  Representations of population models in terms of countable systems of
particles are constructed, in which each particle has a `type', typically
recording both spatial position and genetic type, and a level. For finite
intensity models, the levels are distributed on $[0,\lambda ]$, whereas in the
infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the
joint distribution of types and levels is conditionally Poisson, with mean
measure $\Xi (t)\times \ell$ where $\ell$ denotes Lebesgue measure and $\Xi
(t)$ is a measure-valued population process. The time-evolution of the levels
captures the genealogies of the particles in the population.
  Key forces of ecology and genetics can be captured within this common
framework. Models covered incorporate both individual and event based births
and deaths, one-for-one replacement, immigration, independent `thinning' and
independent or exchangeable spatial motion and mutation of individuals. Since
birth and death probabilities can depend on type, they also include natural
selection. The primary goal of the paper is to present particle-with-level or
lookdown constructions for each of these elements of a population model. Then
the elements can be combined to specify the desired model. In particular, a
non-trivial extension of the spatial $\Lambda$-Fleming-Viot process is
constructed.

Probability