Re: Felsenstein and reproductive excess


"Joe Felsenstein" <joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message news:d7jalg$1t3d$1@xxxxxxxxxxxxxxxxxxxxxx
> In article <d7a6u9$1pbb$1@xxxxxxxxxxxxxxxxxxx>,
> Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote:
> >
> >"Joe Felsenstein" <joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in
> >message news:d78kfg$16l0$1@xxxxxxxxxxxxxxxxxxxxxx
> >[snip]
> >> ReMine has been told many times that my calculation of cost is not
> >> the reproductive excess required to allow substitution (William Morse,
> >> you are wrong about that). He has even occasionally acknowledged that
> >> this is so (and immediately declared my cost useless). The cost I
> >> define is the reproductive excess necessary to avoid extinction.
> >> And for that it is simple and obvious that a beneficial mutation
> >> creates no risk of extinction. [snip]
> >
> >It is not obvious to me that a beneficial mutation creates no risk of
> >extinction.
> >
> >Some, perhaps most, beneficial mutations make the possessor a better
> >competitor against conspecifics. In a density dependent model, these
> >beneficial mutations will result in a lower population count after
> >fixation than before. You might claim that this is not a risk of
> >extinction, but then you are not using a density dependent model in
> >your own model of a deteriorating environment.
>
> In that 1971 model I was investigating whether Haldane's cost could be
> thought of as a cost which, if too large, led to extinction. It turned
> out that it could, and the formulas were very close to Haldane's,
> if one assumed that the substitutions were responses to environmental
> deteriorations, and the cost was calculated as the reproductive excess needed
> to prevent extinction.

The phrase "environmental deterioration" is somewhat ambiguous. In
discussions of Fisher's Fundamental Theorem and the riddle this creates
with its picture of fitnesses increasing over time without bounds, one
occasionally sees the claim that the increase in the fitness of your
conspecifics constitutes a deterioration in the environment for you.
Van Valen makes the same point with his Red Queen hypothesis. So, I
have to ask, when you talk about "environmental deterioration", are you
limiting this to the physical environment? Is ReMine so limiting it? And
most importantly, did Haldane intend to limit it in this way?

I cannot claim to know Haldane's intentions, but ISTM that he was trying
to come up with a mathematically tractable, but biologically realistic,
model which would allow fitness to "increase" over time without population
counts (unrealistically) increasing (and accelerating!) to match. That is,
he needed some population-limiting assumption to counteract the population-
expanding continual increase in "fitness".

The question that Haldane sought to address is difficult to handle under
the usual methods of population genetics. Pop gen, when it computes the
increase in the fitness of a population (i.e. the increase in the average
of the fitnesses of individuals) does not usually project the consequences
of this increase forward over evolutionary or geological time. ISTM that
Haldane avoided this problem (correctly) by adding the assumption that
the fitness increases are exactly matched by deterioration of the environment,
and hence that population densities do not increase. But I think that he
intended to include in this deterioration the increasingly effective
competition of conspecifics.

Haldane's question is difficult for another reason as well. Haldane
considers selective substitutions taking place at many loci simultaneously.
Under the standard assumptions of weak selection and independence, you simply
multiply allele fitnesses (or add selection coefficients) to get the
composite effect at two loci. But this mathematical model begins to
break down when there are so many loci that the total selection is no
longer weak. For this reason, ISTM that ideas like soft selection (or as
Tyler might put it, "escalating damage") can neither be blindly accepted
nor rejected out of hand as "cheating". By the same token, when considering
so many loci that they outnumber the number of available linkage groups,
the Mendelian assumption of independent segregation also breaks down.
That is why Haldane's question strikes me as so difficult and interesting.

> The condition needed is the reproductive excess needed to allow the population
> to grow when its density is low. That is the very situation when density-
> dependence is not relevant.

I understand and agree, IF the environmental deterioration is understood to
take place independently of population density. However, I am not sure that
this is either realistic or what Haldane intended.

> So my discussion left out density-dependence,
> deliberately. It might occur when the population sizes got larger.
>
> Your more realistic model allows a class of substitutions whose effect is
> on the intensity of density dependence. First, note that if the density
> is driven down to a low value, substitutions whose only effect is to
> compete more effectively with the other members of the population will
> have very little effect. As density comes to be low, these substitutions
> would basically stall out, and their rate would drop to near zero.
> Only the beneficial mutations affecting the growth rate would still be
> occurring (among all beneficial mutations).
>
> Second, we have to distinguish between mutations that make an individual
> lower its conspecifics' survival or reproduction, and those that instead
> more effectively resist competition. Here is the standard Beverton-Holt
> discrete-generations density-dependent growth equation:
>
> 1
> N' = N R -------------
> 1 + (R-1) N/K
>
> (use a constant-width font to display that)
> so that R is the total rate of reproduction at low densities, and K is
> the carrying capacity of the population (the reproductive excess is (R-1)).
> In that model, a mutation can either alter R, or K, or both (or neither, in
> which case it is neutral). In the case of R-selection, a beneficial mutation
> affects growth rates at low densities. A mutation that is purely K-selected
> has greater resistance to density-dependent effects, and will achieve a higher
> carrying capacity once it is frequent. In this model, your class of
> mutations would be these latter, and their substitution would stall out
> once low density was reached as a result of other events.
>
> The class of mutations you posit may not actually fit into this
> equation. Does it decrease everyone's K? Or just the K's of those who
> do not have the mutation?

I'm not sure that the standard "rK" model is appropriate for thinking about
this issue. Under this model, a beneficial K mutation apparently decreases
the organism's requirement for resources. When the mutation becomes fixed
in the population, the equilibrium population size has increased to a new,
higher, level (the new K).

But I also doubt that directly answering your question "Does it decrease
everyone's K?" is the right approach either. Consider a mutation which
transforms a population of small bushes into a population of large trees
with spreading canopies. I suppose you could say that it decreases everyone's
K. But it also probably is constrained to defer the age of reproductive
maturity, so it negatively affects r as well.

Perhaps the simplest model would be one in which each organism (indexed by i)
has a competitive ability C_sub_i. The organism achieves a fraction
C_sub_i / Summ (C_sub_i)
of the available resources. But, as an independent trait, each organism
also has a resource requirement or need N_sub_i. The effective growth rate
of a type (positive or negative) can be taken to depend linearly upon the
ratio between resources needed and resources achieved. ISTM that one still
comes up with a two parameter model, like the rK one, but it doesn't have
the objectional (to me) property that K mutations are ineffective at low
densities and r mutations are ineffective at high densities.

[snip remainder]


.

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