Posted by: Jeremy Fox | June 8, 2012

Ecology is mostly not like billiards (but lots of people think it is) (UPDATEDx3)

Billiards is all about sequences of causal events. Your cue strikes the cue ball, causing it to roll into another ball, causing that ball to roll into the corner pocket.

Falling dominoes are sequences of causal events. You knock over the first domino, which knocks over the second, which knocks over the third.

Rube Goldberg machines are sequences of causal events. The toy car is pushed into a line of dominoes, the last of which falls onto another toy car, which rolls down a ramp and runs into a ball, which rolls down another ramp…[skipping ahead]…which causes a piano to fall…[skipping some more]…which causes paintball guns to fire at a rock band.*

When humans think about causality, they find it natural to think in terms of sequences of events. That’s why colliding billiard balls are a paradigmatic example of causality in philosophy.

But ecology is mostly not like billiards, or falling dominoes, or Rube Goldberg machines. Like history, ecology is (mostly) notjust one damned thing after another.” But it’s hard not to think of it that way, and to teach our students not to think of it that way.

(UPDATE: I’m not saying that ecology, or dynamical systems in general, aren’t causal systems. They are! I’m just saying that the nature of that causality is such that it’s misleading to think about it as “Event A causes event B which causes event C which causes event D…”)

(UPDATE#2: Nor am I saying that ecological systems are “nonlinear” or “nonadditive”. They are, but that’s not my point here. For instance, you can have a sequence of causal events in which the magnitude of the effect is nonlinearly related to the magnitude of the cause. See the linked post from Nick Rowe, below, for further clarification. Sorry the original post wasn’t better, it’s clear that I did a lousy job of anticipating the ways in which readers might misunderstand what I’m trying to get at here).

Ecology is about dynamical systems. Stocks and flows, not falling dominoes. Inputs and outputs, not colliding billiard balls. Simultaneity, not sequences. Feedbacks, not one-way traffic.

Here’s an example. It’s a population ecology example, but not because population ecology is the only bit of ecology that’s about dynamical systems. It’s just a bit of ecology I know well. I could equally well have picked an example from physiological ecology (e.g., to do with individual growth), or from community ecology, or from ecosystem ecology, or from island biogeography, or conservation biology, or spatial ecology, or macroecology, or etc.

The example is predator-prey dynamics. You’ve got some prey that reproduce and die, and some of those deaths are due to predators. Predators convert consumed prey into new predators, and they die. Purely for the sake of simplicity (because it doesn’t affect my argument at all), let’s say it’s a closed, deterministic, well-mixed system with no population structure or evolution or anything like that, so we can describe the dynamics with just two coupled equations, one for prey dynamics and one for predator dynamics. And again for the sake of simplicity, let’s say it’s a constant environment and there’s no particular time at which organisms reproduce or die (e.g., there’s no “mating season”), so reproduction and mortality are always happening, albeit at per-capita and total rates that may vary over time as prey and predator abundances vary.

You cannot think about this dynamical system in terms of sequences of causal events. For instance, let’s say the system is at equilibrium, meaning that predator and prey abundances aren’t changing over time. That does not mean nothing’s happening! In fact, there’s a lot happening. At every instant in time, prey are being born, and prey are dying, and those two rates are precisely equal in magnitude but opposite in sign. And at every instant in time, predators are being born and predators are dying, and those two rates are precisely equal in magnitude but opposite in sign. Inputs and outputs are in balance. You cannot think about equilibria in terms of sequences of causal events, it’s like trying to think about smells in terms of their colors, or bricks in terms of their love of Mozart. What “sequence of events” keeps the system in equilibrium?

Or, let’s say the predators and prey exhibit cyclic dynamics. For concreteness, let’s say it’s a limit cycle in the Rosenzweig-MacArthur model. Why do the predators and prey cycle? This is a case where it’s sooo tempting to think in terms of sequences of events; I know because my undergrad students do it every year. “The prey go up, which causes the predators to go up, which causes the prey to crash, which causes the predators to crash.” In lecture, even I’ve been known to slip and fall back on talking this way, and when I do the students’ eyes light up because it “clicks” with them, they feel like they “get” it, they find it natural to think that way. And it’s wrong. Not “wrong in the details, but basically right”. Not “slightly wrong, but close enough.” Wrong. Births and deaths are happening instantly and continuously. There are no sequences of events here.

Now I can hear some of you saying, ok, that’s true of the math we use to describe the world, but it’s not literally true of the real world. In the real world one could in principle write down, in temporal order of occurrence, all the individual birth and death events in both species. But my point would still hold. A prey individual was born, which caused prey abundance to increase by one, which caused…what, exactly? What’s the next domino to fall in the sequence? Another prey birth? No. A prey death? No. A predator birth or death? No. What that increase in prey abundance did was slightly change the expected time until the next birth or death event, by increasing prey abundance and (in any reasonable model) feeding back to slightly change the per-capita probabilities per unit time of giving birth and dying. Now, you could try to drill down even further, down to the underlying physiological (or whatever) causes of individual births and deaths, and the underlying mechanisms linking per-capita birth and death probabilities to species’ abundances. But you’re never going to find something that lets you redescribe predator-prey dynamics in terms of sequences of events, each causing the next. (UPDATE #3: And to clarify further, no, I’m not trying to argue against the notion that population dynamics are ultimately a matter of individual organisms giving birth, dying, and moving around. I actually heartily believe that! My point is to do with how to interpret the causality of what’s going on, whatever level of organization (individuals or populations) we choose to focus on.)

Our deep-seated tendency to think in terms of causal sequences of events rather than in terms of rates of inputs and outputs (i.e. rates at which the amount of something increases or decreases) doesn’t just make it hard to teach ecology. I think it also makes it hard for professionals to do ecology. For instance, to preview a future post, much of the appeal and popularity of structural equation models (SEMs) that they let researchers take causal diagrams (variables connected by arrows indicating which ones causally affect which others) and turn them directly into fitted statistical models. That is, SEMs mesh with and reinforce our natural tendency to think about causality in terms of colliding billiard balls. Which I think makes them positively misleading in many circumstances (as I say, much more on SEMs in a future post).

This post was inspired by a post on the same topic by Nick Rowe. Nick’s post is about economics. His post is way better than mine. You should click through and read it (no training in economics required; stop when you get to the bit at the end about “concrete steppes”, which is where the post segues into technical economics issues).

*Click the link to see what I’m talking about. 😉

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Responses

  1. I think you could re-title all of this, “Ecology is mostly not additive or linear.” Which is definitely true. And to save you the trouble in the future (you knew I would jump on this, even on a Friday afternoon, right), neither must SEM’s be additive or linear.

    Just sayin’.

    Although, to throw my hat in the ring for a more object-based analogy, I’ll say that “Ecology is mostly like a lava lamp.” I’ll leave you to think about that one.

    (on read of Nick Rowe’s post, yes and yes.)

    • Nice try, but you’re not pre-empting the SEM post. 😉 My point is actually not that ecology is nonadditive and nonlinear (though it is). For instance, you can perfectly well have sequences of causal events in which the size of an effect is nonlinearly related to the size of the cause.

      Don’t worry, I’m not going to trash SEMs. I’m going to pose questions about them, not argue that they’re a Bad Thing. 😉

      “Ecology is mostly like a lava lamp.”

      So, ecology is mostly attractive to hippies? Ecology only seems cool and interesting if you’re stoned? If you drink ecology, you get really sick? You lost me. 😉

      “on read of Nick Rowe’s post, yes and yes”

      Hey, when I write something self-deprecating, I’m setting up my commenters to complement me! Not insult me by agreeing with me! 😉 Just kidding, I really do think Nick’s post is better than mine. Nick’s a great explainer, I’m not on his level.

      • So, ecology is mostly attractive to hippies? Ecology only seems cool and interesting if you’re stoned? If you drink ecology, you get really sick? You lost me.

        No, I was more thinking about the dynamics of what goes on in there – and agreeing with you. Seemingly simple system that can be described with some cause-effect kinds of equations, but, there’s a lot of stochasticity….ok, so, it was a strained analogy. And the more I write it out, the sillier it seems. Ah well.

        In the end, though, and after reading through this thread, the central point seems to be that we can describe a dynamical system all we want, but it’s still an estimate of the mean behavior of stochastic processes. One domino might make another fall. Or it might not. But the probability of that next domino falling is changed…only there’s a great deal more going on than just those two dominoes!

        Closer? Or still off the mark?

      • Closer, but not quite. It’s not really stochasticity vs. determinism that I was trying to get at (maybe I should do a post on that?), it really is the problem with thinking about any dynamical system, deterministic or stochastic, as being analogous to a series of falling dominoes. As you say, in a stochastic system one domino falling might change the probability that the next domino will fall. But not only that–you can’t even identify the next domino in the chain, either in advance or after the fact. And the same is true in a deterministic dynamical system. One domino falls, but you can’t point to the next domino in the chain, because what that “falling domino” (i.e. a change in one of the state variables) did was not knock down the next domino in the chain (i.e. cause some other change in some other state variable) but change the left side of the equation (i.e. change the rate of change in the state variable).

        I think part of the problem folks are having is that they don’t think of dynamical systems as like falling dominoes, and haven’t really run into anyone who does. So people are having trouble wrapping their minds around exactly what problematic way of thinking I’m trying to correct here. Which perhaps means that this post is attacking, if not a straw man, at least a fairly rare non-straw man, if that makes any sense.

        But I’ve done about as much as I can to try to clarify what I’m getting at. I’ll just say that I’m flattered you and others have taken the time to try to grapple with this post, and sorry if in the end it didn’t really turn out to be worth the time.

    • Shorter version of my previous comment: you’re saying SEMs can handle nonadditivity and nonlinearity in the usual senses (e.g., a nonlinear function is one that’s not a straight line). I’m questioning whether SEM can handle them when those concepts are defined in the sense that Nick defines them in his post.

      • SEMs aren’t generally set up to handle feedbacks, no (although they can be adapted to handle some forms of it). That’s not the same thing as saying that you can’t have accounts of causality that incorporate feedbacks; you just need to explicitly incorporate the order that events affect each other into a model (Hidden Markov Models are one technique for this that have seen a lot of use). Further, there’s nothing that says that causality has to be a purely deterministic; I would argue that “If we remove all the seedlings from this area, the deer here are less likely to successfully raise a new offspring” is a perfectly valid causal statement.

        Finally, say that ecological dynamics are a lousy example of a non-causal system; for any given ode model I know of in ecology, there’s an equivalent master equation you can derive that model from, where the rates in question indicate specific causal events, such as a predator meeting a prey and producing a second predator (see A.J. Black and A.J. McKane’s 2012 paper, “Stochastic formulation of ecological models and their applications”). I don’t see any reason why we can’t say, as a causal statement, “when prey levels are low, predators encounter fewer prey, and as a result, have fewer offspring”. When a population is at equilibrium, it’s because the causal processes pushing a system in one direction are balanced out rapidly by causal processes operating in the other direction. I think this is one case where using odes as a tool disguises the underlying processes going on, since it lets us ignore the fact that the equilibrium is maintained by many rapidly acting causal processes.

      • Ok, clearly my post wasn’t very good if both you and Jarrett didn’t get it.

        To clarify, I’m not saying that ecology, or dynamical systems, don’t involve causality! They do. It’s just that I think that thinking in terms of sequences of events, where each event causes the next in the chain, is the wrong way to think about that causality.

        I was about to try to offer an example, but then I realized I was about to simply repeat the example in the post. You already read the post, so repeating that example wouldn’t be helpful. Let me try to think of a different way of putting the point that hopefully will “click” with you. And do go read Nick’s post if you haven’t already. He makes the same point but in a different way that might “click”.

  2. Thanks Jeremy!

    Actually, I’m doubly glad you wrote this post, because an economist (one who, IIRC, does serious work on computer simulations of dynamic systems) totally didn’t get my point, and said so on another economics blog. So I told him you were a real scientist, who did real work on predator-prey dynamics, and that he should read your post to see if he could get the point when you explained it.

    • Thanks Nick!

      Given that the first two real scientists to comment on this post (Eric and Jarrett) didn’t really get my point (which is surely my fault), I doubt that your economist colleague will either. 😉

      • Damn!

        OK, try a really simple example. Tie 2 rubber bands together at one end, and pull them one metre apart with your fingers around the other ends. We need a theory to explain where the knot will be located.

        We have two equations, one for each rubber band, relating force to length. And an equilibrium condition which says that the 2 forces must be equal at the knot. And we can solve the 2 equations simultaneously for the 2 lengths.

        In that simple model, what causes what? Does the length of one band cause the force on that band, which causes the length of the second band, which causes the force on that band, which causes the length of the first band,….like a sequence of billiard balls hitting each other?

        No. The two lengths and two forces all cause each other at the same time.

        And a predator-prey model, even if there is change at the level of the individual, or even if there is change in the aggregate numbers, is fundamentally the same. It’s simultaneous causation.

      • Thanks Nick, that’s a good analogy (at least I think it is).

  3. By the way, I notice (I think) you use “dynamics” in a different way from the way economists use that term. (And I’m wondering if your use of the term isn’t better, because it’s less misleading.)

    Let’s take you first, very simple model, with a constant environment etc., so the equilibrium doesn’t change over time. Economists would call that a “static” model. And if you used that model to show that a different environment (notice I did not say a “change” in the environment), would result in a different equilibrium, economists would call that “comparative statics”, because you would be comparing two different static equilibria.

    But you call that a “dynamic” model. Is it because at the level of individual critters there is birth and death? Or is it because there is interaction between predator and prey numbers?

    Now take your second, more complicated model, where the predator-prey ratio is changing over time. We would call that a “dynamic” model, simply because the predator/prey ratio is changing over time. And we sometimes distinguish between “endogenous dynamics”, where you get (say) cycles in the predator/prey ratio even with a constant environment and no exogenous shocks, and the case where the environment does a step change, and it takes time for the predator/prey ratio to converge to the new static equilibrium (if it’s stable).

    To my mind, there is no fundamental difference between what we call a static and what we call a dynamic model. One is just where the solution (expressed in some way) happens to be a function of time. And that itself is arbitrary, because if we took your ‘static” model, and wrote down the solution for the cumulative total number of critters that had ever lived, that number would be a function of time.

    It has taken economists a long time to escape the idea that “equilibrium” means “at rest”, or “demand equals supply”. I translate “equilibrium” as “what the interactive model says will happen”.

    • Good lord, ecologists economists call a model with a stable equilibrium a “static” model?! No wonder they find it hard to teach this stuff–they’ve named equilibria after the most common *misunderstanding* of equilibria! 😉

      As you say, they’re both dynamical models. For instance, the model with a stable equilibrium also tells you how the system behaves when it’s not at equilibrium. It’s just that, as time goes to infinity, the model with a stable equilibrium happens to approach a solution that doesn’t vary over time.

  4. […] A interesting piece about ecology dealing with feedback loops and non-linearity. Something that Economists should take a greater note of. And journalists. (click here) […]

  5. Interesting posts, you guys (Jeremy and Nick). I have a different take on it, though. Firstly, populations are comprised of individuals. Sticking with the predator-prey example, given the number of predators and prey at any instant, and given the parameter values of the system, I think there is a corresponding distribution of the number of individuals that have experienced a sequence of the possible events in a particular order. Understanding this distribution is potentially usually as an explanatory tool, but it should be noted that this is a distribution, and a distribution that changes as the dynamics of the predator-prey system move through their trajectories as governed by the model.

    While it would lead to a cumbersome system of eqns, whatever predator-prey system Jeremy, you have in mind, could also be rewritten in a way that tracks the sequence of events that happen to an individual. For the prey equation you would need to define state variables such as ‘was born and then died of natural causes’; ‘gives birth once then died of natural causes’; ‘was born and then eaten by a predator’. Yes, there are an infinite number of possible sequences of events, but this could be well approximated using any ODE solver if you left off some of the more unlikely but theoretically possible sequences (‘gives birth 1000 times and then dies of natural causes’ and ‘never dies’, which is also possible but extremely unlikely).

    As I said earlier, it is important to note that the number of individuals that experience any particular sequence of the events is going to change and the dynamic system changes. When the number of predators is large, the number of prey that experience a sequence ending in predation is going to be large. Here, I think you could say that most prey in the system die before they reproduce and this is one way to understand why the prey popn subsequently declines. That being said, this is not true of every prey, and this is only a statement of the distribution of individuals among the possible sequences at that particular point in time.

    Thoughts? Am I missing the point here?

    • Thanks for your comments Amy. If you and Eric and Jarrett are all missing my point, then clearly my post was crap.

      Yes, you can describe sequences of events happening to individual organisms, and to the entire population. But you can’t link those up into causal chains. For instance, consider your example of a prey individual that gives birth once, then gives birth again, then dies. Did giving birth the first time *cause* it to give birth the second time, which then *caused* it to die? Or at the population level, if individual A gives birth, and then the next event is that individual B dies, did the first event cause the second?

      • Thanks. Yes, I agree that the you don’t want to say anything about what caused this and that. I completely agree and so I didn’t have much to say on that point.

        I thought maybe you and Nick were coming down fairly hard on trying to understand the population dynamics by trying to think about what happens to individuals. I think there is some value in trying to explain it this way.

        I wouldn’t say the post is crap! I learned something here which is to distinguish between causation and re-interpretations of understanding.

      • Population dynamics is ultimately all about what happens to individuals! Good gravy, if I gave the opposite impression than this post really was rubbish (and you’re very kind not to say so)!

      • I meant to say ‘reinterpretations for understanding’. Sounds like us commenters are causing you a headache today!

    • (I am worse at math than most economists).

      Take the limiting case of Amy’s model, as the individuals get smaller and smaller, and more and more numerous, so we are talking about masses of predators and prey, rather than numbers of predators and prey. Then we can no longer talk about discrete events that either happen or do not happen, probabilistically. Call that model the “smooth” version of the model. In the smooth version, we can’t really talk about individual events. We have smooth deterministic “supply” and “demand” curves, that cross at the equilibrium point, and jointly determine that equilibrium point.

      To my way of thinking, Amy’s model imposes integer constraints on that smooth model, and also adds uncertainty. But if you talked about the *expected* equilibrium predator/prey ratio, the answer should be approximately the same (it presumably won’t be exactly the same unless both curves are straight lines).

  6. Sorry about the proof-reading: ‘potentially usually’-> ‘potentially useful’ and ‘and the dynamic system’ -> ‘as the dynamic system’.

  7. Ok everybody, maybe it would help to get away from debating the appropriate causal description of individual birth and death events that, in aggregate, underpin population dynamics.

    Go back to the simpler, population-level example of predator-prey cycles. If an undergrad says, “Predator-prey cycles happen because prey go up, causing predators to go up, causing prey to decline, causing predators to decline,” how would you correct that (and if you don’t think it needs any correction I *really* mucked up this post!)? Or you’re studying competitive exclusion, say in a Lotka-Volterra model, and the student says “Exclusion happens because species A increases, causing species B to decrease, when then causes species A to increase further, which then causes species B to decrease further, etc.”

    I’m arguing that this way of verbally describing what’s going on is misleading. It’s based on a faulty analogy between the dynamical system, and things like falling dominoes. And I’m arguing that the way to correct that mistake is not to (for instance) get into the derivation of deterministic dynamics as an approximation to some underlying stochastic model. The root cause of the student’s mistake is not a misunderstanding of how the underlying stochastic, event-based model works. Plus I don’t think it’s useful to think of even the underlying stochastic event-based model as analogous to sequences of falling dominoes.

    I don’t know, maybe it’s only me who constantly encounters students (and non-students) who want to think about dynamical systems as being just like falling dominoes. Maybe part of the problem is that this post just doesn’t resonate with people who understand dynamical systems perfectly well themselves, and who don’t often encounter people who don’t?

    Ok, one final try to make clear the mistake I’m trying to get at: Dynamical systems models are often called “bucket models”, so let’s try this: think of a literal bucket model, a bucket of water into which water is being pumped in at some rate, and flowing out through a hole in the bucket at the same rate. Yes, there are causes here–the inflow and outflow combine to cause the amount of water in the bucket to remain constant. Or, if, say, the outflow exceeds the inflow, that causes the amount of water in the bucket to decline over time. Whatever. Here’s my question: do you think it’s at all helpful to describe what’s happening in our literal bucket model in terms of sequences of events in which each event in the sequences causes the next event to happen, in the same way that one domino’s fall causes the next domino to fall?

    If that doesn’t help, then I fold; I’ll just concede that I can’t make clear what I’m trying to get at.

  8. Your bucket model works for me. And to answer your question, no I don’t think it’s helpful to describe the bucket model in terms of a sequence of events.

    I would draw a graph with stock of water on one axis, flows of water on the other, one curve showing flow in and a second curve showing flow out, point to where the “supply” and “demand” curves cross, and say that the stocks and flows are jointly determined by those two curves. It’s the two curves, and not any pair of points on those curves, that determine/cause what happens.

    It’s supply and demand (the curves), that determine both price and quantity (stock and flow).

    • You economists do love your isoclines, don’t you? (That’s what ecologists call them) I actually love them too, primarily as a teaching tool. But whenever I’m trying to think through a problem, my first instinct isn’t usually “draw isoclines”, whereas it seems like that is the first instinct of a certain sort of economist. This isn’t a criticism at all, just an observation.

  9. I see your point, Jeremy, I just think I disagree with you. I for one would consider it perfectly valid to say “the deer population increases, causing the wolves to go up, causing the deer to go down”. Even though its changes in propensities to occur, doesn’t mean its not a causal chain of events. Let’s look at the discrete version of the same system; in that case, it’s very easy to say that the drop in deer from time t-1 to t will cause a drop in the wolves’ population from time t to t+1. Why should I change how I describe it when switching to continuous systems? You only need ask: what happens if we fix the deer or the wolf population at a given level (or a given time course)? Imagine we find out, for a given population cycle, that instead of births and deaths accounting for changes in a deer population, a game warden has been adding extra deer or taking them away so as to keep the deer cycle going in an aesthetically pleasing way; however, they say they never bothered trying to alter the wolf population, they just cycled on their own. Would you be comfortable in saying then that the rise and fall in the deer population was causing the rise and fall in the wolves?

    Or to take the dripping bucket analogy: imagine instead of steady flows of water, we have a stream of drips in the inlet, and a stream of drips from the outlet. The surface of the water will fluctuate up and down, and each fluctuation we can attribute to an individual drip. As we speed up the rate of the drips, it becomes harder and harder to tell which drip is causing the fluctuation (and the amount of fluctuation will go down), but that doesn’t make the original causal story wrong. I don’t see why we should reject causal stories just because we’ve reduced the rate of change to an infinitesimally small amount.

    • Even in discrete time, I don’t think that’s the best way to think about things Eric, sorry. We’ll have to agree to disagree.

      Having said that, I don’t think it’s an especially serious disagreement in the grand scheme of things. It’s not as if I can’t see any point at all to your way of looking at things, and from the sound of it it’s not as if you can’t see any point at all to my way of looking at things.

      Now, if you want to start arguing in favor of the IDH, then I’ll have to fight you. 😉

    • Eric: “Would you be comfortable in saying then that the rise and fall in the deer population was causing the rise and fall in the wolves?”

      I wouldn’t be comfortable saying that. Because the deer population is not an exogenous variable.

      Instead, I would say (something like) “The game warden is causing the deer supply *curve/function* to shift up and down, and it is those up and down shifts in the deer supply curve/function that is causing the rise and fall in the number of wolves.”

      • I don’t actually see why endogenity is a factor here. Simply because a is caused by b, it doesn’t mean that b can’t in turn cause c. In that case, I think both statements “a causes c” and “b causes c” are true; it’s just that b is the mechanism that allows a to cause c. Going back to the domino case, all the dominoes except the first one are endogenous; however, if we remove one domino from the middle of the chain, it prevents the others from falling. How, the can we say that its presence didn’t “cause” the subsequent dominoes to fall? I think every formal definition of causality I know of (Pearl’s do-calculus, Granger, Rubin’s potential outcomes) imply that intermediate variables in a chain are still causal.

        Cosma Shalizi has a good discussion of this in the notes for one of his courses: http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch22.pdf

  10. I agree with much of this post, but not the predator-prey cycle example. You wrote, “This is a case where it’s sooo tempting to think in terms of sequences of events; I know because my undergrad students do it every year. “The prey go up, which causes the predators to go up, which causes the prey to crash, which causes the predators to crash.” … And it’s wrong. Not “wrong in the details, but basically right”. Not “slightly wrong, but close enough.” Wrong. Births and deaths are happening instantly and continuously. There are no sequences of events here.”

    You’re conflating two types of “events” here. There are the micro-events of births and deaths, which occur continuously. I agree with you there. But then there are also the macro-events of (prey high, predator rising), (prey declining, predator high), (prey low, predator falling) and (prey rising, predators low). When speaking in terms of these events, it’s absolutely correct to say, “The prey go up, which causes the predators to go up”.

    • Woohoo, somebody who mostly agrees with me! 😉

      Re: individual vs. population-level “events”, I see what you’re getting at, but I don’t agree. At the population level, predator and prey densities are both changing simultaneously. You can’t point to, say, a particular increase in prey density and say which particular increase in predator density it caused.

      • But that’s true of any continuous process. Think of the regulation of blood sugar by insulin. You can’t point to any particular decline in glucose levels and say it was caused by a given rise in insulin, but it’s still true that insulin lowers glucose levels (and high glucose levels raise insulin).

  11. […] a confusing blog post? Fighty crab tells […]

  12. I thoroughly enjoyed this post Jeremy (as I have enjoyed many previous ones as well!), and think many would benefit from reading it and the following comments. It’s been rewarding for me. While I must admit that I’m a bit torn between the two different views presented by yourself and Eric Pedersen, I lean towards how I originally thought about these things (e.g. like Eric’s view on the dripping bucket analogy). But the post has got me thinking about these issues again. Thanks!

  13. I recall a much earlier post where you commented that if we readers were thinking in terms of biomass instead of growth rates, we’d better think again. Changed the way I think of ecology.

    • Yes, in my original “zombie ideas” post.

      This is something I think every commenter on this post could get behind! To think about a dynamic, changeable world, you need to think in terms of rates (expected rates and their associated statistical moments, in a stochastic world), not in terms of amounts.

  14. […] designed elements of our world and how they are affecting me. After a recent discussion on the Oikos blog, I have been thinking about growth rates. So now, when I walk by a field or by the ocean, I’m […]


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