, we show that the solutions of the model either exist globally in time or may exhibit finite-time amplitude blow-up. We illustrate numerically the various patterns displayed by the model: dispersive aggregations, finite-size groups and blow-up patterns, the latter corresponding to aggregations which may collapse to a point. The transition from finite-size to blow-up patterns is governed by the magnitude of the social interactions and the random turning rates. The presence of these types of patterns and the absence of shocks are consequences of the biologically relevant assumptions regarding the form of the speed and the turning rate functions, as well as of the kernels describing the social interactions.