Re: Felsenstein and reproductive excess


"Joe Felsenstein" <joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message news:d7nsl2$dkl$1@xxxxxxxxxxxxxxxxxxxxxx
> In article <d7m79s$2t5s$1@xxxxxxxxxxxxxxxxxxx>,
> Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote:
> >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?
>
> Neither I nor Haldane (ReMine can speak for himself) limited the source
> of environmental deterioration to physical, abiotic factors (see
> http://www.blackwellpublishing.com/ridley/classictexts/haldane2.asp
> where you can actually read Haldane's paper on-line, for free).
>
> >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".
>
> I do not see this intention in his paper. Where in that PDF do you see that?

Ah! Calling my bluff, are you. OK. I fold.

I wrote that I cannot claim to know Haldane's intentions. In fact, I cannot
claim to have read his paper! My knowlege of his argument is entirely from
secondary sources. (No, not from ReMine). I suppose I should rectify this
before sticking my nose into this debate again.

> >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.
>
> Most population genetic arguments use relative, not absolute fitnesses.
> They thus do not actually address the absolute fitness of the population.
> By using relative fitnesses they make their argument work either with or
> without population density regulation, provided only that any mortality
> or lowered fecundity that is caused by density-dependence falls equally on
> all genotypes, lowering absolute fitnesses by multiplying them by some
> factor.

Yes, I understand this. And the assumption that density dependence
falls equally on all genotypes is clearly preferable to the assumption
that density dependence doesn't exist. Hence, arguments using relative
fitnesses are to be prefered over arguments using absolute fitnesses.
(Sorry about that, Mr. Edser!).

But ISTM that your argument (regarding a risk of extinction caused by a
deterioration in the environment that lowers the fitness of some common
genotypes) must then be one of these deprecated absolute fitness arguments.
THAT is what I find disturbing about your position. It is an absolute
fitness argument without any provision for density dependence.

[Snip more "bluff calling". You win, Joe! I haven't read Haldane.]
>
> [JF:]
> >> 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:
> [... snip details]
> >> 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?
>
> [Perplexed:]
> >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.
>
> A mutation could increase K and reduce r at the same time. That fits into
> the Beverton-Holt equation framework. In the Beverton-Holt population
> dynamics the equilibrium population size achieved is a function of both.

Yes, I see that. After studying this model a little more, I see that it
is a pretty good model - much better that I thought. Maybe the best possible
two-parameter absolute fitness model. But it doesn't allow an allele to
reduce everyone else's K. No big deal if we were using relative fitnesses.
But this is an absolute fitness model.

> >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.

(Incidentally, after looking at the rK model some more, I withdraw this
objection.)

> OK, interesting model. Let's take a look. If we have two genotypes with
> numbers n_1 and n_2 (I use the LaTeX notation of underscore for subscripts
> rather than your "_sub_") and a total amount A of available resources,
> and if competitive abilities per individual are then C_1 and C_2 for these
> two genotypes, and if their "needs" are N_1 and N_2 and growth rate is
> the ratio (upside down from your statement because I think you misspoke)
> of resources achieved to N_1 or N_2, ... [snip math]

No, I didn't misspeak, though I did fail to speak clearly enough. I meant
the ratio (needs/achievements) but I failed to say that the growth rate
depends negatively upon this ratio. And it depends linearly (with a constant
term) rather than proportionately. The idea was that if the ratio is 1, then
the growth rate is zero. If the ratio is greater than 1 (representing
a resource shortage) then the growth rate is negative. As the ratio approaches
positive infinity (complete resource starvation), the growth rate approaches
negative infinity. But, as the ratio approaches zero (resource overabundance)
the growth rate reaches some fixed maximum. This maximum is not taken as being
subject to selection. But it does constitute a third parameter - Rmax

growth rate r = Rmax (1 - needs/achievements)

Incidentally, thank you for taking the model seriously, even if you did
misinterpret it.


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