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Does averaging overestimate or underestimate population growth? It depends

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  • Logofet, Dmitrii O.

Abstract

When a matrix population model is nonautonomous, i.e., when it represents a set of single-time-step ("annual") PPMs, L(t), t = 0, 1, …, T – 1, each corresponding to a fixed life cycle graph, then each of the annual matrices generates its own set of model results to characterize the population. In particular, λ1(t), the asymptotic growth rate, varies with t and may result in alternating predictions of population viability. A logical way to characterize the population over the total period of observations is to average the given set of T PPMs, and I have proved the correct mode of averaging to be the pattern-geometric average. It means finding a matrix, G, such that its Tth power equals the product of T annual matrices (in the chronological order), while its pattern does correspond to the given life cycle graph. In practical cases however, the mathematical problem of pattern-geometric average has no exact solution for a fundamental mathematical reason. Nevertheless, the approximate solutions have revealed a fair precision of approximation in recent case studies of alpine short-lived perennials (Eritrichium caucasicum and Androsace albana), resulting in quite certain predictions of population viability by means of λ1(G), the dominant eigenvalue of the average matrix.

Suggested Citation

  • Logofet, Dmitrii O., 2019. "Does averaging overestimate or underestimate population growth? It depends," Ecological Modelling, Elsevier, vol. 411(C).
    • Handle: RePEc:eee:ecomod:v:411:y:2019:i:c:s0304380019302522
    DOI: 10.1016/j.ecolmodel.2019.108744
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    References listed on IDEAS

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    1. Logofet, Dmitrii O., 2017. "Aggregation may or may not eliminate reproductive uncertainty," Ecological Modelling, Elsevier, vol. 363(C), pages 187-191.
    2. Klimas, Christie A. & Cropper, Wendell P. & Kainer, Karen A. & de Oliveira Wadt, Lúcia H., 2012. "Viability of combined timber and non-timber harvests for one species: A Carapa guianensis case study," Ecological Modelling, Elsevier, vol. 246(C), pages 147-156.
    3. Logofet, Dmitrii O., 2013. "Projection matrices in variable environments: λ1 in theory and practice," Ecological Modelling, Elsevier, vol. 251(C), pages 307-311.
    4. Logofet, Dmitrii O., 2008. "Convexity in projection matrices: Projection to a calibration problem," Ecological Modelling, Elsevier, vol. 216(2), pages 217-228.
    5. Logofet, Dmitrii O., 2013. "Calamagrostis model revisited: Matrix calibration as a constraint maximization problem," Ecological Modelling, Elsevier, vol. 254(C), pages 71-79.
    6. Politi, Tiziano & Popolizio, Marina, 2015. "On stochasticity preserving methods for the computation of the matrix pth root," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 110(C), pages 53-68.
    7. Logofet, Dmitrii O., 2016. "Estimating the fitness of a local discrete-structured population: From uncertainty to an exact number," Ecological Modelling, Elsevier, vol. 329(C), pages 112-120.
    8. Sanz, Luis, 2019. "Conditions for growth and extinction in matrix models with environmental stochasticity," Ecological Modelling, Elsevier, vol. 411(C).
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    Cited by:

    1. Dmitrii O. Logofet & Leonid L. Golubyatnikov & Elena S. Kazantseva & Nina G. Ulanova, 2021. "“Realistic Choice of Annual Matrices Contracts the Range of λ S Estimates” under Reproductive Uncertainty Too," Mathematics, MDPI, vol. 9(23), pages 1-15, November.
    2. Dmitrii O. Logofet, 2023. "Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools," Mathematics, MDPI, vol. 11(14), pages 1-12, July.
    3. Dmitrii O. Logofet & Leonid L. Golubyatnikov & Nina G. Ulanova, 2020. "Realistic Choice of Annual Matrices Contracts the Range of λ S Estimates," Mathematics, MDPI, vol. 8(12), pages 1-15, December.
    4. Logofet, Dmitrii O. & Golubyatnikov, Leonid L. & Kazantseva, Elena S. & Belova, Iya N. & Ulanova, Nina G., 2023. "Thirteen years of monitoring an alpine short-lived perennial: Novel methods disprove the former assessment of population viability," Ecological Modelling, Elsevier, vol. 477(C).
    5. Romanov, Michael S. & Masterov, Vladimir B., 2020. "Low breeding performance of the Steller’s sea eagle (Haliaeetus pelagicus) causes the populations to decline," Ecological Modelling, Elsevier, vol. 420(C).

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