Dialectics in nonautonomous matrix population models: accuracy of calibration versus certain prediction

Logofet D.O.

A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences Russia, 119017, Moscow, Pyzhevskii Per. 3, +7 916 6286229,, Institute of Forest Science, Russian Academy of Sciences Sovetskaya Street 21, Uspenskoe, Moscow region, 143030, Russia, +7 495 634 5257

A great advantage of the matrix model for a discrete-structured population, x(t) Î Rn, is the possibility to calibrate the "projection" matrix L(t) form the data of only two consecutive counts (at time monts t and t + 1) and to calculate l1(L(t)), the adaptation measure of he local population under study [1]. This is the power of matrix models as a tool for comparative demography, but here also a methodological problem arise when we have a time series of data and it is necessary to summarise the outcome of the entire observation period. The nonautonomous matrix model represents a finie set of one-step matrices, L(t), each yielding its own set of quantitative characteristics of the population, which may even be contradictory in the forecast of its fate.

The contradictions are eliminated by averaging the set M of nonnegative matrices that forms the basic model equation

x(t + 1) = L(t) x(t), t = 0, 1, …, M – 1, (1)

and model logic leads to the problem of geometric average [2]. Defined by the life cycle graph for the individuals of a given species, the fixed pattern of these matrices deprives this problem of exact solution, so that the approximate pattern-geometric mean becomes the correct mode of averaging [3] – a novel concept in the theory and practice of modelling biological populations.

For the case where the data bear “reproductive uncertainty” [2] and the calibration yields the whole family of matrices, {L(t)} = T(t) + {F(t)}, at each time step, a heuristic method of averaging is proposed, namely, TF-averaging. It enables calculating uniquely the pattern-geometric mean of the transition matrices, T(t), hence gaining certain age-specific traits from the stage-structured model and, in particular, answering the question how specifically short a short-lived perennial lives.

The research is supported by the RFBR, grant № 16-04-00832.


1. Logofet, D.О. and Ulanova, N.G. Matrix Models in Population Biology. A Tutorial. – M.: MAKS Press, 2017. 128 pp. (in Russian).

2. Logofet, D.О, Kazantseva, E. S., Belova, I.N., and Onipchenko, V.G. Coenopopulation of Eritrichium caucasicum as an object of mathematical modelling. II. How short does a short-lived perennial live? // Zhurnal Obschei Biologii 78, № 1, 2017. Pp. 56–66 (in Russian).

3. Logofet D.О. Averaging the population projection matrices: heuristics against uncertainty and nonexistence // Ecological Complexity, 2018 (accepted for publication).

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