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

Abstract

The mathematics of matrix models for age- and/or stage-structured population dynamics substantiates the use of the dominant eigenvalue λ 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 λ 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.

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Correspondence to Dmitrii O. Logofet.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 6, pp. 41–63, 2011/12.

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Logofet, D.O. Projection matrices revisited: a potential-growth indicator and the merit of indication. J Math Sci 193, 671–686 (2013). https://doi.org/10.1007/s10958-013-1494-3

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Keywords

  • Matrix Model
  • Vertex Versus
  • Projection Matrice
  • Nonnegative Matrice
  • Sandbar Shark