Macrobius
06-25-2010, 07:54 AM
http://www.physicsforums.com/archive/index.php/t-137543.html
I found this 2006 post interesting.
The Malthus-Verhulst equation (yes, that Malthus) is a well known equation of Statistical Mechanics, a solution to the 'Master Equation' (http://en.wikipedia.org/wiki/Master_equation)(a good example of an application of the Master Equation is the Fokker-Planck equation (http://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation)).
M-V is a solvable case of the Master equation, for a birth-death process that has two solutions -- birth rate less than death rate, and the species goes extinct. (birth rate - death rate)/gamma sets the equilibrium carrying capacity of the population to a fixed number.
It is also important as an example of a critical phenomenon that shows 'universality' having radically different characteristics from typical.
Look up 'Malthus-Verhulst Process' (MVP) for a modern take on Malthus. ;)
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.
More at the link.
I found this 2006 post interesting.
The Malthus-Verhulst equation (yes, that Malthus) is a well known equation of Statistical Mechanics, a solution to the 'Master Equation' (http://en.wikipedia.org/wiki/Master_equation)(a good example of an application of the Master Equation is the Fokker-Planck equation (http://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation)).
M-V is a solvable case of the Master equation, for a birth-death process that has two solutions -- birth rate less than death rate, and the species goes extinct. (birth rate - death rate)/gamma sets the equilibrium carrying capacity of the population to a fixed number.
It is also important as an example of a critical phenomenon that shows 'universality' having radically different characteristics from typical.
Look up 'Malthus-Verhulst Process' (MVP) for a modern take on Malthus. ;)
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.
More at the link.