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Convexity in projection matrices: Projection to a calibration problem

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  • Logofet, Dmitrii O.

Abstract

Convexity, as a fundamental property of sets and functions defined on convex sets, plays an important role in many mathematical and applied disciplines, including extremal and optimal-control problems. We prove the set of all feasible projection matrices in a general class of matrix models for stage-structured population dynamics to be convex and the dominant eigenvalue (λ1) of any projection 2×2 matrix to be either a convex, or a concave function on a simplex of the matrix first-row entries (i.e., stage-specific reproduction rates). The latter is also conjectured for the general n×n case. Though looking far from practical needs of matrix population models, this mathematical result has appeared to be quite useful in solving a practical problem to calibrate the projection matrix, i.e., to estimate all the stage-specific vital rates, from empirical data. The data from monitoring of individual life histories of marked plants on permanent sample plots during successive years enable direct calculation of the stage-specific survival and ontogenetic transition rates, but the rates of reproduction do remain uncertain as far as the parent plants can hardly be determined for the (not yet marked!) recruitment.

Suggested Citation

  • Logofet, Dmitrii O., 2008. "Convexity in projection matrices: Projection to a calibration problem," Ecological Modelling, Elsevier, vol. 216(2), pages 217-228.
    • Handle: RePEc:eee:ecomod:v:216:y:2008:i:2:p:217-228
    DOI: 10.1016/j.ecolmodel.2008.03.004
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    Citations

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    Cited by:

    1. 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.
    2. Frisman, E.Y. & Neverova, G.P. & Revutskaya, O.L., 2011. "Complex dynamics of the population with a simple age structure," Ecological Modelling, Elsevier, vol. 222(12), pages 1943-1950.
    3. Logofet, Dmitrii O., 2019. "Does averaging overestimate or underestimate population growth? It depends," Ecological Modelling, Elsevier, vol. 411(C).
    4. 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.
    5. Logofet, Dmitrii O., 2017. "Aggregation may or may not eliminate reproductive uncertainty," Ecological Modelling, Elsevier, vol. 363(C), pages 187-191.
    6. Logofet, Dmitrii O., 2013. "Projection matrices in variable environments: λ1 in theory and practice," Ecological Modelling, Elsevier, vol. 251(C), pages 307-311.
    7. Dmitrii O. Logofet & Valerii N. Razzhevaikin, 2021. "Potential-Growth Indicators Revisited: Higher Generality and Wider Merit of Indication," Mathematics, MDPI, vol. 9(14), pages 1-15, July.
    8. Logofet, Dmitrii O., 2013. "Calamagrostis model revisited: Matrix calibration as a constraint maximization problem," Ecological Modelling, Elsevier, vol. 254(C), pages 71-79.
    9. Picard, Nicolas & Ouédraogo, Dakis & Bar-Hen, Avner, 2010. "Choosing classes for size projection matrix models," Ecological Modelling, Elsevier, vol. 221(19), pages 2270-2279.

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