  A 5-night asteroseismic observation of the F8V star HD 203608 was conducted
in August 2006 with HARPS, followed by an analysis of the data, and a
preliminary modeling of the star (Mosser et al. 2008). The stellar parameters
were significantly constrained, but the behavior of one of the seismic indexes
(the small spacing d01) could not be fitted with the observed one, even with
the best considered models. We study the possibility of improving the agreement
between models and observations by changing the physical properties of the
inner parts of the star (to which d01 is sensitive). We show that, in spite of
its low mass, it is possible to produce models of HD 203608 with a convective
core. No such model was considered in the preliminary modeling. In practice, we
obtain these models here by assuming some extra mixing at the edge of the early
convective core. We optimize the model parameters using the Levenberg-Marquardt
algorithm. The agreement between the new best model with a convective core and
the observations is much better than for the models without. All the
observational parameters are fitted within 1-sigma observational error bars.
This is the first observational evidence of a convective core in an old and
low-mass star such as HD 203608. In standard models of low-mass stars, the core
withdraws shortly after the ZAMS. The survival of the core until the present
age of HD 203608 provides very strong constraints on the size of the mixed zone
associated to the convective core. Using overshooting as a proxy to model the
processes of transport at the edge of the core, we find that to reproduce both
global and seismic observations, we must have alpha_ov = 0.17 +/- 0.03 Hp for
HD 203608. We revisit the process of the extension of the core lifetime due to
overshooting in the particular case of HD 203608.
