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Derivatives of the stochastic growth rate

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  • Steinsaltz, David
  • Tuljapurkar, Shripad
  • Horvitz, Carol

Abstract

We consider stochastic matrix models for population driven by random environments which form a Markov chain. The top Lyapunov exponent a, which describes the long-term growth rate, depends smoothly on the demographic parameters (represented as matrix entries) and on the parameters that define the stochastic matrix of the driving Markov chain. The derivatives of a–the “stochastic elasticities†–with respect to changes in the demographic parameters were derived by Tuljapurkar (1990). These results are here extended to a formula for the derivatives with respect to changes in the Markov chain driving the environments. We supplement these formulas with rigorous bounds on computational estimation errors, and with rigorous derivations of both the new and old formulas.

Suggested Citation

  • Steinsaltz, David & Tuljapurkar, Shripad & Horvitz, Carol, 2011. "Derivatives of the stochastic growth rate," Theoretical Population Biology, Elsevier, vol. 80(1), pages 1-15.
    • Handle: RePEc:eee:thpobi:v:80:y:2011:i:1:p:1-15
    DOI: 10.1016/j.tpb.2011.03.004
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    References listed on IDEAS

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    1. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
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    1. 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.
    2. 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.

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