Disturbances and environmental fluctuations are ubiquitous. You’d think they’d matter for all aspects of ecology, including for coexistence of competing species. And indeed they do–but not for the reasons many people think they do. In part 1 of this series, I cleared the decks by explaining how disturbance and environmental fluctuations *can’t* affect long-term coexistence. Now in part 2 I’ll explain one way in which they can.

This post builds on the previous post in the series, so if you haven’t read part 1 yet, go read it first, then come back here.

In part 1 I introduced the notion of a “linear additive” model, and showed how, in such a model, disturbances and environmental fluctuations are totally irrelevant to coexistence. In other words, disturbances and environmental fluctuations only matter for coexistence in a nonlinear and/or nonadditive world. This post explains how nonadditivity can combine with disturbance or environmental fluctuations to allow coexistence that wouldn’t otherwise occur.

Nonadditivity means that long term average per-capita growth rates depend nonadditively on temporal variation. I’ll illustrate what this means with a simple modification of the competition model from part 1. Instead of allowing per-capita mortality rates to vary over time, as in part 1, let’s allow the *a _{i}* values to vary over time, for any reason. That is, environmental fluctuations affect the per-capita rate at which individual consumers acquire resources. Such environmental fluctuations will cause species’ growth rates, and thus densities, to fluctuate, leading in turn to fluctuations in resource density

*R*. When

*a*and

_{i}*R*both fluctuate, we have to use rule 5 from part 1 to calculate the average per-capita growth rate of species

*i*, because we need the time average of their product. The average per-capita growth rate of species

*i*is:

*i*

*= E*(

*)*

*a*_{i}*E*(

*) + cov(*

*R**a*,

_{i}*R*) –

*m*

_{i}_{,}where

*E*(

*x*) denotes the time average (=expectation) of

*x*and cov(

*x*,

*y*) is the (population, not sample) covariance between

*x*and

*y*. (note: I didn’t use the

*E*(

*x*) notation in part 1, but I could have; I just changed notation here because I felt like it).

The fact that the time average now involves a covariance term is what makes this a nonadditive model. Covariance is a simple example of nonadditivity. You may wish to go back to part 1 and compare this time average to the equilibrium solution. You’ll note that the two don’t resemble one another; the equilibrium solution doesn’t involve any covariance terms. That means that a model in which species’ *a _{i}*

*values vary over time behaves qualitatively differently from an equilibrium model. Here, the covariance term provides the potential for a nonadditivity-based coexistence mechanism known as the “storage effect”. One species may be the superior competitor on average (i.e. have a higher value of*

*E(*)

*a*_{i}*E(*)-

*R**m*) but suffer from negative, or less positive,

_{i}*a*–

_{i}*R*covariance (i.e. grow worst [low

*a*] during times when

_{i}*R*is high). The other species may be the inferior competitor on average but benefit from more positive

*a*–

_{i}*R*covariance.

The “flip-flop competition” model of Klausmeier (2010) provides an example. This is a two-species version of the competition model I’ve been using in this post and the previous post in the series. In Chris’ model, the environment alternates between longer periods in which *a*_{1}>*a*_{2}, thereby favoring species 1, and shorter periods in which the reverse is the case. *R* fluctuates over time, increasing after environmental change reduces the growth rate of the previously-dominant species, and then decreasing to a low level as the new dominant increases. *R* is thus lower on average during the lengthy periods when species 1 is favored, causing cov(*a*_{1},*R*)<0. In contrast, species 2 grows best during the short periods when *R* is high on average (cov(*a*_{2},*R*)>0). Species 1 can increase when rare by virtue of its superiority on average, while species 2 can increase when rare because the rarer it becomes, the more positive the *a*_{2}–*R* covariance becomes. (More could be said about how the covariance terms change as a function of species abundances and model parameters, but I think and hope that I’ve said enough to give you the gist)

This model makes some of the same predictions as Hutchinson’s zombie idea about the IDH, for instance predicting that intermediate frequencies of environmental change will promote coexistence most strongly, but does so for completely different reasons. Intermediate frequencies of environmental change that favor different species at different times promote coexistence *not* because they match the timescale of competitive exclusion, as Hutchinson claimed, but because they generate resource dynamics that lead to appropriate patterns of covariation between *a _{i}* and

*R.*

Technical aside: the covariance term here is basically what Peter Chesson calls an “environment-competition covariance”. I’m not going to go any further into what that means, but in a future post I’ll be including a list of further readings for people who want to dig into the technical details.

I’ll conclude by emphasizing that this is merely one example of a storage effect (aka “nonaddtivity-based coexistence mechanism”), chosen purely because it’s easy to explain, not because it’s empirically plausible. Don’t mistake the specific example for the general principle. Indeed, the most-discussed examples of the storage effect in the literature have rather different features, in particular involving organisms with long-lived life history stages like resting eggs, seed banks, or hard-to-kill adults. It’s those long-lived life history stages that give the storage effect its name, because those stages can be thought of as “storing” the gains that the population makes during those periods of time when environmental conditions favor it. But personally I prefer the simple example discussed here, because it highlights that those sorts of life histories, though they can make it easier to get a storage effect, are by no means essential. In particular, they aren’t part of the *definition* of the storage effect. Storage effects crop up in all sorts of models incorporating disturbance and/or environmental fluctuations, with all sorts of life histories and other ecological features. I’ve come to the conclusion that storage effects are actually harder to *avoid* than to obtain! Basically, *any* model where you have to use “averaging rule #5” to calculate the time-averaged behavior is a strong candidate to exhibit a storage effect. Again, a future post will have a list of references for those of you who want to dig into the literature and convince yourselves of this.

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