Vincent Jansen is Professor of Mathematical Biology at Royal Holloway University of London. He uses mathematical models to understand how things work in biology, evolution and epidemiology. This is an edited version of a contribution to The Scientists’ Scribe, the student led magazine of the Department of Biological Sciences of Royal Holloway University of London.

Barking Beetle Patterns

hat does a mathematical biologist do? A colleague ends his emails with a quote from Asimov: "The most exciting phrase to hear in science, the one that heralds the most discoveries, is not Eureka but That's funny..." Spot on! That is science for me: I want to know -need to know- how things work, even trivial things. Why are the nuts always on the top when I open the muesli box? I want to know why. That is what I do; that makes me a scientist. And I use mathematics to find out that why.

A while ago, when walking home I spot a pile of wood, ready to be taken to the tip. The wood has strange markings on it. I pull up some of the dried bark; under it are wriggly lines, making an intriguing and beautiful pattern. I see a frazzled wheatsheaf, a crumpled fleur-de-lys and a confused looking double headed eagle with rather ruffled feathers.  It looks as if the wood has been in the hands of a designer on I-don’t-know-what who carved it with organic, semi-symmetric psychedelic patterns; Laura Ashley on acid.

These funky patterns are the galleries of bark beetle larvae. Mother bark beetles drill a tunnel between the bark and the wood. In this tunnel they lay eggs, nicely spaced out. When the eggs hatch, the larvae feed themselves on the bark and the wood. Whilst eating, they drill their own little tunnels. Initially they move away from mother’s tunnel. As the larvae grow, the tunnels get wider, and the tunnels seem to radiate out, going their own way, but hardly ever cross other tunnels. Once the larvae are fully grown they pupate, and after that they eat themselves through the bark to discover that there really is light at the end of their tunnels.

But why the crazy pattern? All these larvae munch their way independently through the wood, all minding their own business. How can such an intricate and beautiful pattern emerge? Why do the larvae’s paths hardly ever cross and why are the galleries so recognisable the same (each species of bark beetle has their own, typical pattern)? If all the larvae would mind their own business, why don’t we get one big fuzzball of a pattern?

It must be that the larvae obey some rules, probably very simple rules that allows them to create the pattern. The rules must be genetically determined, as mummy beetle is not there to tell them. Now the modelling part of my brain kicks in.  What could these rules be? Can the larvae smell each other and use that to keep distance? This seems unlikely in a tunnel in a piece of wood. Can they hear each other? You can sometimes hear woodborers at work in a forest, munching wood creates sound. All these little larvae munching through wood would make quite a racket if you get close enough.

But how do they hear? Do insect larvae have ears? It seems they don’t but it is not impossible that they perceive sound. Even if they can’t hear sound, they might pick up vibrations. I find it hard to imagine that insect larvae are able to detect the direction that the sound is coming from without ears. But by simply moving their heads from left to right they probably can detect which side is louder, or they could do the same things by having simple sensors each side of their heads. What would happen if they have a tendency to move away from the loudest side? ¹

Here we go, this is mathematical biology. I work out the gradient of the sound in a sound scape. It isn’t that hard really, basically A-level physics. I then write a little computer programme in which I let virtual larvae move, a step at a time. They can deviate a little to the left and to the right so that they don’t just go in a straight line. But if they hear a sound that is louder on their right they move a little more to the left and the other way around. I let the programme run on my laptop.

First, I try to see what happens if a larva has two sound sources ahead of it with a gap in the middle. On my screen the virtual larva navigates beautifully in a straight line straight down the middle. Once it has passed the sound sources it starts to wander about aimlessly. It seems to work. If the larva is close to the sound sources the path is very straight. I have learned something, with clear sources the larvae go straight and their paths wouldn’t easily cross.

Then I put a lot of virtual larvae together on my computer screen. I let a line of larvae start off, evenly spaced, and facing in two different directions. They all try to move away from the others and radiate out from their starting points. It works! A regular pattern emerges as the larvae move away from the place where they have hatched. The larva at the end of the line of eggs has most free space and curves out very fast. The larva next to it follows, but can’t quite curve as much so has a little bit of a straighter line, and so on. Thus the wheat sheaf, fleur-de-lys and double headed eagle patterns emerge. So from the simple rule that larva move forward, and turn away from the vibrations emitted by the other larvae I can construct similar pretty patterns.

But the patterns are very regular, and not quite like the ones on the logs. On logs some of the paths curl around crazily; none of that happens in my simulations. What I see is perhaps fit for the windows of Laura Ashley, but on acid it is not. I fiddle with the parameters, and I can’t get the frazzled look. I give up. I know that my simple rule sort of works and am happy to leave it there.

But then I want to do one more thing: I place the eggs on a W shape and see what happens. And to my surprise, the larvae now at first create a regular shape but, particularly at the bottom of the W do all sorts of crazy curls, the paths cross and I get the ruffled feathers. It seems that in the W shape the larvae get sounds from all directions and don’t really go in one direction, but swirl around in the acoustic confusion.

So I have figured out how the pattern can emerge from a simple rule. By changing the parameters in my programme I get slightly different patterns:  so species could differ in the parameters that are genetically programmed into them. It all makes some sense. Would different types of wood give different parameters, and therefore different patterns? Could you make an identification key to determine the species by figuring out what the parameters are? If you wanted to go further you could probably do experiments and build a whole research programme around it. But this is not for me, I am content with the insight I have gained. This wasn’t a very important problem, but it was fun and I have scratched my scientific itch.

I varnish the log; it is beautiful and looks like a sculpture with esoteric etchings on it. The barking W will make a nice initial for this blog.

Vincent Jansen, April 2019.

1. I later find a paper that suggests the larvae might indeed be guided by sound: de Jong and Saarenmaa 1985, Ecological Modelling, 27 (1985) 109-138 109, who in turn cite Schmitz, R.F. and Rudinsky, J.A., 1968. Res. Pap. 8, Forestry Research Laboratory, School of Forestry, Oregon State University, as a source for this idea.

                   ------------------- Read the next blog >>>>