  The stability and contraction properties of positive integral semigroups on
Polish spaces are investigated. Our novel analysis is based on the extension of
V-norm contraction methods, associated to functionally weighted Banach spaces
for Markov semigroups, to positive semigroups. This methodology is applied to a
general class of positive and possibly time-inhomogeneous bounded integral
semigroups and their normalised versions. The spectral theorems that we develop
are an extension of Perron-Frobenius and Krein-Rutman theorems for positive
operators to a class of time-varying positive semigroups. In the context of
time-homogeneous models, the regularity conditions discussed in the present
article appear to be necessary and sufficient condition for the existence of
leading eigenvalues. We review and illustrate the impact of these results in
the context of positive semigroups arising in transport theory, physics,
mathematical biology and signal processing.
