Abstract
When we admit the background independence of fertility decline and aspects of diffusion, we can construct a reaction-diffusion model to describe fertility decline as a stochastic process independent of its background. Once we postulate a reaction-diffusion process for this phenomenon, we can estimate the velocity of a progressive wave of diffusion. By means of the estimated velocity, we can estimate where a singularity of fertility decline was and when it appeared. The singularity existed in a French district Aquitaine basin. From Lot-et-Garonne in Aquitaine the reaction-diffusion of fertility decline began to diffuse to all Europe maintaining independence of socio-economic conditions.
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Notes
- 1.
Let p be a probability of a conception at a moment. \(\int _0^t p dt\) is the number of children for t time. \(\int _0^t p dt \equiv p(t+C)\).
- 2.
These occurrences are also supported empirically. Even Hutterites have a maximum, a minimum, a mean, and a variances of their number of children.
- 3.
Actually I selected Tokyo (the capital of Japan) area and Osaka (the centre of west Japan) and their surroundings. Japanese census recoded the means of number of children ever born of ever married women at 1960 and at 1970 by five-year cohorts for city, town and village. I draw circles with each radius (5, 10, 15, 20, 25, 30, 35 km) from the centre where fertility decline began. And I select some cities or towns or villages nearest to each circumference. I set an average of these \(\overline{c}\) for a representative \(\overline{c}_i\) at each radius. From 1896–1900 (born) cohort to 1921–1925 cohort, \(c(x_i, t_k)\) was recoded for each cities.
- 4.
I believe this is because of the size of the area that data cover, with Tokyo and Osaka being much smaller regions than Hokkaido. In order to demonstrate the perfect proportionality, in line with the argument by J.G. Skellam I have resorted to use Hokkaido case, roughly 9 times as large as those in Tokyo, Osaka.
- 5.
The famous article in mathematical ecology by J.G. Skellam [61] reported the same linearity of diffusion of muskrats from Bohemia to mid-Europe. In his article, the space is widened from Munich to Breslau—in a circle with 300 km radius.
- 6.
If we set Hamburg as a point to draw a circle, as Hamburg started fertility decline around 1880 (from Coale and Treadway [13]), it took about 110 years to arrive, so we drew a radius of 1100 km. The intersection is still on Aquitaine.
- 7.
Geneva’s \(I_g\) was 0.458 in 1860. This figure is far more than Lot-et-Garonne. Low fertility of Geneva was caused by the low level of \(I_m\)—low level of the rate of ever married. Swiss still remained in the medieval fertility control stage.
- 8.
Strictly speaking, smaller is not better. There is a lower limit of the space lattice and time pitch widths when the numbers of children of each set of parents are treated in a stochastic manner. Of course these widths are much larger within statistical data.
- 9.
“Kein Sieger glaubt an den Zufall.” (F. Nietzsche ) I think fertility decline in the European modern period was the coincidence.
- 10.
The commune of Agen did not disappear. It remains a beautiful garden city, as portrayed on the website http://www.agen/fr/. The decline of fertility did not endure.
- 11.
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Ike, S. (2016). Reaction-Diffusion of the Number of Children. In: Fertility Decline and Background Independence. SpringerBriefs in Population Studies(). Springer, Tokyo. https://doi.org/10.1007/978-4-431-55151-5_2
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DOI: https://doi.org/10.1007/978-4-431-55151-5_2
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