Spatial self-organization in three-component mixtures

We study a self-driven many-particle complex system of three types of particles (components, species) with self- and cross-interactions. The interactions are modeled using game—theoretical payoff matrices. In the continuum description of the model, the system of partial differential equations describing evolution of the densities is discussed, taking into account diffusion of the components of the mixture. Linearization of the densities around the homogeneous state provides information about the stability of the solutions. Several stability criteria of the homogeneous states are obtained. In the discrete, microscopic description, a cellular automaton model of the system is considered. The particles are distributed in the nodes of a one-dimensional lattice, with periodic boundary conditions. The dynamics of the system allows the entities to jump, in discrete time steps, to a neighboring node if the expected success is larger in that node. The success functions depend on the payoff matrices for binary interactions, and on the densities of interacting species. Various interesting phenomena related to spatial self-organization are investigated. In particular, the small perturbations of the two-component system by the third component are studied, as well as specific features of self-organization phenomena in the systems of three components. Spatio-temporal evolution of the model is investigated for different payoff matrices and diffusion coefficients.