  A systematic study of the central depletion of proton density has been
performed in the isotonic chains of nuclei with neutron numbers $N = 20$ and
$28$ using different variants of the relativistic mean-field (RMF) models.
These models include either the non-linear contributions from the mesons with
the coupling constants being density independent or the non-linearity of the
mesonic fields realized through the density dependent coupling strengths. The
central depletion in deformed nuclei tends to disappear irrespective of the
occupancy of $2s_{1/2}$ state in contrast to the spherical nuclei in which the
unoccupancy of $2s_{1/2}$ state leads to the central depletion. Due to the
differences in the strength of spin-orbit potentials in these models, the
central depletions are found to be model dependent. The influence of the
central depletion on the neutron-skin thickness is also investigated. It
appears that the effects of the central depletion do not percolate far enough
to display its finger prints on the trends of the neutron-skin thickness.
