  We provide quantitative estimates in total variation distance for positive
semi-groups, which can be non-conservative and non-homogeneous. The techniques
relies on a family of conservative semigroups that describes a typical particle
and Doeblin's type conditions for coupling the associated process. Our aim is
to provide quantitative estimates for linear partial differential equations and
we develop several applications for population dynamics in varying environment.
We start with the asymptotic profile for a growth diffusion model with time and
space non-homogeneity. Moreover we provide general estimates for semigroups
which become asymptotically homogeneous, which are applied to an age-structured
population model. Finally, we obtain a speed of convergence for periodic
semi-groups and new bounds in the homogeneous setting. They are are illustrated
on the renewal equation.
