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 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 ), using only two interaction regions. 2) if the predictions of earlier models still hold with this updating rule and 3) if 1) is true, explore the possibility that the continuum-level description for this class of models can be reduced to a fractional wave equation for the position of the centroid of the swarm, Given that it can be derived from a generalized master equation with inverse power law memory and Gaussian displacements. The next step would be to tie this up to ecology via an exploration/exploitation tradeoff. But I guess the first part is enough for the summer project.

---

Participant

Michael Raghib