Tag Archives: maths

An ecologist amongst the mathematicians


The skyline of Nottingham looks nothing like that at all.

I’m spending this week at the joint meeting of the European Society for Mathematical and Theoretical Biology and the Society for Mathematical Biology. There are around 850 registered delegates, and it lasts for five days, so it’s a pretty big one. It’s also not home territory for me. Although I’ve drifted into theoretical ecology over the last few years, and increasingly collaborate with mathematicians, my foundation is still very much in quantitative field ecology. I definitely feel like an imposter.

Below are ten personal observations on attending a conference that’s a long way outside my own area. None of these are specific to ECMTB, nor am I qualified to comment on maths conferences in general. It’s also a little mixed up by being on my own campus here at the University of Nottingham, which does tend to change the experience of a meeting.


What’s the collective noun for mathematicians — an aggregation? Swarm? Piss-up?

1. No one knows you so they don’t talk to you. This is particularly striking having just come back from ATBC in Montpellier. I’ve been going to ATBC meetings since 2003. Walking into coffee, or the poster room, or even on the streets outside the venue, it was impossible not to bump into someone I knew. This has a snowball effect as these people then introduce you to others, who you meet again the following year, and before long you become part of a community. Here, standing in the lunch queue amongst hundreds of people, looking around and seeing no-one I know… that was a little unsettling.*

It’s not quite the same as going to a conference as a grad student (I do still remember). Back then, although I didn’t know many people, I was part of a cohort of other students who gravitated towards one another, often assisted by early-career events, and who later became the group of awesome colleagues I have today. Now that I’m mid-career the social groups have already formed. I’m acutely aware that my own regular conference cliques must appear similarly exclusive, even though we’re not deliberately so.

2. Despite the above, one thing no-one can ever accuse me of is shyness. Not knowing someone doesn’t mean I won’t just walk up and talk to them regardless, and everyone I’ve spoken to has been warm and friendly. But I’ve come across another problem, which is that my usual elevator pitch doesn’t work. Put me in a room with an ecologist and I know how to describe what I do clearly and concisely; in no time we will find common ground. My attempts to do the same here have met with confusion, largely because we define our problems in different terms and I’m not good at expressing my research in language that a mathematician can relate to and find relevant. I also often struggle to get my head round what they’re telling me about their own work. I’ll keep trying though.

3. Connected to this is that mathematicians ask hard questions. They’re probably not tough for a mathematician, but to me they’re challenging and unpredictable. I can usually anticipate what another ecologist might ask me, and I’ve often thought about the answer myself. What makes mathematical questions difficult is that they come from an entirely different perspective and way of thinking about systems. This is great of course, and one of the main reasons that I’m here.

4. All this means that it’s hard to sell your research as an outsider. Without wanting to sound arrogant, I’m confident at BES that if I have a talk scheduled, people will come. Some in my specific area will make a special effort. Here there were a reasonable number of people in the room for my talk, but I can’t pretend that I was the draw as it was in the middle of a long session in between established speakers. Was this because it’s a fringe topic, because I’m an imposter, or because I’m not familiar with how to pitch my title and abstract to make them enticing for this audience? I suspect a bit of all three.** On the other hand, it was refreshing to talk to a room of complete strangers with no expectations or preconceptions.

5. A pattern I’m picking up from the conversations I’ve had so far is that the whole conceptual structure of the field is different. My guess is that mathematicians self-identify with a particular field or approach, which can be applied and tested in a variety of contexts. Ecologists, on the other hand, fix on a particular system or problem and bring in a portfolio of methods that will help them to study and understand it. I find this really invigorating: it’s forcing me to rethink the links between areas that I don’t normally associate with one another, and exposing me to research I wouldn’t otherwise see.

6. This is held back, alas, by the realisation that I don’t even know the basics. In a number of talks, the speaker has glossed over some logical step or series of equations as being commonly understood by everyone in the room. Often that leaves me floundering, although I’m picking up a wish-list of things I need to learn. Are ecology talks as impenetrable to outsiders for the same reason? When we flash past a slide on Janzen-Connell processes or SARs, do we lose a significant fraction of the audience?

7. One real difference in behaviour I’ve noticed is that mathematicians interrupt talks — and the speakers don’t seem to mind at all! I’ve seen several talks stopped while someone asks for clarification of a particular parameter or equation. This is unheard of in ecology, but I think it’s great, because on most occasions I didn’t understand it either.

8. Ecologists (and biologists in general) are often insecure about their grasp on mathematics. Yet I’ve heard a similar sentiment expressed in several talks here; theoretical ecologists worry about their lack of field experience. To me this is very reassuring, and makes me feel more comfortable about being out of place. We can all help one another.

9. Call this a loss of attention span on my part, but I don’t like 15-minute talks any more (and 25+5 is even worse). Ten years ago the 15+5 model was standard in ecology, but the growing size of the meetings has meant that we now run on 12+3 as standard. This not only allows more to be packed in (or fewer parallel sessions) but forces effective speakers to focus rigorously on their core points. With 15 minutes there’s room to digress, and for the audience’s mind to wander. Even I was getting bored of the sound of my own voice by the end.*** If I did have one specific suggestion for the organisers it would be to change this; many in BES resisted at first but now I don’t want to go back.

10. I’m not attributing any of the above comments to inherent differences between mathematicians and ecologists, and that’s not a deliberate choice on my part, just that I don’t see any evidence of ‘two cultures’. The dynamics are exactly the same as any other conference I’ve attended: people gather and talk excitedly about research, ask interested and penetrating questions of speakers, greet old friends with delight… it’s all very familiar****. There’s no need for me to trot out any cliches about mathematicians because, so far as I can see, none of them are true. Everyone has been very welcoming and genuinely pleased to talk. Maybe I’ll come back next time and know a few more people…


Like me, Robin Hood is more at home in the forest than at a maths conference. Posted by @ECMTB2016

* The flip side of being unknown here myself is that I also don’t really have a clue who I should be speaking to. I’ve been taking a haphazard approach so far.

** That said, I’m very grateful to the organisers for giving me a slot in a session on the very first day, despite being an outsider. Many thanks :o) The talk seemed to go fine, though my confidence largely stemmed from having already published most of it, and presented it before at the BES in Lille. It was a safe bet.

*** Please insert sarcastic rejoinders in the comments. I know I’m a bigmouth.

**** OK, there are some differences. For one, there’s much less booze involved. I’m told by people who work in catering that the worst conferences are ecologists, geologists and archaeologists. There’s something about academics who spend large amounts of time in remote locations with nothing to do other than drink. Also, mathematicians are better dressed than ecologists. I’ve yet to see any sandals or — that mainstay of ecology meetings — the barefoot speaker. This is A Good Thing.


Two lumps please

Here’s a quick thought experiment. Imagine you have a spare flowerbed in your garden, in which you scatter a handful of seeds across the bare ground. You then ignore them, and come back some months later. What will have happened?* Your expectation might be that you will have a healthy patch of plants, all about the same size. Some might be larger or smaller than average, but overall you’d expect them to be pretty similar. This is known as a unimodal size distribution. They have after all experienced identical conditions.

You’d be wrong. In fact, it’s more likely that your plants will have separated into two or more size groupings. There will be a set of larger plants, spread apart from one another, and which dominate the newly-formed canopy. In between them will be scattered other plants of smaller size. This results in a bimodal (or multimodal) size distribution. There isn’t a standard, expected size; instead there will be different size classes present.


A normal, unimodal distribution of sizes (left) is what you might expect to see when all plants are the same age and growing in the same conditions. In fact it’s more common to see a bimodal size distribution (right), or something even more complicated.

This observation is nothing new. Much was written about the issue from the 1950s through to the 70s, particularly in the context of forest stands. The phenomenon was widely-recognised but remained paradoxical.

I stumbled upon this old literature back in 2010 when I published a small paper based on a birch forest in Kamchatka which showed a clearly bimodal size distribution. I didn’t need to go all the way to Kamchatka to find a stand with this feature; but since I had the data it made sense to use it. I used the spatial pattern of stems to infer that the bimodality was the result of asymmetric competition (i.e. that large trees obtain disproportionately more resources than small trees, which is definitely true in terms of light capture). All the trees were the same age, but the larger stems were spread out, with the smaller stems in the interstices between them. Had the bimodality been the result of environmental drivers we would expect there to be patches of large and small stems, but in fact they were all mixed together.

White birch forest, central Kamchatka

This is the stand of Betula platyphylla with a bimodal size distribution that was described in Eichhorn (2010). If it looks familiar, it’s because the strapline of this blog is a picture of us surveying it. The white lights on the photo aren’t faeries, it’s the reflectance of mosquito wings from the camera flash. So many mosquitoes.

Three things struck me when I was reading the literature. The first was that hardly anyone had thought about multimodal size distributions in cohorts for several decades**. This was a forgotten problem. The second was that the last major review of the phenomenon back in 1987 had concluded that asymmetric competition was the least likely cause — which conflicted with my own conclusions. Finally, I had no difficulty in finding other examples of multimodal size distributions in the literature, but authors kept dismissing them as anomalous. I wasn’t convinced.

Analysing spatial patterns is all well and good but if you want to really demonstrate that a particular process is important, you need to create a model. Enter Jorge Velazquez, who was a post-doc with me at the time but now has a faculty position in Mexico. He built a simple model in which trees occupy fixed positions in space and can only obtain resources from an the area immediately around themselves. Larger trees can obtain resources from a greater area. When two trees are close to one another, their intake areas overlap, leading to competition for resources.


When there are two individual trees (i and j), each of which obtains resources from within a radius proportional to its size m, the overlap is determined by the distance d between them. Within the area of overlap the amount of resources that each receives depends on the degree of asymmetric competition, i.e. how much of an advantage one gets by being larger than the other. This is included in the model as a parameter described below.

This is where asymmetric competition is introduced as a parameter p. When = 0, competition is symmetric, and resources are evenly divided between two trees when their intake areas overlap. When = 1, each tree receives resources in direct proportion to its size  (i.e. a tree that’s twice as large will receive two thirds of the available resources). Increasing makes competition ever more asymmetric, such that the larger competitor receives a greater fraction of the resources being competed for. In nature we expect asymmetric competition to be strong because a taller tree will capture most of the light and leave very little for those beneath it.

We applied the model to data from a set of forest plots from New Zealand which have already been well-studied. Not only did we discover that two thirds of these plots had multimodal size distributions, but also that our model could reproduce them.

We then started running our own thought experiments. What if you changed the starting patterns, making them clustered, random or dispersed? That turned out to have very little effect on size distributions. What about completely regular patterns? That’s when things started to get really interesting.

By testing the model with different patterns we discovered three important things:

  • Asymmetric competition is the only process which consistently causes multimodal size distributions within simulated cohorts of plants. Nothing else we tried worked.
  • Asymmetric competition is the cause, not the consequence of size differences in the population.
  • The separation of modes is determined by the length of time it takes for competition in the cohort to start, which usually reflects the distance between individuals.
  • The number of modes reflects the effective number of competitors that each individual has.

What does all this mean? Given that asymmetric competition is normal for plants, I would argue that we should expect to see multimodal size distributions everywhere. In fact, seeing unimodal size distributions should be a surprise. Don’t believe me? Grab some seeds, give it a go, and tell me if I’m wrong.

You can read our new paper on the subject here. If you can’t get hold of a copy then let me know.

* Luckily this is a thought experiment, because in my garden the usual answer is ‘everything has been eaten by slugs’.

** I should stress here that I’m specifically referring to multimodality in size distributions of equal-aged cohorts. When several generations overlap then the distribution of sizes reflects the ages of the individuals. If multiple species are present this adds additional complications, and in fact size distributions of species across communities have been a hot topic in the literature of late. This is very interesting but a completely different set of processes are at work.

We’re all stupid to someone

I spend an increasing proportion of my time collaborating with engineers and theoretical physicists. It keeps me on my toes and I’ve had to adjust to very different research cultures. The engineers, for example, get particularly excited by designing a technical solution to a problem. The long haul of data collection and statistical analysis has less appeal; once they’ve proven it can be done then they’re itching to move on to the next challenge. Likewise physicists genuinely do spend meetings in front of whiteboards sketching equations, which leaves me feeling a bit frazzled. Nevertheless, I’ve learnt that if an idea can’t be expressed mathematically then it hasn’t been properly defined. That turns out to apply to a lot of verbal models in ecology.

Both engineers and physicists are ready to publish at an earlier stage than most ecologists would, and their papers are a model of efficiency in preparation. Not for them a lengthy waffle of an introduction, followed by an even more prolonged and rambling discussion. Cut to the point, make it clearly, then wrap up. It makes me wonder whether we’re doing something wrong in ecology. I certainly don’t enjoy either reading or writing long papers, and I can’t fully justify our practice.

I also find myself fielding questions or tackling issues that would never come up when chatting to an ecologist. One of the misapprehensions I’ve had to counter is that trees are not lollipops. It might be more computationally efficient to assume that trees are spheres of leaves on a stick, and it can lead to some elegant mathematical solutions, but the outcomes are going to depart from natural systems pretty rapidly. Our disciplinary training leads us to consider particular assumptions to be perfectly reasonable, despite them sounding ridiculous to others or bearing little resemblance to the real world. (Even within their own field, forest ecologists are not immune to this syndrome).

Understanding how another researcher arrived at their assumptions can be informative — sometimes it boils down to analytical frameworks, computational efficiency or technological limitations, all of which are valid reasons to consider accepting a proposition that on first hearing might sound far-fetched. Likewise it helps to have our own assumptions challenged. Sometimes we are able to justify and defend them. Other times they leave us exposed, which is when we know we’re onto something important.

It’s also a sad but common trait within all social groups to mock outsiders for making mistakes about things that appear self-evident to those on the inside. Ecologists can easily play the same game, but make no friends by doing so. I had a chat with one of my collaborators this week who was itching to find a small tree on campus, scan it using ground-based LiDAR, then strip and record the sizes of all its leaves. It’s a perfectly reasonable idea (if a lot of hard work). The main stumbling block is that it’s the middle of February and we’re a good three months at least from having full leaf canopies to play with. An obvious problem? Only to someone who spends their life thinking about trees the whole time. We had a laugh about it then moved back to our simulations, which have the considerable benefit of not shedding their leaves seasonally.

This kind of interaction only makes me wonder what crazy things I’m responsible for coming out with in our meetings. It also makes me grateful to my collaborators for their patience in humouring me, because I’m pretty sure that I come across as an idiot more often than I realise. This to me is the greatest pleasure of interdisciplinary collaborations. We could all spend the rest of our careers treading the same academic paths, publishing in the same journals, and not need to stretch ourselves quite as far. By heading way outside our comfort zones we all end up learning more than we expected to, so long as we don’t mind feeling stupid every now and again (which happens every time I get tangled in algebra). If you’re not willing to be wrong then you’re not willing to learn. And if I end up the subject of an amusing anecdote at a theoretical physics meeting? That’s fine by me. I hope it raises a good laugh. As a wise man once said, ridicule is nothing to be scared of.