A lattice Boltzmann model for reaction-diffusion systems is developed. The method provides an efficient computational scheme for simulating a variety of problems described by the reaction-diffusion equations. Diffusion phenomena, the decay to a limit cycle, and the formation of Turing patterns are studied. The results of lattice Boltzmann calculations are compared with the lattice gas method and with theoretical predictions, showing quantitative agreement. The model is extended to include velocity convection in chemically reacting fluid flows.

We study the existence or nonexistence of true trajectories of chaotic dynamical systems that lie close to computer-generated trajectories. The nonexistence of such shadowing trajectories is caused by finite-time Lyapunov exponents of the system fluctuating about zero. A dynamical mechanism of the unshadowability is explained through a theoretical model and identified in simulations of a typical physical system. The problem of fluctuating Lyapunov exponents is expected to be common in simulations of higher-dimensional systems.

When Ca ²⁺ is released from internal stores in living cells, the resulting wave of increased concentration can travel without deformation (continuous propagation) or with burst-like behavior (saltatory propagation). We analyze the “fire–diffuse–fire” model in order to illuminate the differences between these two modes of propagation. We show that the Ca ²⁺ release wave in immature Xenopus oocytes and cardiac myocytes is saltatory, whereas the fertilization wave in the mature oocyte is continuous.

Recently solutions to a simple reaction diffusion system have been discovered in which localized structures (spots) make copies of themselves. In this Letter we analyze the one-dimensional analog of this process in which replication occurs until the domain is filled with a periodic array of spots. We provide a heuristic explanation of why this replication process should occur in a broad class of systems. Time dependent solutions are developed for model systems and their analytic structures investigated.