How disturbance and environmental fluctuations actually affect coexistence: part 3 (UPDATED)

This post is part 3 in a series. You should read parts 1 and 2 if you haven’t already done so, otherwise this post won’t make much sense.

In part 2 we saw how disturbances or environmental fluctuations can affect coexistence if average per-capita growth rates depend in a nonadditive way on environmental conditions. A second way for temporal fluctuations to create stable coexistence is if average per-capita growth rates depend nonlinearly on fluctuations rather than linearly.

For instance, as a mathematically-simple illustration, imagine a two species resource competition model in which the dynamics of species 1 are given by (UPDATE: typo fixed):

(1/N₁)dN₁/dt=a₁R-m₁

where R is resource density, m₁ is the per-capita mortality rate, and a₁ is a parameter describing the linear dependence of per-capita growth rate on resource density. In contrast, let’s assume that the instantaneous per-capita growth rate of species 2 depends linearly on R and R², meaning it depends nonlinearly on R:

(1/N₂)dN₂/dt=a₂(R+R²)–m₂

If R varies over time for any reason (say, because of disturbances), the average per-capita growth rate of species 1 is given by the following equation, derived using the averaging rules from part 1:

average per-capita growth rate of species 1 = a₁E(R)–m₁

where E(R) denotes the average (=expected) value of R. You can see that this looks just like the formula for the instantaneous per-capita growth rate, just with E(R) in place of R. For species 1, fluctuations in R are irrelevant; all that matters is the long-term average R. However, when R varies over time, the average per-capita growth rate of species 2 equals (UPDATE: typo fixed)

average per-capita growth rate of species 2 = a₂[E(R)+(E(R))²+var(R)]–m₂

where var(R) is the (population) variance in R. This equation is derived using “averaging rule #6” from part 1.

Now both species can coexist stably, because the average per-capita growth rate of species 2 depends not just on the mean resource density (or the square of the mean of resource density), but also on the variance around the mean. This temporal variance acts like a second “resource”, which only species 2 can “consume”. A trade-off between growth rate under average conditions, and ability to take advantage of the variance around average conditions, can allow multiple competitors to stably coexist. This coexistence mechanism is known as “relative nonlinearity”, because it depends on interspecific differences in the nonlinearity of species’ per-capita growth rates as a function of shared limiting factors. That is, the important thing for coexistence is not how nonlinear any species’ growth rate function is an absolute sense, but how nonlinear it is relative to how nonlinear other species’ growth rate functions are.

Very few studies have attempted to test for coexistence via relative nonlinearity in nature. This is an important gap in empirical work, because key prerequisites for coexistence via relative nonlinearity are common. For instance, the most common type of predator functional response is type II. A predator with a type II functional response likely will have a per-capita growth rate that is a nonlinear, decelerating (saturating) function of prey density (as opposed to the nonlinear, accelerating function used in the simple example discussed above). When predators have type II functional responses, coexistence of competing predators could occur if predators that grow fastest at the average prey density also have the most nonlinear functional responses, so that their growth rates are greatly reduced by fluctuations in prey density.