Most reactions proceed smoothly, at varying rates, to a final state of equilibrium. Some, however, do not. They oscillate in time: reactant, product, or intermediate species' concentrations fluctuate wildly, often leading to easily observable oscillations in time of these concentrations. (Other reactions are known that produce oscillations in spacewaves of reactants, products, or intermediates show up.)
These reaction systems are of great interest. Many natural phenomena, from firefly flashes to your heartbeat, are oscillatory chemical systems, and spatial oscillationstiger stripes, zebra stripes, and so onare also familiar. While oscillating biological reactions are incompletely understood, there are many oscillating reactions known among simple inorganic or organic molecules. Many years of research, both experimental and theoretical, have uncovered the detailsthe mechanismsof these reactions.
One of the simplest mechanisms has served as the paradigm of many others. It has been studied theoretically in great mathematical detail (and in fact, no known chemical reaction follows this mechanism exactly), and it has been adopted in several other branches of science, most notably ecology, as we will see. This mechanism is known as the Lotka-Volterra mechanism after the two scientists, the physical chemist Lotka and the mathematician Volterra, who first described it, independently, around 1925.
The net L-V reaction is quite simple: compound A turns into compound B. The interesting kinetics comes about from the mechanistic steps that carry out this simple net reaction. There are only three steps. The first is
where X is an intermediate. Note that this elementary reaction is not written A -> X. It is important to the kinetics to express what happens on a molecular level in each mechanistic step, and this first step requires A and X to interact to produce two X molecules. This is an example of an autocatalytic process: some of the product must be present to generate more product.
The second step takes place between two intermediates X and Y:
Note that it is also autocatalytic.
The final step is the unimolecular conversion of Y to the final stable product B:
Add these three steps, and you'll see they yield the net reaction A -> B.
We can understand this mechanism better if we turn now to its application to ecology, where it has become known as one of the simplest ways to describe predator-prey populations. (In fact, it was Volterra's father's study of fish populations that lead Volterra to describe these mechanistic steps mathematically.) Remember the story of Peter and the Wolf? (If you don't, there is a good web site that tells the story and plays some of the music Sergei Prokofiev wrote to illustrate it. This site tells a kinder version of the original story, however. At the end, instead of killing the wolf, Peter convinces the hunters to capture it and take it to a zoo!) The key part of the story for us concerns the duck, who, alas, is swallowed whole by the wolf.
Let A be grain (duck food) and X be a duck. Then the first step says, in effect, "ducks eat grain, thrive, and produce more ducks." Let Y be a wolf, and the second step says "wolves eat ducks, thrive, and produce more wolves." But wolves are at the top of this simple food chain, so the final step says "wolves die," and B is a dead wolf. The net reaction is simply the conversion of a food source (grain) into biomass (dead wolves).
A little thought shows how the duck and live wolf populations (our reaction intermediate concentrations) oscillate in time. At first, the ducks thrive and their numbers grow. But an increased duck population allows the wolf population to grow, and as it does, the voracious wolves go too far, and the duck population eventually collapses. This removes the wolf food source, and in time, it collapses as well, allowing the duck population to recover, leading eventually to more wolves, and the cycles repeat as long as the grain supply holds out.
The kinetics of the L-V mechanism is not too difficult to study through computer simulation. The output of one such simulation is shown below:
The oscillations are obvious, but look closely at the way the wolf population slightly lags behind duck population, and notice the way the peaks are slowly falling in size over time, as the grain (the initial amount of A) slowly falls.
It is a very worthwhile exercise to assign rate constants to each elementary process in the mechanism and to work out the rate laws for each species, particularly for the intermediates X and Y. Give it a try! Let each step operate in the forward direction only with rate constants k1, k2, and k3, respectively. Find expressions for d[X]/dt and d[Y]/dt, and apply the steady state condition to X and Y, assuming [A] = a constant for all time (always plenty of grain for the ducks to eat). Show that these two assumptions predict [X] and [Y] to have constant steady-state values, which, since we know [X] and [Y] oscillate in time, means this reaction system is never in a steady state. Once you've tried this yourself, you can check your answers here.