  The bifurcation theory of ordinary differential equations (ODEs), and its
application to deterministic population models, are by now well established. In
this article, we begin to develop a complementary theory for diffusion-like
perturbations of dynamical systems, with the goal of understanding the space
and time scales of fluctuations near bifurcation points of the underlying
deterministic system. To do so we describe the limit processes that arise in
the vicinity of the bifurcation point. In the present article we focus on the
one-dimensional case.
