We start with a hypothetical country (region) with overall population *N*. The population is distributed among two sexes and five age groups: “children” (0–19), “young adults” (20–39), “middle aged” (40–59), “old” (60–79), and “very old” (80–99), with no survivors at 100. The age-sex distribution is captured by a 10-element vector *n* = [*n*
_{
x
}], with *x* = 1 to 5 representing the female age groups, *x* = 6 to 10 the male age groups. All children have young adult parents—mothers in group *n*
_{2}, fathers in *n*
_{7}. The fertility rate for young adult females is thus equivalent to the conventional total fertility rate. In addition, restricting fathers to the same age range allows the calculation of a male fertility rate. (The model is a one-sex model in the classical sense of female lineage, and the male fertility rate is an easily calculated concomitant of the female rate.)

In the absence of migration, in or out, the population is augmented by births, depleted by deaths, has a 10 × 10 projection (Leslie) matrix *Q*, and a stable age distribution. The female fertility rate is set at the natural replacement level, survival rates are unchanging, and *n*
_{
t + 1} = *Qn*
_{
t
} = *n*
_{
t
} for all *t* where the time interval is 20 years, the same as the age intervals. For convenience, we shall refer to this interval as a *generation*. (To keep notation as simple as possible, we add a time subscript when necessary but avoid doing so otherwise.)

Now introduce immigration (still with no emigration, which we shall take to be zero) and assume that immigrants have the same stable age-sex distribution, proportionately, as the original non-immigrants and the same projection matrix *Q*. In our simulations below, immigration may be one-time or repeated, but to develop the framework, assume for the present that it is a one-time event with immigrants arriving at *t* = 0. The question of interest is how rapidly will the populations of immigrants and non-immigrants mix where mixing in our context means cross-parenting—initially, the bearing of children with one immigrant parent and one non-immigrant parent, although the descendants of such a union will also be regarded as mixed. (It is perhaps well to recognize, before proceeding, that realistically in virtually any place in the world, all of the population will be descended from immigrants if one goes back far enough in time. In what follows, we will simply identify *t* = 0 as a point at which new entrants will be classified as immigrants and the existing population as non-immigrants.)

We identify then three separate populations within the overall population *N*, each evolving in its own way: (1) the original non-immigrant population and its non-mixed descendants *H*; (2) the population of immigrants and their non-mixed descendants *M*; and (3) a mixed population *U*, including all children of mixed lineage—children of mixed parents, grandchildren of mixed parents, and, in general, all persons with lineage traceable back to a mixed union. (Mnemonically, *H* is for “home,” *M* for “migrant,” *U* for “union.” Where the meaning is clear, we shall sometimes use the word “immigrants” to refer to members of the *M* population, thus including both those who immigrated originally and their descendants.) These three populations have age-sex vectors *h*, *m*, and *u* corresponding in structure to *n* (and aggregating to *n*). They also have the same projection matrix *Q*. While initially immigrants and non-immigrants have the same proportionate age-sex distribution, they may differ in other characteristics. The non-immigrant population will be augmented in each generation by births and depleted by deaths, but those births to non-immigrant mothers mated with immigrant or mixed fathers will be transferred (reclassified) to the mixed population. Similarly, the immigrant population will be augmented by births and depleted by deaths, but all births to immigrant mothers mated to non-immigrant or mixed fathers will be transferred to the mixed population. The mixed population will be augmented by births, depleted by deaths, and augmented also by the cross-parenting transfers from the other populations. If cross-parenting continues freely and indefinitely—if individuals choose to mate randomly and bear children without preference as to population membership—the non-immigrant and immigrant populations will vanish in the limit; all residents of the country will eventually be of mixed lineage. The proportion of mixed population in the total population of the country serves as an indicator at any given time of the degree of mixing that has occurred. (Note that the overall population *N* continues to have the same stable age-sex distribution; that is not affected by transfers among its component populations.)

The foregoing assumes one-time immigration. If immigration is repeated—at a constant rate proportional to the total population, let us say—the framework is the same as before except that the immigrant population will now be augmented by new immigrants each generation. The non-immigrant population will still vanish, in the limit, under random parenting, but the immigrant population will be continuously replenished, and the mixed proportion in the overall population will always be less than one.

The accounting relations for the process with repeated immigration can be stated informally as follows. The change in the non-immigrant population from one generation to the next can be represented as

$$ \Delta H=\mathrm{Births} - \mathrm{Deaths} - \mathrm{CPT}\left(H,M\to U\right) - \mathrm{CPT}\left(H,U\to U\right) $$

CPT stands for a cross-parenting transfer of newborn children, and the arrow indicates that the direction of transfer is to *U*, the mixed population. The transfers result from a non-immigrant/immigrant (mother/father) pairing (*H*,*M*) in the first case and a non-immigrant/mixed population pairing (*H*,*U*) in the second. (The first letter is always the population of the mother.) Using similar notation, the changes in the immigrant and mixed populations can be represented as

$$ \Delta M=\mathrm{Births} - \mathrm{Deaths} - \mathrm{CPT}\left(M,H\to U\right) - \mathrm{CPT}\left(M,U\to U\right)+\mathrm{New}\ \mathrm{Immigrants} $$

$$ \Delta U=\mathrm{Births} - \mathrm{Deaths}+\mathrm{CPT}\left(U\leftarrow H,M\right)+\mathrm{CPT}\left(U\leftarrow H,U\right)+\mathrm{CPT}\left(U\leftarrow M,H\right)+\mathrm{CPT}\left(U\leftarrow M,U\right) $$