  A recently developed model of random walks on a $D$-dimensional
hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is
extended to include the possibility of creating and annihilating random
walkers. Steady-state distributions of random walkers are obtained for all
dimensions $D>0$ by solving a discrete eigenvalue problem. These distributions
exhibit dimensionally dependent critical behavior as a function of the birth
rate. This remarkably simple model exhibits a second-order phase transition
with a nontrivial critical exponent for all dimensions $D>0$.
