One of the most important conceptual advances in community ecology over the last couple of decades has been the development of modern coexistence theory: a quantitative, rigorous theoretical framework that exhaustively defines, and quantifies the strength of, the classes of mechanisms by which species coexist (e.g., stabilizing vs. equalizing mechanisms). Chesson (2000) is the most accessible summary of this theoretical framework. Adler et al. (2007) is an even more accessible overview of some of the key ideas. Folks like Jon Levine, Peter Adler, Janneke Hille Ris Lambers, Steve Ellner, and their colleagues are now applying modern coexistence theory to real data, showing that it leads to practical real-world insights.
But most ecologists only care about coexistence mechanisms as a means to the end of understanding species diversity. And as various folks have noted (including me here on the blog), a theory of coexistence isn’t necessarily the same thing as a theory of species diversity. The question is, how are those two things related?
I’ve been thinking about that question, have chatted about it with various people, and have seen various people mention it in talks. I’ve been struck by the divergence of opinion as to what the answer is. But obviously, my anecdotal experience probably isn’t representative of the broad views of ecologists. Hence my little poll below: do you think more species-rich communities are those with stronger coexistence mechanisms? Choose the answer that best matches your views.
I may decide to do my ESA talk on this topic if the early poll responses are all over the map or if the modal answer is one I seriously disagree with. So please vote! 🙂
In the comments, I encourage you to explain your vote.
9 votes in the poll so far, 7 different answers!
It would be fun sometime to do a massive poll on ecologists’ beliefs about a bunch of topics. And for each topic, also ask the respondents to say how much they know about it (I research it; I last learned about it as an undergrad; etc.).
First define coexistence then I’ll answer. Note that Chesson is not entirely consistent on this, which is a symptom of a wider problem. I have serious problems with several definitions, particularly those requiring increase from rarity as the key diagnostic feature.
“First define coexistence then I’ll answer.”
I was waiting for someone to say this. 🙂 Let’s go with “every species has a positive long-term average per-capita growth rate when rare”.
As far as I’m aware, Chesson’s always been consistent as to how he defines coexistence, though I’m told he has made subtle technical changes over the years in how he defines particular classes of coexistence mechanism such as the storage effect. Have I overlooked something?
Can you say more about why you don’t like ability to increase when rare as a coexistence criterion? Is it because you worry that it doesn’t really work when applied to communities with alternate stable states?
Glad to have taken the bait 🙂 I had a partially-written review of this somewhere, but can’t track it down right now. I got tired of writing it because semantics irritate me.
My concern about rarity is that it imposes an arbitrary criterion which is itself poorly-defined. What does rare mean for a given species? It can’t be relative to other species (within what — a guild, or functional group?) because the default state of any community is not the equal abundance of all species, even though some mathematical approaches implicitly assume this. Some species are stably, consistently rare. To lift an example from your doorstep, Linnaea borealis is a widespread species of boreal forest understorey. Nowhere is it dominant, or even particularly abundant, even on the plot scale. Yet it is clearly a persistent element (1–2% ground cover) of forest communities. Then might rarity mean ‘relative to the long-term expected population size of a given species’? Again, nice in theory, but once again it assumes the existence of an equilibrial state which might never in practice occur.
In earlier papers, Chesson was happy with the idea that coexistence meant the long-term persistence of a species in a community, allowing for stochastic processes. Later work narrowed this to systems in which a theoretical equilibrial state existed in which their abundance remained positive. We could argue back and to for days on whether this is mathematical convenience or necessary truth, but this isn’t the right forum, and I’ll need to do some revision.
“In earlier papers, Chesson was happy with the idea that coexistence meant the long-term persistence of a species in a community, allowing for stochastic processes. Later work narrowed this to systems in which a theoretical equilibrial state ”
Hmm. Afraid you’re going to have to point me to quotes, because based on what I’ve read I don’t think that’s correct. Chesson never assumes an equilibrium in any paper of his I’ve read, unless my memory is really going.
Re: ability to increase when rare its applicability to species that are consistently rare, that conflates two senses of “rare” that admittedly are easy to conflate. “Rare” in the sense of Chessonian coexistence theory means vanishingly rare. Rare enough that your abundance is effectively zero, so that the dynamics of the resident community are the same as if your abundance were literally zero. Might this criterion be difficult to apply in practice to a species that’s consistently at low relative abundance, so that there’s not much difference between the species’ typical abundance and the hypothetical “vanishingly rare” abundance? Yes, I suppose it might be.
Leaving aside what Chesson has or hasn’t said, one certainly could imagine other coexistence criteria than positive long-term average per-capita growth rate when (vanishingly) rare. For instance, expected persistence time or probability of not instantly going extinct, given that you’ve started out vanishingly rare. In population genetics, they calculate various sorts of measures like these all the time. Probability that a new mutation will become abundant enough that it likely won’t be lost to drift, for instance. I haven’t thought a lot about the strengths and weaknesses of different possible measures of coexistence, beyond that they are different. They’re capturing different aspects of the behavior of the system, so in general I’m sure one shouldn’t expect them to all behave in the same way. I doubt every plausible definition of “strength of coexistence” behaves in the same way as species richness increases.
@Markus Here’s the idea of “invasibility analysis” (sorry if you already know this). “Rare” is defined in this case as approximately zero: so close to zero that the focal species experiences no intraspecific density-dependence, so that its dynamics is either exponential growth or decline. One of the great things about invasibility analysis is that it DOESN’T assume that there is a stable coexistence equilibrium. There could also be a non-equilibrium attractor (either internally generated or externally forced) but it doesn’t matter: only growth rate when rare does.
“Unprotected coexistence” (the existence of an attractor with positive density when the invasion criterion is not met) is theoretically possible, but the argument is that if a species can’t rebound from approximately zero density, then it will not be maintained in the face of demographic stochastic accidents.
I’m sure there are some cases where it isn’t perfect, but it’s proven to be quite useful in theory and experiments as a definition of coexistence. Not sure what’s better!
@Jeremy A student in our adaptive dynamics course of some years ago had a great name for such a rare invader in the theoretical sense: “a ghost”.
So, what did you vote and why?
@Jeremy I voted for “No, there’s no relationship between species richness and strength of coexistence mechanisms”. My real answer is “Maybe” but that seemed too weaselly of an answer. I’m still not completely clear on the definition of “strength of coexistence mechanisms” — particularly in >2sp communities, which seems key when you’re talking about species richness — to have a good idea.
I’m also not a fan of the invasibility criterion, but for me it’s because of Allee effects. I think it’s totally reasonable that some species that fail to invade and persist in a community starting at very low abundance could do so if they started at high abundance. Some actual reintroduction attempts seem to support this (e.g. Sinclair et al. 1998 doi:10.1111/j.1523-1739.1998.97030.x).
Yes, as I understand it that’s more or less Mark Vellend’s reason for not being totally on board with “ability to increase when rare” as *the* coexistence criterion.
I personally tend not to worry much about the possibility of Allee effects as I haven’t seen much evidence that they’re sufficiently common and strong to be worth worrying about in most cases. And more importantly, because I’m not aware of an alternative coexistence criterion that could handle the Allee effects case while also remaining even semi-tractable theoretically. I think having theoretically-tractable definitions of key concepts (here, “coexistence”) leads to more and better *empirical* work as well as more and better theory. That is, I don’t think that redefining “coexistence” in some theoretically-intractable way would actually aid empirical work on coexistence, even if that redefinition applied to a greater range of empirical cases. I think empiricists tend to overrate the value of “universal applicability” of a concept, definition, or theory *in the absence of other desiderata*. “Universal applicability” of a definition often is of little value on its own.
We’re going to have to leave it here… its not an argument I have time for now. One point of clarification though: for equilibrial state, read ‘fixed attractor’. Got to go…
I don’t understand what “Stronger” mechanisms means for n(n-1)/2 pairings of n species. It seems clear to me what “stronger stabilizing mechanisms” means for 2 competing species. Is what you mean that many pairs of species (with all other species hypothetically removed) experience strong stabilizing mechanisms as an individual pairs? Perhaps this is defined in the community modeling literature (coexistance theory isn’t one of my areas).
Strength of coexistence mechanisms needn’t be quantified in a pairwise fashion. You can think of it as an average across all species in the community. For instance, Chesson has a formula in one of his papers for how to calculate the community-wide strength of the storage effect for a community of N species. I don’t want to spoil the poll by giving away whether/how that formula involves N… 🙂
So, with 46 votes in, the most popular options are becoming clear: “Yes, assuming the species are coexisting” and…”No, there’s no relationship”!
Clearly I’d better give a talk about this at ESA. 🙂
Cool, now you’ve got <7 hours to write your abstract!
“Cool, now you’ve got <7 hours to write your abstract!"
Hold my beer, I got this. 🙂
I also like that so far “Yes, that’s the case by definition” and “No, species rich communities actually have WEAKER coexistence mechanisms” have gotten the same number of votes.
One could of course argue that all this disagreement reflects some combination of me asking a vague question, and people casually answering based on their own intuitions. No doubt there’s something to that. But honestly, I don’t think that’s most of what’s going on. Think back to our old post on whether species interactions are stronger and more specialized in the tropics. That comment thread became a polite but serious argument among people who work on that topic for a living! So if the disagreement here reflects “vagueness” in the question or people just going on their own intuition, well, I don’t think that “vagueness” and reliance on intuition are easily eradicable.
Definitely your last sentence. My intuition is so bad that I had actually read the Chesson paper (about 2 years ago) that I think you refer to below my comment and I still failed. Clearly, my reliance on intuition was not easily eradicable. I think your comment probably already gave it away. The correct answer (I think) seems to be increasing in frequency.
Personally, I’m actually with Chris Klausmeier (“lowendtheory”)–I don’t think there is a single correct answer. But yeah, if you look at, e.g., equation 6.2 or 6.5 in Chesson (2008), it sure looks like that at least in some scenarios, the answer is that coexistence mechanisms get weaker as species richness increases.
I haven’t looked up the equations you are referring to (having graduated means you can’t make me read papers anymore!) but it seems inappropriate, or at least purely semantic, to say that finding that the strength of coexistence mechanisms is lower in more species rich communities means that species rich communities are those with weaker coexistence mechanisms. Thinking of a simple model of partitioning a resource gradient, adding more species will fill in the space and reduce the strength of stabilizing mechanisms. Maybe the way the strength of multi-species coexistence is calculated accounts for this…
I respectfully disagree that that’s “trivial”. And I have the entire old literature on “species packing” on my side. 🙂 Not that that literature is without its flaws and limitations, as Peter Abrams long ago pointed out. But I don’t think even Abrams regarded its conclusions as trivially obvious.
I wasn’t trying to say that the finding is trivial. Maybe I misunderstood something. I was thinking about the interpretation of the question you are posing and whether the finding that more species decreases the strength of coexistence is a valid answer to the question I assumed (I would guess most people assumed) you were asking or just an answer to a very literal reading of the question. If, for convenience, we think of scope for coexistence mechanisms as a resource of the system that is used up as new species are added, then species rich communities are likely those that had a lot of that resource but have little remaining. A system with only two resources and two species that completely specialize will have very strong coexistence but it feels inappropriate to say that that “system” has stronger coexistence mechanisms that a community where those two resources are on a gradient which has been partitioned among >2 species.
“it feels inappropriate to say that that “system” has stronger coexistence mechanisms that a community where those two resources are on a gradient which has been partitioned among >2 species.”
Hmm. Will have to think more about how to convince folks who feel as you do that this isn’t “inappropriate” (or trivial, or whatever negative-connotation word you want to use). Indeed, far from being “inappropriate”, it’s the whole point of species packing arguments.
@Jeremy I don’t know you to be vague but as you have suspected, you have asked a question where there should be no answer (but you might get one) and it is quite obvious from your lead to the question where you mentioned:
“a quantitative, rigorous theoretical framework that exhaustively defines, and quantifies the strength of, the classes of mechanisms by which species coexist (e.g., stabilizing vs. equalizing mechanisms)”.
You betrayed the futility of the question (and the need for a corresponding theoretical framework) when you acknowledged that people’s interests and understanding (as confirmed by the comments) of what they are quantifying differ. A theory that does not derive its configuration from sound and coherent data is doomed from the start and candidly, a lot of theories that are derived this way are just academic exercises.
“A theory that does not derive its configuration from sound and coherent data is doomed from the start and candidly, a lot of theories that are derived this way are just academic exercises.”
Can you give some examples of the sorts of theories you’re thinking of? Because here, I wouldn’t say that Chessonian coexistence theory is “derived from data” (whatever that might mean)–it’s derived mathematically. But it can be and has been applied to data, so I certainly wouldn’t call it a mere academic exercise.
I think the question I posed in the poll is an interesting little case study in operationalizing concepts. I acknowledge that different investigators might prefer different operationalizations of “coexistence” and “strength of coexistence” while also believing that there are some very strong (not totally decisive, but very strong) reasons for operationalizing those concepts as modern coexistence theory does.
This question hurts my brain because I have a hard time separating the axes of coexistence mechanisms (e.g. “strong vs. weak” and “stabilizing vs. equalizing”). My inherent inclination is to conflate equalizing and weak. Or maybe it’s that I can easily conceptualize strong and weak for stabilizing mechanisms, but not for equalizing mechanisms. Maybe it’s time to go re-read some Chesson again.
Re: conceptualizing strong and weak for stabilizing and equalizing mechanisms, a maximally-strong equalizing mechanism is one that completely equalizes species’ fitnesses. A maximally-strong stabilizing mechanism is one that reduces interspecific interactions to zero, so that species X has the same rate of increase when rare whether or not there are other species present.
Sorry to make your brain hurt. 🙂
This is going to sound cheap but theory of evolution should be a gold standard. Both its verbal and mathematical representations are firmly rooted in sound and coherent data. Theory of island biogeography comes close. I’m not sure if your argument is that a theory that is derived solely from mathematical intuition is not just an academic exercise because the whole point of ecology is to understand nature. Though it is rarely the case, I’m not taking a position that theory cannot precede data but I strongly believe that a theory that serves any meaningful purpose should have basis in reality. To be clear, I’m not taking a position on Cheeson’s theory of coexistence, I’m just drawing attention to a fundamental conflict in our collective thoughts.
It doesn’t sound cheap. I’m a fan of evolutionary theory. I think you may be over rating the extent to which it, or island biogeography theory, was inspired by or derived from data (as opposed to being tested by data) but perhaps I’m misunderstanding you.
I’m on my phone so can’t easily link. So I’ll just suggest that you search for our old post on whether ecologists should have evolution envy, the one on the most successful and unsuccessful big ideas in ecology, and the one on what metacommunity ecology and population genetics can learn from each other. I think you’d like them.
I’m aware and comfortable with some of the ambiguities surrounding those theories that I mentioned because those theories are for the most part consistent. This cannot be said for those in the other category.
I think I remember the post that you referred me to and I don’t think it is relevant here. I’m quite comfortable with the fact that ecology is complex and we may not have a unifying theory. I’m just not comfortable with pretentious theories. I acknowledge that you and others have been bold in putting those theories and hypotheses in their places. But some of your past arguments for theories have unintentionally exalted mathematical expediency at the expense of ecological realities. This is not peculiar to you and this to me, is the new reality. We have many people conducting ecological research now than ever before. Access to data and mobility are unprecedented. It is reasonable to expect that mathematical intuition and ecological realities in any theoretical framework would be congruent to a large degree.
Afraid I don’t quite follow. Can you give some examples of ecological theories you consider “pretentious”? And examples of cases where you see me as having exalted mathematical expediency at the expense of ecological reality? Not offended, just trying to understand where you’re coming from. For instance, I’d say Peter Chesson’s work is an example of doing challenging math for the sake of better matching reality. If you want to do math about a very broad class of cases, so that your theory is sufficiently broad as to encompass many different real world special cases, the math may well be really hard and inelegant. Peter Chesson is the LAST person who should be accused of ignoring reality for the sake of mathematical convenience!
I’d also say that math has various uses. In my experience many claims that this or that model is oversimplified stem from a misunderstanding of what the model is for. See for instance my old post on how false models are useful because they’re false.
My intention is not to offend you and I’m not sure how my post came across as such. I stated that you “unintentionally” exalted mathematical expediency. This suggests a matter of perception. If you require me to dig out situations where you advanced the need for mathematical interpretations of theories and models, we are not going to have a meaningful conversation because you have expressed opinions against what you described as “verbal models”. But be sure that we are on the same page here. My position however, is that we should be able to match theories with data to an acceptable standard and I don’t have a sense that we disagree here. We can find some theories to thrash. What I’m pressing for in addition, is that we should simultaneously evaluate the elegance and ecological realities of theories and models. Again, I take no position on Cheeson’s theory as I’ve not critically looked at the elegance of its maths and the reality that it seeks to portray. Further, I do not take for granted the difficulties associated with theoretical ecology. My comments are about things that we need to consider as we seek a better understanding of nature.
Ok, thank you Tobi, sounds like we may actually be on the same page, at least mostly. Apologies for any misunderstanding.
I do think mathematical models have some useful roles to play even when they can’t be linked to data very tightly, or at all. Perhaps this may be something on which we disagree a bit (or perhaps not). For instance, one role of mathematical models is as a check on our pre-mathematical intuitions. May’s stability-complexity model is a famous example. Not a very testable model–but that’s fine, because the purpose of the model was to undermine the vague pre-mathematical intuition that “stability” and “complexity” (or “diversity”) always go hand in hand. That is, the model doesn’t generate a testable hypothesis–it *is* the test (of the intuitive hypothesis). Caswell (1988 Ecol Model) is good on the data-independent uses of mathematical theory. You probably know that paper but I never miss an excuse to plug it. 🙂
I voted no link between strength of coexistence and species richness:
a) many (~1/3) of species are transient (not in the sense of moving from point A to point C and happen to be in your community box including point B but in the sense regularly fluctuating in and out of the community with stochastic and environmental events. They are not coexisting in any of the senses used here be it a interior attractor or invasibility
b) as several have pointed out strength of coexistence is an n(n-1)/2 dimensional variable. yes you can take an average. But I don’t know of any theory to say that is going to give the best indicator of richness. Could be minimizing coexistence strength does it. Could be that maximizing the variance does it. or maybe bet hedging-like where a slightly higher mean with a lower variance does it. Or … ? I could cook up any scenarios and I don’t think we have a lot to choose from. There were some attempts to figure out what community arrangement would lead to a community matrix with stable eigenvalues but it never got very far.
c) And is there even the shred of a theory about trade-offs. Can all species have strong coexistence indices with each other or does it behave like a covariance matrix where there are limits to the number and strength of negative relationships. Certainly the more niche separation the less equaliztion you have. I’m not going to speculate on how that constrains putting together “coexistence strength matrices”
I think your a) is incorrect. At least, in any particular case it could be incorrect. A species that has a positive long-term average rate of increase when rare might very well sometimes go extinct. And then if it recolonizes it might increase and persist for a while, only to go locally extinct again, and so on. Of course, a species that’s only occasionally present in the community might *not* have a positive long-term average rate of increase when rare. The point is that you can’t really tell the difference between those two sorts of species just by looking at whether or not they’re usually present in the community. Don’t misunderstand, I have no doubt that lots of transient species in nature aren’t coexisting locally. But in a stochastic world, expected long-term average behavior won’t always be observed. A species with a positive expected rate of increase when rare will sometimes exhibit a negative rate of increase when rare and so go extinct (or fail to invade).
“I could cook up any scenarios and I don’t think we have a lot to choose from. ”
I think so too. But if that’s right, does that mean that the whole question posed by the post is a boring question? Perhaps I should’ve done a poll on that too!
Also important to keep in mind that a species that lacks a long-term average positive rate of increase when rare in one particular patch might well have a long-term average positive rate of increase when rare when you take a spatial average over some larger spatial scale. That’s the whole point of spatial coexistence mechanisms like the spatial storage effect.
“Can all species have strong coexistence indices with each other or does it behave like a covariance matrix where there are limits to the number and strength of negative relationships. ”
Yes, that’s the intuition behind why strength of coexistence might well often get weaker as species richness increases. At least, that’s *my* intuition, and I *think* it’s the intuition behind results like eqs. 6.2 and 6.5 in Chesson (2008). But take that with a large grain of salt, because I haven’t yet studied the issue carefully.
“Certainly the more niche separation the less equaliztion you have.”
Hmm. Whether that’s true or not depends on what you mean by “niche separation”. There certainly are scenarios in which one can strengthen stabilizing mechanisms without affecting the strength of equalizing mechanisms. And scenarios in which tweaking some model parameter affects the strength of both stabilizing and equalizing mechanisms.
I’d like to make the case that (at least in experimental communities) diversity and coexistence strength are negatively correlated. For the sake of explicitness i’ll describe what i think you mean by coexistence strength- some average or other summary of the growth rate when rare of all species currently in a community. For concreteness I’ll assume that species compete, though i suspect the argument would work for other interactions.
When diversity is small, it is extremely easy for a new species to invade the community and so the average invasion growth rate of species in the community is high. When diversity reaches saturation, it is more difficult for any given species in the community to invade and so the average invasion growth rate of species in the community is low.
That’s the gist of a classic species-packing argument, though to flesh it out you’d obviously need to spell out what “saturation” means and where it comes from. Although I’m not entirely sure that argument is cut-and-dried, even within some deliberately-simplified theoretical setting. I say that because of work from Ted Case and later Giorgos Kokkoris in the mid 90s and early oughts, simulating community assembly using the Lotka-Volterra competition model. Briefly, you randomly generate a species pool with species’ r values, K values, and competition coefficients chosen at random from specified distributions. Then you simulate community assembly by randomly picking a species, starting it at K, then randomly picking a second species to try to invade the first. You simulate that invasion until equilibrium (which of course might lead to coexistence, or exclusion of one species or the other). Then randomly pick another species not already in the community to invade. And so on. Keep doing that until you either get a final community uninvasible by any species left in the species pool, or you find an “assembly cycle” (e.g., community A can only be invaded by species X, which leads to community B, which can only be invaded by species Y, which leads back to community A). One little-noted result of that line of work is that the final communities generally end up with interspecific competition coefficients *weaker* than the average for the species pool as a whole. And IIRC (which I may not), the more species-rich the final community, the lower the mean competition coefficient tends to be. I’d need to go back and translate that into the relevant Chessonian measure of community-wide average strength of coexistence. But it at least seems possible to me that, in that scenario, community “saturation” comes about even though as species richness goes up the strength of coexistence *also* goes up. (EDIT: the intuition behind this result is that, once you have one or more resident species, species that have very strong interspecific competition with the residents aren’t likely to be able to invade. So the sequential process of invasion and subsequent competition ends up selecting for a set of resident species that don’t compete that much with one another, but that collectively compete enough with all the non-residents to keep the non-residents from invading.)
Put another way, those simulations suggest that community assembly does not invariably push communities to the edge of being neutrally stable. Which is interesting, because Peter Adler’s “embarrassment of niches” paper finds that the two perennial grassland communities he’s looked at are indeed very far from being neutrally stable. That is, the coexisting species aren’t barely coexisting–they can each increase from rare quite rapidly, not too much slower than they could increase if each was invading an empty habitat.
You can try to explain this observation of Peter’s (and the corresponding theoretical results of Case & Kokkoris) more loosely by appealing to a sort of ecological “anthropic principle” (unstable communities won’t last long enough to be observed; https://dynamicecology.wordpress.com/2016/03/21/unstable-communities-as-10-bills/), but I’m not sure that argument actually works, as opposed to merely sounding plausible.
Hoping to have some coherent and interesting answers on some of this by the time ESA rolls around!
I’d be really interested to see what happens in simulations (or in real life). In some ways my intuition is just that the more competitors there are in a community, the harder it is for new species to invade when rare (on average, in a community with modest amounts of positive density dependence). I would happily bet a beer on that, though you might have to make it to New Zealand to cash in.
My other thought is that the strength of species’ coexistence may be strongly confounded with existing species’ diversity, in a way that makes intuition difficult.
Clearly I’m late to this party, but FWIW, I just want to say:
I understand the question to ask “To what extent is observed variation in species richness/diversity attributable to variation in the strength of coexistence mechanisms?” My comment is just to point out that this is different from “To what extent is observed variation in species richness/diversity related to variation in the strength of coexistence?”
My sense from the comments is that several people are responding to the second question. In a speciose community where (potentially) interspecific competition is high, strong coexistence MECHANISMS might result in weak coexistence (species weakly increase when rare). I wonder how much of the variation in response is due to these two readings of the question.
Afraid I don’t follow the distinction you’re drawing. Can you give an example?
Suppose that very strong Janzen-Connell effects (stronger than in the temperate zone) account for tropical tree richness. We might very well still observe that the tendency for tropical species to increase when rare is very weak (for example, if most of the stems in the community belong to rare species, as should be characteristic of hyperdiverse communities in the presence of strong Janzen-Connell effects).
Now we have a situation where the strength of the coexistence mechanism (the Janzen-Connell effect) is decoupled from the strength of coexistence (the tendency to increase when rare).
A cleaner, purely hypothetical example:
Imagine (big leap of faith here) that communities out there in the world vary in the strength of their coexistence mechanisms, but eventually become saturated. Communities with stronger coexistence mechanisms become saturated at higher species richness. In all of these communities, regardless of their species richness, the ability to increase when rare (i.e. the strength of coexistence) is low by definition: in a saturated community, invaders cannot increase when rare, or cannot do so except at the expense of an established species decreasing when rare (to extinction)*. But in this hypothetical world, we would observe variation in species richness. And this variation is properly attributed to variation in the strength of coexistence mechanisms.
My point is not that communities are often saturated, or that Janzen-Connell effects necessarily drive variation in tree species richness. My point is that the strength of coexistence mechanisms (as I understand the term in the context of the question) is conceptually distinct from the strength of coexistence (as I understand the term in the context of your comments on this thread). Personally, I would answer the poll differently depending on which question it was asking (yes, stronger mechanisms are probably/usually associated with species-rich communities; I have no idea what the association might be between the strength of the tendency to increase when rare and the species richness of a community). Based on personal anecdote, I wonder how much of the variation in response can be attributed to this issue.
*Obviously this occurs on average, and not in one-to-one fashion as written; also, saturated communities of this sort might never exist, and this treatment of them glosses over red queen dynamics etc. This is a stylized example to illustrate a narrow point.