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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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##### References:
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