  This paper studies countable systems of linearly and hierarchically
interacting diffusions taking values in the positive quadrant. These systems
arise in population dynamics for two types of individuals migrating between and
interacting within colonies. Their large-scale space-time behavior can be
studied by means of a renormalization program. This program, which has been
carried out successfully in a number of other cases (mostly one-dimensional),
is based on the construction and the analysis of a nonlinear renormalization
transformation, acting on the diffusion function for the components of the
system and connecting the evolution of successive block averages on successive
time scales. We identify a general class of diffusion functions on the positive
quadrant for which this renormalization transformation is well-defined and,
subject to a conjecture on its boundary behavior, can be iterated. Within
certain subclasses, we identify the fixed points for the transformation and
investigate their domains of attraction. These domains of attraction constitute
the universality classes of the system under space-time scaling.
