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Projection matrices in variable environments: λ1 in theory and practice

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  • Logofet, Dmitrii O.

Abstract

Perron–Frobenius theorem for nonnegative matrices, a mathematical foundation of matrix population models, applies when the projection matrix is not decomposable (or equivalently, when it is irreducible), the application yielding the dominant eigenvalue λ1>0 as a measure of the growth potential that a population with given demography possesses in a given environment. In practice, however, the projection matrix often appears to be decomposable (reducible); to calculate λ1 in this case, a principal submatrix should rather be used that corresponds to the reproductive core of the life cycle graph. I call it the reproductive submatrix and demonstrate that, when the reproductive submatrix does not coincide with the projection matrix and if this discrepancy is neglected in a case study, the resulting λ1 may happen to be overestimated. Averaging over a number of annual projection matrices eliminates the false growth rate but raises the problem of choice among the modes of averaging in the estimation of the stochastic growth rate in a stochastic environment. Computer simulation gives a method that avoids the both kinds of problem.

Suggested Citation

  • Logofet, Dmitrii O., 2013. "Projection matrices in variable environments: λ1 in theory and practice," Ecological Modelling, Elsevier, vol. 251(C), pages 307-311.
  • Handle: RePEc:eee:ecomod:v:251:y:2013:i:c:p:307-311
    DOI: 10.1016/j.ecolmodel.2012.12.028
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    References listed on IDEAS

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    1. Logofet, Dmitrii O., 2013. "Calamagrostis model revisited: Matrix calibration as a constraint maximization problem," Ecological Modelling, Elsevier, vol. 254(C), pages 71-79.
    2. Logofet, Dmitrii O., 2008. "Convexity in projection matrices: Projection to a calibration problem," Ecological Modelling, Elsevier, vol. 216(2), pages 217-228.
    3. 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.
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    1. Vladimir Yu. Protasov & Tatyana I. Zaitseva & Dmitrii O. Logofet, 2022. "Pattern-Multiplicative Average of Nonnegative Matrices: When a Constrained Minimization Problem Requires Versatile Optimization Tools," Mathematics, MDPI, vol. 10(23), pages 1-15, November.
    2. 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.
    3. 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.
    4. Logofet, Dmitrii O. & Maslov, Alexander A., 2019. "Bilberry vs. cowberry in a Scots pine boreal forest: Exclusion or coexistence in a post-fire succession?," Ecological Modelling, Elsevier, vol. 401(C), pages 134-143.
    5. Kim, Daehyun & Phillips, Jonathan D., 2013. "Predicting the structure and mode of vegetation dynamics: An application of graph theory to state-and-transition models," Ecological Modelling, Elsevier, vol. 265(C), pages 64-73.
    6. Logofet, Dmitrii O., 2019. "Does averaging overestimate or underestimate population growth? It depends," Ecological Modelling, Elsevier, vol. 411(C).
    7. 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.
    8. Logofet, Dmitrii O., 2017. "Aggregation may or may not eliminate reproductive uncertainty," Ecological Modelling, Elsevier, vol. 363(C), pages 187-191.
    9. 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).

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