Simulating chemical waves and patterns

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This brief introduction to the modeling and simulation of chemical patterns focused only on the basic elements of several schemes. Chemically reacting media displaced far from equilibrium are complicated systems. Not only are the underlying reaction mechanisms complex, but the spatial patterns are diverse and arise through many different mechanisms that have their basis in the competition between nonlinear chemical kinetics and diffusion. Most commonly observed chemical patterns occur on macroscopic length scales and evolve on macroscopic time scales. Consequently, they can be modeled by reaction-diffusion equations. We discussed the solution of such equations using simple Euler discretizations of space and time. While this simple method suffices in many instances, if the kinetics occurs on a range of widely different time scales or the spatial gradients of chemical concentrations are steep due to sharp chemical fronts, the more elaborate integration schemes mentioned above must be used. Often even complex patterns have their origin in simple general features of the dynamics that are independent of the specific details of the chemical mechanism. Excitable media are an example where the existence of a stable resting or steady state with a characteristic response to perturbations is responsible for many features of the chemical wave propagation seen in such systems. Cellular automaton models specify the dynamics of the system through a simple set of rules. Although CA models are usually abstractions of any real system, they show that even very simple rules are capable of producing very complicated dynamics. Not only can one construct simple CA rules that reproduce the gross features of chemical waves in excitable media as discussed above, but the exploration of these models in a more general context can show what kinds of behavior excitable or oscillatory media can display. Coupled map lattices are one step closer to the macroscopic description of reaction-diffusion equations since the cell variables are continuous like macroscopic concentration fields. The map that specifies how the cell variables change with time can be varied smoothly by parameter changes. This is in contrast to CA, where the rule specification is often difficult to relate to parameters in the physical system. Like CA models, CML models find their most widespread use in showing what types of behavior are possible in systems driven far from equilibrium and providing a computationally efficient way to quantify generic properties of pattern dynamics. Thus, their strength does not lie in the description of a specific system but rather in exploring pattern dynamics in a general context. Phenomena discovered in this way can then be sought in experiment or models designed to mimic the behavior of a specific system. Ultimately, we would like to understand the origin of patterns on a microscopic level. Not only are there fundamental issues to be understood, but the physical situation itself may demand such a more detailed level of description. The mesoscopic models discussed in the last part of this chapter provide one route to describe the system on more microscopic scales. While these models do not provide a full microscopic description of the dynamics (this is the role of molecular dynamics simulation methods), they do incorporate molecular fluctuations that are absent in the macroscopic models. Fundamental questions, such as the validity of macroscopic rate laws for systems with deterministic chaos, can be explored by using such models. The description of the biochemistry inside the cell likely demands such mesoscopic models since the chemistry is complex, cell volumes are small, and the number of chemical species in important reactions is too small to validate a description in terms of mass action kinetics. Such approaches will likely see significant development in the future as our ability to probe chemical dynamics on small scales increases. Copyright © 2004 Wiley-VCH, John Wiley & Sons, Inc.