Dark matter-dark energy interaction for a time-dependent equation of state

In this work we investigate the interaction between dark matter and dark energy for a coupling that obeys the Wang-Meng decaying law, $\rho_{{\rm DM}}\propto (1+z)^{3-\epsilon}$, and the Barboza-Alcaniz dark energy parametric model, $w=w_0+w'_0z(1+z)/(1+z^2)$. Theoretically, we show that the coupling constant, $\epsilon$, should satisfy the physical constraint $\epsilon\ge0$. We use the most recent data of type Ia supernovae, baryon acoustic oscillations, cosmic microwave background and the Hubble expansion rate function to constrain the free parameters of the model. From a purely observational point of view, we show that is not possible to discard values of the coupling constant in the unphysical region $\epsilon<0$. We show that the uncoupled case, $\epsilon=0$, is in better agreement with the data than any of coupled models in the physical region. We also find that all physically acceptable interaction in dark sector lies in the narrow range $0<\epsilon\le0.034$ ($95\%$ CL).