Re: Remine reproductive excess requirement
- From: William Morse <wdmorse@xxxxxxxxxxxx>
- Date: Thu, 7 Jul 2005 08:46:38 -0400 (EDT)
joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Joe Felsenstein) wrote in
news:da243t$e7e$1@xxxxxxxxxxxxxxxxxxx:
> Let's look at a couple of simple cases to illustrate the issues here
> (note we are here *not* trying to ask what reproductive excess is
> needed to avoid extinction or to keep population size up. Those are
> not what ReMine is talking about if I understand his argument.
Which is too bad, because I have noted elsewhere that is probably the
more interesting question at this moment in history, given the current
increase in extinction rates.
> OK. So let's start with a very simple model (I will avoid the
> complicated models of evolving resource competition discussed here
> before, and stay as simple as I can). So shall we have a finite
> population of size N? Or an infinite population of density N? This
> is not so clear. It will turn out that ReMine's limit to selection
> will need discussion of finite populations, though Haldane's argument
> of 1957 which ReMine invokes is phrased in terms of deterministic gene
> frequency changes, therefore in an infinite population.
>
> So let's have an asexual haploid population, infinite in size but with
> density N. Individuals barely reproduce themselves, with 1 offspring
> per parent.
>
> Now along comes favorable mutations. Each increases the fertility by
> a fraction s. Each starts at gene frequency p0 and their
> fertilities multiply, so that an individual carrying two of them has
> fertility (1+s)^2.
>
> We'll also put in density-dependence of population growth. Using the
> standard Beverton-Holt formulation, let's have T be the total number
> of offspring (per unit area as population total size is infinite as
> we're being deterministic). If we have a fraction p0 of individuals
> who have 0 favorable mutations, p1 who have 1, p2, who have 2, and so
> on, we would find that the total (density of newborns would be)
>
> T = N (p0 + p1 (1+s) + p2 (1+s)^2 + p3 (1+s)^3 + ... )
>
> and the way Beverton-Holt dynamics works is that high density of
> offspring lowers their viability by having it be 1/(1+(T/K)). We
> assume this falls equally on all of these offspring. The effect is
> that the density of surviving adults gets closer and closer to the
> carrying capacity K as the number of favorable mutations increases.
>
> Without going through all the math (I can if needed), then mean number
> of newborn offspring per parent rises roughly exponentially with time,
> so that the number of favorable mutations per individual rises
> exponentially with time in the limit with largish time. If the mean
> number of offspring per parent is, say, M, then the Beverton-Holt
> viability becomes 1/(1+NM/K) so the total density of survivors is
> NM/(1+NM/K) which approaches K more and more closely as M gets large.
> What quantity should we be following as the "reproductive excess"?
> (M-1) ? M times the density-dependent survival, less 1? That would be
> M/(1+NM/K)-1. Should we even be using this deterministic model at all?
> Opinions are solicited. I want to stick to a simple model like this,
> adding later maybe some other simple kinds of events, but avoiding
> complicated models such as the resource competition models which were
> discussed here earlier.
As stated, the model would indicate that selection on the allele under
study (the fertility allele) occurs based solely on number of offspring
at birth (since the density selection occurs equally on all offspring).
That being so, there is no difference in taking reproductive excess as M-
1 or M/(1+NM/K)-1, since from the standpoint of substitution of the
allele they are equivalent., i.e. they both have the same effect on the
fixation of the fertility allele. This would not be the case if there
was a more realistic initial die-off of offspring based on resources
that could be allocated by the parents, followed by a density dependent
selection. (I do not advocate adding a parent resources factor to the
model, as it adds complexity without helping address Walter's question.)
Now I do not know what Walter thinks is the reproductive excess, but to
me many of his statements only make sense if he is envisioning the
reproductive excess as M-1, i.e. as the excess before the selective event
takes place. And he also seems to assume that fertility is relatively
constant, so that alleles only affect viability. I can't say for sure
that is his assumption, since he hasn't given a model, but it seems to be
in line with his general discussion of "scenarios". This may in fact be a
lousy assumption for evolution in general, but it may be reasonable for
certain cases. You have set up your example so that the fertility is the
trait subject to selection. I think to answer Walter's question about
limits on the rate of evolution, the model should include a fertility
factor (F) that is similar to K, i.e. a constant for a given "scenario"
but one that can assume a range of values to check the sensitivity of
the model to that parameter. There should then be a trait that affects
relative viability of offspring carrying the trait, in addition to
density dependent selection that affects all offspring regardless of
whether they carry the trait.
I realize this complicates the analysis, but at least it eliminates the
exponential increase in M, which I have to confess bothers me even though
it has to follow mathematically from your model. I know it is just a knee
jerk reaction from the ecological modelling of my long ago college days,
but we always had limits on increases in any parameter.
Yours,
Bill Morse
.
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