Cellular automaton model of chemical wave propagation on fractals

Abstract

A three-state cellular automaton (CA) model of nucleation and chemical wave propagation on fractal lattices is discussed. In the CA, seeds produce spreading wavefronts of active or transformed sites which mutually annihilate on collision. Many chemical and biological growth processes display this behavior. Pattern formation and growth from isolated seeds and from random initial distributions of seeds are considered. Steady-state behavior for continuous seeding is also discussed. For isolated seeds, surface and volume growth exponents on fractals do not show the simple relation that holds for homogeneous lattices, and lattice topology plays an important dynamical role. For initial seeding, mean-field theory predicts that growth depends on the fractal dimension D (since the minimum path dimension is one). However the fractal gap hierarchy (lacunarity) introduces fluctuation contributions. For continuous seeding entirely new effects arise. On sponge-like fractals oscillators are created; these eventually fill the lattice even at low seeding density. However tree-like fractals cannot support oscillators, and instead exhibit dynamical scaling behavior. For mixed fractals complicated periodic phenomena can arise.