"As a country evolves from a traditional to an advanced society, the part of urban growth that is due to net inmigration follows a simple pattern, which can be described by an inverted U-shaped curve: it first increases, then passes through a maximum, and decreases thereafter. This hypothesis is confirmed by quantitative analysis using time-series and cross-section data. The analysis suggests that in the second half of this century natural increase often provides a slightly higher contribution to urban population growth than net inmigration." (summary in FRE, ITA, JPN, )
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Jacques Ledent (1982) studied this question.