Chemical Reactions and Spatial Effects

Chapter 8 focuses on chemical reactions. We first introduce the law of mass action to obtain rate equations, and consider two classical models for oscillatory chemical reactions, which only involve ordinary differential equations. These models may be simplified into two-dimensional dynamical systems and are amenable to the kind of analysis done in the previous chapters.

So far, we did not discuss or even consider the fact that most models describe quantities which vary not only in time, but also in space. For instance, if $N$ measures the density of a population, it is normal to assume that $N$ depends on the landscape: for humans, $N$ is larger in cities than in mountain or desert areas; for animals, $N$ varies according to the presence of forests, rivers, prey, etc.

In order to include spatial effects in the models we have discussed, we proceed as follows. First, we assume that the nature of the model itself does not change. For instance, if species $u$ eats species $v$, then at any point with coordinates $(x,y)$ in the plane, $u(x,y)$ grows proportionally to the local concentration of $v(x,y)$. One could imagine more complicated situations where the growth rate of $u$ depends on say the spatially averaged density of $v$, but such cases are beyond the scope of these notes. In other words, we only consider local models. Second, we need to describe how each species behaves when it is not uniformly distributed over a region of space. Our intuition tells us that motion should take place away from regions of high concentration, in order to reach a uniform distribution. This is the phenomenon of diffusion, which is discussed in Chapter 9. In the presence of an external flow, other transport terms have to be included, but we will not consider such situations.

Adding diffusion to nonlinear differential equations typically leads to reaction-diffusion equations. These models are systems of partial differential equations, and there is a vast literature on this and related topics. Although a complete discussion is beyond the scope of these notes, we nevertheless give a flavor of the kind of modeling done with such systems. Chapter 9 briefly describes an example of surface chemical reaction, in which diffusion is coupled to the corresponding chemical rate equations. The resulting reaction-diffusion system is able to sustain chemical waves and we encourage the reader to study the articles referenced in the text for additional information. Chapter 10 discusses the general phenomenon of pattern formation in systems driven far from equilibrium. There, we explore the dynamics of a generic pattern-forming model, namely the Swift-Hohenberg equation, and also consider the specific example of a two-dimensional model leading to vegetation patterns.