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Perturbation analysis of nonlinear matrix population models

Author

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  • Hal Caswell

    (Universiteit van Amsterdam)

Abstract

Perturbation analysis examines the response of a model to changes in its parameters. It is commonly applied to population growth rates calculated from linear models, but there has been no general approach to the analysis of nonlinear models. Nonlinearities in demographic models may arise due to density-dependence, frequency-dependence (in 2-sex models), feedback through the environment or the economy, and recruitment subsidy due to immigration, or from the scaling inherent in calculations of proportional population structure. This paper uses matrix calculus to derive the sensitivity and elasticity of equilibria, cycles, ratios (e.g. dependency ratios), age averages and variances, temporal averages and variances, life expectancies, and population growth rates, for both age-classified and stage-classified models. Examples are presented, applying the results to both human and non-human populations.

Suggested Citation

  • Hal Caswell, 2008. "Perturbation analysis of nonlinear matrix population models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 18(3), pages 59-116.
    • Handle: RePEc:dem:demres:v:18:y:2008:i:3
    DOI: 10.4054/DemRes.2008.18.3
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    Citations

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    Cited by:

    1. Barabás, György & Meszéna, Géza & Ostling, Annette, 2014. "Fixed point sensitivity analysis of interacting structured populations," Theoretical Population Biology, Elsevier, vol. 92(C), pages 97-106.
    2. Hal Caswell & Nora Sánchez Gassen, 2015. "The sensitivity analysis of population projections," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 33(28), pages 801-840.
    3. Hal Caswell, 2014. "A matrix approach to the statistics of longevity in heterogeneous frailty models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 31(19), pages 553-592.
    4. Alyson Raalte & Hal Caswell, 2013. "Perturbation Analysis of Indices of Lifespan Variability," Demography, Springer;Population Association of America (PAA), vol. 50(5), pages 1615-1640, October.
    5. Coste, Christophe F.D. & Austerlitz, Frédéric & Pavard, Samuel, 2017. "Trait level analysis of multitrait population projection matrices," Theoretical Population Biology, Elsevier, vol. 116(C), pages 47-58.
    6. Tomasz Wrycza & Annette Baudisch, 2012. "How life expectancy varies with perturbations in age-specific mortality," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 27(13), pages 365-376.
    7. Hal Caswell & Xi Song, 2021. "The formal demography of kinship III: Kinship dynamics with time-varying demographic rates," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 45(16), pages 517-546.
    8. Boyce, Mark S. & Baxter, Peter W.J. & Possingham, Hugh P., 2012. "Managing moose harvests by the seat of your pants," Theoretical Population Biology, Elsevier, vol. 82(4), pages 340-347.
    9. Karsten, Richard & Teismann, Holger & Vogels, Angela, 2013. "Reproductive value, sensitivity, and nonlinearity: Population-management heuristics derived from classical demography," Theoretical Population Biology, Elsevier, vol. 85(C), pages 20-25.
    10. Lee, Charlotte T. & Tuljapurkar, Shripad, 2008. "Population and prehistory I: Food-dependent population growth in constant environments," Theoretical Population Biology, Elsevier, vol. 73(4), pages 473-482.
    11. de Vries, Charlotte & Desharnais, Robert A. & Caswell, Hal, 2020. "A matrix model for density-dependent selection in stage-classified populations, with application to pesticide resistance in Tribolium," Ecological Modelling, Elsevier, vol. 416(C).
    12. Michal Engelman & Hal Caswell & Emily Agree, 2014. "Why do lifespan variability trends for the young and old diverge? A perturbation analysis," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 30(48), pages 1367-1396.
    13. Haridas, C.V. & Eager, Eric Alan & Rebarber, Richard & Tenhumberg, Brigitte, 2014. "Frequency-dependent population dynamics: Effect of sex ratio and mating system on the elasticity of population growth rate," Theoretical Population Biology, Elsevier, vol. 97(C), pages 49-56.
    14. Oli, Madan K. & Loughry, W.J. & Caswell, Hal & Perez-Heydrich, Carolina & McDonough, Colleen M. & Truman, Richard W., 2017. "Dynamics of leprosy in nine-banded armadillos: Net reproductive number and effects on host population dynamics," Ecological Modelling, Elsevier, vol. 350(C), pages 100-108.
    15. Caswell, Hal & Shyu, Esther, 2012. "Sensitivity analysis of periodic matrix population models," Theoretical Population Biology, Elsevier, vol. 82(4), pages 329-339.
    16. Hal Caswell, 2019. "The formal demography of kinship: A matrix formulation," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 41(24), pages 679-712.

    More about this item

    Keywords

    immigration; matrix population models; density dependence; homeostasis; sensitivity analysis; elasticity; matrix calculus; dependency ratios; two-sex models; Tribolium; barnacles; population cycles;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics
    • Z0 - Other Special Topics - - General

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