  For tumor growth, the morphological instability provides a mechanism for
invasion via tumor fingering and fragmentation. This work considers the
asymptotic stability of a free boundary tumor model with a periodic supply of
external nutrients. The model consists of two elliptic equations describing the
concentration of nutrients and the distribution of the internal pressure in the
tumor tissue, respectively. The effect of the parameter $\mu$ representing a
measure of mitosis on the morphological stability are taken into account. It
was recently established in [25] that there exists a critical value $\mu_\ast$
such that the unique spherical periodic positive solution is linearly stable
for $0<\mu<\mu_\ast$ and linearly unstable for $\mu>\mu_\ast$. In this paper,
we further prove that the spherical periodic positive solution is
asymptotically stable for $0<\mu<\mu_\ast$ for the fully nonlinear problem.
