On the spectrum of a Leslie matrix with a near-periodic fecundity pattern

Abstract

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 ⋯ m n ], it is well known that if d = gcd{ i| m i >0} ⩾ 2 (which corresponds to a periodic fecundity pattern), then the matrix has d eigenvalues with moduli equal to its spectral radius. In this paper, 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 ≠ 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.