Spectral properties of a near-periodic row-stochastic Leslie matrix

Leslie matrix models are discrete models for the development of age-structured populations. It is known that eigenvalues of a Leslie matrix are important in describing the asymptotic behavior of the corresponding population model. It is also known that the ratio of the spectral radius and the second largest (subdominant) eigenvalue in modulus of a non-periodic Leslie matrix determines the rate of convergence of the corresponding population distributions to a stable age distribution. In this paper, we further study the spectral properties of a row-stochastic Leslie matrix A with a near-periodic fecundity pattern of type (k, d, s) based on Kirkland's results in 1993. Intervals containing arguments of eigenvalues of A on the upper-half plane are given. Sufficient conditions are derived for the argument of the subdominant eigenvalue of A to be in the interval 2πd,2πd-s for the cases where k = 1. A computational scheme is suggested to approximate the subdominant eigenvalue when its argument is in 2πd,2πd-s.