Convergence of the age structure: Applications of the projective metric

This paper states necessary and sufficient conditions for the convergence of the age structure (in a discrete time, one-sex model of population growth); it also contains a new and simple proof of the weak ergodic theorem of stable population theory. The main tool used to attain these results is Hilbert's notion of the projective metric. This metric provides a way of defining the distance between positive vectors in $\text{R}$ⁿ which has two important features: First, the distance between any two positive vectors depends only on the rays on which the vectors lie; and, second, positive matrices act as contractions in this metric.