Stationary self-organized states in many-particle systems with ternary interactions
We consider a mixture of three different types of particles (entities) with microscopic self- and cross interactions. Besides the pair interactions we introduce third order interactions between particles of different types taking into account nonlinear effects in the relevant success functions. Two levels of description are considered. In the continuum description of the model, the system of PDE describing evolution of the particle densities is studied. Linearization of the densities around the homogeneous state provides information about stability of the solutions. Several stability criteria of the homogeneous states are obtained. The influence of the third order interactions on the stability properties of the homogeneous state and on the finally evolving states is investigated. In the discrete description, a cellular automaton model of the system is introduced. The particles are distributed in the nodes of a one-dimensional lattice, with periodic boundary conditions. The dynamics of the system allows the particles 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 and third order interactions, and on the densities of the interacting components. The coefficients of the matrices describe various types of interactions (attractive or repulsive). The influence of the third order interactions on the time evolution and emergence of self-organized stationary states is studied.