It has been shown by \textit{Scherrer and Putter et.al} that, when dynamics
of dark energy is driven by a homogeneous $k-$essence scalar field $\phi$, with
a Lagrangian of the form $L = V_0F(X)$ with a constant potential $V_0$ and $X =
\frac{1}{2}\nabla^\mu\phi \nabla_\mu\phi = \frac{1}{2}\dot{\phi}^2$, one
obtains a scaling relation $X(dF/dX)^2 = Ca^{-6}$ , where $C$ is a constant and
$a$ is the FRW scale factor of the universe. The separate energy conservation
in the dark energy sector and the constancy of $k-$essence potential are
instrumental in obtaining such a scaling. In this paper, we have shown that
even when considering time-dependent interactions between dark energy and dark
matter, the constancy of $k-$essence potential may lead to a modified form of
scaling. We have obtained such a scaling relation for a particular class of
parametrisation of the source term occurring in the continuity equation of dark
energy and dark matter in the interacting scenario. We used inputs from the JLA
analysis of luminosity distance and redshift data from Supernova Ia
observations, to obtain the modified form of the scaling.
