A Singularity of Fertility Decline Is Lot-et-Garonne

Abstract

Although we generally tend to conceive that fertility decline has been caused by the alterations of socioeconomic conditions, fertility decline in the 19th century in Europe was a reaction-diffusion process independent of them (e.g., see Coale, 1967, 1973; Becker, 1978, 1981). 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 relative independence of socioeconomic conditions.

Keywords:

• Aquitaine
• fertility decline
• Lot-et-Garonne
• progressive wave
• reaction-diffusion
• singularity
• stochastic process

Acknowledgments

I am indebted to Emeritus Professor D. Heise of Indiana University for an English check. It is pleasant to express gratitude to him who advised a lot of intelligent instructions. I am also grateful to the two anonymous reviewers who gave me many thoughtful comments.

Notes

Sources: Bureau of Statistics, Office of the Prime Minister (1963); Festy (1984, pp. 48–50).

¹Reaction-diffusion is a mathematical model of diffusion and pattern formation as described differential equation (1) or (7).

²Let p be a probability of a conception at a moment. is the number of children for t time; .

³These occurrences are also supported empirically. Even Hutterites have a maximum, a minimum, a mean, and a variances of their number of children.

⁴Actually, I selected Tokyo (the capital of Japan) area and Osaka (the center 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 5-year cohorts for city, town, and village. I draw circles with each radius (5 km, 10 km, 15 km, 20 km, 25 km, 30 km, 35 km) from the center where fertility decline began. And I select some cities or towns or villages nearest to each circumference. I set an average of these for a representative at each radius. From 1896–1900 (born) cohort to 1921–1925 cohort, c(x _i , t _k ) was recoded for each city.

⁵I 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.

⁶The famous article in mathematical ecology by J. G. Skellam (1951) 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.

Note. Kanto is the name of the zone around Tokyo. Tokyo is one of the singularities of fertility decline in Japan. Niigata is the biggest of the cities front on the Japan Sea. Tokyo is the city front on the Pacific Ocean; namely, the progressive wave ran cross the Japanese Island. Sapporo is the center of Hokaido which is an isolated island of north Japan.

Source: J. Knodel & Etienne van de Walle, 1979, pp. 221–222.

Note. The boldface implies the approximate date when fertility decline began. Monotonic decline is the manifest sign of the beginning of fertility decline caused by reaction-diffusion. Source: Festy, 1984, pp. 48–49.

⁷If I set Hamburg as a point to draw a circle, as Hamburg started fertility decline around 1880 at the real period date (Coales & Treadway, 1986), it took about 110 years to arrive, so I drew a radius of 1100 km. The intersection is still on Aquitaine.