ORIENTATION-INDUCED PATTERN FORMATION: SWARM DYNAMICS IN A LATTICE-GAS AUTOMATON MODEL

Abstract

Swarming patterns might arise not just at organismic levels (bird and fishes exhibiting particularly striking examples) but even at cellular and intracellular scales whenever “collective motion” of biological or chemical entities is involved. Examples are the swarming of myxobacteria and ants, aggregation and slug pattern formation of the slime mold Dictyostelium discoideum, or intracellular network dynamics of actin filaments. Here a stochastic process — discrete in space and time — is developed, the “swarm lattice-gas automaton”. For some lattice-gas models (in physics and chemistry) it was demonstrated that the limit behavior resembles known master equations by means of expectation values of suitably chosen microscopic variables. In particular, for the Navier–Stokes equation the derivation of a continuous macroscopic description from discrete microdynamic equations was shown. The “swarm lattice-gas automaton” possesses a non-local integral-like interaction operator. Particles (cells, organisms) are assigned some orientation (and fixed absolute velocity) which might change by means of interaction with other members of the swarm within a given “region of perception”. The corresponding microdynamical equation is given and results of numerical experiments are shown. Simulations exhibit a variety of aggregation patterns which are distinguished by means of microscopic and macroscopic variables. The influence of a sensitivity parameter and particle density on pattern formation is examined systematically.