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Group living animals form aggregations and flocks that remain cohesive inspite of internal movements of individuals. This is possible because individualgroup members repeatedly adjust their position and motion in response to theposition and motion of other group members. Here we develop a theoreticalapproach to address the question, what general features -- if any -- underliethe interaction rules that mediate group stability in animals of all species?We do so by considering how the spatial organisation of a group would change inthe complete absence of interactions. Without interactions, a group woulddisperse in a way that can be easily characterised in terms of Fick's diffusionequations. We can hence address the inverse theoretical problem of finding theindividual-level interaction responses that are required to counterbalancediffusion and to preserve group stability. We show that an individual-levelresponse to neighbour densities in the form of Weber's law (a 'universal' lawdescribing the functioning of the sensory systems of animals of all species)results in an 'anti-diffusion' term at the group level. On short time scales,this anti-diffusion restores the initial group configuration in a way which isreminiscent of methods for image deblurring in image processing. We also showthat any non-homogeneous, spatial density distribution can be preserved overtime if individual movement patterns have the form of a Weber's law response.Weber's law describes the fundamental functioning of perceptual systems. Ourstudy indicates that it is also a necessary -- but not sufficient -- feature ofcollective interactions in stable animal groups.

Weber's law based perception and the stability of animal groups