Multiregional Projection and Stable Growth
We have seen that the multiregional projection process focuses on the temporal evolution of a multiregional population by considering survivorship and migration, fertility and reproduction—and in instances of unchanging rates, the steady approach of that
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Multiregional Projection and Stable Growth
5.1 Mathematical Models: The Multiregional Projection Process Once Again In this chapter we review and expand our earlier expositions of the multiregional projection process, its link with the multiregional life table, and the conditions of stable growth.
5.1.1 The Multiregional Projection Process We have seen that the projection of a multiregional population of a given sex forward through time may be carried out by calculating the region- and age-specific survivors of that population and adding to this total the new births that survive to the end of the unit time interval. This growth process may be described by the following system of equations: (t+1)
Ki
(0) =
β−5 m
(t)
bj i (x)Kj (x),
α−5≤x ≤β −5
x=α j =1
i, j = 1, 2, . . . , m, Ki(t+1) (x + 5) =
m
sj i (x)Kj(t) (x),
(5.1)
x = 0, 5, 10, . . . , z − 5
j =1
where we henceforth assume a time and age interval of 5 years. In equation (5.1) we use the following notation:
© Springer Nature Switzerland AG 2020 A. Rogers, Applied Multiregional Demography Through Problems, https://doi.org/10.1007/978-3-030-38215-5_5
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5 Multiregional Projection and Stable Growth
bij (x) the average number of babies born during the unit time interval and alive in region j at the end of that interval, per x- to (x + 4)-year-old resident of region i at the beginning of that interval; sij (x) the proportion of x- to (x + 4)-year-old residents of region i at time t who are alive and x + 5 to x + 9 years old t years later in region j at time t + 1; (t) Ki (x) the x- to (x + 4)-year-old population in region i at time t; α the first age of childbearing; β the last age of childbearing.
5.1.2 Survivorship and Migration Consider, once again, an m-region population system with Ki(t) (x) individuals aged x to x + 4 in region i. As in the life table population, (t) Ki (x)
=
m
(t) j Ki (x),
i = 1, 2, . . . , m
(5.2)
j =1
where j Ki(t) (x) notes the number of j -born individuals who are in region i and aged x to x + 4 at time t. The expected survivors of the multiregional population after a unit time interval of 5 years, say, are Kj (x + 5) =
m
sij Ki (x),
j = 1, 2, . . . , m
i=1
where sij (x) is, as before, the probability that an individual in region i aged x to x + 4 will be in region j when aged x + 5 to x + 9. Substituting the definitional relationship of equation (5.2) into the equation just given, and utilizing the fact that i-born individuals can never become members of a j -born population (and vice versa), we have (t+1) (x + 5) = j Ki
m
ski (x)Kk(t) (x),
i, j = 1, 2, . . . , m,
(5.3)
k=1
or, in matrix form, K (t+1) (x + 5) = S(x)K (t) (x)
(5.4)
S(x) = K (t+1) (x + 5)K (t) (x)−1 ,
(5.5)
whence
5.1 Mathematical Models: The Multiregional Projection Process Once Again
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where, for example, in a biregional system, K (x) = (t)
(t) (t) 1 K1 (x) 2 K1 (x) (t) (t) 1 K2 (x) 2 K2 (x)
s (x) s21 (x) S(x) = 11 s12 (x) s22 (x)
As in the uniregional model, we assume that the survivorship and
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