In this paper, we construct a measure-valued diffusion on
$D\subseteq \mathbb{R^d}$ whose underlying motion is a diffusion process with
absorption at the boundary corresponding to an elliptic operator
[L = 1/2 \nabla \cdot a\nabla + b \cdot \nabla \text{
on } D \subseteq \mathbb{R}^d
and whose spatially dependent branching term is of the
form $\beta(x)z-\alpha(x)z^2,x \inD$,where $\beta$ satisfies a very general
condition and $\alpha> 0$. In the case that $\alpha$ and $\beta$ are bounded
from above, we show that the measure-valued process can also be obtained as a
limit of approximating branching particle systems.
¶ We give criteria for extinction/survival, recurrence/transience of
the support, compactness of the support, compactness of the range, and local
extinction for the measure-valued diffusion. We also present a number of
examples which reveal that the behavior of the measure-valued diffusion may be
dramatically different from that of the approximating particle systems.