  This paper continues the analysis, started in [2, 3], of a class of
degenerate elliptic operators defined on manifolds with corners, which arise in
Population Biology. Using techniques pioneered by J. Moser, and extended and
refined by L. Saloff-Coste, Grigoryan, and Sturm, we show that weak solutions
to the parabolic problem defined by a sub-class of these operators, which
consists of those that can be defined by Dirichlet forms and have non-vanishing
transverse vector field, satisfy a Harnack inequality. This allows us to
conclude that the solutions to these equations belong, for positive times, to
the natural anisotropic Holder spaces, and also leads to upper and, in some
cases, lower bounds for the heat kernels of these operators. These results
imply that these operators have a compact resolvent when acting on $C^0$ or
$L^2.$ The proof relies upon a scale invariant Poincare inequality that we
establish for a large class of weighted Dirichlet forms, as well as estimates
to handle certain mildly singular perturbation terms. The weights that we
consider are neither Ahlfors regular, nor do they generally belong to the
Muckenhaupt class $A_2.$
