Modelling persistence of motion in a crowded environment: the diffusive limit of excluding velocity-jump processes

Abstract

Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behaviour is an agent-based velocity-jump process. In the absence of agent-agent interaction, the mean-field continuum limit of the agent-based model (ABM) gives rise to the well known hyperbolic telegraph equation. When agent-agent interaction is included in the ABM, a strictly advective system of partial differential equations (PDEs) can be derived at the population-level. However, no diffusive limit of the ABM has been obtained from such a model. Connecting the microscopic behaviour of the ABM to a diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. In order to connect the ABM at the population-level to a diffusive PDE at the population-level, we consider a generalisation of the agent-based velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalisation allows us to take a diffusive limit and obtain a faithful population-level description. We investigate the properties of the model at both the individual and population-level and we elucidate some of the models' key characteristic features. In particular, we show an intrinsic anisotropy inherent to the models and we find evidence of a spontaneous form of aggregation at both the micro- and macro-scales.