Anomalous Statistics and Density-dependent Asynchronous Updating in Self-Propelled Particle Models of Animal Swarms

Abstract

Much theoretical ecology is based on the assumption that organisms disperse diffusively. The key prediction of a diffusion process is that the mean squared displacement $\$E[X^{2}(t)\$$ increases linearly in time. Yet, an abundance of field and experimental observations note that this quantity increases faster than linearly, leading to super-diffusion.that the position of the centroid of a self-propelled particle model of a swarm comprising naive and informed individuals shows super-diffusion and the power spectrum for the Kuramoto order parameter shows an inverse power law regime, indicating persistent temporal auto correlations. On this ground I speculate that the continuum- level description for this class of models might be given by a fractional wave equation, which can be derived from a generalized master equation with similar scaling behavior. This has the attractive of being analytically tractable (although not very nice), and it might be possible to derive it formally from the individual level model. The main difference with the classical diffusion approach is that the fractional method explicitly incorporates the effect of memory, and thus persistence in the direction of motion.

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Participant

Michael Raghib