Traveling avalanche waves in spatially discrete bistable reaction–diffusion systems

Infinitely extended two-dimensional reaction–diffusion lattices composed of bistable cells are considered. A class of particular stable stationary solutions, called pattern solutions, is introduced and examples are given. Pattern solutions persist at high diffusion coefficients whereas all other stable stationary solutions, with the exception of the constant solutions, disappear one after the other when the diffusion constant is increased. Furthermore, the new concept of avalanche wave is introduced, where upon a sufficiently large perturbation, a pattern solution is transformed progressively into a constant solution or into another stable stationary solution that exists at a given diffusion constant. These waves exist even for (odd-) symmetrical nonlinearities of the individual cells, whereas it is well known that in this case other waves, as e.g. kinks, do not propagate. The existence of certain classes of avalanche waves is discussed theoretically and the theoretical results are confirmed numerically.