Applications of Perron-Frobenius theory to population dynamics

By the use of Perron–Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result, which is closely related to the Stein–Rosenberg theorem in numerical linear algebra, is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the model is $1$. More generally, we show how to achieve a given growth rate for the model by scaling the fertility matrix. Demographic interpretations of the results are given.