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A renewal-equation approach to the dynamics of stage-grouped populations. (English) Zbl 0585.92020

Summary: The Usher matrix model for stage-grouped populations is studied by using a renewal-equation approach. Recurrent formulas are provided for computing the survival function, which depends on both stage and age. It is used to derive the renewal equation. The stable asymptotic behavior of the population is characterized by its stage-by-age structure, its asymptotic multiplication rate, the generation length, and the mean age at childbirth.
The last two quantities play a prominent part in the sensitivities of the multiplication rate to the parameters, which are derived explicitly and used to estimate the sampling variance of the asymptotic multiplication rate. These results generalize the standard stable population theory. A forestry example is provided.

MSC:

92D25 Population dynamics (general)
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References:

[1] Leslie, P. H., On the use of matrices in population mathematics, Biometrika, 33, 182-212 (1945) · Zbl 0060.31803
[2] Leslie, P. H., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245 (1948) · Zbl 0034.23303
[3] Pollard, J. H., On the use of direct matrix product in analyzing certain stochastic population models, Biometrika, 53, 397-415 (1966) · Zbl 0144.43901
[4] Jagers, P., Branching Process with Biological Applications (1975), Wiley: Wiley London · Zbl 0356.60039
[5] Usher, M. B., A matrix model for forest management, Biometrics, 25, 309-315 (1969)
[6] Lefkovitch, L. P., The study of population growth in organisms grouped by stages, Biometrics, 21, 1-18 (1965)
[7] Usher, M. B., A matrix approach to the management of renewable resources, with special reference to selection forests, J. Appl. Ecol., 3, 355-367 (1966)
[8] Peden, L. M.; Williams, J. S.; Frayer, W. E., A Markov model for stand projection, Forest Sci., 19, 303-314 (1973)
[9] Buongiorno, J.; Michie, B. R., A matrix model of uneven-aged forest management, Forest Sci., 26, 609-625 (1980)
[10] Gantmacher, F. R., Théorie des Matrices, (Tome 2 (1966), Dunod: Dunod Paris), Chapter 12 · Zbl 0136.00410
[11] Cull, P.; Vogt, A., Mathematical analysis of the asymptotic behavior of the Leslie population matrix model, Bull. Math. Biol., 35, 645-661 (1973) · Zbl 0276.92025
[12] Cull, P.; Vogt, A., The periodic limit for the Leslie model, Math. Biosci., 21, 39-54 (1974) · Zbl 0287.92007
[13] Keyfitz, N., Introduction to the Mathematics of Population (1968), Addison-Wesley: Addison-Wesley Reading, Mass
[14] Caswell, H., A general formula for the sensitivity of population growth rate to changes in life-history parameters, Theoret. Population Biol., 14, 215-230 (1971) · Zbl 0398.92024
[15] Keyfitz, N., Linkages of intrinsic to age-specific rates, J. Amer. Statist. Assoc., 66, 275-281 (1971)
[16] Lebreton, J. D., Contribution á la dynamique des populations d’oiseaux, modèles mathématiques en temps discret, (Doctorat-ès-sciences Thesis (1981), Lyon I Univ: Lyon I Univ Lyon, France)
[17] Daley, D. J., Bias in estimating the malthusian parameter for Leslie matrices, Theoret. Population Biol., 15, 257-263 (1979) · Zbl 0401.92014
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