Myopic random walkers and exclusion processes: Single and multispecies

A motility mechanism based on a simple exclusion process, where the probability of movement of an agent depends on the number of unoccupied nearest-neighbor sites is considered. Such interacting agents are termed myopic. This problem is an extension of the famous blind or myopic ant in a labyrinth problem. For the interacting agent models considered here, each agent plays the role of an ant in a labyrinth, where the paths of allowed sites though the labyrinth consist of sites not occupied by other agents. We derive a nonlinear diffusion equation for the average occupancy of the discrete agent-based model for myopic agents. In contrast, interacting blind agents have a constant probability of movement to each of their nearest-neighbor sites, giving rise to a linear diffusion equation. Insight into the various terms in the nonlinear diffusion coefficient is obtained from a study of multiple subpopulations of interacting myopic agents, where an advection–diffusion equation for each subpopulation is derived, and from tracking an individual agent within the crowd, where a motility coefficient is extracted. Averaged discrete simulation data compares very well with the solution to the continuum models. We also compare the behavior of myopic and blind agents. The myopic motility mechanism is biologically motivated to emulate information an individual cell gathers from environment cues. The multispecies model developed and investigated here assists with the interpretation of experimental data involving the tracking subpopulations of cells within a total cell population.