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On the spectrum of a Leslie matrix with a near-periodic fecundity pattern. (English) Zbl 0762.92015
Summary: Leslie matrices are square and nonnegative, and arise in the classical discrete, age-dependent model of population growth. Their eigenvalues are important in determining the asymptotic behavior of the age distributions in the model. Denoting the top row of the Leslie matrix by \([m_ 1 m_ 2\cdots m_ n]\), it is well-known that if \(d=\text{gcd}\{i| m_ i>0\}\geq 2\) (which corresponds to a periodic fecundity pattern), then the matrix has \(d\) eigenvalues with moduli equal to its spectral radius.
We consider Leslie matrices with a near-periodic fecundity pattern (roughly speaking, \(m_ i>0\) only if \(i\) is close to a multiple of some \(d\neq 1)\) and show that such matrices have at least two nonreal eigenvalues with moduli close to the spectral radius. We discuss a specific example of such a Leslie matrix which appears in the demographic literature, and give a numerical example to show that the age distributions in the model can also exhibit near-periodic behavior.

MSC:
92D25 Population dynamics (general)
15A18 Eigenvalues, singular values, and eigenvectors
15A99 Basic linear algebra
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[1] Bernardelli, H., Population waves, J. burma res. soc., 31, 1-18, (1941)
[2] 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
[3] Cull, P.; Vogt, A., The periodic limit for the Leslie model, Math. biosci., 21, 39-54, (1974) · Zbl 0287.92007
[4] Keyfitz, N., Introduction to the mathematics of population with revisions, (1977), Addison-Wesley Reading, Mass
[5] S. Kirkland, An eigenvalue region for Leslie matrices, SIAM J. Matrix Anal. Appl., to appear. · Zbl 0749.15004
[6] Kirkland, S., Spectral regions for Leslie matrices, ()
[7] Lefkovitch, L.P., The study of population growth in organisms grouped by stages, Biometrics, 21, 1-18, (1965)
[8] Leslie, P.H., On the use of matrices in certain population mathematics, Biometrika, 33, 183-212, (1945) · Zbl 0060.31803
[9] Lewis, E.G., On the generation and growth of a population, Sankhyā, 6, 93-96, (1942)
[10] Lopez, A., Problems in stable population theory, (1961), Office of Population Research Princeton
[11] Pollard, J.H., Mathematical models for the growth of human populations, (1973), Cambridge U.P Cambridge · Zbl 0295.92013
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