Microevolution according to
the Poisson distribution[1]
Summary
You expect
the genetic changes in a population are described as random or non random and
thus selective, but this appears not to be the case. Random genetic changes are
nearly always described as genetic drift with changes in the heterozygosis, but
the gene frequencies and so the random genetic changes are not to be reduced
from the heterozygosis. So it is tried to develop a uniform theory with the
random expected genetic change as the neutral theory and the zero hypothesis
for the selection. For the procreation and gene transfer from the individual in
a large, relative unlimited population the Poisson distribution is the
obvious method for calculating the
random expected genetic change. Yet also in small populations this distribution
appears accurate and well applicable. The Poisson distribution appears very
flexible in the sense that the parameters determining the intensity can
describe well the small populations and the population dynamics (change in size
and allele selection). By means of the parameters you can also find indications
for the complicated effect of the selection in the procreation on the allele
transfer. Even with a very simple application of this theory there are
immediate indications that the selection by people has been suddenly stopped
with the entrance of the modern society.
The relevancy in the appearance of the mutations for the dynamics of the
organism and the species is checked. By making simple distributions the transfer
of the alleles through the generations is followed in coherence with the random
variation in the effective procreation. These distributions are superposed over
some generations. This superposition of the distributions is possible by
working systematically with the Poisson distribution, but it is very laborious.
It appears than that the accumulation of the distributions over many
generations can easily be calculated, only for the exponential part of the
Poisson distribution The result of it the P0, the extinction of the neutral
alleles is essential for the random theory. The calculations of the extinction
with the exponential recurrence formula is also possible with population
specific parameters determining the cumulated exponential intensity are interesting.
They give information over the random neutral path and the non random
selection. This is showed by describing the decay of the alleles over many
generation in tables for populations, large and small, increasing and
decreasing in size, with and without selection, with and without inbreeding.
The random path of larger quantities of the alleles and thus allele frequencies
is also described. Important is however that these simple distributions do
describe primarily the decay of the absolute quantities, the “quanta”, of the
offspring and the alleles. In literature investigation later I found that these
extinction is also described plain by Motoo Kimura, but he reduced them not in
this basic way. There is some evidence that the conclusion of Kimura’s and others ware: the calculations of the
extinctions are not relevant, because they are not applicable in a limited
population. Nowise this, the extinction is essential in a logic consistent
theory.
Points of
attention are:
The neutral
theory as the zero hypothesis for the genetic selection concerns exclusively
the direct changes in numbers or frequencies of alleles and/or descendants of
individuals as described here. Changes in the heterozygosis are an indirect
basis for the neutral theory.
The direct
neutral change or allele extinction is also described in the limited and small
population and gives here besides the genetic drift extra information over the
genetic changes.
Selection
always is the result of non random differences in the parities, the offspring
of the individuals.
Preface
After my study medicine and some years later the training in the
epidemiology in tuberculosis and similar things until 1976 I did lose all the
tangible connections with the organisations of science and research. Moreover I
do not anymore practice the profession of a doctor already since my 40^{th}.
Now I am 61 and I looked after my children for a long time, while my wife was
working. My curiosity and interest in different fields and the increasing quantity of spare time results in
a number of hobbystudies. Magazines, books and later on the internet did
provide me afterwards plentiful in information The study of the topic
evolutionary biology was one of my most favourites. No official training or
studies were followed in this. My study activities consist of collection here
and there interesting data and than thinking about it endless with my super
critical dialectical customs, or you may can call it also addiction. Anything
you read than is denied and that negation follows a laborious constructed
meaning, but that meaning again is denied, etc. So the negation of the negation
in order to find at last the all synthesis, the logic, the unity, the truth in
which anything is participating. This in principle is the rational method of
Spinoza, which was described later on by Hegel. By these negations points of
view mostly are not simply accepted or rejected, because often a synthesis is
possible so than the and/or is the best solution. Study in this way is an
endless ruminating, destructing and constructing of meanings and theories.
Using these radical dialectics you do not need a teacher, but it can result in
a stomach ulcer, for with this method of negation you are not a nice teacher
for yourself. The advantage of these primary negations is that it makes you
independent of other people like teachers and authors. Their information it is
not followed and taken over, but negated. So I tried to be no man’s follower and an open minded searcher to the
uniform principles.
Publication by internet is for me the most convenient way to share my ideas
about the evolution with other people. Perhaps it can help starting discussions
and deepening studies to the stirring, interesting and in many aspects so
important topics of the evolution. Furthermore I do hope I also will be able to
publish here more ideas about the macroevolution and the
religiousphilosophical aspects of the evolution as a logical integral.
Some principles of the
evolution theory
The evolution of the organisms and their species still is somewhat
disputed. It is nevertheless without doubt that anything we can observe is
changing and that nothing is able to remain the same forever. If the changes in
this are irreversible there always is a development or evolution. So the
apriori statement is plausible: the
changeable and evolutionary characteristic of the nature, living and not
living, is nothing more than its very existence in the time or properly the
spacetime. There are indeed observations that confirm the evolution.
Already in 1859 Charles Darwin did describe and prove that the similarities and
differences between the yet living species and the died out fossils indicate
evolution, as well as the frame of the embryo’s. Although there now is much
more knowledge and we can fill now probably a bookshelf of more than
Microevolutionary principles
Within a population is a knockout competition between the different
allelic variations on the gene loci. By two factors all the gene variations or
alleles are not transferred through the generations of descents and so the numbers
and frequencies of the alleles will increase or decrease in the following
generations. These 2 factors are:
1^{st} the distribution
of the reproduction. The individual organisms of the parent generation F0 do
have different numbers of effective descendents, that reach adultness and are
able to reproduce themselves.
2^{nd} The endowment of the alleles to the effective
descendants.
The organisms of the F0 that have been able to reproduce effectively in
this way, will pass at most the same, but on the average fewer different
alleles to their total offspring than they do have themselves. Even if all
parents should have an equal number of offspring they will pass different parts
of their genetic variations to the next generations or otherwise not. There are
many mutations and often an individual has a number of seldom mutations. Also
are many mutations seldom and are they in small frequencies in large
populations, or a total species. However the absolute numbers of seldom
mutations are large in large populations, of in the whole species. One percent
of 10^8 yet ever is 10^6. So it is obvious apriori that seldom mutations
practically never will vanish in large populations, unless they are ultimate
seldom and occur in immeasurable small frequencies, or are very unfavourable.
In this it also is obvious that alleles will be practically never be fixed in
large homogeneous populations. In small populations seldom alleles do have
small absolute numbers and by this they can vanish or increase in number and
sometimes be fixed in small populations. One percent of 100 yet only is 1. That
this is apriori at random to be expected may appear from the following:
Pose a bag with 100 marbles. They have a number of different colours,
some colours are singular, some occur on 2 marbles, some on 3 or more. The
marbles all are drawn under replace. The results of the total turn of 100
drawings under replace are recorded and a new bag is composed, so that the
colours of the marbles are distributed following these results. It than appears
that the composition has been changed: Some colours have been disappeared and
some colours that were singular in the first bag now are present in twofold or more. At the
second turn, starting from the results of the first bag again the composition
changes evidently. If these drawing turns are ever repeated more and more
colours will disappear (extinction) and ultimately after a big number of turns
only one colour will remain in the bag (fixation). The same experiment can be
executed as well with the help of a computer in a bag with 10^8 marbles, in
which some colours are present on 10^6 marbles or on two or more times 10^6. It
will be evident that in this bag the composition hardly will change in the
drawing turns; 10^6 may become 9.10^5, but not easy 2.10^6 and practical never
0. So the frequencies will hardly change here and can at most fluctuate
somewhat in the turns. Yet is the change, that a singular allele (marble) is
not drawn and will disappear in a population (bag) of 10^8, nearly equal to
that in a population (bag) of 100 and so the change that all the 10^6 alleles
will disappear, is practically zero. This is in principle the model of the
random or neutral genetic change in a population. Essential in this however is
that the non random genetic change, the selection, comes upon to this as a
parameter of the chance distributions. As well in the case of selection are
these drawing turns valid in the model, but the drawings than are not ‘honest’.
In the selection for instance the red
marbles will have a smaller chance to be drawn and the green ones a larger
chance than at random, because the red ‘marbles’ are unfavourable alleles and
the green ones are favourable for the survival and the reproduction of the
individuals. This is the essence of the
microevolution that is elaborated here further.
Genetic Drift
This process by which the alleles
will vanish or settle totally in a close population with limited size is called
in literature genetic drift. So the allele frequencies always become 0 (in extinction)
or 1 (in fixation) and after a longer period this also occurs in larger
populations. The heterozygosis and thus the genetic variation within a
population is getting smaller and smaller by this genetic drift. By the genetic
drift arise ultimately a population that is genetic total identical, which is
of course also total homozygote if there were no mutations. Theoretically the
population becomes even identical exclusive by descent, after it was already a
long time homozygote and identical in general occurring alleles, but in
practice this event is not likely because the population will dye out before.
From the binomial distribution Sewell
Wright deduced there is a decrease in the heterozygosis [2] by
the drift with the average factor (1 1/2n) per generation. In this is n the
size of the population and so 2n the number of alleles on the loci in a diploid
population with sexual procreation. This decrease is to be calculated with the
formula H g +1 = H g [ 1 – 1/2n ],
in which Hg is the heterozygosis in generation g. This means for instance that
in a population with 50 animals participating in the procreation is a decrease
of 1% per generation. So this is an important problem for many threatened
species. This decrease does not mean however that such a population will be
total homozygote and genetic identical by descent already after 100
generations. It yet is an exponential decrease; in general the heterozygosis
changes by a factor ℮^1 = 0,3679 after 2n generations, so the decrease
than is 63,2%. After a x 2n generations the heterozygosis changes with a factor
℮^a. In this is ℮ the logarithmic base, so ℮ = 2,7183.. This
decrease in the heterozygosis at the genetic drift is based on random
inbreeding. The drift to extinction or fixation of the alleles can be
intuitively apriori approached in two ways:
1^{st} By the inevitable or random inbreeding in a close
population arises homozygosis, so that the heterozygosis decreases, being its complement.
This process implicates imperatively the vanishing of some alleles and the
increase of their alternatives on the loci until it remains only one, but now
is it not easy to guess how this will happen.
2^{nd} By random sampling there ever is fluctuation of the
numbers of the alleles, but if the decrease goes incidentally to zero there is
no way of return. This makes the curve of the chances for the smaller numbers
asymmetrical. This vanishing of some alleles means the increase of their
alternatives on the loci and so also the increase of the homozygosis and
decrease of heterozygosis. This happens in a population with limited size as
well as in the unlimited population. This vanishing or extinction of the
alleles however is limited in a pool with a limited number of alleles, because
not all the alleles can disappear here. There must remain in the limited pool
one of all the possible variations and in the unlimited pool are infinite
variations and so there will remain nothing. If this happens there is fixation
in the limited pool, with a fixation chance 1/2n.
Which allele will be fixed by the drift and which will vanish is of
course not to be predicted. You can pose the allele with the largest frequency
on the datum date at start should have the greatest probability. The
differences however in the probabilities often are very small, because there
are many events with random fluctuations between the datum date and the real
fixation. In a population of some size it will last a very long time till an
allele is fixed, but the increases and decreases of the allele frequencies can
go fast temporarily. Conditions for the
genetic drift as it is described by the formula: H g +1 = H g [ 1 – 1/2n ] are:
1^{st} The close
population without genetic exchange. 2^{nd} The constant size of the
population. 3^{rd} Random breeding, so no more or no fewer inbreeding than at random. 4^{th}
There is no selection. 5^{th} There arise no new mutations after the
datum date. 6^{th} There is no mating between the generations. 7^{th}
Selffertilisation is possible, because the individuals are fertile in both
genders.
Criticism on this model, Disadvantages
1^{st} The great drawback
of this formula is: it describes how the heterozygosis
decreases in a close population, but unfortunately not how do allele frequencies
change in populations, as it is sometimes suggested indeed. The in and
decreases of the frequencies and numbers of the alleles also is not to be
derived from this formula or from this model. Insight in the random and non
random changes however is essential for insight into the micro evolution.
2^{nd} This formula H g +1 = H g [ 1 – 1/2n ] only is valid in very
restricted situations, because of the above called conditions. Further on it
is, I think, disputed if this formula and model fulfil if the population has
more than 2 allelic variations on the locus. If for instance 4 different
alleles a; b; c and d are at start on the locus, the extinction of any of these
alleles should be described as a separate process. Yet the vanishing of the
first allele is not lied with the fixation of the last allele. Evident further
is that a number stochastic processes independent of each other can not be
described as one process with one formula. So this should mean that another
condition for the formula is: there should
be only two allelic variations for the locus in the population.
3^{rd} The formula appears than also not applicable in many real
situations. It can not describe for instance how a new arisen and thus very seldom mutation often disappears
very fast from a large population. Also the fast genetic changes that arise in
populations shortly after they got isolated can not be explained well by this
formula. These fast genetic changes arise
for instance in animals that got isolated in small populations after
people did disturb the ecology of their old life area. If so a mother
population splits into a number of deems there will be initially in these deems
alleles singular, in twofold, in threefold etc and by the small numbers of
these alleles many of them will disappear in a little generations. Also the
stocks in descent of the domestic animals are models of these very isolated
populations, that underwent impressive genetic changes in the course of a
restricted number of generations. The different races of the domestic animals
may origin from source populations of
minimal 50 to about 1000 of animals. According to the formula H g +1 = H g [ 1 –
1/2n ] the
heterozygosis should decrease in these effective populations with ca 1% to ca
0,5‰, per generation, while thus the observation indicates us that the changes
in the genes in these populations must have taken place much faster.
The large advantage of the formula however is that it is simple and
gives good and easy insight in the important aspects of the genetic changes:
the decrease in the heterozygosis and the increase of the homozygosis. This
easy calculation of the heterozygosis in this model, means thus a reduction,
which restricts the flexibility of this model in the different situations. By
this it only is possible to get more specific information with very complicated
further calculations, that than again do not give any more at all the simple
intuitive insight in the biologic events. So it could be useful and is any way
harmless trying to approach this matter in another way with models primary describing
what will happen in general with the allelic variations in a close, limited
population and in the theoretical unlimited population of Hardy and Weinberg.
In search of another model
In this model is started from the generation F1, that is born, or arises
and receives at random alleles from the former generation F0 on a distinct
locus in the genome. The size of the population in generation F0, F1, F2, etc
is constant on n examples. Thus are 2n alleles on the diploid loci, so that the
chance that a distinct allele of F0 comes into the zygote of F1 is 1/2n and the
chance that this allele does not come into the zygote is 11/2n. In this way
for all the n zygotes in F1 are ‘drawn’ 2n alleles for the locus from the
generation F0 alleles. Standard should be drawn in this way all the alleles or
gametes of F0 and so should be passed the total set of alleles from generation
F0 to generation F1. This standard event however is in reality as likely as a
long street in a poker game with a lot (2n) of different cards. Always are
drawn a number of alleles two times or more and an accordingly number are not
drawn. We can follow with the aid of a game with marbles or a computer module
of it what are exactly the fortunes of the genes with their potential and real
allelic varieties in a population. We start with a bag of 2n marbles, that all
have a singular number 1; 2; 3; …2n. These numbers represent the separate, in
generation F0 singular alleles, or potential variations of the genes. Further
on the marbles do have colours so that some marbles have the same colour. The
colours indicate the real existent gene variations. 2n marbles are drawn under
replace and the drawings are recorded. After a turn of 2n drawings the contents
of the bag is replaced by the results of the 2n drawings as recorded. So after
the first turn of drawings the first bag, F0, is replaced by the second, F1,
and so on. It than appears from the recordings that already in the first turn a
lot the of numbers on the marbles is not drawn and that many numbers are drawn
2x and some 3x or more. Also the colours did change in number in this way and
some very seldom colours were not drawn. At the second turn from bag F1 is
formed bag F2. Now also many numbers are not drawn, but less than at the turn
from bag F0, because in F1 not all the numbers are singular. If these turns are
ever repeated the singular numbers the singular numbers of F0 will vanish ever
more giving rise to increase of the frequencies of the remaining numbers and
colours. The numbers on the marbles and their colours are fluctuating in the
further turns The numbers and colours will decrease some turns and than again
increase, but if they go to zero no return is possible and it vanishes. By
these vanishing the remaining numbers and colours are ever increasing and by
this the vanishing becomes ever more seldom in the later turns. After a large
number of turns will remain only one colour and later also only one number. If
the bag contains a small number (2n) of marbles this process of fixation goes
very fast and if there are many marbles the fixation is slow and is only
possible after a huge number of turns. In the beginning however will the
singular numbers on the marble vanish in the large bag as fast as in the small
one. If the remaining numbers on the marbles become somewhat larger to about 50
or 100 than they will only very seldom disappear. The colours will be present
in the large bags already at the first turn mostly in numbers of more than 100,
so that they will scarcely disappear from the beginning.
This model of the marble game is a simplification, a reduced principle,
that does not describe the total biological reality. This hazard game model
than also does not intend to describe the total biological reality of the
genetic changes, but only the hazardous events in this. That is why it must
have indeed intrinsically restrictions: The possible gene changes caused by the biological functions are to be excluded in
the model. Unfortunately the model must have also extrinsically restrictions:
For reasons of clarity and surveyability not all possible extrinsic events can
be included in a model. It is convenient
to describe a model within a standard situation with exclusion, or freezing of
all possible events. Later on some events may be included into the model as a
new parameter. These restrictions are mostly the same as in the general model
of the genetic drift, as they are:
1^{st} The close
population without genetic exchange. 2^{nd} The constant size of the
population. 3^{rd} Random breeding, so no more or fewer inbreeding than
at random. 4^{th} There is no selection. 5^{th} No new
mutations do arise after the datum date. 6^{th} There is no mating
between the generations. 7^{th} Selffertilisation is possible, because
the individuals are fertile in both genders.
The conditions random mating and no selection are largely or totally
intrinsically, as they are causal lied with the biological functions. The
events that may open the population and will change its size can be both
extrinsic and intrinsic. The arise of mutations is seen mostly as an extrinsic
factor, but this may be disputed. The mating between the generations is
intrinsic. The fertility in the genders and their participation in the
procreation is a biological or intrinsic factor.
Most of these restrictions are described as a parameter further on here.
This is not the case in the mating only within the same generation. The
possibilities of allele transfer in genealogic studies indicate however the
influence of mating between the generations on the allele transfer may be
small, because this mating does not cause inbreeding. It can cause however some
fluctuation of the effective size of the population while the real size remains
constant. Making models and calculations with this mating is difficult. The
measure of the mating between the generations depends from the length of
generation time in relation to the period of fertility and reproduction of the
individuals. Mice can mate with much more generations than people. The random
possibility of selffertilisation itself in somewhat larger populations is very
small and thus unimportant. If the individuals of the species have separated
genders and can be fertile in only one gender this also is of no influence if
these both genders participate equally in the procreation. The problem however
is that in practical live the genders do not equal participate. The
observations learn us that more individuals of the gender that ‘invests’ the
most in the next generation participate in the procreation than those of the
minor investing gender, thus mostly the masculine. The phenomenon of biological
functions by which is caused this unequal participation of the genders in the
procreation is called sexual selection. It is possible and often practised to
correct for this intrinsic factor in calculating the effective population size
for these cases, but it is disputed if it always is useful to correct in the
biological data to get the random situation.
The potential most important restriction of this model however is that at 8^{th} the transfer of the
alleles from generation Fn to F(n+1) must be one uniform event. In reality
it yet is a composed event. So this
restriction is very important, but it is not generally acknowledged in the
literature I guess. As pointed out before there is in fact a drawing or
distribution of:
1^{st} The reproduction. The individual organisms of the parent
generation F0 draw different numbers of effective descendents, that reach
adultness and are able to reproduce themselves.
2^{nd} The effective descendants can draw different alleles from
their parents and further ancestors.
It is pointed out here further on[3]
that it is possible and in many situations necessary to make a model with
specification of these two drawing events.
Further more it is possible to extend this model by putting more data
into it in order to get more information about the changes of the genes in the
course of time. So the marbles can have besides their number and colour also
other marks by means of which can be reed
for instance which individual is carrier of the allele and which was the
carrier in the former generation. With data like these the genealogy within the
population can be followed and so you can have much more information about
random genotypic distributions, the measure of homozygosis and especially the
important random changing linking as there is between the allele on a distinct
locus and many other loci of the genes of the ancestors. Many interesting
computer models can be made for the study of these problems. Primary however is
this simple reduced model. But besides of the models it is necessary to
describe in algebraic terms what happens at random to the genes and what
happens in essential in the biological reality:
Deduction why the
Poissonexponential distribution is appropriate.
In a population are n individuals, so 2n diploid alleles are in the
model and they are seen as singular, so that they represent the potential
variations on the loci of the genes in the total population. If the size of the
population remains constant, the chance that one distinct allele is drawn in
one fertilization, so in one descendant, or is transferred from F0 to F1, is
1/2n and its complement, the chance that this allele is not drawn thus is
11/2n. This means that this distinct allele is not drawn on the average in 2n
draws, so in one generation (11/2n)^2n and so it is than not transferred in
one generation. It appears now that this relation (11/2n)^2n converges fast to
1/℮, for if n→∞ becomes (1  1/2n)^2n = 1/℮, in this is
℮ the base of the natural logarithm, so ℮=2,7183.. The conversion
goes fast, if the size of the population n=10, it is (19/20)^20=0,3585 so that
the ‘base’ than already is 2,7895, only 2,6% more than ℮. So is 1/℮,
or ℮^1 the proportion of the singular alleles that is not transferred
from generation F0 to F1. The alleles however only can be transferred in this
way at random in a population with individuals that are fertile in the both
genders. In a population with ½ n
individual of the masculine and ½ n of the feminine gender the alleles are
‘drawn’ or transferred separately for the genders. In both of the genders n
alleles are present and are drawn. At one fertilization one allele is drawn in
both of the genders. In this are the proportions 1/n and 11/n transferred
respectively not transferred and so the proportion not transferred is nearly
1/℮ in somewhat larger populations. So is indeed this principle also
valid in separated genders, if the participation in the reproduction is equal. In
somewhat larger populations (n>ca 10) singular alleles are not transferred
and will vanish in the proportion or in the rate 1/℮=0,3679. In the
smallest possible population, if n=1, so in self fertilizing, this rate is (1
½ )^2=0,25. Further it is obvious in a population with 2n alleles that if the
different alleles are not singular but are present in absolute numbers 1; 2; 3;
or q they are transferred or not
transferred to one descendant in proportions q/2n and 1 q/2n respectively. In
2n drawings, so in one generation they are not transferred in the proportion (1 q/2n)^2n. This is if n→∞ (1/℮)^q=℮^q. It is evident to
that if the effective size of the population is not constant, but changes by a
factor p and the alleles do occur in the number q, these alleles are
transferred of not transferred with chances, or in proportions pq/2n and 1
pq/2n respectively. So in one generation are (1 pq/2n)^2n alleles not
transferred. This is for n→∞
℮^pq. This formula P0 = ℮^qp is easily to be deducted and is than
also applied in many specialities. If a number of events occur in a period of
time and the events appear ‘memory less’ in general is valid: P0(t) = ℮^qt. In this is
P(0)t the chance on no observation or hit of any event within period of time t.
The complement of this, Pi(t) = 1 
℮^qt, is the exponential distribution. So it is the
chance on one or more ‘events’ ‘arrivals’ or ‘hits’ within period of time t. This
period may be a constant, for instance the time of one generation. The events
or arrivals can be drawn or transferred alleles, if they are transferred memory
less at random. This is the case in this
biologic field if any individual has any moment the same chance on effective
reproduction.
This (negative) exponential distribution is used generally in science. It is a statistical
distribution, but you can see it also as an essential natural law. It also is
supplied for instance in the field of epidemiology. An unfortunately realistic
instance for illustration: a group of 10 young people goes to
In the Application of the Poissonexponential
distribution in this field are taken as example of events, drawings, etc the ‘arrivals’ of the numbers of descendants
or alleles into the next generation. The starting lemma’s as condition at this
application are the lemma’s of the neutral hypothesis, the hypothesis of the negation
or the zerohypothesis of the evolution by selection:
1^{st} Any individual has any moment the same chance on
effective reproduction and any allele has any moment the same chance on
transfer.
2^{nd} Differences in reproductive success between parents and
differences in the transfer between alleles are caused by accident.
These apriori and aposteriori conditions, which differ only in meaning
concerning the time of observation: before or after the event, are apparently
not present in the actual life of the organisms. The reproduction is yet in
many species seasonal, so that at once are born a number of cubs as a litter.
So the chances on mating and reproduction are apparently not memory less,
because a large part of the year the animals are not for mating disposed and
not fertile as do the plants in most climates. This however concerns the actual
reproduction we can easily observe in nature, but for the study of the genetic changes and the evolution we have to
observe the effective reproduction in the nature which is much more difficult.
In the effective reproduction are counted only the cubs that grow adult and get
a litter themselves. The observation of the effective reproduction of course is
necessary, because the genes and their variations are transferred only via
individuals of the following generations that survive. The effective
reproduction as the balance of birth minus juvenile mortality and infertility is
much more or perhaps total time independent, because the death hazards are
memory less and the juvenile mortality takes mostly a large part of the birth figures.
The above conditions concern for instance the question if an animal that
becomes a grandparent will have after this event the same chances to get
effective descendants than before this. Further on are differences in chances
if smaller scales are concerned which cause fluctuations as is pointed out here
later, but this may be averaged on the larger scale. For instance in an area or
in a period of drought the litters are smaller than in an area or period of
abundance and this may effect even the effective reproduction and so by this it
is fluctuating. On the larger scales however this may be averaged and than the
condition ‘the same chances in any time’ should also be taken larger. The
neutral hypothesis of equal chances and thus random causes for the differences
in the reproductive results and ultimately for the evolution can be tested by
observations of the results of the effective reproduction. Primary are equal
chances on effective reproduction and not equal reproduction. Real equality of
effective reproduction and allele transfer is excluded, because in nature does
not exist something like rationing systems for mating, birth and dying. So not real
or potential existing equality, but equal chances in effective reproduction and
transfer of the genes should be the (negative) basis for the evolutionary
theory.
The Poisson distribution is used here in the explication of the neutral
random theory for the general populations. This because it concerns here at
first the descendants of one person or the transfer of one or a very small
number of alleles in a relative large population. In very small populations as
pointed out later on here the binomial distribution is taken. The Poisson
distribution can be used best in this cases with small numbers in a large space
and it also has great advantages, I think: its flexibility and its simplicity.
The Poisson intensity does describe the average events in time and this intensity,
or ‘λ’ can easily be defined by
parameters as is done here: For the
distribution can be used the formula P(i) = ℮^λ . λ^i/i! in the standard generation time. In this field the Poisson intensity λ apparently is determined by
some factors, q, or Q, p, r and s, so that λ=qprs
or λ=Qprs. In this is Q the primary quantum the
absolute number of ancestors in the parent generation F0 (which always is
So are explained some basic conditions for the neutral theory, which is
elaborated further in the chapters about the calculations of the Poisson
distributed reproduction and allele transfer. At first however some
observations and practical implications are concerned to show the relevance of
the random neutral theory as the basis of the total evolutionary theory.
Observations
It is possible to examine in what extend the participation in the
effective reproduction and thus the transfer of genetic variations is indeed
Poisson distributed. It is possible but not easy to count numbers of effective descendants
of the living animals and plants in the nature. This is a lot of work, but the
statistic data of the numbers of children people do have in the different
counties are direct available:
Table 1
Table H2. Distribution of Women 40 to 44 Years Old by
Number of Children Ever Born and Marital Status: Selected Years, 1970 to 2004 












Source: U.S. Census Bureau 









Internet
release date: 










(leading dots indicate subparts) 









(Years ending in June. Numbers in thousands) 







Year 
Women 4044 yr x1000 
Women by number of children ever born in % 
Children ever born per woman 

Total 
None 
One 
Two 
Three 
Four 
Five and six 
Seven or more 












All Marital Classes 










.1976 
5684 
100 
10,2 
9,6 
21,7 
22,7 
15,8 
13,9 
6,2 
3,091 
Poisson
λ=3,091 
100 
4,546 
14,051 
21,715 
22,374 
17,289 
16,195 
3,827 


.1982 
6336 
100 
11 
9,4 
27,5 
24,1 
13,8 
10,4 
3,9 
2,783 
Poisson
λ=2,783 
100 
6,185 
17,214 
23,953 
22,22 
15,46 
12,596 
2,373 


.2004 
11535 
100 
19,3 
17,4 
34,5 
18,1 
7,4 
2,9 
0,5 
1,895 
Poisson
λ=1,895 

100 
15,032 
28,485 
26,99 
17,049 
8,077 
4,028 
0,34 












Women Ever Married 










.1970 
5815 
100 
8,6 
11,8 
23,8 
21,4 
14,6 
12,9 
6,8 
3,096 
Poisson
λ=3,096 
100 
4,523 
14,003 
21,168 
22,371 
17,315 
16,254 
3,858 


.1976 
5455 
100 
7,5 
9,6 
22,4 
23,4 
16,4 
14,4 
6,3 
3,19 
Poisson
λ=3,190 
100 
4,117 
13,134 
20,948 
22,275 
17,764 
17,359 
4,396 


.1982 
6027 
100 
7,6 
9,6 
28,7 
25,1 
14,3 
10,8 
4 
2,885 
Poisson
λ=2,885 
100 
5,585 
16,114 
23,245 
22,354 
16,123 
13,776 
2,804 


.1985 
6836 
100 
8 
12,9 
34,2 
24,1 
11,4 
7,4 
2 
2,548 
Poisson
λ=2,548 
100 
7,823 
19,935 
25,397 
21,571 
13,741 
9,976 
1,557 


.1988 
7543 
100 
10,2 
14,7 
37,3 
22,1 
9,5 
5,2 
0,9 
2,28 
Poisson
λ=2,280 
100 
10,228 
23,321 
26,586 
20,205 
11,517 
7,247 
0,896 


.1998 
9995 
100 
13,7 
18,1 
38,7 
19,6 
6,2 
3,2 
0,6 
2,002 
Poisson
λ=2,002 
100 
13,506 
27,04 
27,067 
18,063 
9,04 
4,828 
0,456 


.2004 
10036 
100 
13,2 
17,4 
38 
19,9 
8,2 
2,9 
0,4 
2,046 
Poisson
λ=2,046 

100 
12,925 
26,445 
27,053 
18,45 
9,437 
5,179 
0,513 

The US Census Bureau did collect in her table H2 the data of all the
women from the total American population. The figures of the US Census Bureau
are given here in Table 1 and they
show that particular in the short period from 1980 to 1990 has been a sharp
decrease from about 3 to
We see in Table 1 that the initial
figures of the years ’70 and begin ’80 show some resemblance between the
observation and the expectation following the Poisson distribution. Also in the
group ever married are the figures under no children however larger than
expected, probably because there have been always a minimum number of families
that keep childless by biological infertility of one of the partners. In the
later years the childlessness is more in accordance with the Poisson
distribution and this is an indication for the small difference between the
actual and the effective reproduction especially in the later years. In general
are in the observed figures fewer women with 1 child, more with 2 children and
fewer in the higher parities, although initial were the observed figures the
very high parities larger. These general tendencies are increasing in the years
and all the high parities than have lower figures later on. The aspect than in
the later years of the observed figures in relation to the Poisson is that of a
shift from the extreme values to the average value (=2). So the divergence in
the distribution of the natural parities becomes obvious smaller than in the
distribution of the parities following Poisson. So the genetic differences
between the generations have become smaller the at random. This is evidence for the view there is no (more) selective evolution in
the modern American population. These differences are large and the
divergences are so much smaller than random that it is very unlikely that
figures from the effective reproduction should give another indication. The observed
figures also did differ initially from the Poisson figures. The figures here
are somewhat conflicting: the average values (=3) are about equal, but there
was initial a shift from the moderate to the more extreme values in the observed
distribution. So this is indicative for a little larger divergence and thus is likely
a larger genetic difference between the generations than at random.
The divergence in the observed distributions in relation to the random
Poisson distribution is an important datum, which directly indicates the
changing of the genes and so the evolutionary intensity of a population.
To learn the relevance of this you
should inquire some more characteristics of the population. A population that
is very heterogenic in the reproduction will have a large divergence, but such
a population may be not a real existent social or natural group of individuals
and than its evolutionary intensity is not so relevant. Such a population here
in Table 1b is for instance the population women never
married with 0,88 children per woman on average and a large divergence. The
descendants of this population will of course be genetically different in the
next generation, but that may be not relevant. This population must consist of
women that are real singles and get only a little children and a group that
have about average children and a family life, but they only are not married
for the law and there may be others. So it is necessary to inquire the
composition of the populations and the US Census Bureau gives than also the
figures of the typical American subpopulations as they are called: Whites,
Blacks, Asians and Hispanics of any race. Some of their data are given in Table
1b and I do compare these also with the Poisson distribution. In trying to make
a better comparison and to estimate the evolutionary intensities I used a provisory coefficient in the right
column. This is an instance, this method is not satisfying, I think, but is
must be possible to develop here good exact methods. There may be relative
small differences between the American subpopulations in these aspects of
reproduction and genetic evolution. It is possible that these differences
between the European subpopulations are larger but their data are not, or more
difficult available.
Table
1b












Table 1. Women by Number of Children Ever Born by
Race, Hispanic Origin, Nativity Status, Marital Status, and Age: June 2004 



(Numbers in thousands. For meaning of symbols, see table of
contents.) 





 






(leading dots indicate subparts) 










(Column B is in persons, all others are percents) 








Women by number of children ever born 






age 
total
x1000 
Total

None

One 
Two 
Three

Four

5 and 6

≥ 7 
children

coëf
MP 

women 
women % 
% 
% 
% 
% 
% 
% 
% 
ever
born 

ALL
RACES 








per
woman 

All
Marital Classes 











.40
to 44 
11.535 
100 
19,3 
17,4 
34,5 
18,1 
7,4 
2,9 
0,5 
1,895 
0,868 
Poisson
λ=1,895 
100 
15,032 
28,485 
26,99 
17,049 
8,077 
4.028 
0,34 















ALL
RACES 











.Women
Ever Married 










..40
to 44 
10.036 
100 
13,2 
17,4 
38 
19,9 
8,2 
2,9 
0,4 
2,046 
0,723 
Poisson
λ=2,046 
100 
12,925 
26,445 
27,053 
18,45 
9,437 
5,179 
0,513 















ALL
RACES 











.Women
Never Married 










40
to 44 
1.498 
100 
59,8 
17 
11,2 
6,2 
2,1 
2,9 
0,9 
0,88 
1,973 
Poisson
λ=0,88 
100 
41,478 
36,501 
16,06 
4,711 
1,036 
0,042 
0,001 















WHITE
ONLY 











.Women
Ever Married 











40
to 44 
8289 
100 
13,4 
16,8 
39,3 
19,7 
8 
2,4 
0,3 
2,02 
0,694 
Poisson
λ=2,02 
100 
13,266 
26,796 
27,064 
18,223 
9.203 
4,97 
0,478 















WHITE
ONLY, NOT HISPANIC 










.Women
Ever Married 











40
to 44 
7206 
100 
14,1 
17,2 
39,8 
19,6 
7,1 
1,9 
0,2 
1,959 
0,718 
Poisson
λ=1,959 
100 
14,1 
27,622 
27,056 
17,667 
8,653 
4,498 
0,406 















HISPANIC
(of any race) 










.Women
Ever Married 











40
to 44 
1179 
100 
8,1 
14,3 
36,1 
20,5 
14,6 
5,2 
1,2 
2,437 
0,854 
Poisson
λ=2,437 
100 
8,742 
21,305 
25,96 
21,088 
12,848 
8,8056 
1,251 















BLACK
ONLY 











.Women
Ever Married 











40
to 44 
1.054 
100 
12,3 
20,3 
27,9 
24,7 
9,5 
4,2 
1,1 
2,198 
0,928 
Poisson
λ=2,198 
100 
11,102 
24,403 
26,819 
19,65 
10,797 
6,485 
0,743 















ASIAN
ONLY 











.Women
Ever Married 











40
to 44 
470 
100 
12,9 
20,5 
40,1 
13,4 
6,3 
6,1 
0,7 
2,052 
0,711 
Poisson
λ=2,052 
100 
12,848 
26,364 
27,049 
18,502 
9,491 
5,227 
0,519 















Source:
U.S. Census Bureau, Current Population Survey, June 2004. 




















A situation different from the data of the US Census Bureau shows the
picture of the historical data on Table
2. This population is describes more detailed at Table 9b. In this population is described concretely the effective
reproduction. Used are data from a genealogy of a family of fishermen and
skippers living in the Southwestern of the
Tabel 2
→0 
→1 
→2 
→3 
→4 
→5 
→6 
→7 
→8 
→9 
Poisson,
λ=3,05556 









0,0471 
0,14391 
0,21986 
0,22393 
0,17106 
0,10453 
0,05324 
0,02324 
0,00888 
0,00301 
populatie
n=72 
gemiddeld
3,0555 kinderen per ouder 





→0 
→1 
→2 
→3 
→4 
→5 
→6 
→7 
→8 
→9 
0,09722 
0,194444 
0,22222 
0,11111 
0,06944 
0,13889 
0,08333 
0,06944 
0 
0,01389 
7/72 
14/72 
16/72 
8/72 
5/72 
10/72 
6/72 
5/72 
0 
1/72 
There are many publications about birth and fertility figures in the
various countries. Data about the parities are however much more difficult to
be found and are apparently not collected in most counties, but there are other
countries than the
Figure
2.9Distribution of Women by Number of Children Born by Age
Random or[4]
selective changes
The premise of random mating, which often is pointed in literature, can
be described in plain English as: any individual has any moment the same chance
on mating. This mating can better be specified to its consequence we are
interested in: the effective reproduction. This random effective reproduction
does exist actually in real existing biological populations, if the condition
equal chances have been fulfilled and if they have not been fulfilled, as is
mostly, the random effective reproduction will exist only potentially in the
population. The dialectical neutral theory starts from the idea: there are
possibly random differences and nonrandom differences in the reproductive
results individuals have and in the transfer of the individual alleles. It is
possible with the modern technical tool to observe in all kind of levels and
situations in actual populations differences in these both processes: the
effective reproductive results and the transfer of the gene variations
(alleles). As pointed out further here (page 55) reproduction and allele
transfer are causal linked and in the random form they (or it) even exist
uniformly. The observational results are of course the sum of random and
nonrandom differences. The random differences are wellknown, as they are
given by the Poisson distribution. This
also is ascertained in literature. The observed distributions of the effective
offspring thus will mostly differ from the Poisson distribution. This fact thus
should nowise induce the meaning that the statistical distributions are of none
or very limited importance in the inquiry of genetic changes in the
microevolution. The random events are of course physical present, but what we
observe on living beings never is the result of hazard alone. Children of
people and animals are not born and do not dye only by accident. The results we
can observe like products of effective reproduction and so gene transfer always
are a combination of hazard and biological action or function. The latter of
this, the organic skill to procreate and survive, can be defined as selection.
So selection than also is the causal unity of the nonrandom differences, which
is the dialectic complement of hazard in the struggle for life and so has here
a somewhat more specific meaning than in the pure Darwinist sense. Although..,
there is survival of the fittest and there is survival of the luckiest, but only
the survival of the fittest is survival by selection. It ever is important I
think, in philosophical and scientific research to find the essential detail in
the background of the noise of accidental events.
Random transfer or selection
at work
In the data of Table 1 is for instance an about 30% smaller variation
than Poissonrandom in the parities in the data, the US Census Bureau collected
in the later years, of women getting children in the years about 1970
Another picture gives the reproduction of salmons. A population of
salmons is reproducing far in the upper reaches of a distinct river. The
circumstances in the spawninggrounds are rather uniform. The salmons seldom
reproduce more than once in their lives and dye afterwards. The next generation
hatches from the egg and grows in the
river and further in the sea. A small part of the young salmons return later into
the source river as an adult for reproducing and dying. As for concerning the
parent generation the young salmons will apparently behave as real equal luck
‘Poissons’ (Fr: fishes). The parent salmons did yet produce about the same
numbers of fertilized eggs and in this river the young’s became all about the
same change on growing and survival from their parents, that brought them
hitherto. It is all up to the youngsters now. The eventual nonrandom or
selective differences and so non random allele transfer may appear in the
possible differences in chances to survive and to grow up to adulthood and to
swim at last all the way back to this source river. If some salmons have more
chances on success than others, this will give rise to more variations in the
distribution of the offspring and in the larger genetic differences between the
generations in the population. This possible selection than will also
strengthened by the inbreeding the
salmons will probably have by mating in relative small populations for many
generations in the source rivers.
By birds and many other species again is the situation different. Many species
of animals and even plants will give indeed some form of parent care to their
living children or give them something to survive better. The salmons were
particular in this because they do invest into the offspring already before
mating by swimming up to grounds that are favourable for the survival of the
brood. The real parent care birds give come after mating, brooding and
hatching. The survival of the young generation than depends on the dedication
and the possibilities of the parent(s) and the young itself, so of some more genetic different individuals. The
difference in the success the parents have at the raising of their young’s
often is based on good luck or blind evil. It so can happen that at one year there
is a drought in their living area, so that the parents can find less food for
the young’s than do parents in other living area’s. Such incidental differences
are not important, for the nonrandom distributions of the population in
somewhat larger scales may stay valid, because in more litters and more
generations the incidental differences will compensate each other. It is
important that the differences do exit systematically as capacities in the care
and in finding the food some parents may have more than other parents. Because
the young’s themselves are present at this care the better caring capacities
can be transferred in two ways through the generations: by the genetic transfer
and by the imitation, because the young’s will imitate their parents behaviour
later when they get litters themselves. This imitation can make some stocks of
birds systematically more successful in survival and reproduction. This however
may happen without regard of the characteristics of some genes. The genes of
these more successful individuals, because of their acting well by imitation
will also be transferred more than at random, although these genes did not
attribute to the success. In this can the transfer of some illfunctioning
genes accidentally be promoted it they occur in smart acting birds. This,
however may cause drawbacks later on, because physical defects in the birds can
develop by the accumulation of less functioning genes. So on the larger scales
the nonrandom differences are more consistent if the are transferred indeed by
the genes.
Some evolutionary problems in
people
From the beginning some million years ago the human (hominid) species
have had to give  in relation to average animal species  a very intense and
lasting care for their children and this further is increased during their evolution.
They did have than also more abilities in this care than the animals had and
could transfer their abilities more effective through the generations than the
other species. They were able to do this and many other things, because their brain
is very large in relation to individuals of all other species with about the
same bodyweight. This brain makes people capable to use their sense organs more
efficient and that improvement of their sensory perception was very useful for
the people in their care for the children, their defence against predators and
their search for food. The possibilities of the brain however go much further,
we do know now as modern people. Your brain makes also possible to perceive the
things behind the things. This deeper perception however of the causes etc
behind the things was a huge problem for the primitive people in prehistory and
they generally avoided to gather information about the things behind the
things. If they did inquire these or should invent new possibilities this ever
brought themselves and their tribes in large difficulties. Although the brains
of our direct ancestors, homo sapiens and perhaps also of the other hominids as
homo rectus and his later form the homo Neanderthal could work as well as ours
or perhaps even better, those people ware not able to use their brains as we do,
because of their social situation. An important problem is yet that people do observe
the world indeed much deeper than animals do, because much more efficient receivers
are opened for the information from the outside, but people do have the same anxious
cautiousness animals have for survival in dangerous surroundings. This excess
of info about potential dangers makes ancient people, but often also modern
people, very anxious and also very aggressive. The problems are much aggravated
by the consciousness people have of the things behind the things and the
communication about these with each other. People will so, in lack of modern
knowledge, experience various threatening phenomenon’s and see a whole
threatening world as causes for different events. They can be very anxious for
the thunder, for the shining of the moon, for the strange behaviour of other
people being friends of evil ghosts, etc. The destructive consequences of magic
and fear were moreover not the only disadvantages of the peoples brains. In
this matter also important is the interference between the transfer of the
genes  the genetic information  and the transfer of the ‘neuronic’
information through the generations: More than at the birds the problem here
was if your tribe or stock of people have success by their smart solutions they
may gather illgenes. Drawbacks by this must have happened, but this problem
will be prevented mostly by a smash with the knout on the smart brain that should
not let people behave strange and thus evil. The human evolution has been a
very complicated process, which is only partly unravelled by the scientific
research, I think. The microevolution in this, so the genetic transfer through
the homo sapiens generations in about 200000 years, of course is of special
importance. A question that rises is how fast was the evolution in the sapiens
generations, or how large are the genetic differences between the generations.
The problem is that the indications for the answer to this question are
conflicting:
Apriori there is indication for fast evolution within sapiens, because
the nonrandom or selective differences are made by the biological, social and
whatever functions of the organisms selves. Humans are relatively well equipped
to achieve their targets and do often use aggressive and radical modes to do
so. For the things they can prefer, as are the appearances, it thus must have
gone very fast. So, many people in our part of
Also what we know from the historical and prehistorical data about the
reproduction possibilities in the live of the people is indicative for fast
evolution. The more different possibilities will cause probably more variations
in not modern people than average in most species. You can imagine that when
people became somewhat more knowledge together with increasing social
inequalities more chances for the privileged groups will arise. Also the often
very aggressive tendencies in the social life of people (also in modern!) as in
war making in combination with killing the conquered enemies, burning down
their possessions and violating their wives may cause strong selective
differences, although in the larger scales many of the effects by these
inequalities may be balanced. An important issue however in the historical view
at the human evolution is that the fitness in ancient times was mostly very
different from what fits in our present society and in our present biological
situation. It often was fit to be an aggressive man raging at people of other
ethnicity and violating their women. This now is very criminal behaviour, but
it is no wonder that you can watch this kind of behaviour everyday in the
streets of our cities. These problems were still worse if behaviour was total
inheritable and people could not correct by intelligence for their natural
tendencies. In other situations the historical–evolutionary problems are still
more evident. In the past many alleles have accumulated in the genomes of
people by selection at fitness towards situations that do not exist anymore. Examples
of this are the alleles that make people resistant for specific infectious
diseases that are easily to control now. The sickle cell anaemia is a famous example
of this. There are found many more of these cases and some will never be found
because the infectious agent has been disappeared for a long time. The cause of
the systematic occurrence of this phenomenon is very simple: The pathogen needs
a key to come into some specific cell of the host organism and it is
specialized by genetic selection to use the key, which often is a protein on the
cell wall. If the key does not function the pathogen has a problem, but also
the host organism. The less functioning key at the heterozygote does increase
the resistance, but this does ever mean also a less functioning protein or
total cell, which means non functioning in the case of homozygosis. This
problem is evident in some monogenetic diseases in people as the different
haemoglobin disorders, cystic fibrosis and others. The problem may be much
broader: also many polygenetic inheritable diseases are possibly induced by
genetic selection. Our ancestors did live close together without any form of
hygiene and also got many traumata and this did them so suffer a lot from all
kinds of infections during many ten thousands of generations. No wonder from
the evolutionary view that we now posses a very aggressive immune system that
easily deregulates giving rise to auto immune diseases, in which the immune
system attacks the cell of the own body and also to allergies with the
exaggerated reactions on harmless vectors.
This comes to the conclusions: People may have had a relative fast
evolution. The evolution now suddenly has stopped, but we have definitely not
to worry about this ceasing. We are on the contrary in big troubles by the
genetic variations accumulated by selection in our ancestors. The genes are of
course now not to be removed by natural evolution, also not by sharp artificial
evolution, or as it is called radical eugenics. The results of artificial
evolution is to be seen at the sad genetic state of our house animals. The
races of the house animals have been bred mostly under veterinarian control,
but there are still gigantic genetic problems. There are also very important ethical
objections against eugenics. A free medical advise for a family in specific
cases is of course something else.
Random transfer or sexual
selection
The distribution of descendants and alleles is, as apriori expected, in
the masculine gender different from the feminine. The masculine and feminine
organisms do reproduce with different properties and these differences will
result in different chances on reproduction and so in a different distributions
of their descendants. The more general differences between feminine and
masculine is not always corresponding with the biological sex even in the field
of reproduction. So it may be better to use other words for describing the
sexual characteristics more general and more typical: yin for the feminine and
yang for the masculine reproduction. In the yin the woman’s part is uniform
within the mother’s part and the pure yin will receive the both parts. The yin
does invest maximal into the child to
create maximal effective numbers of
children. This is attended with few sexual competition and even with
cooperation between the females. The yin will limit to the minimum the
variation in the numbers of children the different females have. A more unequal
distribution of the numbers of the descendants would be unfavourable for the
survival. It yet is not ‘efficient’ human economics would say if the hard work
of motherhood is not borne by all the
females in proportion to ability. This minimal variation in the effective
children is given by the Poisson distribution, if the abilities are equal and
if there is large juvenile mortality. This last condition is mostly present in
nature, but in our modern human populations the infant mortality is very small
and so than the most efficient distribution of the children among the mothers is
with a smaller variation than Poisson. By differences between the females in abilities for the motherhood the yin wants
a larger variety than Poisson and it so creates selection. Particularly in more
intelligent organisms there may be systematic differences in abilities.
In the yang the man’s part is distinct from the father’s part and so in
the pure yang can present only the man’s part. The yang invests minimal in the child for the
possibility to create maximal
numbers of children. This brings much competition between the males and even
with females. In that eternal malefemale conflict the ever heard female
argument is: you always are thinking of the one thing and the male argument is:
you cannot make well more than one thing simultaneously. The yang is working on
distance the effectiveness of his procreation (care) is via the yin. The number
of the partners and their possibilities are the results of the yang, whereas
the numbers of the children and their possibilities are the results of the yin.
So the yang has to follow the yin in the minimum variation augmented by the
variation the yang has in the numbers of partners. This extra variation the
yang has in the parities is created by the differences between the male
individuals for mating in competition with each other. These extra variations
are called the sexual selection. The sexual selection generally is present in
nature. The sexual selection however probably is larger in species that do have
large individual differences, as is in intelligent creatures. If there also is
a large social inequality, as it was in historical human societies, sexual
selection can become extreme large. Some men with much power and high
distinction did have a lot of descendants. So nearly everymen in the
neighbourhood descents apparently from the old celebrities as Charlemagne or
Dzengis Khan. It may be obvious that the selective variation by the fatherhood
(or sexual selection) in general is larger and also has other qualities,
because it has been selected on other characteristics (mating ability) than by the selective variation by the
motherhood (care ability). These differences in quality may be of evolutionary importance: systematic selection
on characteristics. In the larger scales may exceed furthermore the female
selection for instance also the quantity of the male selection, if the larger
differences by the male selection in more generations can be more neutralized
in the allele transfer, while the female selection can be more systematic.
So because yin and yang are different biologic functions it is useful to
observe and calculate the male reproduction distinct from the female. This
however is not done in the genetic drift theory; in the literature the distinct
yin and yang selection always are equalized by the formula of the effective
population size. Oh, girls did not I say that you can not do your work well if
you try to fix the different things at the same.
Calculation of the Poisson
distributed reproduction.
Suppose the size of the population is constant on n individuals in the generations F0; F1 and F2. The reproduction
population in study consists of parents of effective children. Only descendants
in the first generation that have become parents themselves are counted as
individuals of the population. So parents with 0 children are parents that did
not become grandparents. There is random reproduction. So any individual of F0
has the same change of 1/n to be the
parent of the new individual of F1 and 11/n
to be not the parent. For 2n effective
children[6]
of F1 the change to be not the parent is (1−1/2n)^2n. This is 1/℮
for larger values of n. In this is
℮ the natural base 2,7183.. In the case n=10 this “base” already is 2,7895.. On this account the
distribution of the effective descendents in the next generation is essentially
according to the Poissonexponential distribution, if the population is not
very small. So the average proportions of the individual organisms in F0 with i descendants in F1 are to be
calculated by substitution in P(i) = ℮^λ .
λ^i/i! In this the intensity λ can be determined
by parameters, so that λ=Qprs or λ=qprs. In this is Q the primary quantum; q is the general quantum;
p the change in the size of the population; s the theoretical or
virtual selection and r is the replacement factor by neutral population
dynamics. Because the children and further descendents of 1 individual in the
generation of the first parents, the F0, are studied in Table 3 Q=1.
This individual has in many cases (proportions) more than 1 descendant in the
F1, These will than also participate in a number (quantum) >1 participate in
the new parent generation. That is why q is indeed variable and has natural
values as 1;2;3, etc. In this example the population size is constant, so p=1.
By sexual reproduction the individual organisms have on the average 2
descendants in the next generation in neutral population dynamics, so r=2.
There is random mating with equal chances, without selection, so s=1. So
it is obvious that p, s and thus λ can have in
principle any values ≥0.
The average random or Poisson distributed offspring of the individual
organisms of the primary parent population F0 is given in Table 3. If you calculate by substitution in the formula with λ=2 it appears that
the proportion of F0 with 0 descendents in F1 is ℮^2 = 0,1353.. This
proportion participates active in the reproduction but will have no descendants
by random mating with equal chances. The proportion 2.℮^2 = 0,2707 of F0
has 1 descendant on the average in F1. The proportion 4/3.℮^2 = 0,1804 has 3
descendants. 2/3 .℮^2 = 0,0902
has 4 descendants etc. So the average of 2 descendants is in this way Poisson
distributed and the sum of the descendants, calculated in this way indeed is 0,2707x1
+ 0,2707x2 + 0,1804x3 + 0,0902x4 +… = 2. These are descendants that individuals
in the F0 will have together with their different sexual partners so that the
population keeps the same size.
In the distribution of the descendants of F0 in F1 there is only one
intensity λ of the expected
number of children of F0 in F1. This intensity is in random mating only
determined by the population dynamics, so what is necessary for maintaining of
changing the effective population size. But if we consider the descendants of
F0 in F2, the grandchildren, there must be different intensities. The children
the F0 individual has in F1 determine by their numbers q the expected number of their grandchildren of F0 in F2 and so
determine the λ of the
distribution for the new generation together with any possible changes. So,
because of the different numbers of children in the F1, the distribution of the
descendants of the primary parents, the F0, in the further generations has no uniform intensity. As a symbol for
this variable intensity is used λ*,
so that : λ*=qprs. By
1 descendant in F1 the expected number of descendants in F2 is in neutral
dynamics : λ*=qprs=1x1x2x1=2. By 2
children there are 4 grandchildren on the average, because than q=2, so
that λ*=2x1x2x1=4. By 3
children there are 3x1x2x1=6 descendants in the F2 etc. So it is possible to
calculate the 4 descendants the individual from F0 has on the average in F2 in
a further Poisson distribution. In this would be not right to consider the
originate of the generations F1 and F2 out of F0 straight away as one process
and do so calculating this as a Poisson distribution with intensity 4. That is
not right, because the origin of the F1 and the F2 are two processes all within
its own space of time. In these the individuals of the F1, the parents and not
the grandparents from F0, are concerned in the events, the “arrivals” that give
rise to the F2. The only overlap of the two processes is that individuals in F0
that have no descendents in F1 will also have no descendents in F2.
The proportion 2℮^2=0,2707 of F0 has one descendant in F1. The
size of the population in F0 and F1 is constant on n. So there are n2℮^2
individuals coming in this way. You can calculate the proportion of this, so the
way of 1 descendant from F0 to F1 and zero descendants from F1 to F2 (notice
→1→0) by substitution with λ*=2 and i=0, than it
is 2℮^2 . ℮^2 = 2℮^4. So is (→1→1): 2.℮^2 . 2.℮^2 = 4 ℮^4.
(→1→2) is 2.℮^2 . 2.℮^2 = 4 ℮^4.
(→1→3) is 2.℮^2 . 4/3.
℮^2 = 8/3 . ℮^4. (→1→4) is 2.℮^2. 2/3 .℮^2 = 4/3 .
℮^4. (→1→5) is 2.℮^2 . 4/15 .℮^2 = 8/15 .
℮^4. (→1→6) is 2.℮^2 . 4/45 . ℮^2 = 8/45 .
℮^4. (→1→7) is 2.℮^2 . 8/315 .℮^2 = 16/315 . ℮^4,
etc. Simply used is as substitution in the Poisson formula is q=1; r=2; λ*=2 en i=0,of i=1, of i=2, etc and
multiplied with the factor 2℮^2, the proportion of the one descendant in
the F1.
So has also the proportion 2℮^2 of F0 two descendants in the F1. These
parents in F0 expect however to get here 4 grandchildren, so q=2; r=2 and λ*=4 and
(→2→0) is 2.℮^2. ℮^4=2℮^6. (→2→1)
is 2.℮^2. 4.℮^2 =8.℮^2. (→2→2) is
2.℮^2. 8.℮^2 =16.℮^2, etc. In these calculations
continually λ*=4 en i=0, i=1, i=2, etc. The total
distribution of the descendants of F0 in F1 and F2 is given in Table 3. The way of descend is showed
with the arrows.
Table 3









Descendants of
F0 in F1 







→0 
→1 
→2 
→3 
→4 
→5 
→6 
→7 
→8 
℮^2 
2.℮^2 
2.℮^2 
4/3. ℮^2 
2/3 .℮^2 
4/15℮^2 
4/45℮^2 
8/315℮^2 
2/315℮^2 
0,135342 
0,27067 
0,27067 
0,18045 
0,09022 
0,03609 
0,01203 
0,00344 
0,00086 
x2/1 
x2/2 
x2/3 
x2/4 
x2/5 
x2/6 



Descendants
of F0 in F2 






→0→0 








℮^2 








→1→0 
→1→1 
→1→2 
→1→3 
→1→4 
→1→5 
→1→6 
→1→7 
→1→8 
2.℮^4 
4. ℮^4 
4 ℮^4 
8/3 ℮^4 
4/3 ℮^4 
8/15 . ℮^4 
8/45 ℮^4 
16/315℮^4 
4/315℮^4 
→2→0 
→2→1 
→2→2 
→2→3 
→2→4 
→2→5 
→2→6 
→2→7 
→2→8 
2℮^6 
8℮^6 
16℮^6 
64/3℮^6 
64/3℮^6 
256/15℮^6 
512/45℮^6 
2048/315℮^6 
1024/315℮^6 
→3→0 
→3→1 
→3→2 
→3→3 
→3→4 
→3→5 
→3→6 
→3→7 
→3→8 
4/3℮^8 
8℮^8 
24℮^8 
48℮^8 
72℮^8 
86,4℮^8 
86,4℮^8 
2592/35℮^8 
1944/35℮^8 
→4→0 
→4→1 
→4→2 
→4→3 
→4→4 
→4→5 
→4→6 
→4→7 
→4→8 
2/3℮^10 
16/3℮^10 
32/3℮^10 
512/9℮^10 
1024/9℮^10 
8192/45℮^10 
242,736℮^10 
277,401℮^10 
277,401℮^10 
→5→0 
→5→1 
→5→2 
→5→3 
→5→4 
→5→5 
→5→6 
→5→7 
→5→8 
4/15℮^12 
8/3℮^12 
40/3℮^12 
44,44℮^12 
111,11℮^12 
222,22℮^12 
370,37℮^12 
529,10℮^12 
661,38℮^12 
→6→0 
→6→1 
→6→2 
→6→3 
→6→4 
→6→5 
→6→6 
→6→7 
→6→8 
4/45℮^14 
16/15℮^14 
6,4℮^14 
25,6℮^14 
82,29℮^14 
184,32℮^14 
368,64℮^14 
631,95℮^14 
947,93℮^14 
→0→0 



→→ 
→→ 
→7→6 
→7→7 
→7→8 
0,135342 





265,59℮^16 
531,18℮^16 
929,57℮^16 
∑
0 F2 
∑
1 
∑
2 
∑
3 
∑
4 
∑
5 
∑
6 
∑
7 
∑
8 
0,042068047 
0,09604 
0,12155 
0,1207 
0,10737 
0,09086 
0,074075 
0,05832 
0,04447 
as
℮ functon 
X
1 
X
2 
X
3 
X
4 
X
5 
X
6 
X
7 
X
8 
[(℮^2^(℮^2)1]/℮^2 
0,9604 
0,2431 
0,3621 
0,42948 
0,4543 
0,44445 
0,40824 
0,35576 
∑
0 F1+F2 








0,177412 








[(℮^2^(℮^2)]/℮^2 








X
0 








→9 








0,00141093℮^2 








0,00019 












































→1→9 

















→2→9 
→2→10 
→2→11 






1,4448℮^6 
0,5779℮^6 
0,2102℮^6 
0,07005℮^6 





→3→9 
→3→10 
→3→11 
→3→12 
→3→13 
→3→14 
→3→15 


1296/35℮^8 
22,217℮^8 
12,118℮^8 
6,059℮^8 
2,797℮^8 
1,199℮^8 
0,479℮^8 


→4→9 
→4→10 
→4→11 
→4→12 
→4→13 
→4→14 
→4→15 
→4→16 
→4→17 
246,579℮^10 
197,263℮^10 
143,464℮^10 
95,643℮^10 
58,857℮^10 
33,633℮^10 
17,937℮^10 
8,969℮^10 
4,221℮^10 
→5→9 
→5→10 
→5→11 
→5→12 
→5→13 
→5→14 
→5→15 
→5→16 
→5→17 
734,86℮^12 
734,86℮^12 
668,06℮^12 
556,71℮^12 
428,24℮^12 
305,89℮^12 
203,92℮^12 
127,45℮^12 
74,97℮^12 
→6→9 
→6→10 
→6→11 
→6→12 
→6→13 
→6→14 
→6→15 
→6→16 
→6→17 
1263,91℮^14 
1516,69℮^14 
1654,57℮^14 
1654,57℮^14 
1527,30℮^14 
1309,11℮^14 
1047,29℮^14 
785,47℮^14 
554,45℮^14 
→7→9 
→7→10 
→7→11 
→7→12 
→7→13 
→7→14 
→7→15 
→7→16 
→7→17 
1446,0℮^16 
2024,40℮^16 
2576,51℮^16 
3005,93℮^16 
3237,16℮^16 
3237,16℮^16 
3021,34℮^16 
2643,68℮^16 
2172,65℮^16 
∑
9 
∑
10 
∑
11 
∑
12 
∑
13 
∑
14 
∑
15 
∑
16 
∑
17 
0,0329265 
0,0238454 
0,0168737 
0,0116831 
0.00788 
0,0052613 
0,00343908 
0,00217468 
0,00135777 
X
9 
X
10 
X
11 
X
12 
X
13 
X
14 
X
15 
X
16 
X
17 
0,2963385 
0,238454 
0,1856107 
0,1401972 
0,10244 
0,0736582 
0,0515862 
0,034795 
0,02308 









→4→18 





1,876℮^10 





→5→18 
→5→19 
→5→20 
→5→21 


41,65℮^12 
21,92℮^12 
10,96℮^12 
5,22℮^12 


→6→18 
→6→19 
→6→20 
→6→21 


369,63℮^14 
233,45℮^14 
140,07℮^14 
80,04℮^14 
43,66℮^14 
22,78℮^14 
→7→18 
→7→19 
→7→20 
→7→21 
→7→22 
→7→23 
1693,33℮^16 
1247,72℮^16 
873,40℮^16 
580,27℮^16 
370,54℮^16 
225.55℮^16 
∑
18 
∑
19 
∑
20 
∑
21 
∑
22 
∑
23 
0,00083899 
0,00046922 
0,000249 
0,000164 
0,00008 
0,00004 
X
18 
X
19 
X
20 
X
21 
X
22 
X
23 
0,0151 
0,00892 
0,00498 
0,00344 
0,00176 
0,00092 
In Table 3 are mentioned the
numbers of descendants as a product of ℮ under the field of the arrows.
So under example →3→6 is noticed in
proportions of F0 through 3 descendants in F1 to 6 descendants in F2. Notice
that the numbers of the sums under ∑, so 0,17740 ; 0,09604 ; 0,12155; etc
are proportions of F0 with totally 0; 1; 2; 3; etc descendants in F2. The
further Poissonlike distribution that is given here is thus more asymmetric
than the normal primary Poisson distribution. Of that totals under ∑ only
the sums for 0 descendants can be expressed fully as products of ℮. The
total of all the proportions or change intensities is indeed 1. The sum of the
descendants in F2 to an ancestor in F0, so 0x 0,17740 + 1x 0,09604 + 2x 0,12155
+ … is indeed in total
The Poisson distributed allele
transfer
In Table 4 are described the
fortunes of the allelic variations. So how many of the unique alleles or
possible gene variations are transferred on the average to the next 3 generations
according to the continued Poisson like distributions. Pose an individual has
on a locus the alleles a and b. The change on transfer of allele a by 1
descendant in F1 is 0,5. By 2 descendants, the replacement in neutral
population dynamics, the transfer of allele a is on the average 2x0,5=1. This
the same for allele b. So the parameters for the intensity of the transfer of
one allele to the next generation in a neutral population are Q =1 p=1, s=1 en r=1. The distribution of the alleles
thus is with Poisson intensity λ=1, so that in the F1 the proportion 1/℮=0,368..
has been disappeared, also 1/℮=0,368 occurs singular, the half of this
0,184 occurs in twofold, etc. Also at
the transfer from F1 to F2 and the further generation there will be on the
average 2 descendants and 1 allele in the distribution. To make the distribution
for F2 and F3 you must for all the proportions, those in singular, in twofold
etc from the former generation, separately calculate how their further Poisson
distribution is, as indicated with the arrows. In this are each of these proportions
different distributed with the intensities λ*=q=1; λ*=q=2; λ*=q=3, etc. The total intensity μ of
these 2^{nd}; 3^{rd} and further degrees Poisson distributions
remains
Table 4
F0 









Q=1 λ=1 









F1










1→0 
1→1 
1→2 
1→3 
1→4 
1→5 
1→6 
1→7 
1→8 
1→9 
℮^1 
℮^1 
1/2.℮^1 
1/6.℮^1 
1/24.℮^1 
1/120.℮^1 
1/720.℮^1 
1/5040.℮^1 
1/40320.℮^1 
2,76.10^6
℮^1 
F2 
μ=1 








→0→0 
→0→1 
→0→2 
→0→3 
→0→4 
→0→5 
→0→6 
→0→7 
→0→8 
→0→9 
℮^1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
→1→0 
→1→1 
→1→2 
→1→3 
→1→4 
→1→5 
→1→6 
→1→7 
→1→8 
→1→9 
℮^2 
℮^2 
1/2.℮^2 
1/6.℮^2 
1/24.℮^2 
1/120.℮^2 
1/720.℮^2 
1/5040℮^2 
1/40320.℮^2 
2,76
10^6.℮^2 
→2→0 
→2→1 
→2→2 
→2→3 
→2→4 
→2→5 
→2→6 
→2→7 
→2→8 
→2→9 
1/2.℮^3 
℮^3 
℮^3 
2/3.℮^3 
1/3.℮^3 
2/15.℮^3 
2/45.℮^3 
4/315.℮^3 
1/315.℮^3 
7,06.10^℮^4 
→3→0 
→3→1 
→3→2 
→3→3 
→3→4 
→3→5 
→3→6 
→3→7 
→3→8 
→3→9 
1/6.℮^4 
1/2.℮^4 
3/4.℮^4 
3/4.℮^4 
9/16.℮^4 
27/80.℮^4 
81/480.℮^4 
81/1120.℮^4 
243/8960.℮^4 
9,04.10^4℮^4 
→4→0 
→4→1 
→4→2 
→4→3 
→4→4 
→4→5 
→4→6 
→4→7 
→4→8 
→4→9 
1/24℮^5 
1/6.℮^5 
1/3.℮^5 
4/9.℮^5 
4/9.℮^5 
16/45.℮^5 
32/135.℮^5 
128/945.℮^5 
64/945.℮^5 
0,0301℮^5 
→5→0 
→5→1 
→5→2 
→5→3 
→5→4 
→5→5 
→5→6 
→5→7 
→5→8 
→5→9 
1/120.℮^6 
1/24.℮^6 
5/48℮^6 
25/144.℮^6 
125/576.℮^6 
125/576.℮^6 
0,18084℮^6 
0,12618℮^6 
0,08073℮^6 
0,04485℮^6 
→6→0 
→6→1 
→6→2 
→6→3 
→6→4 
→6→5 
→6→6 
→6→7 
→6→8 
→6→9 
1/720.℮^7 
1/120.℮^7 
1/40.℮^7 
1/20.℮^7 
3/40/.℮^7 
0,09.℮^7 
0,09.℮^7 
27/350.℮^7 
81/1400.℮^7 
0,03857.℮^7 
→7→0 
→7→1 
→7→2 
→7→3 
→7→4 
→7→5 
→7→6 
→7→7 
→7→8 
→7→9 
1/5040.℮^8 
1/720.℮^8 
7/1440.℮^8 
0,01134.℮^8 
0,01985.℮^8 
0,02779℮^8 
0,03242℮^8 
0,03242℮^8 
0,02837℮^8 
0,02206.℮^8 
→8→0 
→8→1 
→8→2 
→8→3 
→8→4 
→8→5 
→8→6 
→8→7 
→8→8 
→8→9 
1/40320℮^9 
1/5040℮^9 
1/1260℮^9 
2/945℮^9 
0,00433℮^9 
0,00677℮^9 
0,00903℮^9 
0,01032℮^9 
0,01031℮^9 
0,00917℮^9 
∑
0 F2 
∑
1 F2 
∑
2 F2 
∑
3 F2 
∑
4 F2 
∑
5 F2 
∑
6 F2 
∑
7 F2 
∑
8 F2 
∑
9 F2 
0,163584164 
0,195514535 
0,13372015 
0,07295863 
0,036145345 
0,016973463 
0,007630948 
0,003299023 
0,001378136 
0,000357693 
as
℮ function 
as
℮ function 








[℮^(
℮^1) 1]/℮ 
℮^(1/℮2) 








∑
0 F0  F2 
or
℮^(1/℮)/℮^2 








0,53146305 









≈℮^(1/℮1) 









or
[℮^(℮^1)]/℮ 








F3 










→0→0 
→0→1 
→0→2 
→0→3 
→0→4 
→0→5 
→0→6 
→0→7 
→0→8 
→0→9 
→0→10 
0,53146305 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
→1→0 
→1→1 
→1→2 
→1→3 
→1→4 
→1→5 
→1→6 
→1→7 
→1→8 
→1→9 
→1→10 
0,07192577 
0,07192577 
0,035962888 
0,011987629 
0,002996907 
0,000599381 
0,000099896 
1,43E05 
1,78E06 
1,98E07 
1,98E08 
→2→0 
→2→1 
→2→2 
→2→3 
→2→4 
→2→5 
→2→6 
→2→7 
→2→8 
→2→9 
→2→10 
0,018097054 
0,036194108 
0,036194108 
0,024129405 
0,012064702 
0,004825881 
0,001608271 
0,000459608 
0,000114902 
2,55E05 
5,11E06 
→3→0 
→3→1 
→3→2 
→3→3 
→3→4 
→3→5 
→3→6 
→3→7 
→3→8 
→3→9 
→3→10 
0,00363234 
0,010897188 
0,016345783 
0,016345783 
0,012259337 
0,007355603 
0,003677801 
0,001576201 
0,000591075 
0,000197025 
5,91E05 
→4→0 
→4→1 
→4→2 
→4→3 
→4→4 
→4→5 
→4→6 
→4→7 
→4→8 
→4→9 
→4→10 
0,000662025 
0,0026481 
0,005296201 
0,007061601 
0,007061601 
0,00564928 
0,003766187 
0,002152107 
0,001076054 
0,000478246 
0,000191298 
→5→0 
→5→1 
→5→2 
→5→3 
→5→4 
→5→5 
→5→6 
→5→7 
→5→8 
→5→9 
→5→10 
0,000114366 
0,000578315 
0,001429579 
0,002382631 
0,002978289 
0,002978289 
0,002481907 
0,001772791 
0,001107994 
0,000615552 
0,000307776 
→6→0 
→6→1 
→6→2 
→6→3 
→6→4 
→6→5 
→6→6 
→6→7 
→6→8 
→6→9 
→6→10 
1,89E05 
0,000113491 
0,000340474 
0,000680948 
0,001021422 
0,001225707 
0,001225707 
0,001050606 
0,000787954 
0,000525303 
0,000315182 

→7→1 
→7→2 
→7→3 
→7→4 
→7→5 
→7→6 
→7→7 
→7→8 
→7→9 
→7→10 

2,11E05 
7,37E05 
0,000171976 
0,000300958 
0,00042134 
0,000491564 
0,000491564 
0,000430118 
0,000334536 
0,000234175 



→8→3 
→8→4 
→8→5 
→8→6 
→8→7 
→8→8 
→8→9 
→8→10 



3,95E05 
7,89E05 
0,000126242 
0,000168323 
0,000192369 
0,000192369 
0,000170995 
0,000136796 




→9→4 
→9→5 
→9→6 
→9→7 
→9→8 
→9→9 
→9→10 




1,21E05 
2,17E05 
3,26E05 
4,19E05 
4,71E05 
4,71E05 
4,24E05 
∑0
F3 
∑
1 F3 
∑
2 F3 
∑
3 F3 
∑
4 F3 
∑
5 F3 
∑
6 F3 
∑
7 F3 
∑
8 F3 
∑
9 F3 
∑
10 F3 
0,09445047 
0,122378031 
0,095642736 
0,062799424 
0,038774185 
0,023203445 
0,013552238 
0,007751407 
0,004349379 
0,002394518 
0,001291877 
∑0
F0F3 










0,625917694 









∑
11 F3 
≈℮^[℮^(1/℮1)1] 








6,88
E4 
The number of the events as “arrivals” of descendants and genes in generation
F1 is Poisson distributed with the known primary Poisson distribution. This
means a distribution of the primary quantum Q with the uniform intensity λ
into proportions for the quanta i=0; i=1; i=2; i=3, etc, so that the
distribution results in quanta and proportions. The result of the former
distribution, these proportion can of course be distributed Poisson again. Then
however the proportions must be distributed separately each with its own
intensity λ*=q.μ, so the product of the quantum q of the proportion in the former
generation and μ. In this way
also in the further generations the arrivals of the alleles remain to be
Poisson distributed in the same generation time t and this distribution can be calculated by the same substitution
in the formula P(i) = ℮^λ*. λ*^i/i! through the
generations. So the proportions of the old generations are always distributed
into the new generations. In this way the 2^{nd} degree Poisson
distribution arises out of the general known primary distribution, the 3^{rd}
degree out of the 2^{nd} degree and the n^{th} degree out of
the (n1)^{th } degree Poisson
distribution. These further distribution all originate from the normal primary
Poisson distribution with the uniform intensity λ. I do call these 2^{nd} , 3^{rd} and further
degree Poisson distributions, because the same primary quantum Q is distributed
here primary, secondary, tertiary and further. In Table 4 this happens with a
constant total Poisson intensity μ=λ=1 for F(g1)→Fg. The μ in
this is the intensity in which all the proportions will decrease or increase in
total at the distribution F(g1)→ Fg. The μ is the proportional
total intensity of the degree g, so that: μ=0x[P(i=0)] + 1x[P(i=1)] + 2x[P(i=2)] + …qx[P(i=q)], in
this is P(i=q) the result of the distribution according to P(i) =
℮^λ*. λ*^i/i! of degree g1. This μ of the continued
Poisson distribution is constant in these examples, but the Poisson
distribution of the quanta can also be continued in the next degree with another
intensity. The calculation of a large number of degrees are easily practicable,
I guess, with a computer and the right software, but not in this way.
So there are in the graduated Poisson distributions levels of quantities
and the distributions are from the former to the next level of the quantities.
In this application the levels of the quanta are called the generations F0; F1; F2;..Fg. The degrees of the Poisson
distributions are between these levels or generations, so that degree Gg
distributes the quanta of Fg into those of F(g1). See on Table 5.
The accumulating exponential
distribution.
The peculiarity of the P(i=0), this is the negation or the complement of
the Poisson event or arrival, the zeroproportion is exponential distributed, according
to P(i=0) = ℮^λ at
the primary and further Poisson distribution and it is the
complement of the exponential distribution, P(i=n) = 1  ℮^λ. The intensity
λ of these exponential distributions is also within the next degrees equal
to the μ of the continued Poisson distribution, of which it is a
part. With λ*=μ.q and
the quanta q the P(i=0) can be calculated with the superposed Poisson
distributions in the way of Table 3 and 4. If you express than the P(i=0) as an
algebraic function of ℮ it just appears that the remaining quantity, so 1P(i=0) just is negative
exponential distributed through the degrees or generations. So P(i=0) of
generation g simply is ℮^{1P(i=0)} of the former generation
g1. The exponential distribution of the non arrival accumulates in this way.
Through the generations is the intensity λ or σ of the
exponential distribution equal to the remaining quantity and decreases, while
the non arrival, the extinction of the allele increases. In this is λ
the intensity of the primary distribution and is σ the accumulated
intensity of the distributions in the further degrees. The superposition of the
exponential part of the Poisson distribution can be calculated in following the
recurrence and this is noticed in σ(Fg)=ν
In Table 5 the extinct alleles, the P(∑i=0), shortly P0, is
calculated from the intensities λ,
or σ for the generations F0F200, starting from λ=μ=1.
Table 5
F0 
F1 
F2 
F3 
F4 
F5 
F6 
F7 
F8 
F9 
λ=1 
σ=0,6321 
σ=0,4685 
σ=0,3741 
σ=0,3121 
σ=0,2681 
σ=0,2352 
σ=0,2095 
σ=0,1890 
σ=0,1723 
P0=0,368 
P0=0,531 
P0=0,626 
P0=0,688 
P0=0,732 
P0=0,765 
P0=0,790 
P0=0,811 
P0=0,828 
P0=0,842 
F1 
F2 
F3 
F4 
F5 
F6 
F7 
F8 
F9 
F10 










F10 
F11 
F12 
F13 
F14 
F15 
F16 
F17 
F18 
F19 
σ=0,1582 
σ=0,1464 
σ=0,1361 
σ=0,1273 
σ=0,1195 
σ=0,1127 
σ=0,1066 
σ=0,1011 
σ=0,0961 
σ=0,0916 
P0=0,854 
P0=0,864 
P0=0,873 
P0=0,880 
P0=0,887 
P0=0,893 
P0=0,899 
P0=0,904 
P0=0,908 
P0=0,912 
F11 
F12 
F13 
F14 
F15 
F16 
F17 
F18 
F19 
F20 










F20 
F21 
F22 
F23 
F24 
F25 
F26 
F27 
F28 
F29 
σ=0,0876 
σ=0,0838 
σ=0,0804 
σ=0,0773 
σ=0,0744 
σ=0,0716 
σ=0,0692 
σ=0,0668 
σ=0,0646 
σ=0,0626 
P0=,0916 
P0=0,919 
P0=0,923 
P0=0,926 
P0=0,928 
P0=0,931 
P0=0,933 
P0=0,935 
P0=0,937 
P0=0,939 
F21 
F22 
F23 
F24 
F25 
F26 
F27 
F28 
F29 
F30 










F30 
F31 
F32 
F33 
F34 
F35 
F36 
F37 
F38 
F39 
σ=0,0607 
σ=0,0589 
σ=0,0572 
σ=0,0556 
σ=0,0541 
σ=0,0526 
σ=0,0513 
σ=0,0500 
σ=0,0487 
σ=0,0476 
P0=0,941 
P0=0,943 
P0=0,944 
P0=0,945 
P0=0,947 
P0=0,949 
P0=0,950 
P0=0,951 
P0=0,952 
P0=0,954 
F31 
F32 
F33 
F34 
F35 
F36 
F37 
F38 
F39 
F40 










F40 
F41 
F42 
F43 
F44 
F45 
F46 
F47 
F48 
F49 
σ=0,0465 
σ=0,0454 
σ=0,0444 
σ=0,0434 
σ=0,0425 
σ=0,0416 
σ=0,0407 
σ=0,0399 
σ=0,0391 
σ=0,0384 
P0=0,955 
P0=0,956 
P0=0,957 
P0=0,958 
P0=0,958 
P0=0,959 
P0=0,960 
P0=0,961 
P0=0,962 
P0=0,962 
F41 
F42 
F43 
F44 
F45 
F46 
F47 
F48 
F49 
F50 










F50 
F51 
F52 
F53 
F54 
F55 
F56 
F57 
F58 
F59 
σ=0,0376 
σ=0,0369 
σ=0,0363 
σ=0,0356 
σ=0,0350 
σ=0,0344 
σ=0,0338 
σ=0,0332 
σ=0,0327 
σ=0,0322 
P0=0,963 
P0=0,964 
P0=0,964 
P0=0,965 
P0=0,966 
P0=0,966 
P0=0,967 
P0=0,967 
P0=0,968 
P0=0,968 
F51 
F52 
F53 
F54 
F55 
F56 
F57 
F58 
F59 
F60 










F60 
F61 
F62 
F63 
F64 
F65 
F66 
F67 
F68 
F69 
σ=0,0317 
σ=0,0312 
σ=0,0307 
σ=0,0302 
σ=0,0298 
σ=0,0293 
σ=0,0289 
σ=0,0285 
σ=0,0281 
σ=0,0277 
P0=0,969 
P0=0,969 
P0=0,970 
P0=0,970 
P0=0,971 
P0=0,971 
P0=0,972 
P0=0,972 
P0=0,972 
P0=0,973 
F61 
F62 
F63 
F64 
F65 
F66 
F67 
F68 
F69 
F70 










F70 
F71 
F72 
F73 
F74 
F75 
F76 
F77 
F78 
F79 
σ=0,0273 
σ=0,0269 
σ=0,0266 
σ=0,0262 
σ=0,0259 
σ=0,0256 
σ=0,0252 
σ=0,0249 
σ=0,0246 
σ=0,0243 
P0=0,973 
P0=0,973 
P0=0,974 
P0=0,974 
P0=0,974 
P0=0,975 
P0=0,975 
P0=0,975 
P0=0,976 
P0=0,976 
F71 
F72 
F73 
F74 
F75 
F76 
F77 
F78 
F79 
F80 










F80 
F81 
F82 
F83 
F84 
F85 
F86 
F87 
F88 
F89 
σ=0,0240 
σ=0,0237 
σ=0,0235 
σ=0,0232 
σ=0,0229 
σ=0,0227 
σ=0,0224 
σ=0,0221 
σ=0,0219 
σ=0,0217 
P0=0,976 
P0=0,977 
P0=0,977 
P0=0,977 
P0=0,977 
P0=0.978 
P0=0,978 
P0=0,978 
P0=0,978 
P0=0,979 
F81 
F82 
F83 
F84 
F85 
F86 
F87 
F88 
F89 
F90 










F90 
F91 
F92 
F93 
F94 
F95 
F96 
F97 
F98 
F99 
σ=o,0214 
σ=0,0212 
σ=0,0210 
σ=0,0208 
σ=0,0205 
σ=0,0203 
σ=0,0201 
σ=0,0199 
σ=0,0197 
σ=0,0195 
P0=0,979 
P0=0,979 
P0=0,979 
P0=0,979 
P0=0,980 
P0=0,980 
P0=0,980 
P0=0,980 
P0=0,980 
P0=0,981 
F91 
F92 
F93 
F94 
F95 
F96 
F97 
F98 
F99 
F100 










F100 
F101 
F102 
F103 
F104 
F105 
F106 
F107 
F108 
F109 
σ=0,01935 
σ=0,01917 
σ=0,01898 
σ=0,01881 
σ=0,01863 
σ=0,01854 
σ=0,01829 
σ=0,01812 
σ=0,01796 
σ=0,01780 
P0=0,98083 
P0=0,98102 
P0=0,98119 
P0=0,98137 
P0=0,98154 
P0=0,98171 
P0=0,98188 
P0=0,98204 
P0=0,98220 
P0=0,98236 
F101 
F102 
F103 
F104 
F105 
F106 
F107 
F108 
F109 
F110 










F110 
F111 
F112 
F113 
F114 
F115 
F116 
F117 
F118 
F119 
σ=0,01764 
σ=0,01749 
σ=0,01733 
σ=0,01718 
σ=0,01704 
σ=0,01689 
σ=0,01675 
σ=0,01661 
σ=0,01647 
σ=0,01634 
P0=0.98251 
P0=0,98267 
P0=0,98281 
P0=0,98296 
P0=0,98311 
P0=0,98325 
P0=0,98339 
P0=0,98352 
P0=0,98366 
P0=0,98379 
F111 
F112 
F113 
F114 
F115 
F116 
F117 
F118 
F119 
F120 










F120 
F121 
F122 
F123 
F124 
F125 
F126 
F127 
F128 
F129 
σ=0,01621 
σ=0,01608 
σ=0,01595 
σ=0,01582 
σ=0,01570 
σ=0,01557 
σ=0,01545 
σ=0,01533 
σ=0,01522 
σ=0,01510 
P0=0,98392 
P0=0,98405 
P0=0,98418 