When push comes to shove: Exclusion processes with nonlocal consequences

• We consider a lattice-based system of randomly motile agents with volume exclusion.
• Agents can displace other agents by a shoving process causing long range effects.
• Mean-field PDE models are derived and their solutions compare well with simulations.
• Both single species and multispecies systems are considered.
• Multispecies systems with species-dependent shoving behaviours are included.

Stochastic agent-based models are useful for modelling collective movement of biological cells. Lattice-based random walk models of interacting agents where each site can be occupied by at most one agent are called simple exclusion processes. An alternative motility mechanism to simple exclusion is formulated, in which agents are granted more freedom to move under the compromise that interactions are no longer necessarily local. This mechanism is termed shoving. A nonlinear diffusion equation is derived for a single population of shoving agents using mean-field continuum approximations. A continuum model is also derived for a multispecies problem with interacting subpopulations, which either obey the shoving rules or the simple exclusion rules. Numerical solutions of the derived partial differential equations compare well with averaged simulation results for both the single species and multispecies processes in two dimensions, while some issues arise in one dimension for the multispecies case.