# Persistent exclusion processes: inertia, drift, mixing and correlation

## TL;DR

This paper develops and analyzes lattice-based models for motile agents with persistence and crowding effects, revealing nonlinear diffusion behavior and extending to multiple species and drift, with validation against simulations.

## Contribution

It introduces a mean-field framework for persistent exclusion processes, deriving nonlinear PDEs and generalizing to multiple species and drift, advancing understanding of collective motion with crowding.

## Key findings

- Models are well described by nonlinear diffusion equations.
- Persistence and exclusion cause nonlinearity in diffusion behavior.
- Mean-field predictions align with stochastic simulations.

## Abstract

In many biological systems, motile agents exhibit random motion with short-term directional persistence, together with crowding effects arising from spatial exclusion. We formulate and study a class of lattice-based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions, and use a mean-field approximation to investigate relevant population-level partial differential equations in the continuum limit. We show that this model of a persistent exclusion process is in general well described by a nonlinear diffusion equation. With reference to results presented in the current literature, our results reveal that the nonlinearity arises from the combination of motion persistence and volume exclusion, with linearity in terms of the canonical diffusion or heat equation being recovered in either the case of persistence without spatial exclusion, or spatial exclusion without persistence. We generalise our results to include systems of multiple species of interacting, motion-persistent walkers, as well as to incorporate a global drift in addition to persistence. These models are shown to be governed approximately by systems of nonlinear advection-diffusion equations. By comparing the prediction of the mean-field approximation to stochastic simulation results, we assess the performance of our results. Finally, we also address the problem of inferring the presence of persistence from simulation results, with a view to application to experimental cell-imaging data.