Maths makes and breaks nestedness theory

Mathematicians from the York Centre for Complex Systems Analysis (YCCSA) at the University of York and the Biomathematics Research Centre, University of Canterbury in New Zealand have examined data on plant-pollinator networks in an effort to understand the structure and mechanisms behind species diversity.

Ecological communities are characterised by complex networks of species which interact with each other in different ways, both competitive and cooperative. It had been posited that the ‘nestedness’ of these networks was influential in determining species survival and biodiversity. However, the researchers were able to demonstrate with their analysis of 59 empirical datasets that it is the number of positive, mutualistic interactions a species has that is the better indicator of its persistence.

I spoke to Dr Jon Pitchford, from the Departments of Biology and Mathematics at York - who co-authored the paper with Dr Alex James and Dr Michael Plank from the University of Canterbury - about the evolution of this idea and the need to combine theory with practice in attempts to explain both mutualistic networks and species richness…

What is the theory of nestedness, and why has it been so important to ecological theory?
Nestedness is the idea of interactions within interactions, like a set of Russian dolls. We know that mutualistic relationships exist whereby animals pollinate flowers, for example. You can imagine that if one species of animal could only pollinate one species of plant and one species of plant could only be pollinated by one insect species, that would be a very specialised relationship. If one species disappeared, the other would disappear too. This is quite a boring extreme. The opposite boring extreme would be if every insect could pollinate every flower and vice versa.

Imagine a matrix with a list of animal species in the rows and plant species in the columns. Put a cross in the matrix wherever a plant and animal species interact. There would be very few crosses if the only relationships to exist were specialist ones. On the other hand, if every relationship was generalist and everything interacted, the whole matrix would be full of crosses.

With nestedness, what people have noticed when looking at these networks is that there is some interesting structure present. This is the root of the Russian doll idea, whereby the specialists interact with a subset of those species that the generalists interact with. In other words, a generalist will have a whole list of plants that it pollinates, but some of those plants will also be pollinated by specialist insects. Nestedness means that the crosses on the matrix form a roughly triangular pattern, with many crosses on the top row and a decreasing number as the pollinating species become more specialist.

How does the analysis carried out by you and your colleagues challenge the accepted version of the theory?
It means that the world is not random, which is really interesting to me. About eight years ago people noticed that these matrices were nested, and they naturally went on to ask why. What is it about the world that means we can observe nestedness?

There was an absolutely beautiful paper in Nature in 2009 by mathematicians who claimed to have understood a link between a mathematical model for the dynamics of these systems and nestedness. They thought of the simplest possible mathematical model to describe how organisms of the same class compete with each other but organisms of different classes share mutualistic relationships. They did some very elegant maths to show that the stability of that sort of model might be increased by having more nested interactions. It is a great paper that has been quite influential and heavily cited, but what we did proved that it was wrong.

Can we be confident that the findings extend to all networks and not just the plant-pollinator networks covered in the study?
I can say unashamedly that we cannot, but it is an interesting question for future work. People have certainly tried to look at mutualisms in other systems. They occur in many areas, from economics to social networks. If you’re kind to somebody then they might be kind to you. The idea of mutualistic networks can be applied to all sorts of areas, but we don’t yet understand whether they will all work in the same way.

What is the difference between negative and mutualistic interactions with regards to their effect on species richness?
In the models that people have been using, it seems that mutualistic interactions make it harder to have lots of things co-existing while predator-prey interactions make it easier to have lots of things co-existing.

If you believe those models are correct, then in a mutualistic system, if we help each other, we both become super-strong and can out-compete our rivals. There is a rather limited amount of evidence that this occurs in reality. Alternatively, it means that the way we model these networks of interaction as very simple dynamical systems is inappropriate.

Is this supposed to be the case only between species or could it apply at an intra-specific level?
If a result came out of these models which suggested this effect occurred within a species, I would believe the models were wrong. To talk about intra-specific effects, you really need different models that take into account more factors in the ecology and biology of that individual species.

We know the world is a complicated and diverse place and we want to model all of the interactions that take place. Mathematically, that means finding out a number – a rate, if you like – for each of these interactions. You can imagine the expense, the tediousness and the impracticability of working out every single rate in all of your models; it’s just impossible.

What people have tried to do is come up with sensible, random strategies to do it. In a clever way, this has involved more or less plucking numbers out of thin air and examining the properties of the resulting randomly generated networks. That is what previous results have been based on, and that is what we have shown to be inadequate in explaining the role of nestedness in diversity.

What might be the implications of these findings in terms of the way we understand biodiversity and sustainability?
I think the findings are good news for ecologists and mathematicians alike. It is good for ecologists because we argue that simple, random mathematical models as we understand them can’t say enough about the biodiversity in evidence around us. Sometimes there might be much simpler mechanistic explanations for the patterns we see in the world.

To give an example, hummingbirds with long beaks can pollinate flowers of all lengths, whereas hummingbirds with short beaks can only pollinate short flowers. In the matrix that would give us a nested structure, where long-beaked hummingbirds have many crosses for the many species they pollinate and smaller-beaked hummingbirds have fewer and fewer crosses according to the shortness of their beak.

We have shown that matrices really are triangular in shape – they’re not random – but the mathematical models people have used to explain why they are not random don’t work. In reality there could be much simpler explanations. We are saying that the more mutualistic partners an organism has, the more likely it is to persist. This is something that an ecologist can understand, rather than some abstract property of a complicated matrix of interactions.

We really need proper mechanistic explanations for what we’re seeing, so there is a call for doing this fundamental ecological research as well as asking the big questions that mathematics and computer science can do. I think the work we’ve done is important because it is exposing the gaps in our knowledge. The ‘get your hands dirty’ ecology and the very theoretical side are equally necessary.

What are the next steps for your research?
To be honest, we started working on this subject by chance, inspired by a random seminar during my recent Erskine Fellowship at the University of Canterbury in New Zealand. Mostly I try to understand fish and fisheries, looking at variability in marine ecosystems and how to manage uncertainty in this context. It is a fascinating a subject because it’s obviously very important to manage fish stocks sustainably, but it can be quite risky. The maths will tell you how reliable your estimates are, and it can even tell you how vulnerable your management strategy is to those uncertainties. For example, although the details of how marine reserves affect fish stocks are not entirely clear, mathematically we can see that they provide a buffer against the uncertainties of environmental and ecological factors.

In that sense, maths is a useful way of confronting unknown data, which fits with this research on nestedness; it is all about trying to understand the consequences of variability and uncertainty.