I think modern coexistence theory, as developed by Peter Chesson and his co-authors, is one of the most important bodies of ideas in community ecology. And I’m far from the only one who thinks so–interest in modern coexistence theory is taking off. Which is great–the best sorts of bandwagons are those centered on really good ideas! But it’s also a bit scary, since modern coexistence theory is one of the most widely-misunderstood ideas in ecology, even among community ecologists who study species coexistence. Currently, people developing and applying modern coexistence theory mostly understand it really well. But if interest in modern coexistence theory takes off as I think and hope it will, that could change. Best case scenario: people who start working on it mostly get the hang of it. That’s more or less what I think happened with, say, resource competition theory in the wake of Dave Tilman’s early work. Worst case scenario: people who start working on it misunderstand it, and the whole research program goes off the rails, possibly even generating zombie ideas. That’s more or less what I think happened with limiting similarity after Robert MacArthur first proposed it.

I’d like to do what little I can to maximize the odds of the best case scenario. I’m *not* an expert on modern coexistence theory. But I do get the gist of it, and some though not all of the technical details.* I get it well enough to have taught bits of it to undergrads, and to have written a series of posts explaining those bits. I’m sometimes asked to review papers on modern coexistence theory, so other people besides me think I get this stuff.** And several of my best friends get it, which totally counts for something, right?***

**So, got questions about modern coexistence theory? Ask away! **I’ll answer them in the comments, and I’ll turn at least some of the questions and answers into future posts.

To get you thinking a bit–since the worst sorts of misunderstandings are those you don’t even know you have–below are some misconceptions about modern coexistence theory.**** I’m not going to explain why they’re misconceptions right now, but they are. So if something you think is true is on the list below, well, that should be all the incentive you need to ask me a question! 🙂

But before I get to that list, here are some other, more comprehensive resources to help you learn modern coexistence theory. Start with these, not my past or forthcoming posts. My posts are not comprehensive and so aren’t a good entry point into the topic.*****

- Chesson 2000. Peter Chesson’s most accessible summary of modern coexistence theory
- Adler et al. 2007. An even more accessible summary of modern coexistence theory (UPDATE: link fixed)
- Chesson 1994. Chesson’s most formal, general, and least accessible derivation of modern coexistence theory. If you have mathematical chops and want to dig into the technical details of modern coexistence theory, this is the paper to read.

Misconceptions about modern coexistence theory

**It only applies to the specific models that Chesson and others often use as illustrative examples**(e.g., the Lotka-Volterra competition model; the R* model of resource competition; a simple model of competing annual plants with a seed bank). This and the next six misconceptions all arise because most explanations of modern coexistence theory rely on the same few go-to examples. Preventing people from mistaking your specific examples for the broader concepts they illustrate is a difficult pedagogical problem.**It only applies to models in which there are “competition coefficient” parameters.**As opposed to models in which species compete indirectly, via shared resources, shared predators, competition for space, etc.**It only applies to species that interact via competition for shared resources**, as opposed to via interference competition, apparent competition, competition for space, or other direct and indirect mechanisms of interaction.**It only applies to communities with a single trophic level.****It only applies to species with structured life histories that include a long-lived stage such as a seed bank.**A variant of this misconception is that the temporal storage effect (a particular class of coexistence mechanism identified by modern coexistence theory) only works with such species.**It only applies to well-mixed or spatially-homogeneous systems.****It only applies in fluctuating or disturbed environments.**A misconception that may be unintentionally reinforced by leading community ecology textbooks.**It doesn’t apply to species that experience intraspecific density dependence.****It predicts that more species rich communities are those in which species coexist more strongly.**I heard this one in a talk at the last ESA meeting. I worry that this sort of misconception could become very widespread very fast. Modern coexistence theory isn’t currently a theory of species richness, but a theory of species richness is what lots of ecologists want. There’s a high potential for errors to creep in if people who don’t fully understand modern coexistence theory start making intuitive-seeming leaps of logic to try to turn it into a theory of species richness.**It’s the alternative hypothesis to neutral****theory**. In fact, a neutrally stable system is one particular special (limiting) case of the much larger range of cases covered by modern coexistence theory.**It’s of purely theoretical interest because you can’t measure any of its terms in nature.**

A closing bleg

Anyone who knows more about modern coexistence theory than I do (and you know who you are) is *highly* encouraged to participate in the comments. So that you can correct any mistakes I make! 🙂

*The fact that I get some of the technical details is why I also get the gist. To really “get the gist” of any idea–or to be able to tell if there’s even a “gist” to get–there’s often no substitute for getting the technical details. In particular, when an idea is expressed mathematically, you usually *cannot* get “the gist” by just reading the words and skipping over the math. The words are there to aid your understanding of the math, not as a substitute for the math. Reading the words while skipping the math just gives you what Noah Smith calls “the warm glow of understandiness”. The other reason I get the gist of modern coexistence theory is that I get the Price equation. Like modern coexistence theory, the Price equation is a partition (not in the combinatorics sense).

**Man, I sure hope those other people are right.

***If you eat dinner with Robin Snyder enough times you’re bound to get smarter. Sadly, this is not a scalable way to make every ecologist in the world smarter.

****All of the items on this list are misconceptions that I’ve encountered in real life. The length of the list illustrates just how difficult modern coexistence theory is. So although I’m pretty sure I get it–sure enough to write these posts!–there is definitely some non-zero chance that I have failed to understand some crucial details and will write something that will make Peter Chesson and Steve Ellner do this.

*****I thought about writing a comprehensive series of posts, but it’d be too much work. Sorry. If it’s going to happen, it’s only going to happen if I gradually accumulate lots of posts on this topic, which I later organize and edit into something comprehensive.

Hello,

I think these two papers could also be useful to start:

http://www.sciencedirect.com/science/article/pii/S0169534702024953

http://www.nature.com/nature/journal/v456/n7219/abs/nature07248.html

Shea and Chesson 2002, although being more focused on invasion biology, they explain how coexistence arises by the balance of niche and fitness differences in a very didactic way.

Kuang and Chesson 2008 was published on Nature, so it is written without too many jargons. The main objective is to explain how predation can act in a very similar way to competition, but before the authors explain the basic of coexistence theory.

The link to Adler et al. 2007 paper is broken, is it this paper http://onlinelibrary.wiley.com/doi/10.1111/j.1461-0248.2006.00996.x/full ?

Thanks, fixed. Yes, that’s the correct link.

I am thinking in similar term to misconception #9 (although I would not phrase it this way): ie that a system with high coexistence (ie high equalizing and stabilizing forces) will be more resistant to variation from external factors and would therefore display a higher species richness than other system which would be more affected by these variations. What is preventing modern coexistence theory for making predictions in terms of species richness?

“I am thinking in similar term to misconception #9 (although I would not phrase it this way): ie that a system with high coexistence (ie high equalizing and stabilizing forces) will be more resistant to variation from external factors and would therefore display a higher species richness than other system which would be more affected by these variations. ”

Could be; you’d have to model it to see. Peter Adler has an Am Nat paper along these lines that you may have seen, looking at the net effects of environmental and demographic stochasticity on the persistence of rare species. On the one hand, in the model Peter considers temporal variability is stabilizing thanks to a storage effect. But on the other hand, stochasticity creates extinction risk for rare populations (a familiar result from single-species models in population biology). Peter’s paper looks at the balance of those two opposing effects.

I probably should’ve been clearer in remarking that an association between species richness and strength of coexistence mechanisms is a misconception. It’s of course possible that the two could be correlated, for various reasons. But it’s not necessarily the case that they’re correlated–they don’t *have* to go hand in hand. Further, I can think of are plausible scenarios in which they’ll be uncorrelated, or even negatively correlated. So it’s not that they’re positively correlated except in some weird scenarios that couldn’t possibly occur in nature. The misconception is thinking that the two necessarily are positively correlated, or that they’re overwhelmingly likely to be positively correlated.

If one looks at Science Citation Index, its clear that Chesson’s ideas took off some time ago; his major papers have been very well cited since, say, 2008. For example the 2000 ARES paper was cited 143 times in 09;; 204 times in 2013; and may hit 250, or more, times in 2015. These are very high citation counts for an Ecology paper, and clearly indicate Mainstream influence. Question: have you looked to see if the last , say, 400 cits show any deep understanding of his ideas, or are merely a gloss? You have done this for other well cited Ecology papers; Chesson ought to get the FOX treatment !

Yes, it’s fair to say that mainstream interest in modern coexistence theory started to take off with that 2000 review, which is by far Chesson’s most read and cited paper. High profile empirical work from folks like Peter Adler, Jon Levine, Amy Angert, Janneke Hille-Ris Lambers, and others has helped a lot too. Most empirically-minded ecologists aren’t anti-theory, but to get them interested in theory (meaning interested enough to want to start reading up on it with an eye towards using it themselves) you have to demonstrate that the theory can be used to do empirical work. That’s what I mean by modern coexistence theory finally hitting the mainstream. Not just that people are citing Chesson 2000 (though that started the ball rolling), but that a critical mass of people are actually doing empirical research projects based on modern coexistence theory.

There’s no way I’m looking at the last 400 cites of Chesson 2000 to see exactly how he’s been cited! That would be work. 🙂

I could see looking at the last 50 or something. But based on previous similar exercises, I’m pretty sure I already know the outcome: most of the cites will be superficial–“Species coexistence is a central question in community ecology (Chesson 2000).” or some such.

You could try text-mining papers for frequency of occurrence of phrases like “equalizing mechanisms” and “stabilizing mechanisms” and see how if papers containing many instances of those terms has been increasing over time. But I don’t know how to do text mining, plus I believe you need to negotiate with the major publishers for permission to do it.

Q1) Can the decomposition of invader growth rate into “fitness difference” + “stabilizing” be applied to every two-species competition model? If so, how are these quantities calculated in terms of model parameters?

Yes, it’s my understanding that the decomposition applies to every competition model. Though with the caveat that the decomposition is an approximation which for some models involves throwing out terms of higher order than quadratic; see Chesson 1994. And that it can be done for models with any number of species–see some of Roger Nisbet’s recent work. Though with >2 species you run into cases where mutual invasibility doesn’t imply coexistence; that’s one of the key gaps or limitations of current theory as I understand it.

The required calculation has to be customized for the model of interest. At least, that’s how it’s always been. But at the ESA, Steve Ellner proposed a simulation-based method for estimating the temporal storage effect for any model, without having to do any math. He suggested that the approach could be generalized to other classes of coexistence mechanism, though it’s not clear to me if it could be generalized to cover them all (maybe it could, I dunno). And of course , Ellner’s simulation-based approach gives you an answer that depends on the model parameters (since you need to have the parameter values in order to simulate the model), rather than a formula into which you could plug any parameter values.

Thanks! I suppose if these terms are always defined as you say, there should be a general definition of “fitness difference” and “stabilizing force” that would be used to calculate them for a given model. What is that definition that would guide the necessary calculation?

It’s in Chesson 1994, is it not? The equation where he writes the general expression for his partition in terms of variances, covariances and expectations.

Chesson 1994 has lots of examples where fluctuation-dependent coexistence mechanisms can be written in this form (at least approximately). Chesson 2000 reviews some of those examples and adds another good one (resource partitioning a la MacArthur as studied by Chesson 1990, an under-appreciated gem). Still, I have yet to see a definition of “fitness difference” and “stabilizing force” that could be applied generally (maybe I missed it!).

For those who stumble on this blog late, Ellner, Snyder, and Adler’s “How to quantify the temporal storage effect using simulations” is now out for everyone to read

http://onlinelibrary.wiley.com/doi/10.1111/ele.12672/pdf

Unfortunately, to really properly decompose the invader growth rate into fitness differences and stabilizing effects, you need to use the right scaling coefficients. Fluctuation-dependent mechanisms like the storage effect will in general have both fitness difference and stabilizing components. Chesson discusses these briefly in his 2000 Annual Reviews paper and there is a worked example in our 2005 American Naturalist paper with Elizabeth Borer, but that calculation was done by Peter and I’ll be damned if I understand it in general. (I eat dinner with Robin Snyder on a regular basis but it does not appear to have been sufficient to get me to understand the scaling factors.)

But yes, in theory one should be able to do this for every two-species model.

Aha! Thank you Robin, this is very useful. And thank you for the reminder of your 2005 paper with Elizabeth, I had been meaning to look that up again for another purpose and now I have a second reason to do so.

As someone more familiar with Tilman’s R* theory, what is its relationship to Chesson’s coexistence theory?

R* theory is a special case–it’s a particular model of how species compete (and possibly coexist, if there are multiple resources or the right sort of nonlinearities+fluctuating dynamics, or etc.). Chesson’s approach is a broader framework in which R* theory and lots of other models fit.

For instance, the simplest R* model, where there’s just one limiting resource and the species with the lowest R* excludes all the others at equilibrium, is an example of what Chesson would call a model with fitness inequality and no stabilizing mechanisms. Speaking roughly (but only slightly), species have different ‘fitnesses’ (per-capita growth rates when rare) because they have different R* values, with the species with the lowest R* being the fittest. And there’s no stabilizing mechanism in the model to create any return tendency. Nothing that raises the fitness of species if they drop to low abundance, or reduces their fitness if they increase to high abundance. For instance, one way (among many) to introduce a stabilizing mechanism would be to assume that each species experiences a per-capita mortality rate that increases with its own density. Another way to get stabilization into the model would be to assume that each species has its own specialist predator, and there’s a trade-off between R* value and vulnerability to predation. That’s the Grover 1994 model. The effect of predators in that model is to produce stabilization: if a species gets abundant (rare), it’s predators get abundant (rare), and so the species’ per-capita mortality rate increases (decreases). (Obviously, stabilization could work through other demographic parameters besides mortality rates…)

See Chesson & Huntly 1997 for an example applying Chesson’s analytical approach to an R*-type model (it’s early in the paper).

A model in which every species had identical R* values but there were no stabilizing mechanisms would be a limiting, neutrally-stable case. Chesson would call that a case with no stabilizing mechanisms and maximally-strong equalizing mechanisms (equalizing mechanisms in this example being “whatever it is that explains why those species have identical R* values”)

It strikes me that one of the biggest challenges for coexistence theory is to figure out when the mean-field differential equation approximations that underlie it are likely to yield insight. When are predictions made by coexistence theory valid in a stochastic world—or what *kinds* of predictions does it make that are robust to stochasticity? How does the amount of stabilization and equalization in the mean field relate to the probability of coexistence (or exclusion) in practice?

Alternatively, how can coexistence theory be combined with various forms of demographic and environmental stochasticity to make good predictions? What are the relevant forms of stochasticity that need to be incorporated, and what mathematical avenues (ranging from stochastic calculus and linear noise approximations to simulation-based approaches) will prove fruitful?

I’m floating these questions more as food for thought than as requests for definitive answers.

I’m puzzled by your question–you may be confused about what modern coexistence theory assumes. It’s not a mean field approximation applying only to differential equations or assuming the absence of stochasticity. Indeed, revealing the consequences of stochasticity for coexistence is the whole point of the largest chunk of the theory! See Chesson’s work.

EDIT: Just speculating here, but I wonder if by “modern coexistence theory” you mean something quite different than what I meant in the posts. By “modern coexistence theory” I do *not* mean “what current undergraduate textbooks say about coexistence” or anything like that. I mean something quite specific: the theoretical framework developed by Peter Chesson and his collaborators. Is that the source of your confusion here?

My turn to be puzzled, I suppose. Much of Chesson’s coexistence theory is developed by considering long-term average population growth rates, and asking whether or not there exists a rare-species advantage. If the system is stochastic, either in terms of a stochastically fluctuating environment or in terms of demographic stochasticity in finite, discrete populations, then the long-term average population growth rates are mean-field approximations.

More importantly, perhaps, even the short term expectations of the instantaneous population growth rates are mean-field approximations to the demographic stochasticity (the same goes for the long-term and annual population growth rates in a difference equation).

To the best of my knowledge, there is not a robust body of theory linking the strength of coexistence to the probability of persistence/extinction in stochastic systems. This is what I meant when I mused “How does the amount of stabilization and equalization in the mean field relate to the probability of coexistence (or exclusion) in practice?”

(There have been a few steps in that direction for specific idealized models of community dynamics, e.g. Haegeman and Loreau 2011)

One way that we could be talking past each other is if I am missing the point by assuming that the main point of coexistence theory is to make predictions about when populations of competing species are likely to persist. In that case, I would especially appreciate to hear your further thoughts.

And to be clear, I am not trying to suggest that coexistence theory hasn’t been fruitful in terms of understanding these systems–quite the opposite. In fact, it’s precisely because it’s been so fruitful that I am excited to think about how to understand the consequences of equalization and stabilization in stochastic communities.

“To the best of my knowledge, there is not a robust body of theory linking the strength of coexistence to the probability of persistence/extinction in stochastic systems. ”

Oh, ok. What you’re saying is that you care about a different response variable.

Re: mean field approximations, I’m still a bit puzzled, because Chesson isn’t just linearizing around equilibrium. He’s studying long-term average behavior, and doing that using an approximation–but unless I’ve misunderstood, it’s a nonlinear approximation and so it’s a nonlinear average. Surely you aren’t suggesting that we ought to do without approximations?

“One way that we could be talking past each other is if I am missing the point by assuming that the main point of coexistence theory is to make predictions about when populations of competing species are likely to persist”

Yes, that’s the point–and that’s what coexistence theory does. Species with negative average rates of increase when rare will tend not to persist over the long run; species with positive average rates of increase when rare will tend to persist over the long run. Are there other response variables one could look at, such as the full probability distribution of rate of increase when rare? Sure. But I guess I’d want to see what one gains by doing that, and what one has to give up to obtain those gains (e.g., in terms of the narrowness of the assumptions one has to make in order to get analytical results).

Cool. Two quick thoughts:

You’re right that mean-field is a bit of a misnomer in systems with environmental stochasticity, for the reasons that you pointed out. In systems where the stochasticity is purely demographic, then I think the approximation is truly a mean-field approximation.

As for whether it’s possible to get traction on the stochastic systems payoffs without stumbling down a giant rabbit-hole of narrow (or absurd) assumptions, I don’t know. But I do think it’s an important question, because weakly stabilized and weakly hierarchical communities with demographic stochasticity are probably pretty ubiquitous. A good understanding of what affects population persistence and the maintenance of species diversity in these systems should have significant basic and applied implications.

Thanks for the pointers to Chesson’s work, Jeremy, and a way into it – it’s going to keep me busy for awhile. It’s seems that this discussion raises a fundamental underlying question about whether community composition is driven by stable or unstable co-existence. The intuitive appeal of Chesson’s theory – that all you need for co-existence between two species is a fitness difference that is offset by the degree to which intra-specific competition is greater than interspecific competition – is compelling. But is there any reason to believe, a priori, that communities are a result of stable co-existence? I understand that immigration doesn’t mitigate the need for a stabilizing component but isn’t that only true if the fitness difference between two species is the same in space? If species 1 would be the winner at Site A and species 2 would be the winner at Site B couldn’t both species often be found together at both sites if the immigration rates offset the rate at which each species would be driven to extinction? Peter Adler makes the point that the real question around neutral and niche theory should be “to what extent does the diversity we observe in natural communities result from strong stabilizing influences overcoming large fitness differences vs. weak stabilization operating on species of similar fitness”. There seems to be an analogous question here – to what extent do natural communities reflect stable versus unstable co-existence among species? Or am I missing something that implies that unstable co-existence is unlikely in nature? Jeff.

I’d say the answer is scale-dependent. At the small scale of sites A and B, this would not constitute stable coexistence because it relies on immigration from the other patch. But zooming out to the patch A+B landscape, this would be seen as stable coexistence because each species could increase from rarity.

So, we could call this a stable co-existence but the stabilizing component as outlined by Chesson and colleagues (i.e. interspecific competition being weaker than intraspecific competition) isn’t necessary? I think Chesson et al. would say that stable co-existence requires a fitness difference and a stabilizing component. I’ve proposed a scenario where we could assume that the stabilizing component didn’t exist and I could still imagine long-term co-existence driven by spatial variation in fitness difference and immigration. It sounds like you’re willing to call that stable co-existence at the landscape scale – I wonder of Chesson et al. would be willing to call that stable co-existence?

“So, we could call this a stable co-existence but the stabilizing component as outlined by Chesson and colleagues (i.e. interspecific competition being weaker than intraspecific competition) isn’t necessary?”

No–at the level of the whole landscape, intraspecific competition is stronger than interspecific. The stabilization emerges from the combination of spatial variation in fitness and dispersal.

Chesson has a bunch of papers on spatial coexistence mechanisms. Roughly speaking, there are spatial equivalents of temporal coexistence mechanisms. So there’s a spatial storage effect, spatial relative nonlinearity, etc.

What you are describing is actually a spatial storage effect and/or a fitness-density covariance. So, yes, that is stable coexistence on the landscape scale.

What you are calling a “stabilizing mechanism” (i.e., intraspecific competition > interspecific competition within any local area) is what Peter would classify as a “variation-independent mechanisms” or “fluctuation-independent mechanism” (the term has changed over time, the former is what I believe he is using now). This is one form of stabilizing mechanism, but not the only one.

In your scenario, intraspecific competition is greater than interspecific competition one the landscape scale for one of a couple of reasons. First, the fact that spp. X has a higher fitness in “X habitat” means that it will be competing most strongly in “X habitat” (and thus individuals are experiencing high competition where they should be growing best because of conspecifics)- this is a spatial storage effect. Second, spp. X is more likely to be clustered in “X habitat,” which means that they are competing more often with other X individuals- this is a fitness-density covariance. (and it is very likely that both are happening in your scenario)

Re: your earlier point, it is possible that communities are not stably coexisting. For example, take Barro Colorado Islands, Panama. Although many people study how trees there can coexist, it has been about 2-3 generations since BCI became an island, parts of the forest are in different stages of recovery from human damage, and honestly, it hasn’t been all that long (in terms of tree generations) since the climate was different. Plus there have been shifts in the animal community since people started studying them. Maybe there aren’t really 300 trees stably coexisting there, we might be catching many on the way to extinction. However, if that is the case, then you really need to invoke some kind of speciation-extinction balance to explain diversity (or say that diversity is being lost from a major recent event, such as humans). For example, under Steve Hubble’s model, BCI would need about 50 speciation or long-distance immigration events per generation to be maintained (I think, I might be wrong, but it was at least a couple dozen).

I think that it would be possible to include speciation-extinction balance situations into modern coexistence theory, it just hasn’t been done yet.

Simon Stump

As Jeremy & Simon said, yes this would be seen as a “stabilizing mechanism”. One of the papers they refer to is:

Chesson, P. (2000). General theory of competitive coexistence in spatially-varying environments. Theoretical Population Biology, 58(3), 211–37.

Personally, I’m happier thinking about this scenario in terms of Tilman’s resource competition theory. Haegeman & Loreau recently published a nice paper on this topic from that perspective:

Haegeman, B., & Loreau, M. (2015). A graphical-mechanistic approach to spatial resource competition. American Naturalist, 185(1), E1–E13.

“Personally, I’m happier thinking about this scenario in terms of Tilman’s resource competition theory”

Sure. I’m actually happy to think of this sort of scenario from that perspective too. I’d only add that I find the two perspectives complementary. I don’t feel like I need to prefer one or the other. They serve different purposes. Chesson’s is a general framework, of which resource competition theory is a special case. Having the general framework helps you connect the special case of resource competition to other special cases. And having models of various special cases lets you make predictions and construct mechanistic explanations, which a general framework on its own wouldn’t allow you to do.

Hello Jeremy,

I’m starting to study the modern coexistence theory and I would like to know if intraspecific variation can cause species coexistence (and how).

Thanks.

It can cut both ways, in both the short and long term.

Hello, I am an undergrad writing a paper on coexistence theory. I’ve never heard of the theory before choosing it and man do I wish I can take that choice back. My question to you is what are the key differences between modern coexistence theory and plain old coexistence theory? Thank you!

That’s an awfully challenging topic for an undergrad paper!

If what you mean by “plain old” coexistence theory is “Hutchinson”, I’d say the main difference is that these days we rely on mathematical models rather than on seemingly-precise but actually-vague metaphors about “niche overlap”. At least, that’s what we *should* do these days!

If what you mean by “plain old” coexistence theory is Lotka/Volterra/Gause, I’d say there are some similarities as well as some differences:

-Coexistence comes down to whether intraspecific competition is stronger than interspecific competition. That’s a similarity; Lotka/Volterra/Gause knew that.

-But today, we’d define “competition” more broadly than someone like Gause would’ve. If for some direct or indirect reason your per-capita growth rate declines with increasing density of some species (conspecific or heterospecific), all else being equal, that species is your “competitor”. For instance (and this is just one possible example among many), think of apparent competition (indirect competition between prey species, mediated by a shared generalist predator).

-Today we know about classes of coexistence mechanism that Lotka/Volterra/Gause didn’t. In particular, we now know how spatial and temporal variation can contribute to (or inhibit!) coexistence.

-Today we have an overarching mathematical framework, largely due to Peter Chesson, within which particular cases fit. Rather than just having a bunch of models and empirical examples of different cases.

Hope that helps! Good luck with your paper.

This helps IMMENSELY!!! I deeply appreciate the time you’ve taken to answer that for me!