In the comments on a previous post, I got into a discussion about the reasons why one might study short-term (i.e. transient) dynamics as opposed to long-term (i.e. asymptotic) dynamics. For instance, one might study a system’s short-term response to a single disturbance event, as opposed to the system’s long-term dynamics under repeated disturbances. I decided to expand my comments and put them up as a separate post.

There are many reasons why one might study short-term dynamics—and some of the most commonly-stated reasons are pretty weak. Further, even people who have good reason to study short-term dynamics often do so in weak ways. So if you’re interested in short-term dynamics, you may wish to consider the following:

1. Some ecologists who study short-term dynamics mistakenly think they’re studying long-term dynamics. Now, “long-term” does not have a universally-applicable and precise definition. But it does *not* mean “longer than one year” (at least not for most organisms ecologists study), “longer than the duration of a typical grant”, “as long or longer than the typical study in my field”, “long enough for me to see a statistically-significant treatment effect”, or even “longer than a human lifetime”. What’s “long-term” is determined by the question you’re asking and the ecology of your study system. In particular, if you’re testing some theoretical prediction about the asymptotic behavior of a system, you had better either study your system long enough to be confident that you’ve observed asymptotic behavior (i.e., the means, variances, and other relevant system features are no longer changing), or else find some way to reliably infer your system’s asymptotic behavior from shorter-term data. Unfortunately, this is a difficult standard to meet because transients can be very long-lasting, as Alan Hastings often points out. But hey, if ecology was easy it would be boring.

2. If you really are interested in short-term rather than long-term dynamics, please don’t use models that are about long-term dynamics to guide your work. Obeying this dictum means you’ll probably have to develop your own models, because pretty much every well-known and widely-studied model on every topic in ecology (including, for instance, both zombie and non-zombie ideas about disturbance) is about long-term outcomes, not transients. Theoreticians study asymptotic behavior almost exclusively, because it’s much easier to study mathematically.

3. If you think (or hope!) that it’s ok to study transients because they will look sort of like the long-term outcome, I’m sorry, but they typically won’t. Transients are infamously idiosyncratic and highly sensitive to initial conditions (i.e. the initial state the system happens to be in, independent of whatever drives system dynamics). That’s the other reason theoreticians generally don’t bother to study transients—they’re totally wonky (the transients, not the theoreticians) (ok, maybe both). There’s a vast array of theoretical and empirical studies, in all areas of ecology, which have made precisely this point. Just off the top of my head, besides Alan Hastings’ work, see Briggs and Borer (2005), Neubert and Caswell’s (1997) work on ‘reactivity’ (even stable systems can temporarily move *further* from equilibrium after being perturbed before eventually returning), my 2007 Oikos paper on the transient dynamics of trophic cascades, Jim Brown’s classic 1980’s work on the contrasting transient and long-term effects of competing seed predators on one another, and many, many other papers. Unfortunately, I know of no rules of thumb on when transient dynamics will resemble long-term outcomes and when they won’t, and I doubt that any such rules exist (or that they’re very generally applicable if they do exist).

4. Ecologists who conduct short-term studies often justify them by saying “Ideally, I’d like to conduct a long-term study, but it’s only feasible to conduct a short-term study.” Which is sometimes defensible, but often is not—not if what’s “feasible” reflects your own previous choices. If you would really like to study long-term outcomes, but transient dynamics are all you can measure in your system, why don’t you change study systems? That’s why my own doctoral supervisor did—Peter Morin switched from mesocosms of amphibians to microcosms of protists in order to do long-term community ecology. Alternatively, why don’t you stick to asking questions you can actually answer with short-term data? Or, why don’t you find a *rigorous* way to infer something about long-term outcomes from short-term data? (Note that “Collect short-term data, compare to theoretically-predicted long-term outcome” is *not* a rigorous thing to do). For instance, if you’re interested in long-term coexistence, you can do short-term mutual invasibility experiments (i.e. experiments that just test whether rare competitors tend to “bounce back”, which is a short-term question), use them to parameterize dynamical models of your system, and then use the models to project the long-term outcome expected from your short-term data. Folks like Jon Levine have been doing exactly that. That approach isn’t foolproof (e.g., if you’re parameterizing a bad model, the long-term projections will be wrong). But no approach is foolproof, and the potential drawbacks of that particular approach are knowable and checkable in various ways (e.g., you can compare the projected long-term outcome of competition to observed species abundances, or long-term time series data). So if you say “I’d like to study long-term outcomes, but it’s only feasible to measure transients”, you’d better have a good argument why you’re not doing the equivalent of tying your own shoelaces together and then complaining that it’s only feasible to walk slowly. The handicap principle should only apply to our study organisms, not to our studies of them.

5. Ecologists who study short-term dynamics often admit that these dynamics are likely to be different than long-term dynamics, but argue that short timescales are still “ecologically-relevant”. Sometimes this is a reasonable argument; it’s one I’ve made myself on occasion (e.g., “These competitors have coexisted for over 100 generations, which is not forever but is long enough to demand an explanation, making 100 generations an ecologically-relevant timescale”). Short timescales also can be “relevant” for other reasons, such as legal ones (e.g., “The law obliges management decisions to be made on this timescale”) But in my experience, talking about “ecologically-relevant” timescales often is shorthand for “I don’t have long-term data, but I want to publish the data I do have, so I’ll call them data about ecologically-relevant timescales.” That is, “ecologically-relevant timescale” often is shorthand for “a timescale long enough for something or other to happen”. Which is a pretty low bar, given that *something* happens on every timescale.

My inner relativist has a problem agreeing with you on this point. I would argue that any timescale an experiment is conducted on is valid for making inferences at that time scale. I believe the thrust of your argument is that “long time scales are good because they allow accurate extrapolations beyond the time scale at which a study was conducted”. So lets begin with my interpretation of why you say short term dynamics aren’t really interesting. Imagine a mesocosm study lasting one year where you apply some sort of disturbance and see declines in frog population vital rates. You say: “Ahh, disturbance x causes frog declines”. But lets say if you’d carried out the study for 5 years, well there were declines initially but the frogs actually survived and increased their abundance. The argument here is that you’ve extrapolated beyond the time scale of on year on your initial study and made valid but unsound (in rules of logic) conclusion. We are structuring our arguments as follows.

1. Frog populations decline after 1 year of disturbance x

2. Effects within one year have an additive effect over multiple years (i.e. asymptotic)

3. Therefore frog populations will disappear after y years due to disturbance x.

Using a short term study we’ve constructed a valid argument, but our 2nd proposition is false so our argument is unsound. This is your point of course, that with a long term study you can construct both a sound and valid argument because proposition 2 is sound.

The problem though is that no matter what you’re assuming asymptotic dynamics. Let me put forth another example. Northeastern forests are fundamentally different today than 500 years ago (http://www.personal.psu.edu/agl/The%20Red%20Maple.pdf). So imagine a pre-european native ecologists performs a 50 year study of tree demography in New England. I think we might all agree that would qualify as a “long term” study. They would say “Red maples will always be a minor component of the species distribution of New England forests”. Of course fast forward 500 years, and we would say that study was wrong. Why? Because supposition 2 was unsound. You say: “What’s “long-term” is determined by the question you’re asking and the ecology of your study system. In particular, if you’re testing some theoretical prediction about the asymptotic behavior of a system, you had better either study your system long enough to be confident that you’ve observed asymptotic behavior (i.e., the means, variances, and other relevant system features are no longer changing)…” I believe you’re still thinking of “long term” on a human time scale. In my hypothetical study, the means and variances of the abundance of red maple wouldn’t have been changing after 50 years of study. And by that definition there has been a sound long term study, but when we extend the time scale out to something beyond what ecologists think about (500 years), we see that the study was unsound.

It might be at this point where you think that I’m suggesting that there is no such thing as asymptotic dynamics but I’m not. Here’s where the relativist in me comes in. I think asymptotic dynamics exist, but its impossible to know them over an infinite time scale. Its a paradox really. By the time you’ve established that a systems dynamics are asymptotic there is no longer any point in carrying out the study because we live in a stochastic world where external disturbances could always cause a shift in dynamics so you constantly need to be measuring a system to ensure it is truly asymptotic. Let me return to my native ecologists. Who back then could have imagined that a race of technologically advanced, white skinned and weirdly dressed people would come from thousands of miles away, decimate your society and radically change the ecology of your land? If you were alive to study dinosaurs, who would have predicted a giant asteroid would crash into earth? I don’t mean to drift into the absurd and I’m choosing large scale events as an example, but these kinds of external disturbances to systems can at varying scales can totally disrupt asymptotic dynamics. No disrespect to Alan Hastings or any of the other mathematically minded and brilliant theoreticians out there, but their models are based on assumptions that can also get violated by a stochastic world. So here are my final conclusions.

We live in a stochastic world and given an infinite time scale its hard to imagine any ecological system is truly asymptotic. So asymptotic dynamics are possible but need to be constantly measured to truly ensure they are asymptotic. I think in the end we just need to be very careful regarding the temporal extent to which we extrapolate from our results. I understand why short term dynamics might not seem interesting to us as human beings because if we can only make predictions over one year because it means nothing to us. If you present the results from a 20 year study, this makes sense as “long term” because its long term for our own life spans. But to say the results from a 20 year study are asymptotic dynamics just because the means and variances don’t change over that 20 years is a bit silly, because relative to a time scale of 500 years or 1,000 years its nothing. I realize this is a bit absurd as well, because really who cares about what the world will look like in 1,000 years? Lets just be honest that time-scales are all relative and what we’re really interested is in making inferences about systems on a time scale that we the investigator care about.

Hi Ted,

I don’t think you and I are actually all that far apart (and I certainly don’t think you’re a “relativist” in any meaningful sense!). I’m perfectly happy to admit that there may not be any such thing as truly asymptotic dynamics in the strict sense. My choice of example (of someone testing asymptotic theory by naively extrapolating short-term data) was merely intended to be a particularly clear-cut example of an inappropriate short-term study. And many quite prominent researchers have made precisely that mistake, as for instance the Briggs and Borer article points out. Many other examples of mismatches between the timescale over which one

wantsto make inferences, and the timescale over which one cancanmake valid inferences, could not doubt be provided.I don’t think the validity of my argument actually hinges at all on whether the real world exhibits asymptotic dynamics. Short-term transients typically won’t look like longer-term dynamics, even if those longer-term dynamics aren’t asymptotic. Further, I think you’re incorrect to downplay the utility of mathematical models for helping us think about a world in which there are no asymptotic dynamics because of exogenous variation on all timescales. For instance, you’re forgetting about mathematical techniques like separation of timescales. Even if asteroid impacts (or whatever) prevent the world from ever reaching asymptotic dynamics, one can still study the asymptotic behavior of those bits of the world that are fast relative to the frequency of asteroid impacts.

For me, the distinction between short-term vs. long-term time-scales, as perhaps synonyms of transient vs. asymptotic dynamics, is not terribly useful. Moreover, it can be misleading. I do agree with Ed Hart on most of the things he says, and I would add a few more ideas.

Regarding time-scales and asymptotes: We tend to say that “we live in a stochastic world”. But, as I see it, in a strict sense this is not true (I hope being strict here can be helpful). We do live in an “ontologically deterministic world”, but, as scientist, all we can do is “interact” with it through an “epistemologically stochastic world”. I mean, by no means “stochastic” should be taken as a synonym of “random”, and nothing in nature is truly random (ok, some say that things going on at the quantum scale are truly random, but this does not seem to be supported by all physics). The point I want to make is that stochasticity enters our models (whether mathematical or not) simply as a term full of “unknown things/forcings”. And these things can affect the “deterministic skeleton” additively or interactively. Just as an example, it is now well known that an underlying non-linear skeleton of marine fish dynamics can amplify stochastic noise to produce apparently random behavior. If noise is no longer a passive forcing, which is the “expected” asymptotic behavior here? Well, I think the answer is we don’t know. And we’ll never know as long as these “interactive stochastic forces” remain stochastic in our model. Moreover, given that these stochastic forces have their own skeleton which can be non-linear as well, talking about asymptotes no longer makes sense. And if we further add the possibility of intermittent linear and non-linear dynamics (e.g., Hsieh & Ohman 2006, Ecology 87:1932) we end up with a mess. And as another example I recall a paper from Alan Hastings (1993 Ecology 74:1362) showing that, in some spatially-explicit ecological models several attractors can coexist in the parameter space, and the boundary separating these attractors is a fractal; the point here is that we cannot predict the asymptotic dynamics even knowing exactly the initial conditions. It is not a matter of data; we will never know it, and that’s it. There is now a great deal of models showing such “architecture” in many ecology areas.

Regarding transients: As you say, Alan Hastings (again) and other people have shown that transients are not rare in many ecological models. Moreover, they have shown that they can not only last long, say, 50 time steps: they can be extremely (from our human perspective) long: thousands of years. That is, it can take thousands of generations before the “system”settles into its attractor. And what is missing in most ecological models? Yes, evolution. During this long-lasting wandering of the system, it is extremely likely that those biological traits linked to the key parameters defining the model dynamics may undergone evolution (whether adaptive or not). I’m referring specifically to the intrinsic rate of increase, the dispersal probability and so on. So, what do we have after all? We have a model that tells us that, given these parameter values, the expected asymptotic dynamics is, e.g., a limit cycle. Oh, and by the way, all this after 3000 time-steps. Is this really helpful for us ecologist, dealing with a world were change is the only expected outcome? In the future, the emerging field of eco-evolutionary dynamics can provide us with a nice answer, I guess.

I’m not saying that mathematical models are not useful (to say that, is to say that what I do is a crap, which, by the way I use to think from time to time). I am saying that we must recover their original logic (Richard Levins said so many insightful things on this…). At the end, we are left alone to model an ecological system, with possibly non-linear dynamics, amplifying a stochastic noise that in reality is everything but random and may have possibly non-linear dynamics as well, and we further learn that supertransients are sure if our family of models are on the right track, and we know that the phenotype distribution within our population is most likely going to change through time. In this scenario, I argue that the asymptote, as a nice mathematical device, perhaps is not the first thing we must look at. But I confess that I personally still don’t know which is the first think we should look at in general, or if any generality (regularity) might even emerge. Of course, there are a lot of subtleties here, but I agree that all we can do is to be explicit about our time-scale of study, about what can we honestly infer from it, and along this process ask whether, in our study system, the asymptotic behavior is not only mathematically interesting but also biologically relevant.

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