# Reaction-diffusion systems

Curator and Contributors

1.00 - Gregoire Nicolis

In the strict sense of the term, reaction-diffusion systems are systems involving constituents locally transformed into each other by chemical reactions and transported in space by diffusion. They arise, quite naturally, in chemistry and chemical engineering but also serve as a reference for the study of a wide range of phenomena encountered beyond the strict realm of chemical science such as environmental and life sciences.

## The phenomenology of reaction-diffusion systems

Reaction-diffusion systems in a closed vessel and in the absence of external forces evolve eventually to the state of chemical equilibrium, whereby the constituents involved are distributed uniformly in space and each elementary reactive step is counteracted by its inverse. It has long been realized that the approach to equilibrium can be both in the form of a simple exponential decay, or more involved transient behaviors associated with damped oscillations or non-trivial space dependencies including wave-like patterns. While the former found immediately a series of important applications such as the experimental measurement of the rate constants of chemical reactions, the latter were originally regarded as curiosities. The development of irreversible thermodynamics in the 1950’s and onwards provided an explanation of the origin of these two kinds of behaviors by linking them to time evolutions starting close to and far from equilibrium, respectively. Experiment and modeling on laboratory scale reactive systems such as the Belousov-Zhabotinski reaction confirmed this view. Still as long as reaction-diffusion processes were carried out in a closed thermostated reactor there was no way to analyze systematically what was going on since, by virtue of the second law of thermodynamics, the system was bound to reach sooner or later the state of equilibrium.

A major development that opened new horizons in the experimental study of reaction-diffusion systems and stimulated, in parallel, important theoretical developments has been the systematic use of open reactors, whereby the system is maintained in a nonequilibrium state as long as desired, through the pumping of fresh reactants (the rate of which determines the distance from equilibrium) and the outflow of used products. When spatial homogeneity is maintained within the reactor through stirring in CSTRs (Continuously Stirred Tank Reactors), a rich phenomenology was revealed as key parameters were varied, in addition to the steady state extrapolating the familiar equilibrium like behavior: simple periodic, multi-periodic and chaotic oscillations ; multistability, i.e., the coexistence of more than one simultaneously stable states; and excitability, whereby once perturbed a system performs an extended excursion before settling back to its original stable state. A most exciting set of behaviors pertains to space patterning, made possible when stirring is not imposed within the reactor as is the case in many real world situations in chemistry, engineering and biology. Propagating wave fronts, or stabilized ones (like a flame separating fresh and burnt reactants in combustion for instance), are familiar examples. A still different form of spatial organization is the formation of regular steady state patterns arising from the spontaneous symmetry breaking of a spatially uniform state (see Fig. 1).

Figure 1: Two-dimensional reaction-diffusion Turing patterns featuring hexagons and bands obtained by numerically integrating the Brusselator model

## Reaction-diffusion equations

The most familiar quantitative description of reaction-diffusion systems is based on the assumption of decoupling between two kinds of processes occurring on widely different scales : the evolution of the macroscopic variables, $$x_i (i=1,\dots,n)$$ such as the concentrations or mole fractions $$c_i$$ and the temperature $$T\ ;$$ and the dynamics at the molecular level, which merely provides the values of a set of phenomenological parameters $$\lambda$$ entering in the description such as the rate constant $$k_{\alpha}$$ of reaction $$\alpha\ ,$$ the mass or heat diffusivity coefficients $$D_i\ ,$$ or the heat $$\Delta H_{\alpha}$$ of reaction $$\alpha\ .$$ This approach, referred as the mean field description, takes the form of a set of balance equations :

$\frac{\partial x_i}{\partial t}=v_i(\{x_j\},\lambda)+D_i \nabla^2 x_i \ \ \ (i=1,\dots,n)$

where the two terms in the right hand side stand, successively, for the effect of the chemical reactions and of transport. For simplicity it was assumed (as is the case in a wide spectrum of problems of interest) that there is no bulk motion (which amounts to discarding the effects of external forces), that there are no cross effects in transport and that Fick’s or Fourier’s laws describe adequately mass and heat transport (see below for extensions of this description).

Eqs. (1) are referred to as the reaction-diffusion equations. The reaction term in their right hand side is system-dependent and typically nonlinear, as the rate constants depend on the temperature and the law of mass action of chemical kinetics expresses the velocity of a reaction in terms of products of concentrations. The most important form of nonlinearity as far as the onset of complex behaviors is concerned is the occurrence of feedbacks, whereby a constituent affects (positively or negatively) its subsequent evolution and/or the evolution of certain other among the constituents present. In contrast, the transport term has a universal structure and is typically linear to an excellent approximation at least as long as the solution is not close to a phase transition point (unmixing of the constituents, etc).

The reaction-diffusion equations form the basis for the interpretation of the experiments reviewed above. The main point is that the observed behaviors arise through the phenomenon of bifurcation, where new solutions are branching out of the reference equilibrium like solution as the parameters are varied. They can be analyzed to a considerable detail using the methods of dynamical systems theory such as stability and bifurcation analyses, reduction to normal form (amplitude) equations in the vicinity of the bifurcation points using perturbation techniques and/or geometric and symmetry arguments, complemented by numerical simulations.

In the absence of spatial degrees of freedom eqs (1) reduce to a set of coupled nonlinear ordinary differential equations. Nonlinearity appears here in its simplest expression, as a property arising from intrinsic and local cooperative events - the chemical reactions. Complex behavior persists even when few variables are present and, because of this, this reduced form of eqs (1) has provided some of the earliest and most widely used models of bifurcation and chaos theories.

In the presence of spatial degrees of freedom eqs (1) define a set of coupled nonlinear partial differential equations of the parabolic type. Complemented with appropriate boundary conditions they generate a wealth of spatial and spatio-temporal patterns. As the intrinsic parameters $$k$$ and $$D$$ in the equations have dimensions of $$[$$time$$]^{-1}$$ and $$[($$length$$)^{2}/$$time$$]\ ,$$ respectively, these patterns have the potential of possessing intrinsic time $$k^{-1}$$ and space $$(D/k)^{1/2}$$ scales. This places reaction-diffusion systems in the forefront for understanding the origin of endogenous rhythmic and patterning phenomena observed in nature and in technological applications. The following is a partial list of solutions of this kind.

• Wave fronts. The spatial coupling of elements possessing two stable steady states or a stable and an unstable one, gives rise to a wave front propagating from the stable or the most stable state toward the unstable or the least stable one. In the presence of locally oscillating or excitable kinetics the front may take some unexpected forms, from cylindrically symmetric (« target ») patterns to spiral ones.
• Space-dependent stationary solutions. These arise through a symmetry-breaking instability of the homogeneous state, first proposed by Turing as a universal mechanism of biological morphogenesis.
• Synchronization, clustering and spatio-temporal chaos. These dynamical behaviors result from the spatial coupling of local elements each in a regime of periodic or chaotic oscillations, depending on the relative strength of diffusion and reactive terms.
• Composite patterns, such as localized structures and defects. They arise from the interference between two or more mechanisms of instability, as it happens for instance when a Turing instability interacts with one leading to time oscillations or to multiple steady states (see Fig.2).
Figure 2: Space-time dynamics (time runs upwards) featuring a localized Turing pattern embedded into a Hopf oscillating background and obtained by numerically integrating the Brusselator model close to a Turing-Hopf codimension-two point.

All elements at our disposal indicate that there exists no exhaustive list and universal classification of the full set of solutions of reaction-diffusion equations. The design and study of canonical models aiming to clarify the relative roles of chemical feedbacks, of transport and of nonequilibrium constraints in the onset of complex behavior provides here a much needed additional insight. One may quote, among others :

• The Fisher or Kolmogorov-Petrovsky-Piskounov (KPP) equation, a 1-dimensional 1-variable version of eqs(1) with $$v=kx(1-x)\ .$$ This equation models the generation of a wave front, associated with solutions of the form $$x(r,t)=x(r-ct)\ ,$$ and is also used extensively in population dynamics and genetics.
• The Brusselator, a 2-variable version of eqs(1) with $$v_1=a-(b+1)x_1+x^2_1x_2, v_2=b x_1-x^2_1 x_2\ ,$$ $$a$$ and $$b$$ being parameters. It allows one to follow the generation of sustained oscillations, Turing patterns and spatio-temporal chaos. In a similar spirit activator-inhibitor models have been designed to provide insights on how such solutions emerge as a « compromize » between a fast diffusing inhibitor and a less mobile activator.

There exists also a wealth of more specialized models for interpreting particular experimental situations, such as those concerned with the Belousov-Zhabotinsky reaction.

## Beyond the classical setting

A number of extensions of the basic setting provided by eqs (1) has been worked out to account for situations encountered in real world problems, from materials science to biology.

A first extension pertains to the inclusion of bulk flow. It amounts to augmenting the right hand side of eqs (1) by a term of the form $$\underline{u} \cdot \underline{\nabla} x_i\ ,$$ $$\underline{u}$$ being the bulk velocity. The resulting augmented reaction-diffusion-advection (RDA) equations feature novel classes of solutions. If the chemical species are passive scalars, the dynamics results from the entrainment of reaction-diffusion modes by the complexity of the flow. If, on the contrary, spatio-temporal changes of concentration affect the density or surface tension of the solution, more subtle effects come into play. The RDA equations are then coupled with the momentum and sometimes the heat balance equations which explicitly depend on $$x_i\ .$$ This chemo-hydrodynamic coupling gives rise to a rich variety of behaviors such as buoyancy- or Marangoni-induced deformation and acceleration of chemical fronts for instance.

Parameter variability is yet another effect to be accounted for by extending eqs(1). It is manifested when the reactions and transport take place in a medium like a porous material, or on a catalyst surface where different crystallographic planes may have different affinities toward reaction and/or mobility.

As mentioned already, developments in reaction-diffusion systems contributed significantly to the resolution of the long-standing riddle, how complex evolutionary and organizational processes that are so ubiquitous in biology can find their origin in the basic laws of chemistry and physics. Beyond this conceptual advance, research in reaction-diffusion systems has led to a semi-quantitative interpretation of a wide spectrum of dynamical behaviors in biology such as metabolic oscillations, the cell cycle, the electrical activity of the brain, the immune response, embryonic development, aggregation, food recruitment and building activity in social insects and biological evolution itself.

Reaction-diffusion dynamics in systems involving small numbers of particles is also an area of growing importance. Such systems, whose size is in the nanometer range, include biological regulatory, synthetic and energy transduction machines at the macromolecular or the subcellular level. As one reaches such small scales, fluctuations (the spontaneous deviations from mean-field behavior induced by microscopic level processes) begin to play an important role. In an asymmetric environment as provided, in particular, by nonequilibrium constraints or by reactions converting energy rich reactants to energy deficient products systems of this kind – referred to in this context as ratchets – may exhibit such counter-intuitive behaviors as the fluctuation induced generation of a flux opposing the external force. In a different context, in systems of restricted geometry such as catalytic surfaces or micelles, limited possibilities of mobility or of chemical bonding may favor the generation of strong inhomogeneous fluctuations and force the segregation of homologous particles into small spatial domains that mix poorly to each other.

In both of the above cases an enlarged description beyond the one afforded by eqs(1), incorporating the effect of the fluctuations on the evolution of the macroscopic observables, becomes necessary. It appeals to the tools of the theory of stochastic processes where one derives equations such as the master equation or the Fokker-Planck equation governing the evolution of the probability distributions of the variables involved in the process, which now become the principal quantities of interest. Conditions may also be derived under which this description, referred to as mesoscopic description, can be deduced from the full-scale microscopic dynamics without any heuristic approximations.

## References

General surveys on reaction-diffusion systems

• G. Nicolis and I. Prigogine, Self-organization in non equilibrium systems, Wiley, New York (1977).
• Y. Kuramoto, Chemical oscillations, waves and turbulence, Springer, Berlin (1984).
• I. Epstein and J. Pojman, An introduction to nonlinear chemical dynamics, Oxford University press, Oxford (1998).
• A. De Wit, Spatial patterns and spatiotemporal dynamics in chemical systems, Adv. Chem. Phys. 109, 435-513 (1999).

Mathematical aspects

• J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, Berlin (1983).
• R. Aris, Mathematical modelling techniques, Dover, (1994).
• G. Nicolis, Introduction to nonlinear science, Cambridge University Press, Cambridge (1995).

Applications beyond chemistry

• M. Eigen and P. Schuster, The hypercycle, Springer, Berlin (1979).
• P. Ortoleva, Geochemical self-organization, Oxford University Press, Oxford (1994).
• A. Goldbeter, Biochemical oscillations and cellular rhythms, Cambridge University Press, Cambridge (1996).
• S. Camazine, J.-L. Deneubourg, N.R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in biological systems, Princeton University Press, Princeton (2003).
• J.D. Murray, Mathematical Biology II, Springer Verlag (3rd ed., 2003).

Beyond the mean-field description

• F. Baras and M. Malek Mansour, Microscopic simulations of chemical instabilities, Adv. Chem. Phys. 100, 393-474 (1997).
• D. ben-Avraham and S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000).
• P. Reimann, Brownian motors: noisy transport far from equilibrium, Phys. Rep. 361, 57-265 (2002).
• P. Gaspard, Fluctuation theorem for nonequilibrium reactions, J. Chem. Phys. 120, 8898-8905 (2004).

Internal references