Perturbations of the Perron Vector: Applications to Finite Markov Chains and Demographic Population Models

Abstract

Transition matrices are widely used in spatial demographic modelling. In this paper, perturbations in the transition matrix of a finite ergodic Markov chain are examined. The effects on the limiting probability vector and on the mean first passage matrix are described. Bounds are presented for the absolute and the relative changes in the mean first passage times. It is shown how the direction of the change in the limiting probabilities may be obtained from the original mean first passage matrix. The results are obtained by application of a single lemma, which expresses the effects of perturbations in a matrix on the Perron vector (that is, the eigenvector associated with the dominant eigenvalue). It is shown how the same approach may also be used to analyze the long-run properties of demographic population models.