We consider non-conservative positive semigroups and obtain necessary and
sufficient conditions for uniform exponential contraction in weighted total
variation norm. This ensures the existence of Perron eigenelements and provides
quantitative estimates of the spectral gap, complementing Krein-Rutman theorems
and generalizing probabilistic approaches. The proof is based on a
non-homogenous h-transform of the semigroup and the construction of Lyapunov
functions for this latter. It exploits then the classical necessary and
sufficient conditions of Harris's theorem for conservative semigroups and
recent techniques developed for the study of absorbed Markov processes. We
apply these results to population dynamics. We obtain exponential convergence
of birth and death processes conditioned on survival to their quasi-stationary
distribution, as well as estimates on exponential relaxation to stationary
profiles in growth-fragmentation PDEs
