# Striking Regularities In The World Population Curve

by markwh04@yahoo.com
Tags: curve, population, regularities, solved, striking, world
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.