Confidence intervals for demographic projections based on products of random matrices

Abstract

This work is concerned with the growth of age-structured populations whose vital rates vary stochastically in time and with the provision of confidence intervals. In this paper a model Yt + 1(omega) = Xt + 1(omega) Yt(omega) is considered, where Yt is the (column) vector of the numbers of individuals in each age class at time t, X is a matrix of vital rates, and omega refers to a particular realization of the process that produces the vital rates. It is assumed that (Xi) is a stationary sequence of random matrices with nonnegative elements and that there is an integer n0 such that any product Xj + n0...Xj + 1Xj has all its elements positive with probability one. Then, under mild additional conditions, strong laws of large numbers and central limit results are obtained for the logarithms of the components of Yt. Large-sample estimators of the parameters in these limit results are derived. From these, confidence intervals on population growth and growth rates can be constructed. Various finite-sample estimators are studied numerically. The estimators are then used to study the growth of the striped bass population breeding in the Potomac River of the eastern United States.