A Mathematical Model for Collective Behaviors and Emergent Patterns Driven by Multiple Distinct Stimuli Produced by Multiple Species

Collective migration underlies key developmental and disease processes in vertebrates. Mathematical models describing collective migration can shed light on emergent patterns arising from simple mechanisms. In this paper, a mathematical model for collective migration is given for arbitrary numbers and types of individuals using principles outlined as a set of assumptions, such as the assumed preference for individuals to be “close but not too close" to others. The model is then specified to the case of two species with arbitrary numbers of individuals in each species. A particular form of signal response is used that may be parameterized based on experiments involving two or three agents. In its simplest form, the model describes two species of individuals that emit distinct signals, distinguishes between them, and exhibits responses unique to the type by moving according to signal gradients in various planar regions, a situation described as "mixotaxis". Beyond this simple form, initial conditions and boundary conditions are altered to simulate specific, additional in vitro as well as in vivo dynamics. The behaviors that were specifically accounted for include motility, directed migration, and a functional response to a signal. Ultimately, the paper’s results highlight the ability of a single framework for signal and response to account for patterns seen in multi-species systems, in particular the emergent self-organization seen in the embryonic development of placodal cells, which display chase-and-run behavior, flocking behavior, herding behavior, and the splitting of a herd, depending on initial conditions. Numerical experiments focus around the primary example of neural crest and placodal cell “chase-and-run” and “flocking” behaviors; the model reproduces the separation of placodal cells into distinct clumps, as described in the literature for neural crest and placodal cell development. This model was developed to describe a heterogeneous environment and can be expanded to capture other biological systems with one or more distinct species.