• 1 Unité de modélisation mathématique et informatique des systèmes complexes [Bondy]
• 2 Université de Tokyo [UTokyo]
• 3 The University of Tokyo [UTokyo]
• 4 Tōkyō teikoku daigaku = University of Tokyo [Tokyo] [UTokyo]
• 5 Département de Mathematiques [Tlemcen]

We study a mathematical model for the demographic transition. It is a homogeneous differential system of degree one. There are two age groups and two fertility levels. Low fertility extends by mimicry to adults with high fertility. When the mimicry coefficient increases, the system crosses two thresholds between which the population increases or decreases exponentially with a stable mixture of the two fertility rates. This partial demographic transition is reminiscent of the situation in some countries of sub-Saharan Africa.