ON THE USE OF MATRICES IN CERTAIN POPULATION MATHEMATICS

• 成果类型: Article
• 署名作者: LESLIE, PH
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/33.3.183
• 发表日期: 1945
• 页码: 183212

摘要

The [female] age distribution at a given time t for a given population may be represented by a vector [xi] (t) whose components are the number of [female] [female] alive between age x and x + 1 at time t. The unit of time is arbitrary. If Px denotes the probability that a [female] aged x to x + 1 at time t will survive to time t + 1, if Fx denotes the number of daughters which are born between t and t + 1 to [female] [female] aged x to x + 1 at time t, and which survive to time t + 1, and if Fk is the last non-vanishing F, then the vector [xi](t + 1), thought of as a one column matrix of k + 1 rows, can be written in the form A[xi](t), where A is a non-singular (k + 1) X (k + 1) matrix whose first row is F0, F1,--------, Fk, whose subdiagonal is Pa, P1,[long dash][long dash][long dash][long dash][long dash], Pk - 1, and whose other elements are 0. The distribution at time t can be written At[xi](0). The age distribution can be extended, if necessary, beyond the end of the reproductive cycle, but the main features of the variation of population are detd. in this restricted age interval. If Hi = PiPi+i......Pk - 1, i = 0-----k = 1, and H is a diagonal matrix with multipliers Ho------Hk, then the computation is facilitated by introducing a related distribution vector [psi](t) = H[xi](t) and a new transformation matrix B = HAH-1 having elements Bi[long dash]FiH0/Hi(j = 0,-----, k) in its first row, l''s in its subdiagonal and 0''s elsewhere. Then [psi](t) = Bt[psi](0). The latent roots of the matrices A and B are the roots of characteristic equation [A - [lambda]I] - [B - [lambda]I] = (-1)k+1 ([lambda]k +1 - Bo[lambda]k-1 - Bi[lambda]k-l-----Bk)----0. To each root [lambda]a, corresponds a vector [psi]a and a vector [xi]a such that B[psi]a = [lambda]a[psi]a A[psi]a = [lambda]a[xi]a. There is just one positive root [lambda]1, and this corresponds to a stable age distribution [xi]1. For [lambda]1 > 1 the population increases, and for [lambda]1 < 1 it decreases. The general age distribution may be expanded as a linear combination of the characteristic distributions corresponding to the latent roots of B. The stable vectors [psi], suitably normalized, can be arranged as columns of a matrix Q. The rows of Q-1 define a set of row vectors [PHI], and the1 vectors [eta] = [PHI]H are such that [eta][xi] is an invariant. A numerical example is given, based on an imaginary rodent population of Rattus norvegicus, subdivided into one month time intervals with k + 1 = 21. Starting with a stable Malthusian distribution [xi](0) at t = 0, the new distribution [xi](1) at t = 1 is computed first by multiplying by [lambda]1 = 1.561505, then independently by operating with the matrix A. The 2 results are in close agreement. The author also studies the effect on the roots [lambda] of choosing k + 1 = 7 with 3-month intervals and some discrepancies are noted between the 2 results.