  We investigate the tachyonic cosmological potential $V(\phi)$ in two
different cases of the quasi-exponential expansion of universe and discuss
various forms of interaction between the two components---matter and the
cosmological constant--- of the tachyonic scalar field, which leads to the
viable solutions of their respective energy densities. The distinction among
the interaction forms is shown to appear in the $O_{m}(x)$ diagnostic. Further,
the role of the high- and low-redshift observations of the Hubble parameter is
discussed to determine the proportionality constants and hence the correct form
of matter--cosmological constant interaction.
