Posted by: Jeremy Fox | November 11, 2011

Is Hutchinson’s zombie idea about coexistence not a zombie after all?! (UPDATED)

Is Hutchinson’s zombie idea about coexistence not a zombie after all?!

In my original zombie ideas post, I criticized three distinct theoretical claims about how disturbances and fluctuating environmental conditions can promote competitive coexistence. These claims are widely cited, and appear in many textbooks. So I was very surprised that I didn’t get any serious pushback. Nobody who fully understood my arguments came forward with any direct counter-arguments.

Until now.

In the comments on a recent post, ace theoretician Chris Klausmeier (commenting as “lowendtheory”) argues that one of my three IDH zombies, the one proposed by Hutchinson (1961), is no zombie at all. It’s a great comment, so I wanted to give it, and my response, a post of its own so that everyone reads it. It’s a longish post, but stick with it, because I think there are some important lessons here, not just for the zombie-ness or otherwise of Hutchinson (1961), but for how to use math to help figure out the messy, complicated real world.

Hutchinson (1961) argued that, if the identity of the dominant competitor depends on environmental conditions, and if environmental conditions fluctuate on an intermediate timescale, no competitor will have time to exclude the others and all will coexist. If the environment fluctuates too fast, the competitors will average across the fluctuations and whichever one is fittest on average will exclude the others. If the environment fluctuates too slowly, the best competitor in the current environment will exclude the others before conditions change. Against this, I cited Chesson and Huntly (1997), who analyzed a general class of mathematical competition models (roughly, “linear additive models”, which includes but is not limited to the classical Lotka-Volterra competition model). They proved that, in this class of models, merely changing the identity of which species grows best, with whatever frequency, does not affect coexistence. Rather, what matters in the long run is which competitor is favored on average. For instance, an environment that favors one competitor 51% of the time, and the other competitor 49% of the time, will eventually lead to exclusion of the latter competitor, no matter what the frequency with which it switches from favoring one competitor to favoring the other. And while an environment that favors each competitor an equal fraction of the time will allow indefinite coexistence (barring stochastic drift), even in that (highly unlikely) case, what matters is not the frequency with which the environment fluctuates but the fact that it favors each species an equal fraction of the time. (As an aside, it is important to recognize, as Chris does, that my argument here is merely a verbal summary of the formal mathematics of Chesson and Huntly; I was not countering Hutchinson’s purely verbal argument with one of my own).

Against this, Chris presents and analyzes a fully-explicit mathematical model which he says vindicates Hutchinson’s (1961) claim. I’ll summarize, and in a few cases, slightly elaborate, what Chris says, for the benefit of readers who may find certain features of his model mysterious (I’m sure Chris will correct me if I’m wrong on any of what follows). It’s a deterministic, continuous time model in which two consumers with linear functional responses and constant, identical per-capita mortality rates compete for a single shared resource in a closed system, in which resource bound in dead consumers is instantly returned to the free resource pool (Chris doesn’t mention it, but he’s measuring consumer biomass as the amount of resource bound in consumer biomass, which is a perfectly ok choice that doesn’t affect the results). The system fluctuates between two environmental states, one in which consumer 1 has a higher per-capita resource uptake rate (=functional response slope) than consumer 2, thereby making it the superior competitor, and a state in which the reverse is the case. When favored, each consumer is favored to the same degree (i.e. it’s not like the system sometimes hugely favors consumer 1, and at other times only slightly favors consumer 2). One can vary both the frequency with which the system changes state (which is what Hutchinson says matters), and the overall fraction of the time the system spends in one state vs. the other (which is what Chesson and Huntly, and I, say matters). Chris analyzed the model by numerically simulating “invasions”—that is, starting one of the competitors out at very low abundance relative to the other, to see if the rare one can increase. This is the appropriate test of stable coexistence, and guards against mistaking very slow competitive exclusion for truly stable long-term coexistence.

The result Chris finds is that, when the environment changes states very frequently, Chesson and Huntly are right: either one consumer or the other wins, unless each consumer is favored exactly 50% of the time. But when the environment changes state less frequently, you can get stable coexistence of both species, as long as neither species is favored for too large a fraction of the time (what’s “too large” depends on the frequency with which the environment fluctuates) Finally, when the environment changes very slowly, the currently-disfavored competitor becomes extremely rare and stays extremely rare until the environment changes state, which causes the other competitor to become and then stay extremely rare until the environment changes back. Realistically, this would result in competitive exclusion in nature, a fact which Chris incorporates into the model by setting an extinction threshold. If a species’ abundance drops below the threshold level, it’s declared extinct (the choice of extinction threshold is arbitrary, but that doesn’t affect the results in any important way) Chris also cites a recent paper of his (Klausmeier 2010) which provides mathematical proofs, as opposed to mere numerical simulations, for what happens in the limiting case of low frequencies of environmental change with no extinction threshold (that paper analyzes several models; the model Chris is referring to here is the one the paper calls “flip-flop” competition).

And here’s the real kicker: as Chris briefly notes, the model he analyzes is a member of (or at least, sure looks like it’s a member of) the general class of models analyzed by Chesson and Huntly. Which is really weird, because if his model falls into that class, then stable coexistence should be impossible!

And the puzzle goes even deeper. If you don’t explicitly model any ecology at all, and instead just assume that the relative fitnesses of the competitors fluctuate over time for some unspecified reason(s), then you can prove that the competitor with the highest geometric mean fitness wins in the long run (see, e.g., Wright 1948, Gillespie’s 1991 book The Causes of Molecular Evolution, and p. 301 of Graham Bell’s Selection: The Mechanism of Evolution). Which is really weird as well, at least to me, because that’s another general result that you’d think would apply to the specific case of Chris’ model.

I think these apparent conflicts comprise a really interesting and important conceptual puzzle. If we can resolve it, we’ll learn something, possibly something deep and subtle, about one of community ecology’s most influential ideas.* It’s kind of like the recent physics experiment which apparently found neutrinos traveling faster than light, which if true would violate a very general theoretical result (Einstein’s theory of relativity). Like the physicists, we (meaning Chris, I, and anyone else who finds this puzzling) need to get to the bottom of this. What have we missed that, once we recognize it, will resolve this apparent conflict? C’mon, mathematically-inclined readers—help us out! (I’m looking at you, Robin…)

Note that just picking sides is not an appropriate resolution. Until we understand the reason for this apparent conflict, choosing one side or the other to believe is just guessing (or worse, cherry-picking evidence while ignoring contrary evidence) If we don’t understand the reason for the apparent conflict between these models, we don’t fully understand any of them.

Chris suspects, and I agree, that there is probably some hidden source of nonlinearity or nonadditivity in his model—i.e. some reason why his model doesn’t fall into the general class analyzed by Chesson and Huntly, so that their argument about the behavior of linear, additive competition models doesn’t apply to Chris’ model (indeed, I have some hunches along these lines that I’m currently pursuing, but they’re currently too vague to be worth sharing). If there is some hidden source of nonlinearity or nonadditivity in Chris’ model, and if it can be found, then that would be a really valuable result. On the one hand, it would confirm Chesson and Huntly’s general argument that nonlinearities and nonadditivities are essential for environmental fluctuations to promote coexistence. On the other hand, it would show that nonlinearities and nonadditivities are very hard to avoid—even a model like Chris’, which is far too simple to be realistic, has them. Many folks, especially evolutionary biologists, tend to think of these sorts of nonlinearities and nonadditivities as being fairly special cases, that only arise when the competing species have certain life histories or the system has other non-generic features. Showing that that’s not true would greatly broaden the range of empirical systems in which we expect environmental fluctuations to promote stable coexistence.

What does all this mean for Hutchinson’s original verbal argument? Is it a zombie? Chris thinks his model vindicates Hutchinson, and I can totally see why he thinks so. His model is indeed the sort of thing Hutchinson was probably thinking of. On the other hand, evolution-style models that just say “the relative fitness of different competitors fluctuates over time” seem to me to be at least as much the sort of model Hutchinson was thinking of—and those models contradict Hutchinson rather than validating him. Further, Chris’ own results demonstrate a key role for long-term average conditions, which Hutchinson ignored entirely. And perhaps most importantly, if Chris’ model really does have some sort of hidden nonlinearity or nonadditivity, then discovering that likely would significantly change our understanding of why his model behaves the way it does, so that we’d no longer see it as the sort of thing Hutchinson had in mind.

But I’m actually not too bothered about whether Chris’ model validates Hutchinson or not. This isn’t about, or shouldn’t be about, vindicating something G. E. Hutchinson wrote decades ago. The conceptual puzzle, the apparent conflict between Chris’ results and those of others, would still exist even if G. E. Hutchinson had never lived. What matters is how nature works, not what Hutchinson (or me, or anyone) personally thinks about how nature works. In nature, environmental conditions fluctuate. Under what conditions does that matter, and why? Those are the important questions. To which we still don’t have a complete answer.

UPDATE: Chris and I have figured it out! At least, we think we have. The short answer is that yes, there appears to be a subtle source of nonadditivity in Chris’ model. More soon–unless we decide this is sufficiently interesting and important that we want to write a paper about it!

*I leave aside the possibility that Chris, and/or Chesson and Huntly, have made some technical mistake. Not because that’s impossible, but just because it’s far from the most likely possibility.


  1. Thanks for the great summary, J. If anyone wants to give this model a try, go here . Note that this is my first attempt at making a demo with Mathematica, you’ll be prompted to install the Wolfram CDF Player, and it can take a few seconds for parameter changes to take effect.

    I’ve got a few calculations trying to understand what’s going on, but for now I’ll just look forward to hearing what others think.

    • Hi Chris,

      you didn’t supply the URL in your comment; it looks like there’s a link but there isn’t.

  2. Ugh, another comment garbled by the WordPress comment system. It’s

  3. […] model, but Chris Klausmeier and I have figured out that that’s what’s going on. See this old post for some discussion. I didn’t walk through the details then and I’m not going to now, […]

  4. […] model, but Chris Klausmeier and I have figured out that that’s what’s going on. See this old post for some discussion. I didn’t walk through the details then and I’m not going to now, […]

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