  Let $P$ be a collection of $n$ points moving along pseudo-algebraic
trajectories in the plane. One of the hardest open problems in combinatorial
and computational geometry is to obtain a nearly quadratic upper bound, or at
least a subcubic bound, on the maximum number of discrete changes that the
Delaunay triangulation $\DT(P)$ of $P$ experiences during the motion of the
points of $P$.
  In this paper we obtain an upper bound of $O(n^{2+\eps})$, for any $\eps>0$,
under the assumptions that (i) any four points can be co-circular at most
twice, and (ii) either no triple of points can be collinear more than twice, or
no ordered triple of points can be collinear more than once.
