In a previous post, I suggested that mathematics is a powerful weapon against zombie ideas, ideas that should be dead, but aren’t. But mathematics is a double-edged sword. Unless it’s used carefully, math can create zombies as well as slay them.

Just as our choice of words can mislead us, so can our choice of math. Once you attach a number or mathematical symbol to something, it’s very easy to start thinking of that number or mathematical symbol as a real feature of the world. Sometimes that’s not a problem—for instance, the variable N in a population model, representing population size, is a real feature of the world. But sometimes it is a problem.

For instance, consider the familiar logistic equation:

*dN*/*dt*=*rN*(1-*N*/*K*) (1)

This equation has two parameters, the intrinsic rate of increase *r*, which gives the per-capita growth rate of the population at low density, and the carrying capacity *K*, which gives the equilibrium population size. It’s standard to think of these parameters as real, measurable features of the population, at least if the population grows approximately logistically. Indeed, these parameters often are measured or estimated; I’ve done so myself. And a very important, textbook theory of life history evolution, the theory of *r*–*K* selection (MacArthur and Wilson 1967), treats these parameters as not just real but as evolvable. Species which are “*r*-selected” are thought to have evolved in low-density environments and so been selected for traits which confer high *r*,* *while “*K*-selected” species are thought to have evolved in high-density environments and so evolved traits which confer high *K* (Pianka 1970).

Now consider a second, less familiar model

*dN*/*dt*=*rN*–*αN ^{2} * (2)

This model also has two parameters, *r*, which still gives the intrinsic rate of increase, and *α*, which gives the per-capita strength of intraspecific competition, also known as density-dependence (i.e. the amount by which per-capita population growth rate declines when a single individual is added to the population). These two parameters are of course measurable or estimable, just like the parameters of the logistic equation.

And that’s because (2) *is* the logistic equation. It’s just a disguised version: a different, but mathematically-equivalent parameterization. If you set the right-hand sides of (1) and (2) equal and do a bit of algebra, you’ll discover that * α* in (2) equals

*r*/

*K*in (1). So if you set

*r*in (2) equal to

*r*in (1), and

*in (2) equal to*

*α**r*/

*K*in (1), equation (2) will exactly duplicate the dynamics of (1).

But so what? (I can hear some of you saying) Equation (1) is the *real* logistic equation, and (2) is just a sort of mathematical parasite, which only looks real because it’s designed to duplicate the real thing. Dressing someone up to look like Charles Darwin doesn’t make them Charles Darwin.

But there’s no reason why (1) should be accorded special status. The fact that it’s traditional and familiar doesn’t make it real. If (2) were traditional, would you argue that it was “real” and (1) was a just a mathematical fiction?*

Ok, fine. (I can hear some of you saying) But if they’re mathematically equivalent, why does it matter which one we use? I can always convert estimates of *r* and *K* into estimates of *r* and *α*, so what’s the problem with just sticking with the traditional form?

Glad you asked. There are a number of problems, but here’s a big one: the whole idea of *r*–*K* selection, which is arguably just an artifact of the mathematically-arbitrary choice to write the logistic equation in the form of (1) rather than (2) (or some other mathematically-equivalent form with the same number of parameters).

The ecological reality that the logistic equation describes, however you choose to write it mathematically, is a well-mixed population of identical organisms which grow and die continuously at a per-capita rate which declines linearly with increasing population size (One can derive this model from various underlying “microscopic” assumptions, but that doesn’t affect my point) Graphically, here’s what you’re assuming about reality:

To express that reality mathematically, you need to describe that negatively-sloped straight line with positive y-intercept. You need two parameters to do it. *And which two parameters you choose is arbitrary*. If you’re a traditionalist, you’ll use the y-intercept, and call it *r*, and the x-intercept, and call it *K*. If you prefer (2), you’ll still use the y-intercept and call it *r*, but you’ll also use the slope and call it * α*. You could even use the x-intercept and the slope.

And your arbitrary choice will, if you’re not very careful, shape your interpretation of the real world. Because presumably that’s why you built the model in the first place—to help you understand the real world. For instance, if you choose *r* and *K* as your parameters, you naturally start wondering what happens if one of those parameters changes, for instance due to natural selection. So let’s say selection increased *r* but left *K* unchanged. What’s happened in reality? Again, here it is graphically:

In reality, the y-intercept increased, which is what you wanted. And the x-intercept remained unchanged, which is what you wanted. *And the slope of the line became more negative, indicating stronger intraspecific competition on a per-capita basis*. Is that what you wanted? When you think of “*r*-selected” species, do you ordinarily think of species in which intraspecific competition is strong on a per-capita basis, so that adding a single individual to the population takes a big bite out of the population growth rate? Do you ordinarily think of environments which select for “high *r*” (e.g., disturbed environments, newly-colonized islands) as “selecting” for stronger intraspecific competitive ability? Do you ordinarily even think of *r*-selection as having the “side effect” of strengthening intraspecific competition? If you’re like most people, you probably don’t. For instance, *r*-selected species are classically thought of as small-bodied (Pianka 1970), and you’d think that adding a small individual to the population shouldn’t reduce population growth rate as much as adding a large individual would.

The point is not that (1) is misleading and (2) isn’t. Thinking in terms of *r*–* α *selection (or

*K*–

*selection!) isn’t any better (or worse) than thinking in terms of*

*α**r*–

*K*selection. The point is that,

*if you’re even asking about the effect of selection on any parameter in either of these equations, you’re letting a mathematically-arbitrary choice dictate what questions you ask about reality*.

It’s not just in the context of *r*–*K* selection that this issue crops up. For instance, what if you want to incorporate abiotic environmental fluctuations into the logistic equation? Conventionally, we think of abiotic environmental fluctuations as acting in a density-independent fashion. But if you incorporate such fluctuations into your model by allowing *r* in (1) to vary over time while holding *K* constant, you’re implicitly assuming that the strength of intraspecific competition varies over time. Which seems like pretty much the exact opposite of what you wanted to assume.

It’s absolutely legitimate and interesting to ask whether natural selection will favor different traits in low- and high-density populations. That’s the question MacArthur and Wilson (1967) asked. But that question needs to be asked in a way that makes sense independent of arbitrary mathematical choices. Your ideas should dictate your words and your math—not the other way around.**

*footnote: I actually do think there are situations where a mathematical model can be considered “real”, and another, mathematically-equivalent version merely “phenomenological”. For instance, the Holling type II functional response is mathematically equivalent to the Michaelis-Menten equation for enzyme kinetics. But the parameters of the Holling type II functional response (“attack rate” and “handling time”) both have straightforward mechanistic interpretations; they are natural parameters in the “microscopic” model of the behavior of individual predators from which the Holling type II functional response is derived. In contrast, the Michaelis-Menten equation contains a phenomenological parameter (the “half-saturation constant”), which is simply half of the maximum reaction rate. It’s perfectly estimable, but it would be somewhat awkward and artificial (at least to my mind) to use it as a parameter in a “microscopic” model of the behavior of individual molecules in an enzyme-catalyzed reaction. A more controversial case has to do with alternative, mathematically-equivalent formulations of group and kin selection models in evolutionary biology.

**footnote #2: In doing some background reading for this post, I stumbled on an old article by Eric Pianka (Pianka 1972) He explicitly recognizes that equations (1) and (2) are equivalent, although he seems to prefer (2) as a description of reality. He also argues against the alternative idea of “*b* and *d* selection” (Hairston et al. 1970), saying that this notion is “merely a matter of definition” springing from Hairston et al.’s choice of notation for the logistic equation, which restricts discussion of “biologic reality”. So my complaint about arbitrary notational choices shaping our thinking about real-world issues is hardly original to me, and in fact goes back to the earliest discussions of *r* and *K* selection. I still think it’s worth repeating. Indeed, Rueffler et al. (2008) make much the same complaint, arguing that the parameters in (1) and (2) are population-level parameters which have no straightforward mapping to the individual-level parameters on which selection actually acts, so that it’s misleading to think of selection as acting on *r* and *K* (or, presumably, on *r* and * α*). I wonder how many times this point will have to be repeated in order to kill off this zombie idea, so that discussions of selection in low- and high-density populations are no longer distorted by arbitrary notational choices.

A really excellent post, should be required reading introductory graduate courses!

I must say though, I think the comment that the Michaelis-Mentin dynamics are somehow more artificial then Holling II is jsut another r-k zombie. The Michaelis-Mentin expression can also be derived from basic principles (i.e. http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#Derivation), and is not any more phenomenological than the Holling II function. This raises the question of scale: both in this example and in equation (1) and (2), the math describes a process that can be derived from more detailed description at a lower level, or possibly many such descriptions.

I think this illustrates the same danger as the rest of the article discusses in attaching too much meaning to a variable. Jeremy does such a nice job of illustrating in the rest of the discussion, there are many ways to mathematically represent the same dynamics.

I think another common flavor of these misconceptions arises because many quantities we seek to estimate exist only in a statistical sense, and depend as much on the way we measure as much as they are a property of biology. For instance, “the population growth rate” is an average over space, time, and individuals. Confusing something we measure (statistically) for a fundamentally biological property can lead to substantial errors, such as mistakenly assuming that a population will grow because it has a positive growth rate.

Consider a population that experiences good and bad years with even probability. On good years, all individuals get 4 offspring, on bad years, survivorship is 20%. Measuring the average growth rate over a series of years, we find 0.5*4+0.5*1/5 > 1, but the population goes extinct almost surely.

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cboettigon June 29, 2011at 8:31 pm

Thanks Carl! Although instead of making this post required reading for introductory graduate courses, I’d be in favor of dropping all mention of r and K selection from undergraduate courses. You can certainly introduce the idea that selection will favor different traits at high vs. low density, and even build simple undergraduate-level models of this, without referring to r and K selection. I’ll bet astronomers don’t teach epicycles to undergraduates, or if they do they first make clear that what they’re teaching is the wrong way to approach the problem and is of purely historical interest. Indeed, it’s my sense that the r-K selection zombie doesn’t actually live on among researchers working on life history evolution (maybe I’m wrong, this isn’t an area I really read). The zombie lives on because it’s still taught to undergrads.

I admit that I’ve never actually had a look at a detailed “microscopic” derivation of the Michaelis-Menten equation (I was mainly going on my knowledge of how Holling type II is derived, combined with a passage from an old book of Tom Fenchel’s in which he argues that M-M is a phenomenological version of Holling type II). A quick glance at the link you provided shows that the half-saturation constant in M-M is actually the ratio of two parameters in the underlying “microscopic” model. So if I If I were really inclined to dig in my heels, I think I could still argue that the M-M parameterization is more phenomenological in some sense than Holling type II. But I’m not inclined to dig in my heels, I don’t actually have super-strong feelings one way or the other. I think the larger point you raise, about how many of the quantities we want to estimate (even ones we think of as “real”) are in some sense statistical or phenomenological, is absolutely correct. So maybe ‘What’s real?’ would be a good topic for a future post…

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oikosjeremyon June 29, 2011at 8:58 pm

I think a lot of confusion can be avoided by talking about r, K, and a as real quantities, rather than disembodied variables. They have physical interpretations, and units: “r” is # of individuals per individual per unit time, “K” is number of individuals, and “a” is # of individuals per individual per individual supportable by the environment per unit time (whew). More concisely, if our time step is a year, the units are 1/y, #, and 1/(y*#) respectively.

I see the way r, K, and a are mathematically related as a feature rather than a bug. Neither of these equations is a more “real” representation of the natural world than the other, and combining them allows us to reason much more clearly about populations, competition, and growth. I’d argue we

areused to the idea of r-selected species having high intraspecific competition. If you have thousands of offspring, only a fraction of a percent will grow to adulthood. They out-compete the rest, who die from lack of resources, predation, or straight-up cannibalism.I don’t think r-K selection is a worthless zombie idea, it’s just an incompletely-expressed one. If physical oceanographers, to take one example, can deal with interrelated, seemingly oddball quantities like momentum flux, force per unit mass, and spatial rates of turbulent diffusion, we ecologists can do it with per capita growth rate, carrying capacity, and carrying-capacity-specific growth rates, a.k.a. intraspecific competition.

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ElOceanografoon June 29, 2011at 10:23 pm

Re: measurement units: fair enough, although I don’t necessarily agree. I don’t think the difficulty ecologists have had in using r-K selection to think about life history evolution has to do with lack of awareness of measurement units. Typical theoretical papers in ecology are silent on measurement units, but in most cases that seems not to lead to any confusion. I don’t see how specifying the units of r and K addresses complaints like those of Pianka 1972 and Rueffler et al. 2008.

As to whether the bulk of ecologists are used to thinking of r-selected species as experiencing strong intraspecific competition, I’m not sure but I kind of doubt it. Usually, I think most ecologists think that the many offspring of highly-fecund, “r-selected” species will suffer high mortality because they are poorly provisioned and so very vulnerable to all sorts of mortality risks that aren’t, or at least aren’t very, density dependent. But I can’t say that impression is based on talking to a huge number of people or reading scads of papers on r-K selection.

Yes, it would be nice if ecologists were comfortable switching back and forth between “microscopic” and “macroscopic” models, and had a good sense of how macroscopic quantities map onto the underlying microscopic quantities, so that they could work with macroscopic models without being misled. I agree with you, and Rueffler et al. 2008, and Pianka 1972, that r, K, and a are going to be interrelated and you need to think about how and why you expect them to be interrelated in order to say anything about how you expect them to evolve (or more precisely, to change as a side effect of selection on underlying individual-level life history traits). But I don’t know that ecologists as a whole have yet the comfort level with this sort of thinking that physical oceanographers have. If they did, wouldn’t papers like Pianka 1972 and Rueffler et al. 2008 just be seen as making obvious, well-understood points, rather than being seen as making novel, important points worthy of publication in a leading journal? If one effect of this blog is to help some future ecologists improve their comfort level in thinking about these kinds of issues, I’ll be thrilled.

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oikosjeremyon June 29, 2011at 10:51 pm

Hi Jeremy,

Seeing as it’s the same equation, do you have any idea if the classical metapopulation approach has suffered from similar zombie like afflictions?

My first guess would be a competition–colonization trade-off, which would relate to the extinction and (ummm) colonization rates in the metapopulation formulation, and remains very popular. But this may be a poor undead analogy, I’ve not had my morning coffee yet…

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Mike Fowleron July 7, 2011at 7:24 am

Not quite sure what you’re getting at here, Mike. As far as I know, alternative parameterizations of the Levins metapopulation model don’t exist. (I mean, yes, the Levins model can be converted into the logistic, but in practice nobody who works on metapopulations uses the logistic instead of the Levins model, as far as I know)

Not sure if I would consider the competition-colonization trade-off a zombie idea or not. There are circumstances in which it works at advertised, but those circumstances are quite limited. I guess I’d consider it a zombie if I thought that lots of people thought it applied far more generally than it actually does (‘limiting similarity’ is a zombie for the same reason; I’ll be posting on this at some point soon…) Note that, if it is a zombie I don’t think it’s one that’s arisen from an arbitrary mathematical choice.

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oikosjeremyon July 7, 2011at 12:51 pm

Hmmm, I guess it was a bit too early for me then. I might try and think about this more carefully again, but it was partly based on boiling the logistic equation down to its individual level birth and death rate components (as you mention can be done in the main post):

dN = N[(B0 – D0) – (b + d)N]

where B0 & D0 are the density independent birth and death rates, and b & d are equivalent density dependent rates, giving

a = (b + d) = r/K,

where K = (B0 – D0)/(b+d).

I’m now trying to remember how/why I thought that would correspond to extinction and colonization rates, but that will have to wait until I can concentrate a bit better on it. Roughly put, colonization rate is given as the difference between successful (B0) and unsuccessful (D0) colonization attempts (the latter are probably hard to estimate), extinction rate is based on density dependent parameters – not sure exactly what b and d would correspond to yet. I’m already lost.

By the way, Roughgarden refers to the classical logistic parameterisation of the Levins metapop model as the ‘logistic weed’ in her ‘Primer of Ecological Theory’. Not sure if this is the only place this is done though.

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Mike Fowleron July 8, 2011at 12:37 pm

Part of why you’re struggling is that, in the logistic, density-dependence is one of the assumptions. In the Levins model, it emerges from the fact that patches can only be in one of two states: ‘occupied’ or ‘unoccupied’ (extinct). When a high proportion of patches are occupied, by definition only a low proportion of patches are unoccupied. And if there aren’t many unoccupied patches, there aren’t many patches available to be turned into new occupied patches, hence the rate of increase in occupancy must be low, no matter what the per-patch colonization rate is. Conversely, if very few patches are occupied, then most are unoccupied and even a low per-patch colonization rate will still give a very high rate of increase in occupancy.

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oikosjeremyon July 8, 2011at 12:53 pm

[…] from logical fallacies like the IDH, or over-literal interpretation of mathematical models like r-K selection. For that reason, unimodal empirical relationships were never widely identified with, or worse, […]

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Zombie ideas in ecology: the unimodal diversity-productivity relationship « Oikos Blogon October 21, 2011at 11:28 pm

[…] Zombie ideas in ecology: r and K selection […]

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Recommendations for new readers arriving from Evolving Thoughts « Oikos Blogon October 29, 2011at 2:37 pm

Hi, this is interesting post i found randomly while trying to find traditional support for the r- / K- continuum so that I could teach something sensible.

The problem is with NOT teaching r-K selection to students is that there really is something there: there probably are real tradeoffs between r and what you call a. We expect that putting more resources into reproduction will reduce lifespan or increase susceptibility to crowding.

But, the _problem_ with teaching it is, as you state, focusing too much attention onto the parameter K. Under the r-a model (in your terminology), K is a compound parameter — r/a . So this means that r-selection will occur to maximize K, which can go a long way to explain why Luckinbill’s results seemed so screwy for the general theory. He was just selecting for improved fitness, so when he got a bigger r0 he also improved K. Unexpected under r/K tradeoff, but certainly expected if r is closely related to K.

Counter to your conclusions, I’ve concluded that, although, as you say, the two models are mathematically equivalent, the r-a version is actually

betterand morerealistic. Here’s a few reasons:1) The r-a model doesn’t suffer from “Levins’ paradox”. See Kuno 1991.

2) As pointed out by your correspondent Mike Fowler, the r-a model is micro-derivable from consideration of births and deaths — the r-K model is not.

[Incidentally, Verhulst, Pearl, Lotka and Volterra all used the r-a formulation. Kostitzin in the 1930s was the first to use the microscopic derivation of Fowler above, and also Gotelli, Pastor, and a number of others. It was Gause whose influential book made the switch, helpful for analysis of his empirical experiments with population growth and competition. Gause was I think the person who started people on the wrong route, though he and Witt also used this variable change successfully to sort out the isocline method we all use in teaching Lotka-Volterra]

3) The links with metapopulation theory are confused by the r-K model.

4) K is definitely NOT the “carrying capacity” — it’s just the equilibrium population density. This follows from the realization that there must be empty space opened up by mortality to allow births to have any success. Our naive view of K being the number of available holes in the environment for individuals to occupy is wrong.

5) Both r and K are density-dependent parameters in the r-K model. Only a is in the r-a model. A clean separation of d-d and d-independent parameters seems reasonable, as things that affect crowding seem unlikely to affect the intrinsic d-i birth and death rates (here, I am not including the link due to tradeoffs, which are evolved).

6) Environmental studies indicate that when you treat lakes with a toxin you reduce K as well as r, suggesting that the two are “really” linked via K=r/a. A kill affects density-independent death rate D0, which reduces r0, and so K. As with Luckinbill’s experiments this is not expected if r and K are “independent” parameters. See Hendriks et al.’s metaanalysis and discussion.

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eratosignison February 20, 2012at 3:52 am

A lot to digest in this comment! You clearly know the literature and history on this better than I do, so I won’t push back too much.

I am inclined to stick with my basic argument, though, which is that r-K and r-alpha selection are both suboptimal ways of thinking about life history evolution. I still think you’re best off formulating a proper model of individual-level life history, focusing on the effects of selection on those individual-level life history parameters, and treating any effects of selection on r, K, or alpha as “side effects”. I think that’s the best way to think about the problem even if, as an empirical matter, selection typically leads to trade-offs between r and alpha rather than between r and K.

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Jeremy Foxon February 20, 2012at 4:31 am

Oh I agree with you there! r and K and r and a don’t encapsulate life histories. But this is a different matter from the meaningless of K as a parameter, which as you point out is if you like a prior reason why r-K selection doesn’t really work as a divide.

On life histories, Stearns has given a good diatribe-type treatment, and I also particularly liked Reznick D, Bryant MJ, Bashey F. 2002. r- and K-selection revisited: the role of population regulation in life-history evolution. Ecology 83: 1509-1520.

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eratosignison February 20, 2012at 2:20 pm

By the way, I looked up ‘logistic weed’ in Roughgarden’s book. It’s relating logistic models of population growth to the classic Levins metapopulation model. of course, these are the same equations, really, they are both logistics. This idea was certainly known to Hanski 1991 in the Biol J Linn Soc42: 17-38. I find it hard to believe that Levins didn’t/doesn’t know about it as well.

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eratosignison March 8, 2012at 10:47 pm

Roughgarden’s book is a great resource, I always recommend it to students studying population biology.

I agree with nearly everything you say above and appreciate the references you given, but I’m not sure I agree with this statement. ‘r’ is defined as the difference between density independent birth and death rates: r = B0-D0 (describing the maximum rate of per-capita increase at the lowest density). So ‘r’ is a density independent parameter in my view.

K is (or at least can be) defined as K = r/(b+d), where b and d are density dependent birth and death rates, respectively. And a = (b+d). So I would respectfully suggest that both K and a are density dependent parameters, but r is not.

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Mike Fowleron March 9, 2012at 10:08 am

H

~~i Mike,~~~~Not quite sure what you mean by a “density-dependent parameter”–a parameter is a constant. In calling K and a density dependent parameters, do you mean their values will change with population density? Or merely that their values determine how per-capita growth rate varies with population density?~~~~Also, afraid I don’t really understand that argument, framed in terms of b and d, about which parameters are “really” the density dependent ones. First of all, if b and/or d are density-dependent, then they’re functions of N, not parameters, and you need to specify those functions in order to fully specify the model. Talking about b and d as “density dependent parameters” is a rather confusing way of talking. I’m not just being pedantic here; for instance, it makes no physical sense for per-capita birth rate to be negative, so if you think per-capita birth rates, but not death rates, are density-dependent, then you actually aren’t imagining a model that can be converted into the traditional logistic model. Now, if you are careful you can of course redefine logistic growth in terms of density-dependent per-capita birth and death rates, or even in other terms described with other, still lower-level, parameters (e.g., in a logistically-growing flowering plant, b might reflect seed production, which in turn might reflect success in attracting pollinators, resource allocation to seed development, seed survival in the soil…). But if you do that, you still end up with a model that’s mathematically-equivalent to the traditional logistic equation, with your assumptions about per-capita birth and death rates dictating the behavior of r, K, and/or a. I freely grant all that, and indeed that was the whole point of my post–that you need to be clear about all the implications of whatever biological assumptions you’re making, rather than assuming something about the determinants of r or K that has unintended consequences for the behavior of other parameters. I take it that that’s what you’re saying too, so I’m afraid I don’t see how you’re disagreeing with me.~~My bad, it’s early in the morning, thought you were replying to my post rather than one of the comments. Forget everything I just said.

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Jeremy Foxon March 9, 2012at 1:20 pm

No problem, Jeremy – I’ve been there before😉

But I think I get part of what you were getting at: maybe talking about a ‘density-dependent’ parameter was a bit sloppy. But that’s what blogs are for, innit?

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Mike Fowleron March 9, 2012at 2:14 pm

> In calling K and a density dependent parameters,

> do you mean their values will change with population

> density? Or merely that their values determine how

> per-capita growth rate varies with population density?

I meant it in the latter sense; in my usage, a density-dependent parameter is one for which a change in the parameter causes a change in per capita population growth (I call this R, below) that is dependent on density; a density independent parameter is one for which a change in the parameter has the same effect on per capita population growth regardless of density.

Maybe it’s sloppy, I dunno. But I meant it that way because it’s easy to imagine genetic changes affecting each kind of parameter separately, and even tradeoffs between them

And this is why the r-K model is so screwy! Both parameters are in fact density-dependent, even though the r’s in both formulations *appear* to be the same thing, and numerically will be the same thing when N = 0!

But they cannot be the same, of course, because both models are different formulations of the same logistic, so if K is not equivalent to a (as we all agree), then r cannot be equivalent to r; that’s because, if you like, K actually has a little bit of r’s density-independence hidden within it, and that makes the visible remainder of the r density-dependent in the r-K model.

I made the above somewhat confusing, but my point is easy to show:

If R = r(1-N/K)

§R/§r = 1-N/K where “§” is little delta, for partial derivative

§R/§K = rN/K^2

I found this first in Schoener 1973: Theor Pop Bio 4: 78

Now if R = r – aN

§R/§r = 1

§R/§a = -N

So the r-a model nicely splits up density-dependent and density-independent parameters, whereas the r-K model does not, although lots of people seem to think it does!

Now going back to the original question:

> Thinking in terms of r-a selection (or K-a selection!) isn’t

> any better (or worse) than thinking in terms of r-K selection.

> The point is that, if you’re even asking about the effect of

> selection on any parameter in either of these equations,

> you’re letting a mathematically-arbitrary choice dictate what

> questions you ask about reality.

Actually, for the reasons given above, I tend to think that the r-a formulation is in some sense more transparent, and therefore more useful (even though they are the same model).

It seems to me easier to imagine different selection pressures affecting birth and death rates in crowded populations from those affecting birth rates and death rates in the absence of crowding. In that sense, maybe the r-a model is more insightful for studying potential tradeoffs between the two? So maybe the choice is not TOTALLY arbitrary?

I also tend to think that some of the confusions about r- vs. K-selection are due to this muddling of what’s actually going on in the r-K model. (Maybe it also caused other confusions, such as about whether density-dependent or density-independent factors were more important in controlling populations).

The r-a model certainly doesn’t explain why or if there are tradeoffs any more than the r-K model, of course.

And the r-a model says nothing about lots of other parameters of population growth: as has been pointed out, one can make perfectly good models of life-history tradeoffs among purely density-independent parameters, and that’s actually what had happened independently of the MacArthur/Wilson ideas of r- vs. K- tradeoffs. Then it was MacArthur’s student Pianka who sort of fused these rather different ideas together from the two fields, making the whole theory even more controversial. This has been said before by Stearns, and Reznick et al 1992 (Ecology 83: 1509-1520, which I liked particularly) among others.

I’m new at Harvard, teaching a basic ecology course for the first time while Paul Moorcroft is on sabbatical. With the TA’s help, I tried teaching some of these ideas using mainly the r-a formulation to undergrads, but still mentioning the controversies over “K” and r-K selection etc. Seemed to go ok. So it is possible, even though we’ve been brought up to believe that the r-K model is more “intuitive”.

But I feel that the whole area is still rather misunderstood even among academic ecologists. What do you all think?

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eratosignison March 9, 2012at 4:29 pm

This is probably exhuming a zombie post, but these ideas have just been put together, now, in a formal paper: Mallet, J. (2012). The struggle for existence. How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution and speciation. Evolutionary Ecology Research 14: 627–665.

By:

eratosignison January 12, 2013at 9:12 pm

All of my old Oikos Blog posts are archived over at Dynamic Ecology and comments over there remain open. If you find an old post of mine on which you wish to comment, I encourage you to do so over on Dynamic Ecology. The URL for the post is the same, except just replace “oikosjournal” with “dynamicecology”.

By:

Jeremy Foxon January 12, 2013at 10:43 pm