I’ll be defending my thesis in a few months time, and I’ve been re-reading old papers recently to get myself in the right head-space. A few days ago, I came across a paper that I read at the beginning of my Ph.D., but otherwise forgot about until now – which is a shame, because it’s a good paper. Worse, it’s the sort of paper that should have gotten more attention than it did, or at least more attention than I’ve seen it get. This issue, of the endless treadmill of peer-review publications fading into the twilight (sometimes a mere weeks after publication), has been on my mind lately. But that’s the virtue of having this blog: I can dig this paper up and give it some of the attention that it does deserve.
Of course, since I’ve done this a bit clumsily, you have no idea what paper I’m talking about. So, let’s bring it out into the light:
G. D. Ruxton. Foraging in flocks: non-spatial models may neglect important costs. Ecological Modelling, 82:277–285, 1995.
Behavioural ecology is a discipline that has, to date, been largely focused on problems that can be modelled analytically by solving equations or simple games, or perhaps deriving more complicated equations that were solved through numerical methods. But some behavioural situations are not so amenable to these types of analyses, and so we tend to turn to simulation work to help bridge the gap. One such example is group foraging. Foraging models, be they optimal foraging models (where individuals forage individually, even if they do so in a group) or social foraging models (where individuals use information from other foragers to influence their own decisions), have rarely been situated in a spatially-explicit world. The reason for this is simple: spatial models are hard. Simulations are a great way to handle these types of models, and in recent years this has become more common – my own Ph.D. work is a fair example of this! – but in the early-to-mid-90s, this was just in its infancy.
Enter Dr. Graeme Ruxton, who in 1995 wrote the paper above to make the point that ignoring spatial effects in group foraging models was doing a disservice to the understanding of these behaviours. Because I think it’s an important point, and because 15 years later I’m flogging the same point in some of my own work, I’m going to spend a few minutes explaining Ruxton’s model.
The model is what would now be called an individual-based model (IBM; also known as Agent-Based Modelling, ABM). Foragers moved across a 1000×1000 regular grid, searching for food patches that were scattered across the grid (see the diagram). When an individual found a food patch, every other searcher abandoned what they were doing and moved toward the food find until every forager was at the same patch. This may sound a little odd – what if the patch was exhausted? – but because the goal was to look at how spatial effects played out in terms of search time and movement time, Ruxton didn’t record feeding rates or any such thing (this issue is explored more thoroughly in a paper published the same year with S. J. Hall and W. S. C. Gurney). If you’re familiar with social foraging models, you may recognize this model as a simple example of an Information-Sharing game; I plan on returning to this theme in a later post.
The point of the paper was to ask what effect various movement strategies would have on the time it took for individuals to find patches. In particular, how long would it take to find the first searcher to find a patch, and how long would it take the rest of the group to converge on that patch? To do this, he explored six different movement rules, of which I’ll present four here. (You can read the paper for the other two; the point here is to discuss the importance of the paper, not replicate it in full).
- A simple random walk. The individual moves in any of the eight possible directions, randomly chosen each turn. All eight directions are equally likely to be chosen as the direction for next turn’s movement.
- A group biased walk: individuals would move more often in a direction chosen at the start of the simulation. For instance, they might prefer to move to the right, so more of their moves will be to the right as opposed to the other seven directions. The strength of the preference is tweakable, and the entire group has the same direction preference.
- An individual biased walk: same as a group biased walk, but every individual has their own preferred direction.
- Site marking: individuals mark sites that they’ve been to, something like ant pheromone trails; searches don’t enter a marked site unless they have nowhere else to go. Site markers fade over time, and the decay time is adjustable.
The paper goes on to explore the effect of these rules on the searching and convergence times, presenting results for each rule and contrasting them with the results from classical models. I’ll mention a couple of the results, but because I’m uncertain as to what I can clip from the paper in terms of figures, I can’t get too deeply into it (now there’s something I need to look into!) and I’d recommend giving the paper a read.
One of the more interesting results relates to the time taken to find a patch as a function of the number of searchers. Classical models expected that this was linear: add more searchers, and your search time goes down in a straight line fashion. But Ruxton’s results showed that adding more searchers might not have such a linear effect, because often new searchers would blindly trample over spaces that previous searchers had already covered. Thus, you might get diminishing returns to search efficiency from adding more searchers. (You can see this in Figure 1b from the paper, which I’ve reproduced here). The rest of the paper explores the individual effects of each of the movement rules, and it’s worth reading for that alone.
But why is this important? Well, behavioural ecologists studying the evolution of group behaviours like foraging might incorrectly predict – or have trouble explaining – the size of foraging groups based on these search efficiencies. Worse, they might not build these diminishing returns into their models as a cost of increasing group size when trying to explain why individuals form groups and why groups are of a certain size. Because these effects come directly from incorporating spatial structure into the model, classical non-spatial models can’t easily take them into account.
This paper is especially important because it was one of the first – to my knowledge – to make this point so clearly. It’s not the first to argue that spatial effects are important, by any means, but it does so in a way that highlights the importance of these effects to the study of foraging. Unfortunately, despite fits and starts over the years to follow, this lesson is only now starting to take hold in a big way in behavioural ecology (as far as I can see!). I’ll say it: I’d recommend that anyone studying the evolution of foraging behaviour should read this paper, if only because it foresees issues that we’re grappling with right now.
And hey, it’s only 9 pages. Not too bad, right?