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Striking Regularities In The World Population Curve

by markwh04@yahoo.com
Tags: curve, population, regularities, solved, striking, world
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markwh04@yahoo.com
#1
Oct12-06, 04:58 AM
P: n/a
In statistical physics, one learns of the general concept of
Birth-Death processes which describe systems typified by populations,
such as particle number, bacterial count, or human or animal systems.
The one equation that stands out is the Malthus-Verhulst equation
dp/dt = k p (u-p)/u
which describes a population curve of the form
p(t) = u/(1 + exp(k(t0-t)))
with a minimum of 0, a maximum of u and an inflection time at t0 at
which p(t0) = u/2.

A significant event happened on Earth in 1989 which has yet to receive
wide attention: the point of inflection passed by for the world
population -- t0 = 1989. Furthermore, a new regularity has emerged in
the time since then:
P(1989 + x) + P(1989 - x) = 10386 +/- 5 million
(based on the mid-year population estimates given by the International
Database of the US Census Bureau). According to the IDB projections,
this would hold out to 2011.

HOWEVER...

The world population is NOT following a curve corresponding to a
Malthus-Verhulst process. Instead, what one is finding is that it is
(and has been) closely tracking a logistic curve for the past 30 years
which has a POSITIVE offset:
p(t) = (v + u exp(k(t-t0)))/(1 + exp(k(t-t0)))
with
v = 2.5 billion, u = 7.9 billion.

The accuracy of this curve is 99.9% (+/- 6 million), with the
differences from the actual curve given (in the millions) in the
following table:

1974 1975 1976 1977 1978 1979 1980 1981 1982 1983
-1.2 2.9 3.7 3.3 1.2 0.4 -1.1 -3.5 -3.5 -2.9

1984 1985 1986 1987 1988 1989 1990 1991 1992 1993
-4.5 -5.3 -5.2 -2.9 -0.3 2.2 5.9 5.9 6.1 4.5

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
2.7 1.8 0.0 -1.0 -2.2 -3.1 -3.6 -3.4 -2.3 0.3 4.9

[based on the projection
p_{Log} = 5196.5 + 2684.7 tanh((t-1989.047)/31.847) million
versus the 2005 update of the IDB's midyear estimates].

That's an RMS deviation of only 3.5 million -- about the size of a
large city.

This curve maxes out at under 8 billion, less than 1.5 billion of where
the population is currently at.

In general, what one is finding is that the world population is a
birth-death process that satisfies a differential equation of the form
p'(t) = f(p(t))
where f(p) is piecewise quadratic or linear; each piece associated with
a separate phase. The phase boundaries occur roughly in the vicinity
of 1 billion (corresponding to the time of the dawn of the Industrial
Revolution) and 3.5 billion with the latter boundary being associated
with a transition region between 2.5 - 4.5 billion that corresponds to
the Second World War and the consequent rise of the post-Industrial
era.

The p(1989+t) + p(1989-t) regularity, given the occurrence of the phase
boundary in the vicinity of 1970, should start to show signs of
breaking in the next few years, some time around 2008 (contrary to the
IDB's projections).

A reply received from Peter Johnson back in June (the main contact of
the US Census Bureau's IDB) confirms that this regularity is not an
artifact of the IDB's estimation/projection process and is not an
unwitting reverse-engineering of a projection curve, but represents
(within the resolution of the IDB's estimation process) a bona fide
phenomenon.

In his reply, he produced several tables, including one with a logistic
which, however, was a Malthus-Verhulst curve and (consquently) failed
to reproduce the +/- 6 million accuracy I described above.

The following is an open letter sent to Peter Johnson, in reply to his
last communication, which addresses all the issues in greater depth.

====================

(To Peter Johnson:)
I've had a chance to review the tables you sent me in June, where you
provide some feedback on my earlier observations from April concerning
the unusual regularities in the world population curve I found.

As I mentioned in my earlier letter
>In the mid-year world population estimates provided by the IDB,
>there is a nearly precise regularity [the 1989+/-t regularity]
>for t in the range [1974,2004]; and a near exact match to a
>logistic *that tops off at 7.8 [billion] [...]


there is a near-exact fit of the time range 1974-2004 of the population
curve as given by the 2004 IDB estimates to a logistic curve with the
same accuracy as the linear relation cited above. For the 2005 update
of the IDB estimates, the fit is within 6 million and is of the form
P(t) = P_{Log} = [... the expression listed above ...]
with asymptotes at
v = 2512 million, u = 7881 million.

In your last reply, you pointed out that this is not an artifact of any
modelling the IDB is doing that I might have unwittingly
reverse-engineered -- which means that these two striking patterns are
probably for real. However in the table included in your reply, you
used the wrong logistic curve:
P_{IDB} = u/(1 + exp(k(t0-t));
u = 10331.535312 million
t0 = 1988.66; k = 0.03151
tacitly assuming the lower asymptote, v, to be 0; which is why you
couldn't reproduce the fit I quoted.

A 0-asymptote logistic may be motivated, for instance, by the
Malthus-Verhulst equation
dp/dt = kp(u-p)/u
u = 2 p(1989) = upper asymptote
which arises naturally from an underlying demographic theory, but it
does not even fit the target range (1974-2004) very well, with large
distortions on the order of 20 million, and distortions in the middle
of the range; both of which clearly show that the problem is that the
lower asymptote is placed wrong.

Instead, what you have is a more generalized birth-death process which
satisfies an equation of the form
dp/dt = k(p-v)(u-p)/(u-v)
0 < v < p(1989) < u < 2 p(1989)
u + v = 2 p(1989)
t > c. 1970
involving a quadratic expression on the right with a NON-ZERO constant
coefficient.

The upshot of this is that the upper limit is under 8 billion; not over
10 billion. That was the significance of my earlier remark.

The comparison [made between P_{Log} and P(t)] also highlights the
occurrence of the natural phase boundary c. 1950-1970 putting in clear
relief the existence of TWO demographic transitions involved in recent
times, not just one. The population curve post-1970 is an entirely
independent development from that predating 1950 -- *falsifying* a key
assumption of _Demographic_Transition_Theory_.

So there is no reason to want to fit the lower asymptote to 0.

The phase boundary should start showing up in the next few years with
the breaking of the p(1989-t) + p(1989+t) symmetry; which contrasts to
your suggestion that it may hold all the way out to 2011, based on the
IDB projection.

You'll find distortions centering around 1950-1970 no matter what
segment you try to model, because of this phase boundary. The earlier,
pre-1950, phase does not fit any logistic at all. The best-fitting
curve for any of the intervals [satisfying a differential equation
p'(t) = quadratic in p(t)]
1850-1974, 1850-1950, 1900-1950, 1900-1974, 1950-1974;
has a divergent upper asymptote and tends toward an exponential [with a
postive offset]. Therefore, attempting to model across the phase
boundary using (for instance)
UN estimates 1850-1950 + IDB estimates 1950-2004
will only serve to create large distortions and an exaggerated upper
asymptote nearing 10 billion. You end up with the curve:
p(t) = 5107 + 3639 tanh((t-1987.9)/43.893) million
and asymptotes at u = 8746 million, v = 1468 million.

Even fitting a logistic to the 1967-2011 region, using the IDB
projections out to 2011, has a similar result:
p(t) = 5199.6 + 4130.7 tanh(t-1989.086)/50.43558) million
u = 9330.3 million, v = 1068.9 million.

More generally: trying to do a global fit across the 1950-1970 phase
boundary leads to future projections whose divergence from the actual
population curve has become conspicuously clear even now. Other
projections by the USCB and UN have, likewise, gotten off-track in the
last few years; with even their most conservative estimates now visibly
overshooting the mark.

Looking in your own database, in fact, you will see a dramatic
consequence of this phase change: birthrates throughout the world in
just the past 5 years have dropped like a rock. This is to be expected
if the population curve were following P(t) = P_{Log}, with the current
population now only within 1.5 billion of the projected maximum;
instead of the overestimates provided by the USCB, the UN and even by
your P_{IDB}, which would still leave nearly 4 billion more room for
growth. (And as I pointed out in my last reply, this marked downturn
in birthrates is, in turn, directly connected to the growing female
majority worldwide, even in the Arab world, of the post-secondary level
student population.)

The estimates get even more divergent still if the pre-1850 segment is
included, since there is a second phase boundary in the vicinity of
1850. I've gotten curves that way that go as high as 20 billion -- a
figure once cited by the UN.

The best fit of ranges starting in 1850 requires the ending point to be
around 1940, and yields
P_E = 1044.6 + exp((t-1568.0)/52.211) million
[satifying a differential equation of the form
p'(t) = k (p-v)/v
]

Overall, this paints a picture that is markedly different than one
which would have the historical curve undergoing some kind of
once-and-for-all demographic transition following a long-term
historical exponential climb upwards from 0 and then settling down and
possibly even going down. Instead, what you find are at least 3
independent segments to the population curve separated by the last 2 of
a series of historical phase transitions, these transitions being tied
closely to the 2 major historical watersheds that mark respectively the
boundary between the agricultural & industrial era; and the industrial
& post-industrial era.

The exponential growth of the earth 20th century (P_E) is an entirely
independent phase from that which following 1970 and it came to an end
with the Second World War.

In turn, the larger picture that emerges suggests that other historical
watersheds, such as the Neolithic Revolution, are linked to phase
transitions in the world population curve.

[In turn, this was a point raised by Kapitza in "The phenomenological
theory of world population growth" Physics-Uspekhi 39(1) 57-71 (1996).]

More detail on the dropping birth rate (which is quite dramatic in the
last 5 years) and the link to the female college-level attendance is
provided in

http://groups-google.com/group/sci.e...2?dmode=source
Islamic demographics? [Growth rate dropping like a rock]
sci.econ, sci.anthropology, uk.politics.misc
2005 August 23

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Gerard Westendorp
#2
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#3
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard


Gerard Westendorp
#4
Oct12-06, 05:01 AM
P: n/a
Striking Regularities In The World Population Curve

markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#5
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#6
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#7
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#8
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#9
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

Gerard Westendorp
#10
Oct12-06, 05:01 AM
P: n/a
markwh04@yahoo.com wrote:

> In statistical physics, one learns of the general concept of
> Birth-Death processes which describe systems typified by populations,
> such as particle number, bacterial count, or human or animal systems.
> The one equation that stands out is the Malthus-Verhulst equation
> dp/dt = k p (u-p)/u
> which describes a population curve of the form
> p(t) = u/(1 + exp(k(t0-t)))


I did not know that this had an exact solution! The equation can also
be write as:

dp/dt = k p - (k/u) p^2

So it is sort of the simplest non-linear system: If u is very large,
then you get a linear system, and the quadratic non-linearity is
proportional to 1/u.

I tried complex values of k and u too:
For positive real k, you get an exponentially growing oscillation, but
when the oscillation gets to amplitude 1, it stops growing, and starts
damping, but around the equilibrium value of 1 instead of zero.

Interesting...

Anyway, if you want to model the worlds population, I think you have to
look at the physical meaning of k and u. I think k would be something
like the net effect of birth and death per person. If people acted
independently, (i.e. not be influenced by other people around them nor
by the total amount of people) then you would get an exponential growth
with an exponent of k. But eventually, something will stop this.
Presumably this is parametrized by u.

It would seem reasonable that the factor k is dependent on a "phase" as
you say, in world history, since the death rate can suddenly decline as
a result of the invention of better medicine.

I don't understand what exactly the factor u should mean. As you remark,
the drop in birth rate could be due to education of women. But that
would not seem to be a result of some population density.

To get a better fit of the model, you might try splitting the world up
into regions. Maybe the "first world" behaves different than the "third
world".


Gerard

markwh04@yahoo.com
#11
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.


markwh04@yahoo.com
#12
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#13
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#14
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#15
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#16
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#17
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.

markwh04@yahoo.com
#18
Oct12-06, 05:02 AM
P: n/a
Gerard Westendorp wrote:
> It would seem reasonable that the factor k is dependent on a "phase" as
> you say, in world history, since the death rate can suddenly decline as
> a result of the invention of better medicine.
>
> I don't understand what exactly the factor u should mean. As you remark,
> the drop in birth rate could be due to education of women. But that
> would not seem to be a result of some population density.


The suggestion being made is that the cause and effect are going the
other way around and that the historical changes are, themselves,
adaptive responses triggered by the crossing of critical thresholds in
the world population.

You get a more interesting view of the general picture by plotting
p'(t) vs. p(t) using, say, the historical estimates from various
sources provided in the International Database of the United States
Census Bureau.

Here, you can see much more clearly the natural piecewise segmentation
of f(p) in the equation p'(t) = f(p(t)). One also picks up, in this
way, the natural of the transitional phase (the "baby boom") c.
1950-1974.

The real mystery is not the u or k in the expression f(p) =
k(p-v)(u-p)/(u-v), since that accords with the understanding provided
by the Malthus-Verhulst equation. Going through the usual derivation
from a birth-death process Pr(p(t + h) = p2| p(t) = p1) = D(p2)
delta(p1,p2+1) + B(p2) delta(p1,p2-1) with D(p) = D1 p - D2 p^2; B(p) =
B1 p - B2 p^2 (D2, B2 to account for diminishing returns) ultimately
gives you (for the average value of p(t)) a differential equation with
f(p) = kp(u-p)/u, for suitable u, k. To get a constant term (or,
equivalently, a non-zero v), one needs to add constant terms D0, B0
respectively to D(p) and B(p) -- which goes beyond Malthus-Verhulst.


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