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/*N*_{1})*d N*

_{1}/

*dt*=

*a*_{1}

*R-*

*m*_{1}

where *R* is resource density, *m*_{1} is the per-capita mortality rate, and *a*_{1} 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 *R*^{2}, meaning it depends *nonlinearly* on *R*:

(1/*N*_{2})*d N*

_{2}/

*dt*=

*a*_{2}(

*²)*

*R+R**–*

*m*_{2}

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 = *a*_{1}* E*(

*)*

*R**–*

*m*_{1}

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 = *a*_{2}[*E*(

*)*

*R**(*

*+**(*

*E**))²+var(*

*R**R*)]

*–*

*m*_{2}

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 mean of the square 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.

Great post!

I would love if you could add MathJax to your blog so that the equations would be easier to read. With MathJax, you can type LaTeX-style equations and they render wonderfully in HTML. Instructions for WordPress here: http://www.mathjax.org/docs/1.1/platforms/wordpress.html

Or use the plugin here: http://wordpress.org/extend/plugins/mathjax-latex/

Glad you liked the post.

Unfortunately, since this blog is hosted for free by WordPress.com, rather than me paying for a host and using the WordPress.org software, I don’t think I can use the many WordPress.org plugins. But I will look into it.

I believe wordpress.com doesn’t enable mathjax, but they do provide latex-style equations as images (instead of vector graphics like mathjax). e.g. should work. See http://en.support.wordpress.com/latex/

Jeremy,

Here are a couple of points that I don’t think are well appreciated. We usually think of competition acting via resources, and disturbance as being variation in mortality (more specifically spatiotemporal variation in mortality). What this means is that Cov(E,C)=0 i.e. there can be no storage effect for these models. Storage effect is only mechanism that is capable of explaining coexistence of more than two species. We are left only with relative non-linearity which is good for the coexistence of only 2 species under most models (Chesson 1994; TPB). Adding space does not help. That gets us one more mechanism, growth-density covariance, but this again can only explain the coexistence of two species.

To recover more than 2 spp coexistence via disturbance requires assuming disturbance affects mortality and differentially affects recruitment/competition of the species.

Hmm…Not sure if I’m quite following you here Don, so I’m not sure if you’re right. If, by referring to disturbance affecting mortality, you mean something like the linear additive model I showed in part 1 of this series, then yes, you’re absolutely right. But I already made precisely that point in part 1. Is that just a point you meant to re-emphasize? Or are you trying to make a different point than I made in part 1?

Yep, you did make that point clearly in post 1🙂. Well done. But, you are still talking about disturbances along with the less specific term environmental fluctuations. Does that term (distrubance) imply something to you other than spatiotemporal variation in survivorship/mortality? How do you think about it as a quantitative entity? Seems to me, after we put the model in post1 to bed, we are done talking about disturbance as a serious general mechanism for multispp coexistence.

Yes, I haven’t been explicit about what I mean by environmental fluctuations, as opposed to disturbance. Basically, I mean exogenously-driven changes in model parameters that differentially affect different species, and/or that affect model parameters other than per-capita mortality rates. In the context of the zombie ideas posts, Hutchinson (1961) is an example of a zombie model of how environmental fluctuations, as opposed to disturbance, can affect coexistence.

We’re not necessarily done talking about disturbance after post 1, since disturbances could be one thing that generates fluctuations in resource levels, thereby creating the potential for relative nonlinearity to operate.

Yes, but relative non-linearity due to variation in (abiotic) resources is good for at most 2 species (Grover 1999?, Abrams 2004, Me), if you are lucky (in most models). So, if not a Zombie then at least looking dark under the eyes and entertaining a hankering for organ-meat. The possibility that disturbance affect the non-resource environment to affect species parameters is still an open possibility to facilitate multispecies (>3) coexistence because it allows Cov(E,C)!=0 and potentially n species coexistence.

Hi Don,

See my reply to Florian. I am well aware of the issue you raise, and made the pedagogical decision to gloss over it. Further, I note that part 4 in this series, which I just posted, is a reading list pointing readers to Armstrong and McGehee 1980, Chesson 1994, 2000, and other key references addressing this and many other advanced issues I chose to gloss over for pedagogical reasons. That reading list is by no means exhaustive, but it is not intended to be. The readers who need such a list are looking for just a few things to read, not an exhaustive list.

No, relative nonlinearity as a coexistence mechanism is not a zombie, or anywhere close. It’s logically valid and can work, unlike zombie ideas about the IDH, which aren’t and can’t. Nothing I wrote stated or implied that relative nonlinearity is easy to obtain, theoretically or empirically. As I explicitly stated in the first post in this series, these posts are purely pedagogical. Their only purpose is to explain some key concepts in a way that non-theoreticians can understand. As I’m sure you appreciate, one needs to understand those concepts before one is in a position to discuss more advanced and technical issues such as the one you raise.

More broadly, the relative importance of different classes of coexistence mechanisms is actually something we know pretty much squat about, empirically or theoretically. Empirically, as Mark McPeek has recently emphasized, hardly anyone has ever gone to any system and done all the experiments you need to do to quantify the strength of even

oneclass of coexistence mechanism, never mind all classes. And theoretically, hardly anyone ever analyzes models in which more than one class of coexistence mechanism could be at work. Peter Chesson’s group has just started doing this, with the interesting result that different coexistence mechanisms often (not always) substitute for one another, so that having two of them together doesn’t give you much more coexistence than would either on its own. Personally, before I write off relative nonlinearity as a trivial theoretical curiosity, I’d want to know a lot more about how it interacts with other coexistence mechanisms in theoretical models, and how strong it and other coexistence mechanisms are in nature.I sincerely apologize for being testy, I’m sure you didn’t mean to piss me off. But I’m just not in the mood to spar over these sorts of advanced issues, not in the comment sections on these posts. It’s been hard enough work to just get the basics across. People who are interested in the issue you raise can go read the primary literature, which I just did a whole post pointing them to. In making the comments you’ve been making, it seems to me that you’re ignoring the purely pedagogical purpose of this series of posts, and you’re also reading individual posts out of the context of the rest of the series. Don, you’re a friend, I know you don’t mean to rub me the wrong way, but I’m sorry, that’s what you’re doing. If you disagree with my pedagogical decisions on what material to include in this series of posts, please make an argument as to what other material I should’ve included and why.

What a cool forum for talking about these issues and ideas. Thanks Jeremy.

Hi Jeremy,

that’s a great series of posts! I would like to add one point about this one though: relative nonlinearity requires that resource fluctuations are correlated to the relative frequency of the two coexisting species. Peter Chesson puts it in his 2000 review that way:

“Two species with different values of tau may coexist stably provided the species with the larger value of tau (a) has a mean fitness advantage in the absence of fluctuations in the limiting factor and (b) experiences lower fluctuations in the limiting factor when it is an invader than when it is a resident.” (tau is the degree of nonlinearity)

One situation where this occurs is when you have a strong competitor that, however, is very dependent on resources. If this species becomes too abundant, it depletes resources too strongly, which leads to fluctuations, and this opens the possibility of coexistence with a less competitive, alas, more frugal species.

So, I believe it is important that we must have var(R) ~ N1/N2 in your example to provide stable coexistence. Or am I missing something here and this is included already in some way?

A comment to Don about the relative nonlinearity only being applicable to two species: it’s definitely a good question. I wonder whether someone has seriously looked at this though. Chesson seems to be more positive about the possibility of multiple species coexisting in the 2000 review at any rate.

Hi Florian,

You’re absolutely right about relative nonlinearity, that’s a technical point I chose to gloss over. In general, in writing these posts I had to make a lot of judgment calls about what to explain, and what to leave out. I hope I made the right choices, or at least good choices.

The number of species that can coexist stably via relative nonlinearity is not entirely clear. The conventional wisdom is that, if it’s not just two species (more precisely “two species per nonlinearity”), it’s not much more than that, and that the storage effect is a potentially more-powerful fluctuation-based coexistence mechanism. But there’s a scattering of papers–one is by Anderies and Beisner in Am Nat, if memory serves–suggesting otherwise. I think the jury is still out on this, just because relative nonlinearity isn’t that well-studied theoretically.

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Is the first equation supposed to be

(1/N1)…

instead of

(1/N2)…

???

Also the average per-capita growth rate of species 2 should be

a2[E(R)+E(R)^2+var(R)]-m2

instead of

a2[E(R)+E(R^2)+var(R)]-m2

???

Yes, there are typos, thank you. Fixed now.