This post is the second in my series on mathematical constraints in ecology. In part 1 we saw an example of a non-obvious mathematical constraint that can’t be removed from one’s data. Today, an example of a mathematical constraint that’s totally obvious, but that has non-obvious consequences. And that fortunately can be removed from one’s data (well, worked around). Today’s topic is one I’ve posted on before, but I’m revisiting it for my ongoing comparative study of mathematical constraints in ecology and what to do about them.
Here’s a big ecological question: Is local species richness limited by local species interactions, so that sufficiently species-rich local communities are “saturated” with species? Or are local communities always open to colonization by whichever immigrants happen to arrive from the surrounding area, so that local species richness (and composition) is basically just a random draw from the regional “species pool”? For a while now, a popular way to try to test these alternative hypotheses has been to plot the richnesses of local communities vs. the species richnesses of the surrounding regions, and then fit competing regression models to the data. The idea is that a saturating curve (a line that goes up but asymptotes, like a type II functional response) shows that local communities in richer regions are saturated with species. Whereas a straight line with no asymptote shows that local communities are just equal-sized samples from the regional species pool. Each locality in the dataset just contains X% of the species from its surrounding region, on average. Most applications of this approach find linear local-regional richness relationships, which has been taken as evidence for a big general conclusion: local species interactions usually don’t matter.
All this is very intuitively appealing. When it was first proposed, it was a clever idea well worth pursuing and further developing. It’s in several undergraduate ecology textbooks now. Unfortunately, it turns out it’s wrong; I’d say it’s now a zombie idea. Theory and experiments show that it’s based on faulty intuitions. It’s easy to get a saturating relationship in a model with no local species interactions (Fox and Srivastava 2006). Conversely, you can easily get a linear relationship even when species interactions severely limit local richness by preventing colonization by species not present locally (Shurin 2000, Shurin et al. 2000). And that’s without even getting into all sorts of other technical issues and potential artifacts (Hillebrand 2005). Sadly, this turns out to be yet another failed way to infer process from pattern.
To which some “pattern first” readers might respond “But, the observational data! There’s a pattern there! There must be some interesting biological reason why most local-regional richness relationships are linear!” That response is wrong, because it’s based on a false premise. Most local-regional richness relationships are not linear. They only appear to be mostly linear because of a mathematical constraint.
The mathematical constraint is that local species richness can’t exceed regional richness. A locality is part of the surrounding region, so it can’t contain more species than the surrounding region. So a regression of local richness on regional richness necessarily occupies a bounded operational space: all the data, and the true local-regional richness relationship you’re estimating from the data, have to fall on or below the 1:1 line. It turns out that biases the results of standard regressions in favor of finding linear relationships (Szava-Kovats et al. 2011). When you correct this bias using what Szava-Kovats et al. call a logratio regression, you find that linear and saturating local-regional richness relationships are equally common in published datasets (Gonçalves-Souza et al. 2013, Szava-Kovats et al. 2013). And nobody’s been able to find an explanatory variable that differentiates the ~50% of systems with linear relationships from the ~50% of systems with saturating relationships. So there’s no empirical pattern here, and no theoretical reason to expect one (indeed, there’s good theoretical reason to expect a lack of pattern). Which as I argued in that old post means there’s no longer any reason for anyone to care if the local-regional richness relationship is linear or saturating. Much less try to infer anything from its shape. The idea should be dropped from the textbooks. (Aside: To be clear, there are still plenty of reasons to study beta diversity and scale-dependence of diversity more broadly.)
This is a good example of a mathematical constraint creating an artifact that can and should be corrected for. It’s also an interesting example of a constraint that was universally recognized, but that has consequences that long went unrecognized. In future posts in the series, we’ll see some examples of mathematical constraints that aren’t widely recognized (in some cases because they may not really be “constraints” at all). And we’ll see examples of mathematical constraints that, far from being nuisances, actually lead to profound biological insights.