17: Equations with names: the importance of Lotka and Volterra (and Tolstoy?)

The most famous equations with names – in one case by universal association – seem to come from physics: Newton’s Law of Gravity – the gravitational force between two objects is proportional to their masses and inversely proportional to the distance between them; Maxwell’s equations for electromagnetic fields; the Navier-Stokes’ equation in fluid dynamics; and E = mc2, Einstein’s equation which converts mass into energy. The latter is the only equation to appear in the index (under ‘E’) in Ian Stewart’s book ‘Seventeen equations that changed the world’. While the gravitational law has been used to represent situations where distance attenuation is important, the translation is analogous and not exact. An interesting example, pointed out to me by Mark Birkin, is Tolstoy in ‘War and Peace’: “Meanwhile, the very next morning after the battle, the French army moved against the Russians, carried along by its own impetus, now accelerating in inverse proportion to the square of the distance from its goal.” Penguin edition, 2005. Tolstoy would have written this in the 1860s! The physics equations, on the whole, work in physics and not elsewhere. An exception – that is, it does work elsewhere and has served me well in my own work – is Boltzmann’s equation for entropy, S = klogW (to be found on his gravestone in Vienna). The other equations which have served me well – plural because they come in several forms – are the Lotka-Volterra equations, originally developed in ecology. Because of the nature of ecology relative to physics, they do not deliver the physics kind of ‘exactness’ but this may in part be the reason for their utility in translation to other disciplines.

The Boltzmann entropy-maximising method works for any problem (and hence in a variety of fields) where there are large numbers of weakly-interacting elements where interesting questions can be posed about average properties of the system. Boltzmann does this for the distribution of energy levels of particles in gases at particular temperatures for example. In my own work, I use the method to calculate, for example, journey to work flows in cities. The entropy measure was also introduced by Shannon into information theory and in one sense underpins much of computer science. When he produced his equation to measure ‘information’ he is said to have consulted the famous mathematician von Neumann on what to call the main term. “Call it ‘entropy’”, von Neumann replied (paraphrased): “It is like the entropy in physics and if you do this you will find in any argument, no one will understand it and you will always win!” I would dare to say that von Neumann was wrong in this respect: it can be explained. Cesar Hidalgo in his recent book ‘Why information grows’ makes the interesting point about Boltzmann’s work that it crosses and links scales – the atoms in the micro with the thermodynamic properties of the macro; this was unusual at the time and perhaps still is.

The Lotka-Volterra equations are concerned with systems of populations of different kinds – different species in ecology for example. In one sense, their historical roots can be related back to Malthus and his exponential ‘growth of population’ model. In that model, there were no limits to growth, and these were supplied by Verhulst who dampened growth to produce the well-known logistic curve. (Bob May in the 1970s showed that this simple model has remarkable properties and was the route into chaos theory.) What Lotka and Volterra did – each working independently, unknown to each other – was to model two or more populations that interacted with each other. The simplest L-V model is the well known two-species predator-prey model. There is a logistic equation for each species linked by their interactions: the predator species grows when there is an abundance of prey; the prey species declines when it is eaten by the predator. Not surprisingly, there is an oscillating solution. What is more interesting in terms of the translation into other fields is the ‘competition for resources’ form of the L-V model. In this case, two or more species compete for one or more resources and this provides a way of representing interactions between species in an ecosystem. The translation comes through identifying systems of interest in which populations of other kinds compete for resources. There are examples in chemistry where molecules in a mixture compete for energy, in geography where retailers compete for consumers (as in my own work with Britton Harris) and in security with Lewis Fry Richardson’s model of arms races and wars. There are undoubtedly many more possibilities.

Lotka, Volterra and Richardson were working in the 1930s and 40s and there are interesting common features of their research. None of them worked primarily – at least in the first instance – in ecology. Lotka was a mathematician and chemist, and later an actuary; Volterra was a mathematician and an Italian Senator. Both came to mathematical biology relatively late. Richardson was a distinguished meteorologist and later a College Principal. It is worth looking at their original papers to see the extraordinary range of examples they pursued in each case(with real data which must have been difficult to accumulate) – particularly bearing in mind that there were no computers. Indeed, Richardson, at the end of one of his papers, thanks “…..the Government Grant Committee of the Royal Society for the loan of a calculating machine”! It was also interesting, perhaps not surprising given their mathematical skills, that they explored the mathematical properties of these systems of equations, in various forms, in some depth. Their work at the time was picked up by others – notably V. A. Kostitzin. I picked up a second-hand copy of his 1939 book ‘Mathematical biology’ via the internet after searching for Volterra’s work: Volterra wrote a generous preface of the book!

The Lotka-Volterra equations represent one of the keys to a particular kind of interdisciplinarity: a concept that can be applied across many disciplines because of the nature of what is a generic problem – modelling the ‘competition for resources’. In a particular instance of a research challenge, the trick is to be aware that the problem may be generic and that there are elements of a toolkit lurking in another discipline!

Alan Wilson

June 2015

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