In this paper, we introduce a class of one-dimensional non-autonomous
dynamical systems that allow an explicit study of their orbits,
of the associated variational equations as well as of certain types of
bifurcations. In a special case, the model class can be
transformed into the non-autonomous Beverton-Holt equation.
We use these model functions for analyzing various notions of
non-autonomous transcritical and pitchfork bifurcations
that have been recently proposed in the literature.