From birds to bacteria: generalised velocity jump processes with resting states

TL;DR

This paper develops a mathematical framework for animal movement with two behavioral modes, deriving equations and statistics that match experimental data and reveal how movement parameters influence diffusion.

Contribution

It introduces a generalized velocity jump process with resting states, accommodating arbitrary waiting and running time distributions, and derives a novel diffusion approximation.

Findings

Mean squared displacement matches experimental data from bacteria and gulls.

Effective diffusion constant depends on mean and variance of running times, and mean of waiting times.

Derived a Cattaneo approximation for large-time diffusive behavior.

Abstract

There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. Here a mathematical description of this process is formulated, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.