Many ecologists expect competing species to exhibit compensatory dynamics, meaning that the densities of any two competing species should be negatively correlated over time or across space. If your competitor increases in abundance, you ought to decline, right? After all, to the extent that two species are competing, that means that when one increases, it’s at the expense of the other, right?
Um, no. Or rather, not necessarily. For instance, environmental fluctuations can cause competing species to exhibit positive rather than negative correlations in abundance. Think of a drought which causes the density of every plant species to decline, even though they’re all competing. But there’s a deeper reason why you should not necessarily expect the densities of competing species to all be strongly negatively correlated with one another: in general, it’s mathematically impossible. I don’t think this fact is as well-known as it should be, so I thought I’d post on it.
Say you have just two competitors, each of whose densities you’ve measured at a bunch of different time points, or a bunch of different spatial locations. In this special case, the correlation coefficient (Pearson’s correlation or rank correlation) between the density of species 1 and the density of species 2 can indeed take on any value from +1 to -1. So depending on how strongly the species compete and other factors, it’s possible that their densities could be perfectly compensatory (correlation = -1). So for the sake of illustration, let’s assume that the correlation between their densities is -1.
Now imagine that there’s a third competitor. How will its densities correlate with those of species 1 and 2? Well, to answer that, you’d have to specify more information about the ecology of all three species. But without knowing anything about the ecology, I can tell you what the answer won’t be. Species 3 won’t have a correlation of -1 with both species 1 and 2. Because that’s mathematically impossible. For instance, if species 1 and 3 have a correlation of -1, then by definition species 2 and 3 must have a correlation of +1, i.e. perfectly synchronous rather than perfectly compensatory dynamics. Conversely, if species 3 has correlations of -1 with both species 1 and 2, then by definition species 1 and 2 must have a correlation of +1.
This three species case is a simple illustration of a general principle: the more species you have, the less-compensatory their dynamics can possibly be. It’s mathematically possible for any number of species to all be perfectly in sync with one another. But the more species you have, the less density compensation they can possibly exhibit, on average. In general, we can describe the pairwise correlations among s competitors with a correlation matrix, a square matrix with s rows and s columns, one row and column for each species. The number in row i of column j gives the correlation between species i and j, and of course the same number will appear in row j of column i since the correlation between species i and j is the same as that between j and i. The numbers on the diagonal will all be +1, since by definition any variable is perfectly correlated with itself. Now, as a matter of mathematical necessity, correlation matrices are positive semidefinite. Which turns out to imply that, the larger s is, the less-negative the off-diagonal elements of the correlation matrix can possibly be, on average.
For instance, in the special case when every pair of species has the same correlation, the minimum possible value of that correlation equals -1/(s-1). Here’s the graph for that special case:
As you can see, even with as few as 5 species, in this special case the minimum possible correlation is only -0.25, which is pretty weakly compensatory. In the limit, as s goes to infinity, the minimum possible correlation goes to 0 (i.e. species fluctuate independently of one another).
Of course, in reality the pairwise correlations won’t all be equal, and so even with many competing species it’s possible that some pair of them might have strongly compensatory dynamics. But if they do, that just implies that some other pair of them must have strongly synchronous dynamics. On average, the pairwise correlations can’t be more than slightly negative when you have more than a few species.
Note as well that the same basic point holds for other measures of synchrony. For instance, the exact same points hold if you want to analyze synchrony in the frequency domain by looking at phase differences.
This mathematical fact is certainly familiar to folks who do a lot of work on this stuff, like my collaborator Dave Vasseur. But it deserves to be more widely known. Lots of ecologists have the vague sense that competitors ought to exhibit compensatory dynamics, and so are somewhat surprised to learn that compensatory dynamics are actually quite rare in nature. But the reason they’re rare is mathematical, not ecological. Which means you cannot use the rarity of compensatory dynamics as evidence for anything about ecology. For instance, you can’t say “These species only exhibit weakly compensatory dynamics, so they must not be competing very strongly”. You can’t even say “These species only exhibit weakly compensatory dynamics, so environmental fluctuations must be generating synchrony that overrides the strongly compensatory dynamics that would otherwise occur.”
Just to be clear, there absolutely is scope for the strength of synchrony or compensation to vary among communities, and among different pairs of species, for all kinds of interesting ecological reasons. But if you aren’t clear on what dynamics are possible, you’re liable to misinterpret actual dynamics.