Zombie ideas about disturbance, and the paradox of Achilles and the tortoise (UPDATED)

One common zombie idea about disturbance is that it prevents competitive exclusion by “interrupting” or “setting back” the process of exclusion. The idea is that disturbance, by temporarily reducing every species to low density, perturbs the system away from the equilibrium it was approaching, an equilibrium at which species would be competitively excluded. Repeated disturbances, so the argument goes, prevent the system from ever actually getting to equilibrium, and so prevent competitive exclusion from ever “going to completion”.

I’ve explained before why that idea is wrong (see my forthcoming paper in Trends in Ecology and Evolution, or use our search bar for old “zombie ideas” posts). But often it helps to have different ways of explaining the same thing, in case one way doesn’t “click” with everyone. And I just thought of a fun analogy to the zombie idea that “disturbance prevents competitive exclusion by interrupting it”. That zombie idea is kind of like Zeno’s paradox of Achilles and the tortoise.


Let me explain. 😉 Zeno (c. 490-430 BC) was a Greek philosopher who proposed a number of paradoxes, the most famous of which all have to do with ideas of infinity, continuity vs. discreteness, and wholes vs. parts. The overall intent was to demonstrate the oneness of reality and the impossibility of any kind of change. Zeno was a follower of Parmenides, who famously argued that nothing can move and the appearance of movement is a mere illusion. Zeno’s paradoxes were defenses of Parmenides’ views.

In Zeno’s paradox of Achilles and the tortoise, Achilles is in a footrace with a tortoise. Suppose that Achilles allows the tortoise a head start of 100 m. After some finite amount of time, the fleet-footed Achilles will reach the point 100 m ahead of him, where the tortoise was when the race began. But during that time, the tortoise will have moved ahead a bit (say, 10 m), and so will still be leading the race. It will take Achilles some further amount of time to cover that 10 m–but by the time he does so, the tortoise will have moved ahead some further distance. Thus, whenever Achilles reaches a point where the tortoise has already been, he still has further to go. Meaning that he can never overtake the tortoise.*

The analogy to the notion that disturbances prevent competitive exclusion by “interrupting” it or “setting it back” isn’t perfect, but I think it’s close enough to be interesting. The system approaches competitive exclusion (=Achilles approaches the tortoise). But before it can get all the way there, its progress is set back by disturbance (=the tortoise moves ahead a bit). Ergo, competitive exclusion can never occur (=false conclusion). Further, you can actually tweak a lot of the details without changing the basic nature of the paradox, or the obvious falsity of the conclusion. For instance, Zeno’s “dichotomy paradox” differs in many details from the paradox of Achilles and the tortoise, but at bottom is basically the same paradox. So just as Achilles will in fact pass the tortoise, competitive exclusion will in fact occur despite periodic interruptions or setbacks.

But of course, just pointing out that the conclusion of a chain of reasoning is false doesn’t on its own give any insight into why it’s false (When he first heard Zeno’s arguments about the impossibility of motion, Diogenes the Cynic apparently refuted them by saying nothing, getting up, and walking out) What’s wrong with Zeno’s reasoning about Achilles and the tortoise, and what, if anything, does it tell us about the zombie idea that periodic disturbances can prevent competitive exclusion simply by interrupting it? Surprisingly, there’s no universal agreement on precisely how Zeno goes wrong. And in any case, I suspect that claims about how interruptions can prevent competitive exclusion go wrong for rather different reasons than Zeno does. Zeno does recognize that Achilles gets closer and closer to the tortoise, he just denies that Achilles can ever catch up entirely. In contrast, verbal arguments about how periodic disturbance interrupts competitive exclusion tend to forget that, in between each disturbance, the system gets closer and closer to equilibrium (they also tend to forget that adding disturbances to a disturbance-free system changes the location of the equilibrium, but that’s another issue).

So no big conceptual insight into ecology here, which actually disappoints me a bit. When the idea for this post popped into my head the other day, I thought it would be really fun to show how zombie ideas about the IDH actually predated Christianity and had been refuted by Aristotle or something. 🙂 But while that didn’t quite pan out, I still thought the analogy was sufficiently off the wall, and sufficiently close, to be worth sharing.

See The Stanford Encyclopedia of Philosophy for further discussion of Zeno’s paradoxes, on which I drew for this post.

*Zeno took his paradoxes as proof that time and space are fundamentally indivisible, since assuming that they are divisible leads to absurd conclusions.

UPDATE: xkcd on Zeno’s paradoxes. 😉

6 thoughts on “Zombie ideas about disturbance, and the paradox of Achilles and the tortoise (UPDATED)

  1. The statements on Wikipedia regarding a lack of consensus about the solution of this paradox are a bit weird.

    I remember we were going through this in calculus I as a classical fallacy that originates from the fact that the Greeks hadn’t really grasped the concept of convergence of infinite series yet. Once you accept that you can consider infinitely many steps in finite time, the solution is obvious and is indeed given as a exercise in many introductory math classes. I don’t see much room for a lack of agreement on that beyond resorting to historical / philosophical gibberish.

    Other than that, I agree that the idea that disturbance reduces competition as such is not an explanation at all – if competition is reduced, but relative fitness is constant, the only thing that changes about competitive exclusion is that it may be slightly slowed down.

    However, it’s a different situation if you assume that relative fitness is density or frequency dependent. Then you can get a succession situation, which has a good chance to produce a hump-shaped IDH, depending on what you have in your species pool. I have to admit that I found your discussion on this in the TREE paper quite short, I always thought of sucession as one of the main explanations for the IDH, and I would think it remains a valid one, despite the fact that I share your scepticism about the universality of the IDH.

    • Hi Florian,

      I agree that the Greeks’ failure to understand convergence of infinite series seems like the most straightforward explanation here. And it is indeed a popular one. I’m a bit out of my depth here, but see the Stanford Encyclopedia of Philosophy for discussion of why this explanation might be considered inadequate.

      Re: frequency dependence, the whole point of my TREE paper was to point out that zombie ideas about the IDH don’t generate any frequency dependence! Sorry if that didn’t come through more clearly.

      I think saying “it’s succession” is too brief to be helpful. I’m sure one could imagine models of succession that don’t (in combination with disturbance) generate frequency dependence. But yes, the right sort of succession model, like Pacala and Rees’ “successional niches” model or certain competition-colonization trade-off models, can indeed generate negative frequency dependence via nonlinearities and nonadditivities. Of course, these sorts of models oon’t necessarily produce humped disturbance-diversity relationships, although they can.

      • Hi Jeremy,

        no, your paper was clear on ruling out these three mechanisms as explanation for the IDH through their failure to produce frequency dependence. The only thing that had surprised me a bit is that, prior to reading your paper, and in my mind, I hadn’t filed these three mechanisms as the main explanation for the IDH, but rather this “succession-type” stabilization. A reason might well be that I haven’t really systematically read the literature on the IDH. Anyway, as you say, succession doesn’t guarantee hump-shaped relationships either, although I would still say that they are not particularly unlikely.

      • As to what the “main” explanation for the IDH is, all I can say is, have a look at the citation statistics for papers describing zombie ideas about the IDH vs. non-zombie ideas. I compiled these data in an old post. Connell 1978 has been cited more than 1000 times in the past 5 years, and its annual citation count is growing exponentially, at the same rate as the size of the ecological literature as a whole. Huston 1979 is cited dozens of times a year, and its annual citation count also is growing. Non-zombie theoretical papers on or closely related to the IDH, like Chesson and Huntly 1997, Pacala and Rees 1998, and Roxburgh et al. 2004 are cited at least an order of magnitude less frequently than zombie papers. And the zombie ideas are in the textbooks, while the non-zombie ideas mostly aren’t.

        So yeah, in only paying attention to non-zombie ideas about the IDH, you’re very much in the minority, Florian. When most people think about this stuff, they still think in terms of zombie ideas, sadly. The zombies I’m attacking are very much alive in most ecologists’ minds, especially empiricists.

      • Hi Jeremy, thanks for the references, I guess I should have a closer look at those “classics” and how people cite them. About the textbooks: you know what they say, science advances funeral by funeral 😉

  2. Pingback: Saturday blast from the past: the intermediate disturbance hypothesis vs. Zeno’s paradox of Achilles and the tortoise | Dynamic Ecology

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