FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 6, PAGES 41-63

**Projection matrices revisited: a potential-growth indicator and
the merit of indication**

D. O. Logofet

Abstract

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The mathematics of matrix models for age- or/and stage-structured
population dynamics substantiates the use of the dominant
eigenvalue $$l_{1} of the
projection matrix $$**L** as a measure of the growth potential, or
of adaptation, for a given species population in modern plant or
animal demography.
The calibration of $$**L** =
**T** + **F** on the
"identified-individuals-of-unknown-parents" kind of empirical data
determines precisely the transition matrix $$**T**, but
admits arbitrariness in the estimation of the fertility
matrix $$**F**.
We propose an adaptation principle that reduces calibration to the
maximization of $$l_{1}(**L**)
under the
fixed $$**T**
and constraints on $$**F** ensuing
from the data and expert knowledge.
A theorem has been proved on the existence and uniqueness of the
maximizing solution for projection matrices of a general pattern.
A conjugated maximization problem for a "potential-growth
indicator" under the same constraints has appeared to be
a linear-programming problem with a ready solution, the
solution testing whether the data and knowledge are compatible with
the population growth observed.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k116/k11603h.htm

Last modified: November 21, 2012