## Fanciful mathematics and ecological fantasy

3 05 2010

Bear with me here, dear reader – this one’s a bit of a stretch for conservation relevance at first glance, but it is important. Also, it’s one of my own papers so I have the prerogative :-)

As some of you probably know, I dabble quite a bit in population dynamics theory, which basically means examining the mathematics people use to decipher ecological patterns. Why is this important? Well, most models predicting extinction risk, estimating optimal harvest rates, determining minimum viable population size and metapopulation dynamics for species’ persistence rely on good mathematical abstraction to be realistic. Get the maths wrong, and you could end up overharvesting a species (e.g., 99.99 % of fisheries management), underestimating extinction risk from habitat degradation, and getting your predictions wrong about the effects of invasive species. Expressed as an equation itself, (conservation) ecology = mathematics.

A long-standing family of models known as ‘phenomenological’ models (i.e., because they deal with the phenomenon of population size which is an emergent property of the mechanisms of birth, death and immigration) has been used to estimate everything from maximum sustainable yield targets, temporal abundance patterns, wildlife management interventions, extinction risk to epidemiological patterns. The basic form of the model describes the growth response, or the relationship between the population’s rate of change (growth) and its size. The simplest form (known as the Ricker), assumes a linear decline in population growth rate (r) as the number of individuals increases, which basically means that populations can’t grow indefinitely (i.e., they fluctuate around some carrying capacity if unperturbed).

Yes, it’s simplistic, but highly useful. One well-flogged complication of the basic linear model adds what amounts essentially to a ‘shape’ parameter that gives curvature to the growth response (see figure below). When convex, the growth response remains relatively constant over a wide range of population sizes until it passes a threshold after which population growth declines precipitously; when concave, the growth response declines rapidly with even modest increases in population size, but then doesn’t change much thereafter.

This ‘shape’ parameter has been given the symbol θ (theta) and when θ > 1, you get a convex growth response; when θ < 1, it’s concave. When θ = 1, it reverts to the linear Ricker model (see figure right). This model with the shape parameter is called the ‘theta-logistic’ model.

All sorts of work has been done using this model, with perhaps the most controversial published in 2005 by Richard Sibly and colleagues in the journal Science. They looked at abundance time series (effectively, temporal counts of abundance, N) from thousands of species and concluded that most of them showed concave growth responses (θ < 1, but in their case, using a weird method where θ was negative, but I digress…).

Who cares? Well think about it – if you’re trying to predict how fast a species will recover from, say, harvest, you would conclude that one demonstrating a concave growth response would only increase quickly at very small population sizes. Alternatively, if one has a predominately convex form, then you could hammer the shit out of it over a wide range of densities and be convinced that its rebound potential would remain high. Put simply, get the value of theta wrong, and you predictions about sustainability become completely invalid.

There are all sorts of other evolutionary implications too about the value of theta, but I won’t go into that here (somewhat off-topic).

Sibly and colleagues were more or less hung out to dry about their paper (see the litany of responses here, here, here and here), but they stood by their conclusions. Just recently however, we took it upon ourselves to break open the theta-logistic model and pick it apart to see if it can really deliver a meaningful value of theta (or other parameters, such as the maximum population growth rate rm), or even just provide a rough description of the growth response’s  curvature.

Turns out, it can’t (or at least, hardly ever with real data).

Our paper led by our postdoctoral fellow Francis Clark was entitled The theta-logistic is unreliable for modelling most census data has just appeared online in the new journal Methods in Ecology and Evolution. Using a meticulous simulation dataset where theta was known, the model was quite simply incapable of returning an even remotely meaningful value of theta in almost all cases (and yes, even when using Sibly’s weird negative-theta approach – see Supplementary Results). Not only was the estimated value of theta completely wrong, the theta-logistic couldn’t even tell you about gross curvature of the growth response (i.e., whether concave or convex).

Now, I don’t like to say too often that paper x is completely wrong (i.e., probably most papers have some flaws, but the general approach, main conclusions and implications are more often valid and meaningful); however, in the case of Sibly and colleagues’ work in Science, there’s no other conclusion but that their results were complete ecological fantasy. It’s really surprising the results were published at all, let alone in the reputable journal Science. Don’t get me wrong – Richard Sibly and colleagues have done some good work; it’s just unfortunate they didn’t examine their models more closely and determine whether they were remotely meaningful.

They’re not.

Don’t let the maths scare you, good reader – we’ve written it so that even the mathematically inept should be able to work through the logic.

(and many thanks to my co-authors, Francis Clark, Barry Brook, Steven Delean and Resit Akçakaya; also thanks to Fangliang He and Rob Freckleton)

Clark, F., Brook, B.W., Delean, S., Reşit Akçakaya, H., & Bradshaw, C.J.A. (2010). The theta-logistic is unreliable for modelling most census data Methods in Ecology and Evolution DOI: 10.1111/j.2041-210X.2010.00029.x

### 4 responses

23 09 2015

[…] systematically breaks down just how ridiculous this is, and something that I’ve struggled with for years both theoretically and empirically. He points out that bottom-up theories require that most species be near carrying capacity, even […]

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21 05 2010

[…] Fanciful mathematics and ecological fantasy […]

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5 05 2010

Good stuff Corey – I’m pleased you are giving this a push.

We made a HUGE effort to make this really clear, and we encountered something like eighteen months of resistance in getting it through the peer review “process”.

The POINT I’d really like the dear precious reader to take away is that what we did here can, and should, be done for most modelling work. It’s a basic check – like checking the oil and water. Before telling the world that you have some fantastic result, use the model itself to simulate a pile of data, and then fit the model to this data. If you don’t get the right answer back (because you know what the right answer is in this case), then it’s time to cool off and have a long careful think.

I remain somewhat sympathetic towards Sibly — I think he is one of many who have been part of a culture where it is really hard to back off and admit a mistake. We all make them.

-fc

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5 05 2010

Thanks, Francis – brilliantly worded and my sentiments exactly.

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