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 $$l₁ of the projection matrix $$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 $$F. We propose an adaptation principle that reduces calibration to the maximization of $$l₁(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.

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Last modified: November 21, 2012