Convexity and Concavity of the Perron Root and Vector of Leslie Matrices with Applications to a Population Model

Abstract

This paper considers the Leslie model of population growth and analyzes its asymptotic growth rate and its asymptotically stable age distribution as functions of the fecundity and survival rates of each age group in the population. This analysis is performed by computing first- and second-order partial derivatives of the Perron root and vector of a Leslie matrix with respect to each relevant entry in the matrix, with emphasis on the second partial derivatives. The signs of these derivatives as well as the qualitative implications that the results have for the Leslie model are discussed. Where possible, quantitative interpretations of the results are also given. Throughout, the techniques employ ideas from the theory of group generalized Inverses.