On the Use of Matrices in Certain Population Mathematics

Symmetries between life lived and left in finite stationary populations

“…Certainly, the same values from Table 3 would be obtained in any census carried out in the stationary part of the series at exact times t, t + 1, etc. These observations lead to the following conclusion: If a finite stationary population is observed at regular time-intervals, and age and life left are grouped at intervals of 5 The same approach is used in matrix population models, in which one also considers an age-classified population, and defines the projection matrix ('Leslie matrix') from exact time t to exact time t + 1 for each age class (Caswell 2001;Leslie 1945) the same width, then the Brouard-Carey equality may hold. A more formal statement is presented in the following section.…”

“…However, in matrix population models, death probabilities -or the analogous survival probabilities -usually refer to the period-cohort Lexis parallelogram, instead of the age-cohort parallelogram. For instance, the elements of the sub-diagonal of the Leslie matrix are the probabilities that an individual with age [x, x + 1) at exact time t will be alive in age group [x + 1, x + 2) at exact time t + 1 (Caswell 2001;Leslie 1945). 7 For time-intervals of length n = 1, the conversion from death rates m x to death probabilities is given by q x = m x /(1 − (1 − a x ) · m x ), assuming m x is represented by a step function.…”

A simulation analysis of the longer-term effects of immigration on per capita in-come in an aging population

“…where m is a vector of immigrants (with age-sex elements corresponding to those of n, all nonnegative) and Q is a 10×10 Leslie matrix (Leslie, 1945(Leslie, , 1948; its nonzero elements are determined by age-sex-specific survival rates, the fertility rate, and the male/female birth ratio. If there were no immigration, and all rates were constant,…”

Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates

“…Such populations are described by stage-classified demographic models, of which the age-classified theory is a special case. Stage-classified demography can be analyzed using matrix population models (Leslie 1945;Caswell 2001); see Section B. The discrete-time population growth rate λ is the dominant eigenvalue of the population projection matrix A (guaranteed to be real and positive by the Perron-Frobenius theorem; Caswell 2001).…”

“…To compare (24) with Hamilton's results (1) and (2), consider an age-classified matrix (a Leslie matrix) with fertilities F i in the first row, survival probabilities P i on the subdiagonal, and zeros elsewhere (Leslie 1945;Keyfitz 1968). In this case (24) simplifies to…”