Clustering due to Acceleration in the Response to Population Gradient: A Simple Self-Organization Model

We explore the phenomenon of animal self-organization due to autotaxis, that is, the movement of individuals induced by their own density gradient. There is natural evidence that clustering occurs as a result of the interplay between random and directed movements of individuals due to mutual attraction and repulsion. Classically, it is assumed that taxis velocity is determined by the density gradient of some stimulus. However, it is known that partial differential equation (PDE) diffusion-advection models that rest on this assumption cannot give a realistic representation of a stationary or moving cohesive group of individuals with a uniform interior density and sharp edges. Pioneering work by Okubo and coworkers suggests that the acceleration of individuals (rather than their velocity directly) is proportional to the population density gradient. A PDE model resting on this finding was constructed and investigated. The model demonstrates the formation of steady heterogeneous structures of the required shape. This feature can be interpreted as dynamic self-organization, like fish shoaling or insect swarming. This model is the first to achieve this result while considering an autonomous population in a simple PDE framework. Analytical and numerical studies show that the link between the acceleration and the density gradient is crucial for the appearance of clusters.