  We study the implications of $R^n$ extension of Starobinsky model on
dynamical instability of Vorticity-free axially symmetric gravitating body. The
matter distribution is considered to be anisotropic for which modified field
equations are formed in context of $f(R)$ gravity. In order to achieve the
collapse equation, we make use of the dynamical equations, extracted from
linearly perturbed contracted Bianchi identities. The collapse equation carries
adiabatic index $\Gamma$ in terms of usual and dark source components, defining
the range of stability/insatbility in Newtonian (N) and post-Newtonian (pN)
eras. It is found that supersymmetric supergravity $f(R)$ model represents the
more practical substitute of higher order curvature corrections.
