View source for 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 increases linearly in time. Yet, an abundance of field and experimental observations note that this quantity increases faster than linearly, leading to super-diffusion. An relatively recent class of models for swarms,  the so-called self-propelled particle models, have this superdifussive property, due to the emergenge of an inverse power law regime for the Kuramoto order parameter of the swarm, which is effectively the stochastic velocity that drives the systems centroid. However, the assumptions required to achieve this are somewhat artificial, in the sense that one either needs three different interaction regions (i.e avoidance, alignment and attraction), or the introduction of informed individuals in the population (when the attraction and alignment regions overlap) in order to obtain anomalous statistics, besides the models are updated asynchronously.  I propose the introduction of a density-dependent waiting time distribution, calibrated by the local density around a focal individual.  In this way, each individual has an exponential clock that runs faster or more slowly depending on its local neighborhod. in a crowded region needs to update its avoidance clock faster tso that each individual has two clocks citcsthat 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 inveutionrse 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