Species area curves & neutral theory
How many species are there here? It’s a beguilingly simple question, and a fundamental area of interest. A moment’s thought shows that the bigger here is, the more species there will be. So, if we start from a little patch of my lawn, and take successively larger heres until we’ve included the whole world, we can draw a ‘species area curve’. It generally looks a bit like this:
It’s got three distinct parts: at local scales, the number of species increases sharply as you look at larger areas, then at regional scales it slows down, but at the very largest scales it picks up again. It’s easy to come up with possible explanations in words: at first, the number of species increases as you happen to ‘catch’ more species in your area, then it levels off because you’re mostly finding the same species again, and finally climbs as you encounter ‘exotic’ species that don’t live near your starting point. But can a mathematical model of species come up with the same sort of result?
Enter neutral theory. Laid out in a book by Stephen Hubbell, it tried to model a group of species by ignoring all the differences between them, imagining that every individual has the same chance of dying, the same chance of reproducing, and the same (small) chance of producing a new species. This is, to say the least, controversial, but remember that it’s a model: of course reality’s not like that, what’s interesting is how well such a simple model fits ecological patterns like the species area curve.
The very simplest version of neutral theory completely disregards where individuals are: when there’s a gap to be filled, any individual has the same chance of filling it. An extra development is the idea of a ‘metacommunity’, where individuals die and reproduce within one population, but occasionally disperse from one population to another
That sort of model can’t study the intricacies of species-area curves, though. Both of the studies referenced below used versions of neutral theory that do take account of where each individual is: ‘spatially explicit’ models, in the jargon.
James Rosindell and Stephen Cornell made a computer simulation, in which each individual occupied one square of a grid (it helps to imagine trees in a forest, rather than moving animals). When one dies, its square is most likely to be filled by the offspring of a nearby individual. This led to a species area curve with more or less the right shape, and by running the simulation many times, with different settings, they were able to get a decent fit to real-world data; it turned out that their original model had favoured the nearby individuals a bit too much when filling gaps, and they had to allow slightly more dispersal from farther away.
Computer simulations are unwieldy, though. For every change, the model must be run several times over, and some changes will make it slower: Rosindell and Cornell admit that at least one possibility was too “computationally expensive” to test. So James O’Dwyer and Jessica Green set out to make a mathematical model, a set of equations, based on probabilities, to act as a shortcut between the settings and the result.
They, too, start off with a grid, except that they allow more than one individual to share a square. The equations for this I think I can understand. Then they turn it into a different type of equation (a “moment generating function”), and then work out what happens if you make the squares of the grid infinitesimally small. At this point they use some maths from quantum field theory, the proportion of Greek letters goes up, and curly Z and downward-pointing-triangle put in appearances, so I won’t pretend to understand it at all. The result, however, is a curve with three parts, and realistic numbers for things like the speciation rate do give it the right shape.
So what does this tell us? Well, it seems that the pattern we see for the number of species in different sized areas can be explained without considering either biological factors, such as competition and adaptation, or geographical ones, such as the arrangement of landmasses. And it sets the bar for anyone studying the effect of such things: can they explain the pattern better than a neutral model does?
O’Dwyer, J., & Green, J. (2010). Field theory for biogeography: a spatially explicit model for predicting patterns of biodiversity Ecology Letters, 13 (1), 87-95 DOI: 10.1111/j.1461-0248.2009.01404.x
Rosindell, J., & Cornell, S. (2009). Species–area curves, neutral models, and long-distance dispersal Ecology, 90 (7), 1743-1750 DOI: 10.1890/08-0661.1