Immigration and the rate of population mixing: Explorations with a stylized model

The integration or mixing of immigrants with non-immigrants is an important issue in many countries. There are various forms of mixing. We consider here cross-parenting, the bearing of children with one immigrant parent and one non-immigrant. Our objective is to model cross-parenting as a demographic process and investigate the rate at which such mixing could occur. We identify three populations within an overall total: non-immigrant, immigrant, and mixed. A model is constructed to track the three as they change and interact through cross-parenting. The populations evolve by simulation in accordance with a common stable projection matrix. However, as cross-parenting between immigrants and non-immigrants occurs, the progeny are transferred to the mixed population; the immigrant and non-immigrant populations are thus depleted by the transfers and the mixed population augmented in each generation. The transfers are governed by underlying preferences, but the preference pattern must be modified to recognize constraints imposed by differences in population size. A restricted least-squares procedure effects the modification so that the actual pattern is as close as possible to the preferred one. Simulations are carried out with alternative preferential patterns and rates of immigration. Of particular interest is the proportion of mixed population in the total in each generation and the final steady state. The paper develops a new framework and model to show the rate at which population mixing could occur under alternative assumptions about the immigration rate and preferences for cross-parenting.