  We consider a class of nonautonomous parabolic competition-diffusion systems
on bounded radial domains under Neumann boundary conditions. We show that, if
the initial profiles satisfy a reflection inequality with respect to a
hyperplane, then global positive solutions are asymptotically (in time)
foliated Schwarz symmetric with respect to antipodal points. Additionally, a
related result for (positive and sign changing solutions) of a scalar equation
with Neumann boundary conditions is given. The asymptotic shape of solutions to
cooperative systems is also discussed.
