Continuum models describing probabilistic motion of tagged agents in exclusion processes

Lattice-based random walk models are widely used to study populations of migrating cells with motility bias and proliferation. Crowding is typically represented by volume exclusion, where each lattice site can be occupied by at most one agent and conflicting moves are aborted. This framework enables simulations that yield both population-level spatiotemporal agent density profiles and individual agent trajectories, comparable to experimental cell-tracking data. Previous continuum models for tagged-agent trajectories captured trajectory information only, and overlooked any measure of variability. This is an important limitation since trajectory data is inherently variable. To address this limitation, here we derive partial differential equations for the probability density function of tagged-agent trajectories. This continuum description has a clear physical interpretation, agrees well with distributional data from stochastic simulations, reveals the role of stochasticity in different contexts, and generalises to multiple subpopulations of distinct agents.