  Non-gravitational interaction between dark matter and dark energy has been
considered in a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)
universe. The interaction rate is assumed to be linear in the energy densities
of dark matter and dark energy and it is also proportional to the Hubble rate
of the FLRW universe. This kind of interaction model leads to an autonomous
linear dynamical system, and depending on the coupling parameters, could be
solved analytically by calculating the exponential of the matrix, defining the
system. We show here that such interaction rate has a very deep connection with
some well known cosmological theories. We then investigate the theoretical
bounds on the coupling parameters of the interaction rate in order that the
energy densities of the dark sector remain positive throughout the evolution of
the universe and asymptotically converge to zero at very late times. Our
analyses also point out that such linear interacting model may encounter with
finite time future singularities depending on the coupling parameters as well
as the dark energy state parameter.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics