  Studies of neutron stars are at their peak after the multi-messenger
observation of the binary merger event GW170817, which strongly constraints the
stellar parameters like tidal deformability, masses and radii. Although current
and future observations will provide stronger limits on the neutron stars
parameters, knowledge of explicit interior solutions to Einstein's equations,
which connect observed parameters with the internal structure, are crucial to
have a satisfactory description of the interior of these compact objects. A
well known exact solution, which has shown a relatively good approximation to a
neutron star, is the Tolman VII solution. In order to provide a better fitting
for the energy density profile, with the realistic equations of state for
neutron stars, recently Jiang and Yagi proposed a modified version of this
model which introduces an additional parameter $\alpha$ reflecting the
interplay of the quadratic and the newly added quartic term in the energy
density profile. Here we study the dynamical stability of this modified Tolman
VII solution using the theory of infinitesimal and adiabatic radial
oscillations developed by Chandrasekhar. For this purpose, we determine values
of the critical adiabatic index, for the onset of instability, considering
configurations with varying compactness and $\alpha$. We found that the new
models are stable against radial oscillations for a considerable range of
values of compactness and the new parameter $\alpha$, thus supporting their
applicability as a physically plausible approximation of realistic neutron
stars.
