Are unstable communities like $10 bills lying on the ground?

Attention conservation notice: this post is old wine in a new bottle. But my old post was a long time ago and didn’t get many readers (or any comments!), so I thought I’d try again.


Here’s an old joke about economists:

Two economists are walking across a college campus. Suddenly the first economist looks down and says “Hey look, a $10 bill on the ground!” The second economist replies “That’s impossible. If it were there, someone would’ve picked it up by now.”

The joke is about the idea that, in a well-functioning market, there are no risk-free profits to be had. No $10 bills just lying on the ground, waiting for someone to pick them up. Opportunities for easy, risk-free profits get exploited very fast (for instance via arbitrage), causing them to vanish. But just because something shouldn’t exist in theory doesn’t mean it can’t exist in practice, which is the point of the joke.*

Having said that, the situation described in the joke is indeed unlikely. You really do hardly ever see $10 bills lying on the ground, for exactly the reason the second economist gives. In general, you don’t expect to observe anything that can’t last very long. That’s why you never see raw sodium floating around in the water. Or see much ununoctium anywhere.

This “no $10 bills lying around” principle often gets invoked, implicitly or explicitly, to explain ecological phenomena. For instance, numerous papers argue that observed food webs, or other ecological “networks” like plant-pollinator interaction networks, are structured so as to be stable in some sense. The implicit argument is that this explains why those networks look the way they do. If they looked any different, they’d be unstable and so wouldn’t persist long enough to be observed. Conversely, many papers on omnivory in food webs are motivated by the observation that, in theoretical models, food webs with ominivory are prone to instability. The commonness of omnivory in natural food webs therefore is a puzzle that needs solving. From the perspective of simple theoretical models, food webs with lots of omnivory are like $10 bills lying on the ground.

Further back, this argument was deployed by advocates of density-dependence in population ecology against advocates of density-independence like Andrewartha and Birch. Any population that doesn’t experience at least weak negative density-dependence will do a random walk to extinction, so we shouldn’t expect to observe such populations. The community ecology equivalent is our focus on conditions for stable coexistence (I say “our” because I certainly share this focus). When analyzing theoretical models, we focus on the conditions that lead to stable coexistence rather than exclusion or priority effects or etc. And when we do empirical studies we expect to find that those conditions hold in nature. Because we shouldn’t expect to observe species that are on their way to being excluded. “If they were there, they’d have been excluded by now” would be the punchline to the coexistence theory version of the economist’s joke. I’m sure this sort of argument has been deployed in many other ecological contexts.

I find this argument appealing as an explanation for why the world is the way it is. But I’m also suspicious of it, as it’s often deployed without much critical evaluation. For instance, you might actually expect to see lots of $10 bills on the ground if picking up $10 bills were really costly. In economics, transaction costs can prevent otherwise-profitable arbitrage opportunities from being exploited. Or (and this is really another way of saying the same thing), maybe $10 aren’t actually worth much, so it’s not worth the effort to pick them up. That’s why you often do see pennies on the ground. Or maybe $10 bills are really hard to pick up because they blow around in the wind. Or maybe you see $10 bills on the ground because for some reason they’re being dropped on the ground even faster than people can pick them up. That does happen on occasion.

Analogous possibilities can occur in ecology. For instance, you might expect to observe density-independent populations even though they’re doomed to eventual extinction because a random walk to extinction is a slow process, at least under some circumstances. That’s an analogue to $10 bills being hard to pick up because they’re blowing in the wind. Or maybe new density-independent populations can be established via immigration or speciation fast enough to balance the rate at which they vanish. That’s an analogue to $10 bills being dropped as fast as they’re picked up. Steve Hubbell made precisely this argument in his “neutral theory of biodiversity”, although his argument has been strongly disputed (speciation is probably too slow to balance even the slow rates of extinction produced by stochastic drift in a neutrally-stable system). As another example, grassland communities established during old field succession are unstable, in the sense that they’re eventually replaced by forests. But nevertheless, we observe them, because old field succession is a slow process. As a third example, many species that are thought to be doomed to eventual extinction due to habitat loss are still around, and will be for decades at least, because that’s often how long it takes to pay the “extinction debt”. As a final example, Kristensen (2008) is a really nice theoretical paper showing that model food webs constrained so that all species must exhibit positive, finite densities don’t actually differ much in their structure from food webs not so constrained. In other words, lots of unobserved food web structures would be just as “stable” as observed ones. So you can’t argue that we only observe the food web structures that we do because the alternatives would be unstable.

Evolutionary biologists have thoroughly studied the various factors that prevent natural selection from always and everywhere producing populations in which all individuals have the same, perfectly-optimized phenotype. There are lots of well-studied reasons why we might observe less-than-maximally-fit individuals, the evolutionary equivalent of a $10 bill on the ground. Same in economics–there’s a huge body of research on why real-world markets might not be perfectly efficient or might fail to “clear”. But I don’t know that ecologists have been as thorough and systematic about studying the analogous problem. There are some papers like Kristensen (2008), but not nearly as many as there should be, I don’t think. As I noted in that old post linked to at the beginning, there are various reasons why ecological systems might look the way they do. “If they weren’t this way, they wouldn’t persist” isn’t necessarily the tightest constraint on what we observe, or even a constraint at all. Maybe the ecological world is actually full of $10 bills on the ground, and our task is to explain why. Or maybe it doesn’t even matter if there are $10 bills on the ground or not.

*There’s an old Bloom County cartoon about this. In it, boy genius Oliver Wendell Jones discovers the fundamental theory of physics, which predicts that penguins shouldn’t exist. Thereby causing Oliver’s friend Opus, a penguin, to vanish. But then Oliver realizes he made a mistake (“Forgot to carry the two”), causing Opus to reappear. 🙂 But I’ve been unable to find the strip online.

9 thoughts on “Are unstable communities like $10 bills lying on the ground?

  1. Classic joke and a nice post too.

    There are at least three reasons why ecologists often focus on equilibrium behavior. 1) If there’s a globally stable equilibrium, that’s where the system will go eventually, so this provides at least a target for the dynamics. 2) Initial conditions, which are hard to know and an imaginary construct anyhow, don’t matter. This reduces the idiosyncrasy and increases the generality of results. 3) The analytical tools to find equilibria and their stability are the easiest ones to use.

    These aren’t necessarily bad reasons, which explains the continued popularity of the equilibrium approach, but most ecologists recognize its limitations. A common response is to look to larger spatial and temporal scales, where local instability can translate into global stability. DeAngelis & Waterhouse (1987) lay out a roadmap of early uses of this idea and Hastings (2004) based his MacArthur Award lecture on the topic of transients. Metacommunity theory (Leibold et al. 2004) is a contemporary version of this concept. We developed an method for modeling plankton succession where annual successional cycles can be thought of as a series of unstable community states (Klausmeier & Litchman 2012).

    On the other hand, I think there’s been less work focusing on the small-scale, short-term dynamics (as in your old field succession example). Many of applied problems require this info, not some mythical limit where time and spatial scale go to infinity. Direct simulation is always possible but the challenge will be formulating appropriate general concepts to capture these short-term responses in a way that isn’t completely system-specific and idiosyncratic.

    DeAngelis, D. L., & Waterhouse, J. C. (1987). Equilibrium and nonequilibrium concepts in ecological models. Ecological Monographs, 57(1), 1-21.

    Hastings, A. (2010). Timescales, dynamics, and ecological understanding. Ecology, 91(May), 3471–3480.

    Klausmeier, C. A., & Litchman, E. (2012). Successional dynamics in the seasonally forced diamond food web. American Naturalist, 180(1), 1–16.

    Leibold, Mathew A., et al. 2004. The metacommunity concept: a framework for multi-scale ecology. Ecology Letters 7: 601-613.

    • This is really interesting. In economics, there has been a great contribution from theoretical computer scientists pointing out that the sort of equilibria that are usually assumed, although possible in the infinite limit, cannot be found by any reasonable (I.e efficient) algorithm like the economy. That means there is a purely logical reason for why these equilibria cannot be achieved (in general) if the system was away from equilibrium at any point. Are there similar results using cstheory and the computational complexity of equilibria in ecology? If not then I would be interested in showing some such results.

      • For a precis of why mainstream economics modeling takes the approach it does, the limitations of this approach, and brief comments on the alternatives, see here:

        The piece is from a very sharp physicist-turned-economist, who is very aware of the limitations of standard economic models and the approaches used to analyze them. But who is also skeptical that the alternatives are sufficiently well-developed to be very useful.

        I’ll note in passing that the CS work to which you refer hasn’t actually had much impact on mainstream macroeconomics as far as I understand. Whether it should have, I have no idea, I don’t know nearly enough about either economics or CS to say.

  2. Thanks Chris.

    Re: lack of theoretical study of small scale, short term transient dynamics, vs. empiricists’ desire to know about such dynamics, yup. I have an old post on this:

    Re: the reasons why theoreticians often focus on equilibrium behavior, yup. It’s interesting that other fields, like economics, share that focus, and mostly for the same reasons. Which leads to similar sorts of worries about what we might be missing. Does a focus on the behavior of linearized systems in the neighborhood of equilibrium just serve to cover up our lack of understanding of nonlinear, nonequilibrium, nonstationary dynamics? Maybe in economics it does; in ecology I think we’ve actually gotten increasingly good about studying nonlinear, nonequilibrium, nonstationary dynamics, at least in our theoretical models.

  3. Just to add some empirical data – and this is a true story – just last week I was walking across campus to lunch and there was a $10 bill lying in the grass. I didn’t pick it up because I didn’t want to steal from somebody who might be coming back. However, when I walked by an hour later and it was still there, I picked it up. So empirically $10 bills lying on the ground do exist but they are transient and not equilibrial

    Increasingly I am subscribing to the view now gaining vogue in hydrology and climate. It was announced in a series of papers with the phrase “stationarity is dead”. Stationarity being a technical term from stochastic process theory assuming roughly that the mean and variance are constant over time. This roughly matches to the stochastic equivalent of a noisy equilibrium (something has to keep the data centered around the mean). The alternative is that the mean and possibly variance are constantly changing. This is roughly the red noise view of the world. You’re always in the middle of a trend, no matter what time scale you look at. Or as Heraclitus would put it “change is constant”. It is too bad because as Chris noted the math of equilibria is much easier to work on.

    • That story about the $10 bill is best-timed anecdote I have ever heard! It’s a good thing you have a trustworthy face, or I’d never believe you. 😉

      Re: nonstationarity, I agree that that’s a really key frontier for ecology. Not so much describing it, but figuring out how to think about it and its consequences. As you hint, in a nonstationary world it’s not even clear what it *means* to talk about, e.g., the stability of coexistence. Empiricists often don’t realize that theoreticians actually have quite a good handle on how to think about issues like stable coexistence in a *stationary* stochastic world. Conceptually, we have quite a good handle on how stationary stochastic systems work. Non-stationary systems, not so much, at least as far as I know.

      Fortunately, Peter Chesson is on the case. Based on past experience, the optimal thing for the rest of us to do is just wait until he figures the problem out. 😉

    • The nonstationarity you describe is related to the “pink” noise models (1/f noise), which was highlighted in ecology by Halley (1996) and have been studies in relation to environmental variance and extinction (e.g. Ruokolainen et al 2007. For much ecological variability pink noise models seem to be a good fit, but you also run into the issue of how variability in environmental drivers are filtered thought species (e.g. Greenman & Benton 2005).

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