  We consider the history of the universe from a possible big bang or a bounce
into a late period of a unified interacting dark energy - dark matter model.
The model is based on the Two Measures Theories (T.M.T.) which introduces a
metric independent volume element and this allows us to construct a unification
of dark energy and dark matter. A generalization of the T.M.T. gives a
diffusive non-conservative stress energy momentum tensor in addition to the
conserved stress energy tensor which appear in Einstein equations. These leads
to a formulation of interacting DE-DM dust models in the form of a diffusive
type interacting Unified Dark Energy and Dark Matter scenario. The deviation
from $\Lambda$CDM is determined by the diffusion constant $C_2$. For $C_2=0$
the model is indistinguishable from $\Lambda$CDM. Numerical solutions of the
theories show that in some $C_2 \neq 0$ the evolution of the early universe is
governed by Stiff equation of state or the universe bounces to hyper inflation.
But all of those solutions have a final transition to $\Lambda$CDM as a stable
fixed point for the late universe.
