Anomalous Statistics and Locally Regulated Asynchronous Updating in Self-Propelled Particle Models of Animal Swarms

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CSSS 2006 Santa Fe


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. A relatively recent class of models for swarms, the so-called Self-Propelled Particle models (SPP's), have this superdifussive property, due to the emergence of an inverse power law regime for the temporal autocorrelation of the Kuramoto order parameter of the swarm, which is effectively the stochastic velocity driving the system's centroid.

However, the assumptions required to achieve this are somewhat artificial, in the sense that: 1) one either needs three distinct interaction regions (i.e avoidance, alignment and attraction), or 2) the introduction of informed individuals (when the attraction and alignment regions overlap), and 3) the models are updated synchronously. I propose the modification to asynchronous updating by assigning a waiting time distribution to each individual in the population, calibrated by the local density. In this way, each individual has two exponential clocks controlling avoidance and alignment/attraction events that run faster or more slowly depending on its local neighborhood. For instance, if the individual is in a locally crowded region, it would need to update its orientation more frequently to avoid collisions.

I want to explore three things: 1) whether this local regulation in the updating rule leads to an inverse power law for the temporal autocorrelation of the Kuramoto parameter, and hence super-difussion, without informed individuals and with only two types of events: a) avoidance and, b) attraction/alignment. This is motivated by earlier work in locally regulated spatial-temporal point processes, where the spatial correlations built by the dynamics lead to an inverse power law in the autocorrelation function for the population size time series; 2) possible changes in the predictions of the three and two zone models (the latter with informed individuals) under this updating rule and, 3) if case 1) is true, and the displacement statistics turn out to be Gaussian, the continuum-level model for this class of SPP's might be reduced to a fractional wave equation for the PDF of the position of the centroid of the swarm at time t, given that this equation can be derived quite naturally from a generalized master equation with a separable transition kernel composed by velocity-velocity temporal autocorrelations following a power law and Gaussian statistics for the displacements. The next step would be to tie this up to ecology via an exploration/exploitation tradeoff, and evolve the parameters with some sort of GA, but I guess the first part is enough for the summer project.


Michael Raghib