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 time units, and an individual is said to be in the
th age group if its exact age falls in the interval
, for some
. The corresponding Leslie matrix is given by
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where for each ,
is the proportion of individuals in the
th age group who survive one time unit (this is assumed to be positive), and for each
,
is the average number of individuals produced in one time unit by a member of the
th age group. Let
be the average number of individuals in the
th age group at time
units, and let
be the vector
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Then , and since the conditions of mortality and fertility are assumed to persist,
for each integer
.
If some is positive, then
has one positive eigen value
which is a simple root of the characteristic polynomial. For any eigenvalue
of
,
; indeed
has exactly
eigenvalues of modulus
, where
is the greatest common divisor of
. Corresponding to the eigenvalue
is the right eigenvector
given by the formula
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where . A left eigenvector corresponding to
has the form
, where for
,
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The quantity is interpreted as the reproductive value of an individual in the
th age group.
Suppose that there are indices ,
such that
, and both
and
are positive. If
, the sequence of age-distribution vectors,
, is asymptotically periodic as
, and the period is a divisor of
depending on
. When
, then as
, the sequence of age-distribution vectors converges to the eigenvector
, 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
, where
is an eigenvalue of
distinct from
; a containment region in the complex plane for these quantities has been characterized (cf. [a2], [a5]). The sequence of vectors
is asymptotic to
, where
is a positive constant depending on
; hence
is sometimes called the rate of increase for the population. The sensitivity of
to changes in
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 under the hypotheses that
and
(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) |
Leslie matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leslie_matrix&oldid=49901