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

Immigrants can mix with the population of a receiving country in various ways. We consider demographic mixing by which we mean cross-mating, and more particularly the bearing of children where one parent is of immigrant descent and the other is not - cross-parenting as we term it. We consider a hypothetical country with an initial stable population and introduce immigration. The results of cross-parenting are taken into account by identifying three separate populations within the overall total: non-immigrant population, immigrant population (immigrants and their descendants), and mixed population. We develop a stylized model to track the three populations, with interest focusing in particular on how the proportion of mixed population changes through time as it moves toward a steady state. The model has a stable projection (Leslie) matrix that holds for all three populations and moves them forward from generation to generation as each evolves in its own way. As cross-parenting occurs the resulting progeny are transferred from the other populations to the mixed population. The pattern of cross-parenting is determined in the first instance by a matrix representing preferences among the three populations and alternative preferential patterns are experimented with, ranging from complete isolation to indifference as to cross-parenting choices. However the matrix must be modified to recognize supply constraints imposed by the sizes of the available populations and a restricted least-squares procedure is employed to effect the modification while remaining as close as possible to the original preference pattern. Alternative rates of immigration are experimented with also.