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Striking Regularities In The World Population Curve 
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#1
Oct1206, 04:58 AM

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In statistical physics, one learns of the general concept of
BirthDeath 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 MalthusVerhulst equation dp/dt = k p (up)/u which describes a population curve of the form p(t) = u/(1 + exp(k(t0t))) 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 midyear 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 MalthusVerhulst 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(tt0)))/(1 + exp(k(tt0))) 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((t1989.047)/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 birthdeath 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 postIndustrial era. The p(1989+t) + p(1989t) 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 reverseengineering 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 MalthusVerhulst 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 midyear 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 nearexact fit of the time range 19742004 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 reverseengineered  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(t0t)); 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 0asymptote logistic may be motivated, for instance, by the MalthusVerhulst equation dp/dt = kp(up)/u u = 2 p(1989) = upper asymptote which arises naturally from an underlying demographic theory, but it does not even fit the target range (19742004) 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 birthdeath process which satisfies an equation of the form dp/dt = k(pv)(up)/(uv) 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 NONZERO 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. 19501970 putting in clear relief the existence of TWO demographic transitions involved in recent times, not just one. The population curve post1970 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(1989t) + 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 19501970 no matter what segment you try to model, because of this phase boundary. The earlier, pre1950, phase does not fit any logistic at all. The bestfitting curve for any of the intervals [satisfying a differential equation p'(t) = quadratic in p(t)] 18501974, 18501950, 19001950, 19001974, 19501974; 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 18501950 + IDB estimates 19502004 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((t1987.9)/43.893) million and asymptotes at u = 8746 million, v = 1468 million. Even fitting a logistic to the 19672011 region, using the IDB projections out to 2011, has a similar result: p(t) = 5199.6 + 4130.7 tanh(t1989.086)/50.43558) million u = 9330.3 million, v = 1068.9 million. More generally: trying to do a global fit across the 19501970 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 offtrack 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 postsecondary level student population.) The estimates get even more divergent still if the pre1850 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((t1568.0)/52.211) million [satifying a differential equation of the form p'(t) = k (pv)/v ] Overall, this paints a picture that is markedly different than one which would have the historical curve undergoing some kind of onceandforall demographic transition following a longterm 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 & postindustrial 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" PhysicsUspekhi 39(1) 5771 (1996).] More detail on the dropping birth rate (which is quite dramatic in the last 5 years) and the link to the female collegelevel attendance is provided in http://groupsgoogle.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 


#2
Oct1206, 05:01 AM

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markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#3
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#4
Oct1206, 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 > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#5
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#6
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#7
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#8
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#9
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#10
Oct1206, 05:01 AM

P: n/a

markwh04@yahoo.com wrote:
> In statistical physics, one learns of the general concept of > BirthDeath 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 MalthusVerhulst equation > dp/dt = k p (up)/u > which describes a population curve of the form > p(t) = u/(1 + exp(k(t0t))) 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 nonlinear system: If u is very large, then you get a linear system, and the quadratic nonlinearity 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 


#11
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#12
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#13
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#14
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#15
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#16
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#17
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


#18
Oct1206, 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. 19501974. The real mystery is not the u or k in the expression f(p) = k(pv)(up)/(uv), since that accords with the understanding provided by the MalthusVerhulst equation. Going through the usual derivation from a birthdeath process Pr(p(t + h) = p2 p(t) = p1) = D(p2) delta(p1,p2+1) + B(p2) delta(p1,p21) 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(up)/u, for suitable u, k. To get a constant term (or, equivalently, a nonzero v), one needs to add constant terms D0, B0 respectively to D(p) and B(p)  which goes beyond MalthusVerhulst. 


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