One way among many others by which a theoretician might develop a mathematical model of one scenario is by analogy with some other scenario that we already know how to model.
The effectiveness of this approach depends in part on how loose the analogy is. At the risk of shameless self-promotion, I’ll highlight a physical analogy that my own work draws on (the analogy isn’t originally mine): dispersal synchronizes spatially-separated predator-prey cycles for the same reason that physical coupling synchronizes physical oscillators. Here’s a standard, and very cool, demonstration involving metronomes sitting on a rolling platform. The analogy between the ecological system and the physical system is actually fairly close, though for reasons that might not be immediately apparent (how come coupling via dispersal works like coupling via a rolling platform?) The closeness of the analogy is why it works so well (Vasseur and Fox 2009, Fox et al. 2011, Noble et al. 2015, and see Strogatz and Stewart 1993 for a non-technical review of coupled oscillators in physics, chemistry, and biology).
But it’s more common for physical analogies in ecology to be quite loose, justified only by verbal argument. Hence my question (and is is an honest question, not a rhetorical one): can you think of any examples in ecology in which models based on loose physical analogies have worked, for any purpose? Sharpening of intuition, quantitative prediction, generation of hypotheses that are useful to test empirically, etc.? Because I can’t.
Here’s a familiar example of an ecological model based on a loose physical analogy: MacArthur’s broken stick model for the species-abundance distribution, and its variants. Whatever it is that determines the relative abundances of species in ecological communities is analogous to randomly breaking a stick into pieces according to some algorithmic procedure. This analogy has some superficial plausibility. In some loose sense you can think of the species comprising a community as “dividing up” all the available “resources”. But it turns out the analogy is just too loose to be useful. Not only does the resulting model not fit the observed data very well (either in an absolute sense, or compared to other candidate models), the analogy on which it’s based is so loose as to make the model difficult or impossible to learn from. For instance, you can’t really check its assumptions in addition to checking how well it fits observed species-abundance distributions, because its assumptions can’t be operationalized into quantities you can measure in real communities. You can’t check its predictions about anything besides the shape of the species-abundance distribution, because it doesn’t predict anything else. The stick-breaking analogy lends itself to only a very limited range of extensions or modifications. Finally, whether or not the model fits the data, what do you learn from that fit (or lack thereof)? That whatever it is that determines the species-abundance distribution has (or doesn’t have) the same effects as a stick-breaking algorithm. Which seems…unhelpful. At least to me.
Contrast a more successful example of modeling via physical analogy in biology (though in genetics, not ecology): the Haldane mapping function describing the relationship between recombination frequency and map distance on chromosomes. Wimsatt (1987) has a wonderful discussion of the Haldane mapping function and other (much less successful) physical models of chromosomes, from which my discussion is cribbed. Haldane actually considered a series of physical analogues to chromosomes: an “infinitely flexible” string, a “rigid” chromosome like a metal rod, and an intermediate case of a stiff but somewhat flexible rod. Haldane’s mapping function was successful in capturing some but not all aspects of the empirical data on frequency of crossing-over, and its successes and failures provided fruitful guidance for further theoretical and empirical work (e.g., why is it, biologically, that chromosomes work more like somewhat-flexible rods than like infinitely-flexible strings?) Haldane’s physical analogy was successful because it wasn’t all that loose. By which I mean that Haldane’s physical analogy was sufficiently biologically interpretable that one could learn from its failures (see this old post for much more on how we learn from false models). In contrast, other physical analogical models of chromosomes lacked any biological interpretation, so when they turned out not to fit the data the only thing that was learned was “welp, guess that was a bad analogy”.*
I thought of this because of a new example of an ecological model based on a loose physical analogy: Colwell et al. 2016. Colwell et al. model the distribution of species’ geographic ranges in a 1-dimensional habitat with hard boundaries by analogy with pencils (=species’ ranges) laying parallel to one another in a pencil box (=1-D habitat with hard boundaries). The pencils vary in length (=some species have big ranges, others have small ranges). Each pencil also has a steel ball embedded in its midpoint that tends to attract it to a magnetic field of some specified strength, centered at a random point within the box (=ranges have some tendency to cluster together because of species’ similar responses to some sort of environmental gradient). I confess that this physical analogy is too loose for my taste. Something like the model of Connolly (2005), or that of Rangel & Diniz-Filho (2005), strikes me as a much more promising starting point for research on this topic. Those models explicitly describe real biology and it’s easy to imagine how they could be modified to incorporate additional real biology.** But time will tell. As we’ve discussed before the fruitfulness of any given research approach depends a lot on the skill and creativity of the investigator applying it. So perhaps future work will refute my initial skepticism (wouldn’t be the first time…). In any case, I’m not trying to pick on Colwell et al.; it just so happens that their paper was the one that got me thinking about the broader issue here.
My argument here is not that models should always be “mechanistic”, or even that they should always be at least a bit mechanistic. Good phenomenological models that implicitly summarize lots of unspecified underlying biology in a few “high-level” parameters definitely have their place (e.g., this and this). I’m just reflexively suspicious of trying to have your cake and it too. I don’t like trying to give a phenomenological model a thin veneer of mechanisticness*** by motivating it with a loose physical analogy. I think it leads to bad phenomenological models that are difficult to learn from, because the parameters of the physical model aren’t necessarily going to map onto the underlying biology in an interpretable way.
But maybe I’m wrong. Wouldn’t be the first time! Can you think of any successful ecological models based on loose physical analogies?
*There’s a connection here to critiques of “rational reconstructions” in philosophy of science. For instance, a philosopher of science who takes Bayesian reasoning as the sine qua non of rationality might explain how, say, geologists reasoning their way to the theory of plate tectonics could have done so via Bayesian reasoning. And if that’s not how the geologists themselves actually reasoned, well, that’s ok, because they behaved as if they were using Bayesian arguments even if they didn’t actually do so. The hypothetical Bayesian reasoning thus serves as a “rational reconstruction” of (and thus a justification of) their actual line of reasoning. Deborah Mayo criticizes such rational reconstructions, comparing them to a paint by numbers version of the Mona Lisa. The fact that you can come up with a paint by numbers version of the Mona Lisa doesn’t give you any insight into why the techniques actually used to paint it are good ways to paint. Saying that Leonardo Da Vinci painted “as if” he was filling in a paint by numbers of the Mona Lisa is just completely unhelpful. Analogously (ha ha), a loose physical analogy to some ecological system doesn’t necessarily teach you anything even if it fits the ecological data.
**That both models describe very simple biology–surely too simple to fit real-world data with any quantitative accuracy–is totally fine. The point here is just that both models explicitly describe (simple) biology, rather than trying to implicitly summarize (simple or complicated) biology via a loose physical analogy. Of course, any mathematical model of anything is in part an implicit summary of unspecified, lower-level underlying mechanisms (well, except quantum mechanics). But at least if you’re explicitly describing higher-level biology, you’re implicitly summarizing lower-level biology. In contrast, if your model is an implicit summary of all of the biology in it, it’s often going to be difficult to interpret biologically.
***I apologize for making you read that horrible word I just invented.
Interesting post, Jeremy! But I guess I’ll be the guy who points out that you really haven’t defined “loose” (other than quite loosely, if I can be a bit meta). I would worry that this becomes circular, if we see a model not working well as evidence that the analogy was too loose. I don’t think you’re doing that in the case of the broken stick, mind you; intuitively, I agree with you that that’s “loose”. And I don’t think I can do any better!
Yes, this post would be better if I had a definition of “loose” that wasn’t itself too loose to be useful.
Rather than looseness, how about this: a physical analogy is unhelpful/bad if its parameters lack any “natural” mapping onto the unspecified underlying biology. So for instance, models of coupled oscillators work in ecology because they map neatly onto a natural-seeming way to “carve up” the ecological world. The oscillators are the cycling populations, the coupling is dispersal among them.
Whereas in the mid-domain effect model of Gotelli et al., you’re dividing up the unspecified underlying biology into (i) stuff that determines the sizes of species’ ranges, and (ii) stuff that determines where those ranges are located within the bounded domain. Which seems like an “unnatural” division, insofar as (i) and (ii) involve the same stuff.
Yes, I think that’s better and it maps pretty well onto my intuition about both the oscillators and the broken stick. (I’ll reserve judgement on the pencils). I mean, it’s still a bit loose, but good enough for me 🙂
Just remembered that I made this same argument about “natural” vs. “unnatural” mappings using different words in an old post: https://dynamicecology.wordpress.com/2012/08/30/why-do-our-null-models-nullify-some-effects-and-not-others/
I guess one needs to distinguish between an analogous (lateral) transfer of scientific knowledge from one science to another and a loose analogy with something that is not scientific knowledge. Though the example is not physical, game theory was a theory (model) built for the economic behaviour of humans and has then successfully been transferred to animal behaviour. Likewise, your example, Jeremy, seems to involve a horizontal transfer of scientific knowledge about physical oscillators. I do not see how likening species to broken sticks or pencils can be described as horizontal knowledge transfer from one science to another.
How about those diagrams used to demonstrate fitness landscapes and alternative stable states? I.e., equating forces required to propel the ball into a new domain of attraction with mutations or perturbations. Both (the former, especially) seem to have inspired some pretty successful research and have even been quantified.
Interesting suggestion. Let me think for six second, as if that will substantially improve my response…
*Fitness* landscapes are a special sort of landscape. The whole “landscape” metaphor runs into some subtle technical problems when it’s a fitness landscape. Especially if fitnesses are frequency-dependent.
The analogy between stability analysis and balls (=state variables) rolling around on a landscape is a good one as far as I know. It’s main limitation is that there are some dynamical phenomena that don’t have any obvious physical analogue to balls rolling around landscapes. How do you talk about demographic stochasticity in the context of a “balls rolling on a landscape” analogy? (Environmental stochasticity is actually pretty doable…)
What I don’t know is whether those landscape analogies have ever been used as anything other than pedagogical tools. It seems to me that there’s an important difference between using a physical analogy to explain a mathematical model to students, and basing your model on a physical analogy.
In one of his books, Murray Gell-Mann argued that fitness landscapes should be turned upside down, organisms (intuitively aided by an equivalent of gravity) falling down into fitness hikes which are then hard to climb out of and escape.
On the other hand, while oscillators (including coupled oscillators) have been an immensely useful metaphor in my field, I have previously warned against falling for it too firmly: https://blogs.scientificamerican.com/a-blog-around-the-clock/the-clock-metaphor/
Autocorrect: fitness holes, not hikes.
Or maybe analogies are more often used as a simple communication/pedagogical tool, to give the reader has an instantly recognizable mental model with movable parts to play around with, and therefore help evaluate the ecology model. Then the analogy itself shouldn’t be held responsible for a bad ecological model. But I guess idea generation and idea communication are too tangled to figure out when and why researchers bring in analogies.
I loved learning the analogy between Nicholson-Bailey parasitoid-prey models and collisions between randomly moving gas particles. It taught 1st-year undergrad TH that ecologists think about animal life as equation variables, and aren’t always Jane Goodalling in the forest.
Good point that a physics analogy (randomly-colliding particles) underpins every dynamical model that assumes a well-mixed system.
Re: analogies as a pedagogical tool, see my reply to Dave Armitage. I do think there’s a difference between analogies as pedagogical tool vs. analogy as basis for a model, but I agree it can be a fuzzy difference.
It is perhaps worth pointing out that the “loose analogy”of MacArthur’s broken stick was always from the first paper recognized as a loose analogy that was put out in a “what if its this simple” mode and the author was happy to accept it when the answer was “no its not this simple”. As always, it is probably more some of the later jumper-ons to the broken stick that treated it like a strong analogy that could be ridden into the ground (which they proceeded to do)
I was also going to suggest balls on landscapes, but since that’s already out there, cut me some slack and let me mention MONIAC , a physical model of the economy from 1949. Not exactly what you’re asking for, but cool nonetheless!
Yes, we’ve linked to a video of that thing in operation in the past. Very cool, though as you say not really quite what I’m thinking of.
How about Circuit Theory to analyse landscape connectivity?
Don’t know anything about that. Can you give an overview?
Let me try. Circuit theory in landscape connectivity analysis borrows the idea of an electrical circuit to examine the multiple pathways that an organism can take to move between/among habitat patches. This approach is better than least cost path where a single pathway is considered. Circuit theory is enriched by the progress in the application of Graph Theory in connectivity analysis where the habitat patches are considered as the ‘nodes’ and their connections as ‘edges.’ You can read McRae et al (2008) http://bit.ly/2ppNbRm for an overview.
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