Amplified Stochastic Cycles

Many systems show sustained oscillations. This is especially true in biology. For instance, it is obviously desirable that metabolic reactions should be self-sustaining, with the system returning near to its original state on a periodic basis. But oscillations are also found in data on epidemics and on predator-prey interactions. The argument for the latter is that if prey numbers are high, then predator numbers will increase, so leading to a decrease in prey numbers, and so a decrease in predator numbers, and so on. This seems to imply that predator-prey numbers should oscillate out of phase. The problem is that while many computer simulations of metabolic reactions, epidemics and predator-prey interactions show cycles, most models show the system tending to a fixed point (and not a limit cycle) at large times.

A possible resolution of the paradox was proposed long ago: the deterministic models which are commonly studied do not take account of the stochasticity caused by the discreteness of the constituents (so-called intrinsic or demographic stochasticity). This will have the effect of converting oscillatory decay to a fixed point into sustained oscillations. However, this raises more questions: when the number of constituents, N, is large the deterministic equations should work well, and in any case fluctuations should only be of order the inverse square root of N, and so negligible for large N. Yet computer simulations show cyclic behaviour when N is quite large.

To gain an understanding of these questions, the first thing that has to be done is to carefully formulate the model that is describing the system of interest. Surprisingly, this is rarely done systematically. Instead many authors study the stochastic aspects through computer simulation, and the deterministic aspects through a (set of) nonlinear differential equation(s), without checking that the latter can be derived from a more basic stochastic formulation. Together with Tim Newman I have advocated a starting point for describing systems of the type discussed above (for specific examples, see later) in which the constituents interact with each other through specified probabilistic rules and at given rates. Of course, this is exactly the usual starting point for the efficient simulation of such processes, through the Gillespie algorithm, for instance. However the key point is that, when formulated in this way, they may also be studied analytically as master equations. Once this is achieved, the deterministic equations (formally valid when N tends to infinity), and stochastic equations which provide the corrections to them, can be found. This approach is described in Phys. Rev. E70, 041902 (2004).

If this idea is applied to model a community consisting of n predators and m prey, in the limit where the total number of individuals becomes very large, the most basic model can be shown to go over to the simple, and deterministic, Volterra equations describing the predator-prey dynamics of these two species. The Volterra equations have a fixed point, but no limit cycle, yet a simulation of the original individual-level description does show predator-prey cycles of the kind which are well-known (such as the hare-lynx cycle). Again together with Tim Newman I have applied the van Kampen system size expansion to the master equation for the system. This technique is ideally suited to investigate this problem, since at leading order it gives a set of deterministic equations (in this case, the Volterra equations) and at next-to-leading order a linear Fokker-Planck equation which gives a simplified version of the stochastic dynamics valid for moderately large systems. Since this Fokker-Planck equation is linear it can be straightforwardly analysed. In Phys. Rev. Lett. 94, 218102 (2005) we show that by converting the Fokker-Planck equation to a set of linear Langevin equations and carrying out a Fourier analysis, large cycles can be found analytically, which perfectly match those found numerically. These cycles vanish as the size of the system goes to infinity like the inverse of the square root of N, and so do not appear in the corresponding deterministic theory. The inverse square root of N is multiplied by a large numerical factor due to resonance effect, and so even for quite large values of N the cycles have a large amplitude.

This approach may be applied to several other areas, as mentioned above. One of these, where data is more abundant, is in epidemiology, especially childhood diseases such as measles, rubella and whopping cough. The analogous model to Volterra is the SIR (Susceptible-Infective-Recovered) model. Again, in a deterministic framework, there is oscillatory decay to a fixed point, but taking into account the individuals who comprise the population changes this decay into sustained oscillations. This has been analysed in detail in some work carried out in collaboration with David Alonso and Mercedes Pascual. The situation is a little more complicated in this case, since individuals will be subject to external effects which have a period of one year (for example, term times for schoolchildren). Nevertheless, we have shown in J. R. Soc. Interface 4, 575-582 (2007) that the amplified oscillations are still present and provide an explanation for some of the observed cyclic behaviour.

Another area where this mechanism has an important role --- as indicated earlier --- is in biochemical oscillations within the cell. The number of molecules involved in reactions in a cell may be relatively small, and it is certainly debatable whether the "thermodynamic limit", where the number of molecules is assumed to be infinitely large, is relevant. As a consequence it may be expected that the predictions of rate equations would not be reliable and, as mentioned above, this is frequently found to be the case. In J. Stat. Phys. 128, 165-191 (2007) the formalism we have been discussing above is applied to simple models of self-regulatory gene expression and glycolytic oscillations. Amplified oscillations are found which give rise to cycles of significant amplitude, even if the numbers of molecules are not too small.

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