Genetics and Fertility by Algebra:

The curve in figure 1, and 2 and 4 for that matter since they are all alike, invites finding the algebraic formula that would generate the curve.  This is not a strength of mine, but we can make a start.  I shall assume we are talking about humans, but this is a general theory that should apply to any higher organism that has been optimized so that evolution is in abeyance.

To begin with, the vertical or y axis is the growth rate.  Let growth rate be GR.  GR is total births, minus births that would produce an equal population in the next generation divided by population size.  Let RN be the replacement number, TB be total births and PS be population size.

Thus

GR = TB – RN
PS

RN equals PS, so we have

GR = TB – PS
PS

GR can thus be positive or negative.

Let BPW be births per woman and W be the number of women.

Then, since half the population are women,

W = PS/2

TB = BPW X W

TB = BPW X PS/2

So

GR = BPW X PS/2 – PS
PS

Let PM, pregnancies maximum, be the maximum number of offspring a woman could possibly have.  That is to say assume every time she ovulated she produced a fertile child.  In humans, a woman ovulates about 13 times in a year, so if she is fertile from age 15 to 45 she is fertile 30 years or ovulates (13 X 30) 390 times.  That would be quite a brood, right up there with what fruit flies actually accomplish, which is about 200 maggots I understand.

But, unlike flies, there is a fertility penalty for a woman.  She won’t get pregnant if she is already pregnant.  So let PMA be the maximum number of pregnancies available minus about a years worth of ovulations in a woman (assuming she nurses for a few months, which also suppresses ovulation at least in some mammals).  So let PP be the pregnancy penalty or number of missed ovulations per pregnancy, which we take to be equal to the number of offspring.

PMA = PM – PP X BPW

Of those ovulations, some do not become pregnancies because she was not having intercourse, or was using birth control.  So if we let those considerations reduce the maximum number of pregnancies, and we count no intercourse and birth control in years, we can define PMP as the maximum number of pregnancies practical after considering both infertility from pregnancy and from absence of potentially effective fertilization.  Let RS be total reproductive span, as we said about 30 years for a woman. YNF be years without potentially effective fertilization.  So we reduce PMA by the fraction of her potential reproductive life that is unproductive because of choice.

PMP = PMA X RS / YNY

In humans, a woman does not become pregnant every time she ovulates and is potentially fertilized.  In fact, we say that roughly 1 in 6 fertilized ovulations will establish a pregnancy.  In clinical medicine we assume that sperm got to egg well enough but that the fertilized egg, the zygote, did not develop into a pregnancy; it probably never divided at all, but that is a hard fact to get at.  We generally recon that the reason it did not develop is that it did not come up with a lucky combination of genes.  In other words, if you came up with an even more maniacal form of Russian Roulette, in which five of six chambers were loaded, you won against such lethal odds once, or you would not be here.  (Don’t try it again.)  So we let PL be pregnancies lost for genetic reasons.  Now BPW equals PMP minus PL.

BPW = PMP – PL

We can now substitute this into our formula for growth rate.

GR = (PMP – PL) X PS/2 – PS
PS

So far, this is not a theory but a simple set of definitions.  What we are interested in is PL, the number of pregnancies lost for genetic reasons, and how it affects GR at different values of PS.

There are, in the model, 2 components to PL.  Some infertility is due to combining genes that are recessive lethal mutations, genes where 2 bad copies prevent development.  We can call it PLRL, pregnancies missed because of recessive lethals. Some infertility is due to the detuning of genes, one against the other.  We shall call it PLDG.  Thus

PL = PLRL + PLDG

And substituting in the formula

GR = (PMP – PLRL – PLDG) X PS/2 – PS
PS

Now we are ready to go back to the graph on figures 1, 2 and 4 and make some guesses.  When the population is close to zero, a small increase in population size results in a large increase in growth rate (unless, as we know, the population has recently crashed, but ignore that for now).  The way to get a really steep line near zero is to invoke a hyperbola.  This is a curve with this form (in which k, l, m, n and p) are constants

kx X ly + mx + ny = p

In our formulation, the x axis is population size and growth rate is the y axis.  PLRL is the loss of growth because of recessive lethal mutations. PS is our independent variable.  We substitute and get

kPS X lPLRL + mPS + nPLRL = p

Eyeballing the graphs it looks like m and n are both zero.  So

kPS X lPLRL = p

We solve for PLRL so

PLRL = p / klPS

And we go back and substitute it into our formula.

GR = (PMP – p / klPS – PLDG) X PS/2 – PS
PS

Going back to the graphs, it looks like population sizes near zero are dominated by PLRL, but it fades away at moderate population sizes.  So we look at the curve and try to guess what it is.  It cannot be very complicated, because our number of parameters that we enter into the computer program is small.  The curve descends.  It descends ever more slowly, but it never quite levels off.  That would be the shape of a parabola.  A parabola is of the form (with q, r and s being more constants)

y = qx2 + rx + s

But that gives us a curve that gets steeper and steeper.  What we want is a curve that gets less and less steep.  Taking t, u and v as more constants, we want

y = tx-2 + ux + v
in which x-2 is the square root of x.

In terms of our definitions, that comes out to

PLDG = tPS-2 + uPS + v

Our formula now looks like this

GR = (PMP – p / klPS – (tPS-2 + uPS + v)) X PS/2 – PS
PS

Glossary:
It is assumed that all pregnancies become potential parents in the next generation.  Thus pregnancy rate and birth rate are identical with number of offspring.
GR, growth rate of a population in a single generation
RN, replacement number, just enough offspring to replace the parental generation
TB total births in a generation
PS population size, the number in the population
W number of females in a population
BPW number of births per female
PM maximum number of offspring a female could possibly have if every ovulation event produced a pregnancy
PP pregnancy penalty or the number of potential ovulations that do not occur because of pregnancy and lactation if any
PMA maximum number of pregnancies available allowing for the infertility of pregnancy
PMP maximum number of pregnancies practical after considering both infertility from pregnancy and from absence of potentially effective fertilization
RS her reproductive span, the number of years during which she might possibly become pregnant
YNF years without potentially effective fertilization
PL pregnancies missed for genetic reasons
PLRL pregnancies missed because of recessive lethal mutations
PLDG pregnancies missed due do the detuning of genes
k, l, m, n and p numerical constants
q, r, s, t, u and v more numerical constants

The formula looks cumbersome, but I assure you it is not so cumbersome as wading around lost in source code.  And it is there in broad daylight for all to see.  At this point I see 4 challenges.

1. Figure out a way to derive the formula from first principles.  Just looking at a graph and saying, “It must be this equation,” may have been good enough at one time, but Newton taught us better.
2. The computer model and the Iceland data both agree that some areas of the curve are more stable than others.  It would be nice to have an algebraic solution that made the same prediction.
3. Solve the equation in a way that predicts figure 5.
4. If, indeed the algebraic formula given here is right, and frankly it is only a guess, it might be possible to solve for the relevant constants.  The product kl which dominates at population sizes close to zero could probably be calculated from data on inbreeding.  That only leaves 3 unknowns, t, u and v.  We really only need three numbers to nail them down.  One is the fact that it takes a woman six months to get pregnant.  A second number would be the fact that figure 5 together with historical data imply a 10 generation cycle.  It should be possible to fill in the blank by a proper analysis of the Iceland data at moderate degrees of relatedness.

As with all the things I am saying on this site, excluding the main page, I do not propose to stand by this with my last strength.  I quite possibly have messed up.  That would not bother me, nor should it you.  (If the main page were in substantial error, I would be horrified, mortified and mystified to the limits of human endurance.)  But I think it is clear that there is a “right” formula and if you can find it, tell the world.

If you don’t find all this helpful, just ignore it.

27 visitors so far. This is research not advice.  Linton Herbert