On stability and mobility of shapes in the local attraction model

Many animal groups, for example schools of fish or flocks of birds, exhibit complex dynamic shapes while moving cohesively in the same direction. The main theoretical tools used to study the formation and dynamics of these shapes are so called self-propelled particle models. However, even the simplest models typically require computer simulations for their analysis, especially when the number of particles is small. For example, this is the case for the local attraction model with a blind angle. Here we explore three geometrical ideas based on transferring the attention from the particles themselves to the local center of mass they detect and the shapes they constitute. We use these methods to investigate the persistence and mobility of shapes in a local attraction model with a blind zone. More specifically, we address the persistence/stability of the mill shape. Then we investigate how the dynamics of the detected local center of mass relate to the shapes we observe, including a moving figure of eight shape generated by the model. Finally, we provide some insight into why some rotating chains exhibit translational motion and some do not. Although this work is in its infancy we believe that these ideas have potential and may facilitate analysis of similarly complex models.

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