Hello all:

Thank you Hans for contributing to this discussion...  I take it that the growth rate (Y axis of growth domain graph) should be expressed in 1/year and not 1/day?

Regarding your question about whether to fit data in the growth/time domain or in the population/time domain, I would speculate that since the growth rates calculated from the census data correspond to the geometric means of the respective inter-census periods (for example, 1950-1961), matching them up to the instant growth rates at a specific census time (say 1950 or 1961, for the example) would not be a good idea.  At any rate, since you're working with geometric means during a whole time period, some error is added to the fit and furthermore, you'd have to decide on the point at each interval at which to match the growth rates (say 1955.5?).  I'd say this is why the blue curve on the growth domain graph seems skewed with respect to the red curve, although on the left graph it is evident that the blue curve has a better fit on the population/time domain.

Your other question is very important- what is the model for?

Originally, my intention was just to do a model fitting excercise on the venezuelan census data set.  Then I started reading about demographic transition theory and a lot of questions came up.  For example, on a book titled "Beyond Malthus ..." (download at www.worldwatch.org/system/files/EWB110_0.pdf), the authors state that a country is considered to have reached the 4th stage of demographic transition (population stabilized with low fertility and death rates) when its inter-anual growth rate is 0.4 percent or less.  Given the best fit model, when would Venezuela reach the 4th stage? The interseting thing about this is that the question would be answered by considering the total population levels and not the mortality and fertility rates separately.  And of course there is the million dollar question: at how many inhabitants does the population stabilize? ...

I continued to play around with Thomas' 2 variable model and found that an even better fit resulted when both variables were asumed to have logistic growth:

model <- function(t, y, p) {
   with(as.list(c(y, p)), {
     dy1 <- -a * y1 + r1 * y1 * (1 - y1 / K1)    # "inactive" part of population
     dy2 <-  a * y1 + r2 * y2 * (1 - y2 / K2)    # growing population
     list(c(dy1, dy2), pob = y1 + y2)            # total pop = y1 + y2
   })
 }

:
:

> parms
          r1           r2           K1           K2            a 
5.285589e-01 5.440895e-02 1.923383e+06 3.326471e+07 1.577931e-03 

The sum of squares weighed by std deviation for this model was SC=0.003876696, for Thomas' model it was 0.005744879.  This model seems to have fared well in the parameter collinearity analysis, for the five parameters, the collinearity was 13.3, as compared to Thomas' 3 parameter model for which it was 10.3.  Strangely enough, when I considered exponential growth for the y1 subpopulation (without the logistic reduction term), the model was slightly better sum-of-square wise, but the collinearity was higher than 20.

I didn't run this model too far into the future, but it seems that the total population would stabilize around K1+K2=35188093; for Thomas' 2-var model the saturation point was only slightly higher (35805760).  So according to both models, Venezuelan population would stabilize at around 35 million inhabitants.  Can this be taken as a good prediction? One can only speculate...

Another question: why would a 2 population model with each group growing logistically offer a better fit than the generalized logistic growth model? I took the first order and second order differences for this data and saw that more than one inflection point was present in the data: one between 1891-1920, another between 1920-1926, yet another between 1926-1936 and the final one between 1990-2001.  From a historic perspective, my surmise is that during Guzman Blanco's long regime (1870-1887), his project of establishing a central "bourgeois" state and the modernization of the productive apparatus in conjunction with international capital created a situation that favored "faster than usual" population growth (as can be seen between the three census in the 1873-1891 period).  After Guzman Blanco, Venezuela was again in the throes of power strife between local caudillos which resulted in the so called Revolución Libertadora one of the bloodiest civil
 wars after the Federal Wars of the mid 19th century in Venezuela.  Add to that the scourge of malaria (with its sterilizing effects on the population) and as a result the population growth momentum that started during Guzman Blanco suffered a setback.  However, as Venezuela became an oil producing country after the 1920's, the "modernization" that this brought about had the effect of putting the population back into the fast growth track.  This would account for at least one of the inflection points.  The other inflection point during the 1990-2001 period I would attribute to the "normal" population dynamic during this latter petroleum-modern phase in Venezuela.  At any rate, the census growth curve exhibits more than one inflection point and that would make the one variable logistic models (generalized or not) bad candidates for this data set, as they only have one inflection point.  This would also explain why all the one variable logistic
 models overestimate the population levels during the 1891-1961 period when all this "inflection point disorder" was taking place.

Previously I had interpreted the y1 and y2 populations as subgroups with pre-modern and modern growth dynamics, respectively.  Now, I don't think they have any interpretation as such and are just merely an artifice to appropiately "capture" the more than one inflection points present in the real data.  What do you all think?

It is also interesting to note that when you assume a 2 variable model with each variable following the simple Verhulst logistic growth model (with the excess y1 population trickling up to y2), you don't need to adjust the inflection point by adding parameters to the Verhulst equation as you do when you obtain the generalized logistic model.

Regards,

jose loreto romero palma
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