## 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?

### References:

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

Neutral models do seem to work well in many cases. I think this is especially so for plants, which is no surprise given the origins of the theory in the data set from Barro Colorado. I can also see why its threatening to many ecologists, because it suggests you can explain a lot of observed biodiversity patterns without resorting to interspecific interactions, and that suggests that whole branches of ecology are less important than maybe we thought they were.

Having said all that, I think its far from a universal law in ecology. Sometimes its hard to fit communities – I’m thinking particularly of parasite assemblages – into the neutral model categories you described above (simple, meta and spatial), though they seem closest to the metacommunity. See these for futher discussions:

doi: 10.1645/GE-677R.1 “Defining parasite communities is a challenge for neutral theory”

DOI: 10.1016/j.pt.2006.09.008 “Species accumulation curves and their applications to parasite ecology”

Thanks for a thought provoking post.

Al

Al Dove7 June, 2010 at 1:40 am

Thanks, Al, I’d not thought about parasite communities before. Presumably it also depends on what sort of parasite you’re thinking of: animal parasites which only move between hosts by direct contact are probably rather different to a

Strigaplant which can produce thousands of tiny seeds. Off the top of my head, I’d guess it would be easier to make a neutral model for the latter case.Thomas Kluyver19 June, 2010 at 5:48 pm

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Great post Thomas, you have explained species area curves and the relevance of neutral theory very well. Since you mention my work I would like to add a couple of points that I think are relevant.

How far away exactly will an offspring be from its parent in these neutral models and in reality? In the real world the distance is not fixed, but depends on many random factors. In the models there is a function (known as a dispersal kernel) giving us the probability of dispersal between patches, which will vary depending on their distance apart. In general the larger the distance, the less likely dispersal is. There are many different possible dispersal kernels one could consider but those who have studied dispersal in nature suggest that “fat tailed” dispersal kernels are the more realistic for many systems. These fat tailed dispersal kernels increase the chance of dispersal over long distances dramatically (although it is still more likely to travel a shorter distance than a longer distance, the penalty associated with long distances gets less severe). One biological example may be a wind borne seed, once it has been caught by the wind and taken up to a fair altitude its chances of travelling 20Km are very small, but not much different from its chances of travelling 10Km. There are many different kinds of fat tailed dispersal kernel: the more fat tailed the dispersal kernel, the more likely dispersal over long distances will be.

What I showed in my paper with Stephen Cornell is that in order for the model to match species area curves of real world data these “fat tailed dispersal kernels” are needed, in agreement with the expectations of other studies on dispersal. We found that the more ‘fat tailed’ the dispersal kernel was, the better the fit to data and the more biologically reasonable the parameters. Unfortunately, it was also true that for technical reasons, the more fat tailed the dispersal kernels were, the more computationally intensive the simulations became so was a limit where we could not realistically go further.

O’Dwyer and Green published an analytical solution for the spatially explicit neutral model using an ingenious method. As you say analytical solutions are preferable to simulations, but you have to compare like with like. Fat-tailed dispersal kernels are typically awkward analytically and the results of O’Dwyer and Green do not apply to fat tailed dispersal kernels at all. It is also unclear whether or not their approach can possibly be extended to incorporate fat tailed dispersal kernels or not. So we were able to simulate ecologically important versions of the model that were not covered by the analytical solution and the cases we could not simulate were not covered by the analytical solution either. This type of situation is mirrored all over science: there are analytical solutions for some cases, but not all and simulations are used these other cases. In general I think that both simulations and mathematical solutions both have an important role to play in this field with analytical results being preferable, but only covering special cases. Simulations are also useful as a means of toying with a number of models to see which are sufficiently interesting to warrant seeking an analytical solution.

Thanks for an interesting post,

James Rosindell

James Rosindell18 September, 2010 at 2:08 am

Thanks, James! I confess I hadn’t picked up that you’d been able to study something that wasn’t possible by the other method–the maths is a bit over my head.

Thomas Kluyver26 September, 2010 at 6:30 pm