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. 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 in the temporal autocorrelation of the Kuramoto order parameter of the swarm, which is effectively the stochastic velocity that drives 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) in order to obtain anomalous statistics, and 3) the models are updated synchronously. I propose the modification to asynchronous updating for the SPP model where the aligment and attraction regions overlap. This asynchronous updating rule will be based on a density-dependent, waiting time distribution for each individual in the population, calibrated by the local density. In this way, each individual has an exponential clock that runs 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) if this density-dependent updating rule leads to and inverse power law in the temporal velocity-velocity autocorrelations, without informed individuals ( as is the case with locally regulated spatial-temporal point process ), and overlapping the alignment and attraction regions; 2) if the predictions of the three and two zone models ( the latter with informed individuals) change with this locally regulated updating rule and, 3) in case 1) is true, and the displacement statistics are Gaussian, explore the possibility that the continuum-level model for this class of SPP's can 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 inverse power law velocity-velocity temporal autocorrelations 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.

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Participant

Michael Raghib