ON THE USE OF MATRICES IN CERTAIN POPULATION MATHEMATICS
ON THE USE OF MATRICES IN CERTAIN POPULATION MATHEMATICS
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DOI：
10.1093/biomet/33.3.183
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发表时间：
1945-01-01
期刊：
影响因子：
2.7
通讯作者：
LESLIE, PH
中科院分区：
文献类型：
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作者：
LESLIE, PH
给定人口在给定时间t的[女性]年龄分布可以由向量[xi](t)表示，其分量是在时间t时年龄x和x+1之间存活的[女性][女性]数量。时间单位是任意的。如果 Px 表示在时间 t 年龄为 x 至 x + 1 的[女性]存活到时间 t + 1 的概率，如果 Fx 表示在时间 t 时年龄为 x 至 x + 1 的[女性]在 t 和 t + 1 之间出生并存活到时间 t + 1 的女儿的数量，如果 Fk 是最后一个非零 F，则向量 [xi](t + 1)，被认为是一列 k + 1 行矩阵，可以写成 A[xi](t) 形式，其中 A 是非奇异 (k + 1) X (k + 1) 矩阵，第一行为 F0, F1,--------, Fk，下对角线为 Pa, P1,[长划线][长划线][长划线][长划线][长划线], Pk - 1，其他元素为 0。 时间 t 的分布可以写为 At[xi](0)。如有必要，年龄分布可以延伸到生殖周期结束之后，但人口变化的主要特征是确定的。在这个有限的年龄区间。如果 Hi = PiPi+i......Pk - 1，i = 0-----k = 1，并且 H 是乘数 Ho--------Hk 的对角矩阵，则通过引入相关分布向量 [psi](t) = H[xi](t) 和新的变换矩阵 B = HAH-1（其元素为 Bi[长破折号]FiH0/Hi(j = 0,-----, k)）来简化计算 第一行，l 位于其下对角线，0 位于其他位置。那么ψ(t)=Btψ(0)。矩阵A和B的潜根是特征方程[A-λI]-[B-λI]＝(-1)k+1(λk+1-Boλk-1-Biλk-l-----Bk)----0的根。对于每个根λa，对应于向量ψa和向量xia，使得Bψa＝λaψa Aψa＝λa[xi]a。只有一个正根λ 1 ，这对应于稳定的年龄分布xi 1 。当λ1>1时，人口增加，而当λ1<1时，人口减少。一般年龄分布可以扩展为对应于 B 的潜在根的特征分布的线性组合。经过适当归一化的稳定向量 psi 可以排列为矩阵 Q 的列。Q-1 的行定义一组行向量 [PHI]，并且向量 eta = [PHI]H 使得 eta[xi] 是不变量。给出了一个数值示例，基于假想的褐家鼠种群，细分为一个月的时间间隔，k + 1 = 21。从 t = 0 时的稳定马尔萨斯分布 [xi](0) 开始，首先通过乘以 lambda1 = 1.561505 计算 t = 1 时的新分布 [xi](1)，然后独立地与矩阵 A 运算。这 2 个结果是 非常一致。作者还研究了以 3 个月为间隔选择 k + 1 = 7 对根 λ 的影响，并且注意到两个结果之间存在一些差异。给定人口在给定时间 t 的[女性]年龄分布可以用向量 [xi] (t) 表示，其分量是在时间 t 时年龄 x 和 x + 1 之间的[女性]存活人数。时间单位是任意的。如果 Px 表示在时间 t 年龄为 x 至 x + 1 的[女性]存活到时间 t + 1 的概率，如果 Fx 表示在时间 t 时年龄为 x 至 x + 1 的[女性]在 t 和 t + 1 之间出生并存活到时间 t + 1 的女儿的数量，如果 Fk 是最后一个非零 F，则向量 [xi](t + 1)，被认为是一列 k + 1 行矩阵，可以写成 A[xi](t) 形式，其中 A 是非奇异 (k + 1) X (k + 1) 矩阵，第一行为 F0, F1,--------, Fk，下对角线为 Pa, P1,[长划线][长划线][长划线][长划线][长划线], Pk - 1，其他元素为 0。 时间 t 的分布可以写为 At[xi](0)。如有必要，年龄分布可以延伸到生殖周期结束之后，但人口变化的主要特征是确定的。在这个有限的年龄区间。如果 Hi = PiPi+i......Pk - 1，i = 0-----k = 1，并且 H 是乘数 Ho--------Hk 的对角矩阵，则通过引入相关分布向量 [psi](t) = H[xi](t) 和新的变换矩阵 B = HAH-1（其元素为 Bi[长破折号]FiH0/Hi(j = 0,-----, k)）来简化计算 第一行，l 位于其下对角线，0 位于其他位置。那么ψ(t)=Btψ(0)。矩阵A和B的潜根是特征方程[A-λI]-[B-λI]＝(-1)k+1(λk+1-Boλk-1-Biλk-l-----Bk)----0的根。对于每个根λa，对应于向量ψa和向量xia，使得Bψa＝λaψa Aψa＝λa[xi]a。只有一个正根λ 1 ，这对应于稳定的年龄分布xi 1 。当λ1>1时，人口增加，而当λ1<1时，人口减少。一般年龄分布可以扩展为对应于 B 的潜在根的特征分布的线性组合。经过适当归一化的稳定向量 psi 可以排列为矩阵 Q 的列。Q-1 的行定义一组行向量 [PHI]，并且向量 eta = [PHI]H 使得 eta[xi] 是不变量。给出了一个数值示例，基于假想的褐家鼠种群，细分为一个月的时间间隔，k + 1 = 21。从 t = 0 时的稳定马尔萨斯分布 [xi](0) 开始，首先通过乘以 lambda1 = 1.561505 计算 t = 1 时的新分布 [xi](1)，然后独立地与矩阵 A 运算。这 2 个结果是 非常一致。作者还研究了以 3 个月为间隔选择 k + 1 = 7 对根 λ 的影响，并注意到两个结果之间存在一些差异。
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.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.