  Representations of branching Markov processes and their measure-valued limits
in terms of countable systems of particles are constructed for models with
spatially varying birth and death rates. Each particle has a location and a
"level," but unlike earlier constructions, the levels change with time. In
fact, death of a particle occurs only when the level of the particle crosses a
specified level $r$, or for the limiting models, hits infinity. For branching
Markov processes, at each time $t$, conditioned on the state of the process,
the levels are independent and uniformly distributed on $[0,r]$. For the
limiting measure-valued process, at each time $t$, the joint distribution of
locations and levels is conditionally Poisson distributed with mean measure
$K(t)\times\varLambda$, where $\varLambda$ denotes Lebesgue measure, and $K$ is
the desired measure-valued process. The representation simplifies or gives
alternative proofs for a variety of calculations and results including
conditioning on extinction or nonextinction, Harris's convergence theorem for
supercritical branching processes, and diffusion approximations for processes
in random environments.
