Zombie ideas about disturbance: a dialogue (UPDATED)

UPDATE: Just to be clear, disturbance, and environmental variability more generally, can promote stable coexistence. They just can’t do so via the mechanisms that the Professor is trying to teach in this dialogue. I’ve been clear about this in previous posts, but it was suggested to me that anyone who only reads this post (and doesn’t read the comments) could get the wrong idea.


The scene: An undergraduate ecology lecture. The Professor has been teaching students about the effects of disturbance on competitive exclusion.

Professor: In summary of this section of the course, the great diversity of species to be found in a community is one of the puzzles of ecology. In an ideal world the most competitive species (the one that is most efficient at converting limited resources into descendants) would be expected to drive less competitive species to extinction. However, this argument rests on two assumptions that are not necessarily always valid.

The first assumption is that organisms are actually competing, which in turn implies that resources are limiting. But there are many situations where disturbance, such as predation, storms on a rocky shore, or frequent fire, may hold down the densities of populations, so that resources are not limiting and individuals do not compete for them.

The second assumption is that when competition is operating, one species will inevitably exclude the other. But in the real world, when no year is exactly like another, the process of competitive exclusion may never proceed to its monotonous end. Any force that continually changes direction at least delays, and may prevent, an equilibrium or a stable conclusion from being reached. Any force that simply interrupts the process of competitive exclusion may prevent extinction and enhance diversity.*

Clever student: [raises hand]

Professor: Yes, a question in the back.

Clever student: I’m confused about both of those assumptions. I just don’t understand how they prevent competitive exclusion.

Professor: Can you be more specific? What exactly don’t you understand?

Clever student: Well, with the first assumption, if a species is experiencing really high mortality rates from predation or fires or whatever, how come it doesn’t just go extinct?

Professor: Because if its density is low, then resource levels will be high, which allows the species to have a very high reproductive rate.

Clever student: So there’s really rapid mortality, but really rapid reproduction, and the two balance out?

Professor: Yes.

Clever student: So then wouldn’t anything that reduces reproduction even a little, or increases mortality even a little, like just a little bit of competition, still lead to extinctions? I mean, in that simple resource competition model from that Tilman guy that you showed us to teach us about competitive exclusion, there were per-capita mortality rate parameters for each competitor. You can jack up those mortality rate parameters, and yeah, you reduce the equilibrium abundance of the dominant competitor, and you raise the equilibrium resource level, but you still get competitive exclusion. The species that can reduce the limiting resource to the lowest equilibrium level relative to its competitors still wins. It’s just that everybody’s R* value increases as mortality rates increase.

Professor: [pauses for thought] Hmm… I see what you mean. But remember, we’re envisioning a situation in which mortality is so high that resources aren’t limiting, so there’s no competition at all.

Clever student: You mean, species are there and they’re consuming resources, but their densities are so low that their consumption doesn’t reduce resource levels at all? How can that be? If they’re there, they have to be consuming some resources, right? And if they’re consuming some resources, then they’re surely at least slightly reducing resource levels, which means there’s at least some competition, right? Plus, isn’t it super-unlikely that mortality rates would be just high enough to reduce species to near-zero density, so that there’s no competition, but no higher? So that species can still persist rather than just being totally wiped out because they’re getting killed faster than they can possibly reproduce?

Professor: I think you’re over-thinking things. Rather than thinking about hypotheticals, think about real natural systems. Think of harsh environments like rocky shores and alpine meadows. There is in fact a lot of disturbance and mortality in those environments, which does reduce population densities, and species do coexist in those environments. So there you go.

Clever student: Yes, I know all that, but that doesn’t answer my question. I want to know why they coexist. I mean, how do we know they’re not just coexisting for some reason that doesn’t have anything to do with their densities being low? ‘Cause I read some microcosm papers where they manipulated mortality rates of competing species and found that you got just as much if not more competitive exclusion at high mortality, and if you didn’t it wasn’t just because competitor densities were low.

Professor [not sure how to answer so he bails out]: We’re running a bit short of time, and I’m not sure I’m familiar with the papers you refer to, although I don’t know that microcosm studies are relevant to what happens in nature. Why don’t you come to my office hours later and we’ll talk more about it? Now, you said you also had a question about the other assumption?

Clever student: Yes, I don’t understand the whole “interruption” thing. Like, if I give my buddy $2 today, and he gives me $1 tomorrow, then I give him $2 the next day, and we keep alternating like that, eventually I’ll run out of money and he’ll end up with all the money. Even though every other day my losses are interrupted by him giving me $1.

Professor: Ok, I see where you’re confused. The point of the second assumption is that the interruptions slow down the rate of exclusion, here the rate at which you go broke, compared to what would happen if there were no interruptions. If you gave your friend $2 every day, with no interruptions, you’d go broke a lot faster.

Clever student: Yes, that’s true, but why is that the right comparison? I mean, I’d also go broke slower if I only gave my friend $0.50 every day, with no interruptions. I’m losing the same amount of money every day, I’m just losing less than if I give my friend $2 every day. That’s why I go broke slower when I give him $2 every other day, and he gives me $1 every other day–on average I’m only losing $0.50 per day in that case. So I go broke at the same rate, whether I give him $0.50 every day, or we alternate between me giving him $2 and him giving me $1.  It seems like how much money I’m losing on average is all that matters. The interruptions are just noise, aren’t they? They don’t actually have any effect on anything.

Professor: Well, remember that we learned that the frequency of environmental change matters. Hutchinson said so. The environment has to switch from favoring one competitor to favoring another with intermediate frequency, on the same timescale as competitive exclusion. In your example, you and your friend are favored on alternate days, which is a really high switching frequency, not an intermediate frequency. In that kind of case, Hutchinson says that species just average across the fluctuations.

Clever student: But what if I give my friend $2 every day for, like, a week, and then he gives me $1 every day for a week, and so on? Or a month, whatever. I’m still losing $0.50 per day on average, so I’m still going broke at the same rate.

Professor: Hmm, yes, I see what you mean. But at least when you and your friend are favored for longer periods of time, those periods of time are economically relevant. I mean, at the end of a week when you were favored, you’ll have saved up enough money to buy a beer. [laughs] As ecologists, it’s often difficult to study what happens in the long term, so we just focus on shorter, ecologically-relevant timescales.

Clever student: [does not laugh] But I thought we were trying to explain coexistence. Like, real coexistence, not just temporary blips that sort of look like coexistence. I mean yeah, sure, the long run is hard to study–but I’d still have the same question even if the long run were really easy to study. So I’m even more confused now. Are you saying that species will go extinct no matter what the frequency of disturbance, but it’s ok, because along the way they’ll sometimes increase in density? Isn’t that totally changing the question?

Professor: Ok, you’re right, average conditions do matter. So consider a case in which you give your friend $2, and the next day he gives you $2, and so on. In that case, neither of you loses or wins any money on average.

Clever student: But that’s just the same as if we each give the other no money. Or we each give each other the same amount of money on the same day rather than alternating days. Or there’s a continuous steady flow of money from his bank account to mine, and an equal continuous steady flow of money from my account back to his. The day-to-day fluctuations in who gives money to who don’t matter at all, all that matters is the fact that we’re each breaking even on average.

Professor: You seem very focused on the average conditions and the long-term outcome, to the exclusion of the fluctuating dynamics that disturbance generates. Those fluctuations are a very interesting part of ecology, you can’t just ignore them. You can’t just ignore dynamics.

Clever student: [getting frustrated] I’m not ignoring dynamics–competitive exclusion is a kind of population dynamic. The abundance keeps going down until it hits zero. And I’d be happy to pay attention to fluctuating dynamics if you gave me a reason to. It’s you who said that “Any force that simply interrupts the process of competitive exclusion may prevent extinction and enhance diversity”. You didn’t say “Any force that simply interrupts the process of competitive exclusion creates interesting fluctuations in species abundances on the way to exclusion, which is something we can’t ignore even though it has no effect on diversity.”

Professor: Ok, I see your point on dynamics, you certainly do have a way with words. I think where you’re confused is that you’re trying to separate two effects of disturbance that just can’t be separated. Adding disturbance to a disturbance-free system both changes the average conditions, and interrupts approach to equilibrium. Those two things always go hand in hand, and you don’t really need to worry about separating their effects. It’s all just effects of disturbance.

Clever student: But they don’t always go hand in hand–there are ecological systems with similar average conditions and different amounts of variability around the average. And there are ecological systems with different average conditions but similar amounts of variability. Plus, we can do experiments to manipulate average conditions and variability independently of one another, can’t we? Don’t scientists do that all the time–do experiments to tease apart effects that usually co-occur?

And if you don’t separate effects of changes in the average from effects of changes in the variability around the average, how do you know that the variability is what matters? Because that’s how you explained it–you talked about variability, you talked about “interrupting” the approach to equilibrium. I mean, that’s why we call them “disturbances”, right? They disturb what we think of as the “normal” course of events. But based on what you’ve told us, disturbances don’t actually matter as disturbances at all, since what they do isn’t really to disturb the normal course of events, it’s to change what constitutes the normal course of events. That is, change the average conditions. Which of course you could also change without having any disturbances. So I don’t see what’s so special and unique about disturbances as opposed to just any old change in average environmental conditions.

So I guess that’s really my big question: what can introducing disturbances do that can’t be done by just making the equivalent change in average environmental conditions. Why does variability per se matter?

Professor: Umm…


This post is directed at the many readers who are still on the fence about whether the intermediate disturbance hypothesis (IDH) is a zombie idea. My goal with this post is to try to force those fence-sitters to come down on the correct side, by reminding those readers that this isn’t a purely intellectual debate among academics. This is about what we teach to our students. And as teachers, we need to be able to answer our students’ questions. I don’t think the questions I’ve put in Clever Student’s mouth are at all unreasonable. Indeed, and in all honesty, they’re exactly the sort of questions that I’d expect my University of Calgary undergraduates to ask. They’re certainly the sorts of questions any undergraduate who reads this blog would ask (and there are many undergrads who do read this blog, at universities around the world). Maybe most students wouldn’t ask these questions in quite so articulate or pointed a manner, but they would ask them. They’re perfectly natural questions, that arise from the standard way in which the IDH is explained in textbooks.

So, for those of you who are still on the fence about whether the standard, textbook explanations of the IDH are zombie ideas: How would you answer Clever Student’s questions?

Note that you can’t answer Clever Student’s questions by claiming that the IDH actually has to do with competition-colonization trade-offs, or trade-offs between disturbance tolerance and competitive ability, or successional niches, or the storage effect, or other factors not mentioned by the Professor in his opening lines. Yes, those other factors are relevant to thinking about the effects of disturbance on coexistence. They’re also irrelevant here, because the way in which the Professor explains the IDH is the standard explanation that actually appears in numerous textbooks, and in the Introduction sections of many, many papers. If you think the way to answer Clever Student’s questions is to redefine the IDH by dropping both of the Professor’s assumptions and explaining the effects of disturbance completely differently, then you’re admitting that the standard, textbook understanding of the IDH is 100% wrong. Which I suggest ought to bother you as much as it bothers me. Yes, textbook explanations have to simplify and gloss over technical details–but surely not the point of inviting the sorts of questions Clever Student asks!

Note as well that Clever Student is very alert to attempts to change the question, which the Professor tries and fails to do several times. Note as well that all those attempts to change the question were actually tried by commenters on my original zombie ideas post, or folks who’ve corresponded with me privately. The Professor in this post is just trying to answer Clever Student’s questions the way commenters and correspondents have tried to respond to my original post. The Professor here is no straw man.

If you’re tempted to respond by arguing that Clever Student’s questions are somehow ambiguous or otherwise flawed, please be aware that Clever Student’s questions can be put much more rigorously and precisely. In particular, I would discourage you from trying to argue that “students exchanging money is nothing like species competing for resources”, unless you’re prepared to explain why the analogy is a bad one. Because in every relevant respect, the analogy is perfectly consistent with the Professor’s second assumption. So if you think the monetary exchange analogy is a bad analogy to the IDH, then what you actually think is that the standard, textbook explanation of the IDH is a bad explanation.

The Professor here is not an unreasonable or ignorant person. He’s smart, and he’s doing his best to answer Clever Student’s questions. But those answers just don’t cut it. Hence my curiosity whether any readers can come up with better answers. Our students–real students, not Clever Student–deserve no less.

Of course, I think Clever Student’s questions don’t have good answers. I think the only legitimate response to those questions is to stop teaching the Professor’s zombie ideas in the first place.**

*These lines of the Professor’s dialogue are an abridged quote from p. 740 of the second edition of Begon, Harper, and Townsend’s textbook Ecology: Individuals, Populations, and Communities. My abridgements are minor and do not alter the meaning of the passage.

**I wonder if anyone will try to argue for “teaching the controversy”–teaching both standard ideas about the IDH, and counterarguments. Personally, I think that’s about as good an idea as “teaching the controversy” between evolution and creationism. Remember, this isn’t a controversy between alternative logically-valid claims, which simply make different assumptions about how the world works, and which we can decide between by conducting an appropriate experiment. It’s a controversy about the logical validity of one set of claims. There are scientific controversies which can be usefully taught in science classes. But this isn’t that kind of controversy.

36 thoughts on “Zombie ideas about disturbance: a dialogue (UPDATED)

  1. I think that the students exchanging money example is analogous to a population dynamic that has no density-dependence because you are assuming that $2 on any given day has the same value. There is a proof by Tim Reluga (iirc, I think this is it: http://imammb.oxfordjournals.org/content/22/2/187) regarding a related problem. The argument that I am thinking of goes something like, if fitness in habit a is ra and in habit b is rb then if the orgaanism sees habitat in this order a-b-a-b we have n1 = ra*n0; n2 = rb*(ra*n0); n3 = ra*(rb*ra*n0), etc, and so from that example, it’s not hard to see that at time t, what matters is how many times you’ve seen habitat a versus habitat b and the order in which the organism sees the environments doesn’t matter at all because multiplication commutes, i.e. ra*ra*rb*rb = ra*rb*ra*rb.

    • Yes, I was anticipating a comment about density dependence. But I don’t think that undermines the point of the monetary exchange analogy (not sure if you’re saying that it does…). After all, it’s not as if the standard textbook explanation for why “interruptions” matter (or the original paper by Hutchinson 1961 on which those explanations are based) says anything about density dependence (or the multiplicative nature of population growth, or the importance of overlapping generations, or all sorts of other technical issues). The point of the monetary exchange analogy is just to put pressure on people who think, incorrectly, that Hutchinson’s (1961) argument somehow “captures the essence” of how coexistence works in a temporally-varying environment. On the contrary, Hutchinson’s way of thinking is positively misleading–it invites the sort of questions that Clever Student asks. As I’m sure you know (and as Clever Student has come to realize!), just having each species favored some of the time isn’t even close to sufficient for coexistence, no matter what frequency with which the environment changes. What you need is the right sort of nonlinearity or nonadditivity, so that time-averaged per-capita growth rate depends on something besides just average conditions. See Chesson and Huntly 1997 Am Nat and Chesson 2000 ARES.

      The Reluga paper does look related, I’ll have a look.

  2. sorry, my browser was giving me trouble and I wasn’t quite finished… the previous example was meant to get at the question of when averaging the environment is a reasonable approx.

    In the context of density-dependence and competition then doesn’t the analogy go something more like ‘and every day you take your money and use it in an auction to buy a beer’ because then the real value of your money depends on how much money everyone else has as much as your ability to reproduce depends on crowding or competition.

    If I was going to read up on disturbance and coexistence in community ecology can you suggest something I should read?

    • If you’re only going to read one thing on disturbance and coexistence in community ecology, Chesson and Huntly 1997 Am Nat is the paper to read. There’s a lot of math, but that won’t be an obstacle for someone like you. Chesson 2000 is a more general explication of how to think about coexistence, and probably Chesson’s most accessible paper.

      For better or worse, Peter Chesson is a totally brilliant and very rigorous guy, but not always the best explainer, which has severely limited his influence. Basically, what I’m trying to do in these posts is explain (one piece of) Peter’s math in terms anyone can understand.

  3. Okay, you convinced me that the IDH is not a valid explanation for coexistence of competing. So we can as well ask what might enable coexistence regardless of the question whether conditions are at equilibrium or not. For simplisity then, let’s just assume equilibrium conditions and go back to Hutchinson’s phytoplankton paradox.

    How can more than one phytoplankton species coexist in equilibrium conditions?

    Is it possible that your models (including those of Hutchinson and MacArthur) only consider the resources that are given from the start and what is taken away from them by consumption of competitors?

    Suppose a phytoplankton community in equilibrium conditions with one species exploiting the resouces that are given from the start best. If all that ever happened was competition for these (limited) resources, this best competitor should exclude all other phytoplankton species.

    If, however, this best competitor excretes certain compounds, which another species can use, or if the predators or pathogens (e.g. viruses) excrete or release such compounds, then new kinds of resources are added to the system. Consumption of the best competitor does not simply reduce the given resources but transforms them into another set of resources for which it is no longer the best competitor. And this happens again for all the competitors in the system do not only consume but also excrete (or die or are killed).

    Then we probably end up with a dominant phytoplankton species “carrying” a trail of sub-dominant species that could be described as commensals as well as competitors.

    • Hooray, I convinced someone!

      You’ve asked the right follow-up question too: what *does* explain coexistence? Broadly speaking, there are three classes of answer:

      (i) Coexistence mechanisms that can operate at equilibrium, in a spatially homogeneous and temporally-constant world. The commensalism-based mechanism you suggest is one such example (there are of course many other possibilities). I’m not sure it’s the most plausible possibility for phytoplankton, but it’s known to work for competing bacteria in the lab. Under appropriate culture conditions, bacteria easily evolve such “cross-feeding”; see Grover’s 1997 book Resource Competition for a review.

      (ii) Coexistence mechanisms that can operate only in a spatially-heterogeneous world. For instance, if the world has different kinds of patches, and different species are best-adapted to different kinds of patches, then different species win in each patch and at a global level they all coexist.

      (iii) Coexistence mechanisms that can operate only a temporally-variable world, like relative nonlinearity and the storage effect.

      What all these classes of mechanism do (and which zombie ideas about disturbance don’t) is cause species to compete (directly or indirectly) more strongly intraspecifically than interspecifically, so that rare species (which by definition experience little intraspecific competition) have a relative fitness advantage over common ones. The fatal flaw of all these zombie ideas about disturbance is that they don’t generate any return tendency, any tendency for rare species to “bounce back” rather than going extinct.

      Chesson 2000 ARES is a good review of these ideas.

      • If we assume equilibrium conditions in homogeneous environments, such as phytoplankton is supposed to occupy, then your categroies ii) and three seem to be non-starters. They simply alter the assumption of homogeneity (ii) or equilibrium conditions (iii) rather than being a solution for the assumed conditions.

      • Um, not sure why you think phytoplankton are “supposed” to occupy an equilibrium, homogeneous environment. Indeed, lake environments are well-known for being temporally dynamic, leading to phenomena like seasonal succession of phytoplankton species (a topic on which ‘lowendtheory’ has done some nice theoretical work). Did something I wrote give you the impression that phytoplankton occupy homogeneous, equilibrium environments?

        More broadly, I was trying to respond to your previous comment in a general way, by sketching out all the classes of coexistence mechanisms. I didn’t mean to imply anything about the importance of any particular class of coexistence mechanism in any particular case.

        In practice, coexistence of phytoplankton probably depends on a mix of mechanisms that can operate in a homogeneous equilibrium system (note that those mechanisms don’t *require* equilibrium in order to operate; it’s just that they *don’t* require spatial or temporal variability in order to operate), and mechanisms that require temporal fluctuations in order to work.

  4. When faced with a zombie, the easiest thing to do is blast it with a shotgun. But remember, that zombie was once human, just like you and me. Maybe there’s some soul left that’s worth saving. So put down your shotgun, pick up your exorcism kit, and let’s figure out what we can all agree on here.

    I refuse to step into your professor’s shoes (he lost any sympathy with his “first assumption” nonsense), but allow me to rephrase the conclusion of his second argument as “environmental variability can increase the chance of coexistence”, “in general agreement with Hutchinson’s idea that environmental variability can promote species diversity”*. The key word is your professor’s dialog is “may”; in the rephrase, it’s “can”. Environmental variability CAN promote species diversity. It’s possible. That was Hutchinson’s insight in 1961, and it was a good one.

    Thirty-six years later, Chesson and Huntly refined the notion by showing that environmental variability doesn’t necessarily promote species diversity. Certain conditions need to be met, namely nonlinearity and nonadditivity. This was also a valuable contribution, but let’s not revoke all of Hutchinson’s credit. After all, nonlinear systems are like non-elephant zoology. Intermediate timescale environmental variability can allow coexistence in common resource competition models, even some with Lotka-Volterra form. Newton still gets mad props despite Einstein’s relativistic corrections.

    A similar argument could be made for the IDH. There are certainly models where it works as expected.

    So kill the zombie, but save its soul. The soul of the IDH/environmental variability hypothesis is that temporal variability can allow species to coexist that otherwise wouldn’t. It’s not automatic. It depends on how the temporal variability affects growth. To figure out how, you’re going to have to get your hands dirty with some equations. That’s when you set your undergrads loose in the computer lab with Mathematica and have them investigate it themselves.

    Can we agree on that last paragraph?

    * – Chesson & Warner 1981

    • Hutchinson could hardly ask for a more fair-minded and eloquent defender than you. 😉

      I actually do agree on the last paragraph. It’s exactly how I teach this subject to my undergrads.

      And if you want to credit Hutchinson with the key insight that “environmental variability *can* promote coexistence”, even though he was wrong about *how* that can happen, I don’t necessarily have a problem with that. But the risk of doing that is that people who don’t have a firm grasp of modern coexistence theory will misunderstand and fail to recognize that nonlinearity and nonadditivity are in fact essential for fluctuating environments to promote coexistence. If you tell people that, say, Chesson and Huntly “refined” Hutchinson’s ideas, the natural conclusion many people will draw is “Hutchinson was basically right, and Chesson and Huntly is just technical details, so I’ll just focus on Hutchinson and remember what he said.” Students, and math-phobic ecologists, are especially likely to do this because Hutchinson is in the textbooks, and because Hutchinson expressed himself in words rather than equations. Indeed, in the comments on a previous post, Robin Snyder talks about her students doing exactly that–just sticking with Hutchinson’s idea of *how* coexistence works in fluctuating environments, because Chesson and Huntly seems hard. So when I’m writing these posts, or when I’m teaching my students, I don’t give Hutchinson any credit. That’s because the most important thing I’m trying to get across is not the historical questions of who had exactly what insights first, or who made exactly what mistakes. The most important thing I’m trying to get across is *how* environmental variability can (and can’t) promote coexistence. I feel like unless I maintain a laser-like focus on that question, my audience will misunderstand. Plus, Hutchinson’s idea about how coexistence works is in all the textbooks–it’s not like he lacks for credit! So if my approach risks throwing the baby out with the bathwater and leaving Hutchinson with no credit for anything, well, I think the risk of that happening is actually really small. The much bigger risk, in my view, is that Hutchinson’s and other zombie ideas about *how* disturbance affects coexistence will simply persist, except among the cognoscenti. It is a huge uphill struggle to convince people that there’s *anything* wrong with such a famous paper as Hutchinson 1961 (or Connell 1978, or Huston 1979), which is discussed without *any* criticism in all of our textbooks. You dismiss one of the Professor’s claims as “nonsense”, which it is (although even I steered away from using such strong language!). But how many people, especially students, do you think would so easily dismiss any statement in one of our most popular textbooks as “nonsense”?

      Quibble: I do disagree with the analogy of Hutchinson to Newton. Newton’s laws aren’t so much wrong as a very good approximation to Einstein’s relativity at sufficiently low accelerations. I don’t think Hutchinson’s own proposed mechanism by which environmental variability can promote coexistence is a good approximation to any correct idea. That there are correct models (like yours) that also happen to make some of the same predictions as Hutchinson’s mechanism is just a coincidence, not evidence that Hutchinson was on the right track.

  5. You can count me as someone who’s been convinced, although I was convinced at your first post; IDH was something I took as a vague idea from undergrad, and didn’t really put the thought into it until your post. After that, the logic was pretty clear.
    I not entirely convinced of this point though: “If they’re there, they have to be consuming some resources, right? And if they’re consuming some resources, then they’re surely at least slightly reducing resource levels, which means there’s at least some competition, right?”. There’s a clear counter-example where this isn’t true: coming from a marine background, there’s plenty of species that are space limited. In a given patch, a species could likely happily grow at a constant rate until all the space was used up, and only then face competition. For me, I think this is what people often have in mind with this sort of argument; the intuitive idea that, as long as disturbance “keeps the population away” from that boundary, competition won’t have an effect, and species can coexist. That has a clear problem, though: the only ways disturbance can keep it away from the boundary is if the population is growing linearly and disturbance acts on a per-capital basis (dP/dt = r – D(t)P), or population grows exponentially, and disturbance acts greater than exponentially (dP/dt = rP – D(t)P^a, a>1). In case one, it’s like you said, temporal variance doesn’t matter, just average mortality, and in the second case, you need non-linear disturbance effects.

    • Thanks Eric!

      Re: your counterexample to “they have to be consuming some resources”, space *is* a resource! 😉 Or if you prefer, space can be very usefully thought of as a resource. One useful definition of a “resource” is something that an organism can consume (thereby removing a unit of that something from the environment and preventing its use by some other organism), the consumption of which increases fitness for at least some consumption levels (that last caveat addresses the fact that lots of resources are toxic if you consume way too much of them). For instance, Tilman’s Resource Competition and Community Structure has a chapter on attachment space as a resource for sessile organisms.

      The counterexample that comes to my mind is a stationary consumer taking resources out of the water in a very fast-flowing stream. The consumer can’t really reduce resource availability *in its immediate neighborhood* because the “neighborhood” quickly flows away and is replaced by a new “neighborhood”. If that makes sense…

      In any case, I think it’s pretty clear that neither your counterexample or mine has anything to do with what Begon et al. had in mind. What exactly they did have in mind is less clear. As lowendtheory says, that passage from Begon et al. about how resources aren’t limiting and there’s no competition in disturbed environments is just nonsense if you take it at face value. Which is kind of a problem as it’s supposed to be an overview and summary of a major chunk of one of the chapters! Perhaps I need to do a post on what it means (or should mean!) to say that a “resource” is “not limiting”…

      Re: the notion of disturbance “keeping the population away” from some boundary, you could well be right that that’s the intuition many folks have about sessile organisms.

      • Sorry, should have been more clear; I consider ‘space’ a resource, it’s just that one where populations aren’t necessarily “surely at least slightly reducing resource levels”; or rather, they are reducing resource levels, but it might not change dynamics until total resource use hits some limit. As long as disturbance acts to keep the community “inside” that limit, there would be essentially no inter-specific competition at all. That would require, though, that disturbance acts in a density-dependent fashion, otherwise there would be no way to keep the community’s density below the limit.

      • Ok, I see what you’re getting it. Although I could imagine that, depending on the way in which colonization or growth works, density dependence would start to kick in even below 100% occupancy of the available sites (e.g., colonizing propagules settle at random, and settling on an occupied site is fatal, so that per-colonist probability of successful settlement declines with increasing occupancy even when occupancy is below 100%). It sounds like you’re thinking of a situation in which settled individuals just grow or spread horizontally, and there’s no competition until they bump up against another individual or the boundaries of the habitat?

        But that’s quibbling. Your broad point–that the precise consequences of disturbance are going to depend on the way in which per-capita growth rates vary as a function of resource level–is certainly correct.

  6. I have been reading through your posts on the IDH-as-zombie, and I have found it really fascinating and challenging. It’s left me with a few questions. I am an evolutionary biologist, not an ecologist, so they are probably pretty uninformed…

    First, you quite emphatically state that the IDH is dead, but that disturbance, in concert with nonlinearities and non-additivities can produce coexistence. Does the original formulation of the IDH specifically exclude such things? I was wondering if you could give a few examples of what those might be in nature. It sounds like the IDH was first formulated verbally by Hutchinson, Connell, etc. but then later failed to stand up to mathematical scrutiny. Is it possible that theoreticians have unfairly couched it in unfavorable mathematical terms (e.g. lotka-volterra models)? I am totally convinced by your arguments that in the theoretical framework you describe, the IDH is pointless, I just wonder if that framework is the best approximation of reality…

    That leads me to my second question, which is in line with Eric Pedersen’s point above. You describe the most competitive species as being the one that “is most efficient at converting limited resources into descendants” and your arguments against the IDH rely on the notion that SOME competition occurs between species, even at very low densities (e.g. “Clever student: So then wouldn’t anything that reduces reproduction even a little, or increases mortality even a little, like just a little bit of competition, still lead to extinctions?”) but what if there is literally no competition at low densities? There is the idea of sessile marine organisms, but really, any species for which a discrete territory is the limiting resource might conceivably have identical per capita growth rate, but differ in ability to secure a territory. Under those circumstances, competition might be zero until there are more individuals than territories, and as long as disturbances are clearing territories at a rate faster than individuals are reproducing, coexistence would be predicted. So I guess the question is, isn’t this exactly consistent with the IDH as you describe it in your first post? (“Disturbance reduces species’ densities, thereby weakening or eliminating competition and preventing the competitive exclusion that occurs in undisturbed environments”).

    Thanks for some really thought-provoking writing, and apologies in advance if my questions are too naive…

    • Hi Noah,

      Those are perfectly natural questions.

      There isn’t really a single original formulation of the IDH (which is part of what makes it hard to kill–in some ways it’s more of a hydra than a zombie). The term “IDH” was coined by Connell 1978, but textbooks and reviews that explicate the idea routinely intermingle ideas from Connell’s paper with ideas from other papers, earlier and later (particularly Hutchinson 1961, Grime 1973, and Huston 1979, but also others). You’re absolutely right that these ideas were originally formulated verbally, except for Huston 1979, who misinterprets simulations of a simple competition model with periodic disturbances. And you’re right, those verbal ideas haven’t stood up to mathematical scrutiny. But no, it’s not because the theoreticians have been unfair in their choices of assumptions (interpreting the verbal arguments in some kind of perverse way so as to force them to fail mathematically). Indeed (and perhaps ironically), it turns out that the relevant sorts of nonlinearities and nonadditivities are actually quite easy to get. You can even get nonadditivities in Lotka-Volterra models if you add environmental fluctuations in in the right way. So far from ecological theoreticians choosing perverse assumptions in order to avoid finding effects of disturbance on coexistence, ecological theoreticians have actually found it difficult to produce models in which disturbance *doesn’t* affect coexistence! It’s just that those easy-to-find mechanisms are *not* the mechanisms identified by classic verbal arguments.

      Note that the best theoretical work on this stuff, by Peter Chesson, actually considers extremely broad *classes* of cases, a point which isn’t as widely appreciated as it should be. So when I say “You need the right sort of nonlinearities and nonadditivities for disturbance or environmental fluctuations to promote coexistence”, I’m not generalizing from a specific theoretical model with specific assumptions, or even from comparative study of a bunch of specific models making different specific assumptions. I’m relying primarily on analyses of extremely broad *classes* of models, with very broad and generally-applicable assumptions that cover pretty much any specific biological situation.

      It’s true that the original verbal formulations of the IDH don’t specifically *exclude* nonlinearities and nonadditivities. But they don’t recognize or appeal to those nonlinearities or nonadditivities either (for instance, you will search that quote from Begon et al.’s textbook in vain for any mention of anything that could be interpreted as “nonlinearity” or “nonadditivity”). Now, in fairness, the original verbal formulations of the IDH are as ambiguous as verbal models inevitably are. So you can argue, as ‘lowendtheory’ argues, that modern nonlinear, nonadditive models aren’t inconsistent with classical verbal arguments, and perhaps even “refine” or “clarify” classic verbal arguments. I personally wouldn’t phrase things that way, but that’s just me. As long as you’re perfectly clear on exactly how environmental fluctuations do or don’t affect coexistence, I don’t really care how you parse ambiguous verbal arguments.

      Re: Eric’s point about situations in which there really is no competition at low densities…As I replied to Eric, you do have to be careful to fully specify exactly what sort of biology you’re imagining. For instance, a system in which not all sites are occupied by adults, so that no adults bump up against any other adults and so don’t compete that way, can still have density dependence even at very low adult densities. For instance if adults reproduce via propagules that settle randomly, with propagules perishing if they settle in a site occupied by an adult, so that the per-propagule survival probability (and thus adult per-capita reproductive rate) decreases linearly with increasing adult density (occupancy). But yes, I can imagine situations in which intra- and interspecific competition only kick in once species’ densities exceed some threshold. Broadly speaking, that’s just one way in which you can get nonlinear density dependence–per-capita growth rate depends on conspecific and heterospecific densities, but in a nonlinear way. Here, you’re imagining that per-capita growth rate is high and independent of density below some density threshold, but then declines in some fashion with increasing density above that density threshold (rather than just declining linearly with increasing density as in a Lotka-Volterra-type world). In such a threshold-type nonlinear system, I do think you’d generally expect species to attain densities high enough that density dependence kicks in. And because you’ve got *nonlinear* density dependence, yes, you’d expect environmental variability per se to alter the system’s behavior.

      By the way, as an evolutionary biologist you might be interested to know that evolutionary biologists never fell for these zombie ideas about mechanisms, perhaps in part because in population genetics formal mathematical theory from folks like Sewell Wright preceded any verbal theorizing or empirical work. Which isn’t to say that evolutionary biologists fully “get” this stuff either. Evolutionary biologists tend to think that the necessary sorts of nonlinearities and nonadditivities are much rarer or more unusual than they actually are. So evolutionary biologists tend to be unduly pessimistic about temporal environmental variation maintaining genetic diversity. Steve Ellner and Nelson Hairston Jr. had a nice paper in Am Nat in the mid-’90s that points this out.

  7. Hi Noah,

    I’ll let Jeremy field the first point, but in regards to second question: This was part of what I was trying to get at… even when you have no competition below a certain level, you still need some sort of per-capita density dependence mechanism for disturbance to lead to coexistence. Basically, if in a patch, individuals are growing without per-capita density dependence, each species’ population (P_i) will change exponentially, with some birth rate r_i, and some death rate m_i (dP_i/dt = r_i*P_i – m_i*P_i). If we treat disturbance just as a time-varying rate of mortality (m_i(t) ), without making it depend on population density, the only things that can happen are the species will grow (possibly slightly slower) towards the point where competition kicks in, or they decrease to extinction, and all that will determine that is if the time averaged mortality (m) is less than or greater the time-averaged grow rate. This means that, without some sort of density dependence, it doesn’t matter that mortality rates vary over time (disturbance) or are at some constant high or low level. Now, this all changes if, say, each species faced intra-specific competition even when the population as a whole is under-saturated, or if disturbance affects species at a higher rate when they have a higher density.

    Hope this helps.

  8. So how could the competitive exclusion principle be formulated verbally in the light of the new findings?

    As it stands/stood, it only considers between species competition. The new formulation would have to explain why, under equilibrium conditions, the best competing species should drive the worse competers to extinction despite its within species competition rising and those of the losers sinking all the while.

    • No, the competitive exclusion principle implicitly considers intraspecific competition as well. Insofar as species have a relative fitness advantage when rare, because intraspecific competition is stronger than interspecific competition, that means that there are niche differences. But those niche differences have to be sufficiently large; a sufficiently superior competitor can still exclude a sufficiently inferior one if that’s not the case.

      I encourage you to have a look at Chesson 2000 ARES, or Adler et al. 2007 Ecol. Lett. for good explications of modern ideas about how coexistence works. Simon Levin has a nice Am Nat paper from back in the 70s with a general, formal statement of the competitive exclusion principle.

      • Okay, I looked at Hutchinson 1961, Conell 1976, and Huston 1979 already, but I will look at the above once as well. Just for the record, the way you now explained/explicated competitive exclusion, coexistence seems to be the default expectation and displacement the anomaly requiring special explanation*. At that point, I do not see why I should bother about IDH and its subtleties anymore?

        *For example, Gause’s experiments did not allow the niche differences necessary for coexistence to be realised.

      • I don’t know that I’d say that either coexistence or exclusion is an “anomaly” requiring “special explanation”. Both occur in nature. I think it’s useful to start from the competitive exclusion principle just because that’s a very simple limiting case, not because as an empirical matter we think exclusion is more likely than coexistence. Analogously, when we teach population ecology, we start by teaching exponential growth, even though we don’t expect any species in nature to grow or decline exponentially for long periods (or to not interact with other species, or grow in a constant environment, or lack all of the other realistic complications omitted by the simplest exponential growth model). Exponential growth is just a simple limiting case, which it helps to understand in order to understand the effects of realistic ecological complications (like density dependence, time lags, interactions with other species, etc.).

        Even if you think coexistence is to be expected and competitive exclusion an anomaly, then you have the task of explaining those rare cases of exclusion. To do that, you presumably have to figure out what coexistence mechanisms there are and how to remove them from your models in order to generate a model that can explain those exclusion anomalies. So you still have to identify and understand coexistence mechanisms and figure out how the world works in both the presence and absence of those mechanisms. It’s just that you’ve taken as your starting point some highly complex picture of the world, filled with all sorts of coexistence mechanisms, which from a pedagogical perspective seems like not the best starting point. Pedagogically, it’s easier to start simple and then add in complexities.

  9. As I was reading the post and people’s comments and questions, I was wondering about the role of positive interactions or rather positive indirect interactions in the context of this discussion. I get the feeling that the topic goes around negative interactions (i.e. competition) and that disturbance plays a negative role in species coexistence. I think disturbance is needed for species survival and that goes beyond ecological time scales. So I guess I would recommend to the clever student to read Joseph Connell (1980) Diversity and coevolution of competitors or the ghost of the competition past Oikos 35, 131-138.

    • Positive interactions haven’t been much considered in the context of coexistence theory, which certainly needs to be rectified. In principle, incorporating them isn’t difficult, but someone still needs to do it. Worth noting that they won’t necessarily promote coexistence–anything that raises the per-capita growth rate of one or more species need not have any effect, or any coexistence-promoting effect, on the relative per-capita growth rates of different species.

      As noted in other comments and posts, disturbance can promote, inhibit, or not affect species coexistence. This post merely points out that some ways in which ecologists *think* disturbance promotes coexistence don’t actually work.

      Not sure what you mean about disturbance being necessary for species survival beyond ecological time scales, or the connection to Connell’s classic paper…

  10. I mean is that what we see in a give habitat rich in species composition or high diversity is the result of the evolutionary processes. I think that during the course of that processes the stronger negative interaction of competion did occur so what we see now in our ecological time scale is more mechanisms allowing coexistence such as positive interactions (i.e. facilitations). My point with this is that very few people is including this type of interaction when explaning the coexistence of species.

    • Whether coevolution of competitors generally strengthens coexistence, or tends to strengthen facilitation and weaken competition, are two very interesting questions. The former question has been studied (e.g., the recent paper by Vasseur et al. in Am Nat), but we still don’t have a fully general answer. The second question has been studied in the context of the evolution of mutualism and parasitism, but not more broadly.

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  17. I am a graduate student who is a willing new recruit in the war against the zombies. However, I’m having a hard time fully comprehending what exactly is wrong with this particular zombie(s). I have read most of the related posts on the blog (and the comments), but I admittedly haven’t gotten around to reading Connell (1978) and the others, or even Fox (2013), which I soon will, but in MacArthurian spirit, I decided against waiting until then before posting some of my questions here. Hence, I apologise in advance if they are repetitive or trivial; I just wish to make sure I fully understand the cause I am fighting for!

    My questions are mostly about Hutchinson (1961):

    1. I may have missed something vital, but in your original post, you state that the time-scale of fluctuations doesn’t matter (on its own). However, you describe the two species in the hypothetical being favoured x% of the time (“without worrying about the frequency with which the environment changes”). As I understand it, these two are the same, and the time-scale indeed is important (as Hutchinson wrote). However, I do also gather from the more general definitions of the IDH that the theory places the emphasis on *intermediate frequencies* of disturbances as being important in “interrupting” or “delaying” the competitive exclusion. If this is what the argument is against, then I am inclined to agree that frequency by itself is of not much importance. (And is this what you meant by “some ways in which ecologists *think* disturbance promotes coexistence don’t actually work”?)

    2. Am I correct in thinking that the main counter to Hutchinson’s 3rd point (that if Tc>>Te, species competing the best on average under the full range of environmental conditions excludes the others), is that this is not possible, that a species cannot compete best in both situations (fluctuation in dominant identity, or non-additivity like storage effect)?

    3. It follows, then, that the argument against Hutchinson (1961) is focussed more on his 2nd and 3rd points, while the 1st makes sense and is valid: if Tc<<Te, then the long-term average doesn't matter, as one of the species is excluded before the environment can fluctuate in favour of it (emphasising the extinction threshold in Chris' flip-flop model).

    4. Finally, in these scenarios we assume that the fluctuations favour one or the other species (and to the same extent), but what if this is not so? What if disturbances were not dichotomous or polarising? Or is this is what non-linearity and non-additivity in this respect explore?

    Once more, I apologise if these questions seem either naive/over-simplified or muddled. It is very much possible that I have mixed up multiple points.

    • Thanks for the comments! Fox (2013) does answer some of your questions, but I’ll briefly answer them here.

      The timescale on which the environment fluctuates is different than the % of time the environment favors one species vs. another. For instance, imagine that the environment favors species A for a day, then flips to favoring species B the next day, then back to favoring A the day after that, and so on. It favors each species 50% of the time, and switches states daily. Now imagine the environment favors each species for 1 week at a time before switching. Different timescale of environmental fluctuations, but each species is still favored 50% of the time.

      On Hutchinson’s assumptions, the species that’s the best competitor on average wins no matter what the timescale of the environmental fluctuations. The fluctuations don’t have to be fast.

      For non-additivity and non-linearity, see my old series of blog posts. Starts here: https://dynamicecology.wordpress.com/2012/07/30/how-disturbance-and-environmental-fluctuations-actually-affect-coexistence-part-1/ Scroll down to the comments for links to later posts in the series.

      • I get it now: firstly, although Hutchinson’s first assumption is technically true, it is redundant as exclusion happens regardless of the frequency of environmental fluctuation, and moreover, stating “equilibrium before the environment changes significantly” leads one to think that how often the change occurs does matter, when in reality it doesn’t. And secondly, his other assumptions are mostly invalid. I also guess that in his scenarios, he assumes equal competitive ability for both species?

        Thanks a lot for clarifying and for the link! I’ll read up on the non-additivity.

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