  We study the symmetry properties of limit profiles of nonautonomous nonlinear
parabolic systems with Dirichlet boundary conditions in radial bounded domains.
In the case of competitive systems, we show that if the initial profiles
satisfy a reflectional inequality with respect to a hyperplane, then all limit
profiles are foliated Schwarz symmetric with respect to antipodal points. One
of the main ingredients in the proof is a new parabolic version of Serrin's
boundary point lemma. Results on radial symmetry of semi-trivial profiles are
discussed also for noncompetitive systems.
