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

Abstract

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 crossparenting, the bearing of children with one immigrant parent and one nonimmigrant. 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 leastsquares 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. JEL Classification: J10, J15

Critical patch-size for two-sex populations

“…In addition, G M (M, F) and G F (M, F) are continuous functions with continuous first derivatives of their variables.With these properties established, along with boundary conditions(8,9), this system satisfies the hypotheses of the comparison theorem 1.20 in[2]. This theorem implies that, given two solutions = S M X T F X T ( ( , ), ( , ))…”

“…In our analysis, the population of each sex was considered homogeneous and therefore a homogeneous mixing between males and females was assumed. The subdivision of populations in different subgroups and the consideration preferential mixing between the subgroups have shown to induce major effects on the dynamics of epidemiological models [7][8][9]. We conjecture that the inclusion of such effects can affect the critical-patch size of two-sex populations.…”

“…Solutions for Eq. (12) with negative δ m 2 lead to real-valued exponential functions that do not satisfy the boundary conditions (8). Therefore, δ m is real and can be assumed non-negative without loss of generality, so the solution of (12) can be written as:…”

“…We write the system of Eqs. (6), (7) as: of (6,7) subject to boundary conditions (8,9) such that initially we have…”