Leslie matrix

Matrices arising in a discrete-time deterministic model of population growth [a3]. The Leslie model considers individuals of one sex in a population which is closed to migration. The maximum life span is $k$ time units, and an individual is said to be in the $i$th age group if its exact age falls in the interval $[ i - 1 , i )$, for some $1 \leq i \leq k$. The corresponding Leslie matrix is given by

\begin{equation*} L = \left( \begin{array} { c c c c c } { m _ { 1 } } & { m _ { 2 } } & { \ldots } & { \ldots } & { m _ { k } } \\ { p _ { 1 } } & { 0 } & { \ldots } & { \ldots } & { 0 } \\ { 0 } & { p _ { 2 } } & { 0 } & { \ldots } & { 0 } \\ { \vdots } & { \square } & { \ddots } & { \square } & { \vdots } \\ { 0 } & { \ldots } & { 0 } & { p _ { k - 1 } } & { 0 } \end{array} \right), \end{equation*}

where for each $1 \leq i \leq k - 1$, $p _ { i }$ is the proportion of individuals in the $i$th age group who survive one time unit (this is assumed to be positive), and for each $1 \leq i \leq k$, $m \quad i$ is the average number of individuals produced in one time unit by a member of the $i$th age group. Let $v _ { i , t }$ be the average number of individuals in the $i$th age group at time $t$ units, and let $v _ { t }$ be the vector

\begin{equation*} \left( \begin{array} { c } { v _ { 1 , t }} \\ { \vdots } \\ { v _ { k , t } } \end{array} \right). \end{equation*}

Then $v _ { t + 1} = L v_ t $, and since the conditions of mortality and fertility are assumed to persist, $v _ { t } = L ^ { t } v _ { 0 }$ for each integer $t \geq 0$.

If some $m \quad i$ is positive, then $L$ has one positive eigen value $r$ which is a simple root of the characteristic polynomial. For any eigenvalue $\lambda$ of $L$, $r \geq | \lambda |$; indeed $L$ has exactly $d$ eigenvalues of modulus $r$, where $d$ is the greatest common divisor of $\{ i : m _ { - } i > 0 \}$. Corresponding to the eigenvalue $r$ is the right eigenvector $w$ given by the formula

\begin{equation*} w = \frac { 1 } { s } \left( \begin{array} { c } { 1 } \\ { p _ { 1 } / r } \\ { p _ { 1 } p _ { 2 } / r ^ { 2 } } \\ { \vdots } \\ { p _ { 1 } \dots p _ { k - 1} / r ^ { k - 1 } } \end{array} \right), \end{equation*}

where $s = 1 + p _ { 1 } / r + \ldots + p _ { 1 } \ldots p _ { k - 1 } / r ^ { k - 1 }$. A left eigenvector corresponding to $r$ has the form $[ y _ { 1 } \ldots y _ { k } ]$, where for $1 \leq j \leq k$,

\begin{equation*} y _ { j } = \sum _ { i = j } ^ { k } p _ { j } \ldots p _ { i - 1 } m _ { i } r ^ { j - i - 1 }. \end{equation*}

The quantity $y_j$ is interpreted as the reproductive value of an individual in the $j$th age group.

Suppose that there are indices $i$, $j$ such that $1 \leq i \leq j \leq k$, and both $m_j$ and $v _ { i ,0} $ are positive. If $d > 1$, the sequence of age-distribution vectors, $v _ { t } / \sum _ { i = 1 } ^ { k } v _ { i , t }$, is asymptotically periodic as $t \rightarrow \infty$, and the period is a divisor of $d$ depending on $v_0$. When $d = 1$, then as $t \rightarrow \infty$, the sequence of age-distribution vectors converges to the eigenvector $w$, which is called the asymptotic stable age distribution for the population. The nature of the convergence of the age distributions is governed by the quantities $\lambda / r$, where $\lambda$ is an eigenvalue of $L$ distinct from $r$; a containment region in the complex plane for these quantities has been characterized (cf. [a2], [a5]). The sequence of vectors $v _ { t }$ is asymptotic to $c r ^ { t } w$, where $c$ is a positive constant depending on $v_0$; hence $r$ is sometimes called the rate of increase for the population. The sensitivity of $r$ to changes in $L$ is discussed in [a1] and [a6].

Variations on the Leslie model include matrix models for populations classified by criteria other than age (see [a1]), and a model involving a sequence of Leslie matrices changing over time (see [a4] and [a6]). A stochastic version of the Leslie model yields a convergence result for the sequence $v _ { t } / r ^ { t }$ under the hypotheses that $d = 1$ and $r > 1$ (see [a6]).

References

[a1]	H. Caswell, "Matrix population models" , Sinauer (1989)
[a2]	K.P. Hadeler, G. Meinardus, "On the roots of Cauchy polynomials" Linear Alg. & Its Appl. , 38 (1981) pp. 81–102
[a3]	P.H. Leslie, "On the use of matrices in certain population mathematics" Biometrika , 33 (1945) pp. 213–245
[a4]	N. Keyfitz, "Introduction to the mathematics of population" , Addison-Wesley (1977)
[a5]	S. Kirkland, "An eigenvalue region for Leslie matrices" SIAM J. Matrix Anal. Appl. , 13 (1992) pp. 507–529
[a6]	J.H. Pollard, "Mathematical models for the growth of human populations" , Cambridge Univ. Press (1973)