  Simultaneous deterministic and weakly stochastic dynamics of multiple
populations described by a large system of ODE's is considered in the phase
space of population sizes and ODE's parameters. We show that many practically
interesting problems can be formulated as a low-dimensional phase-space
conservation law and solved either explicitly or with simple iterative methods.
In particular, we consider: non-interacting populations with unbounded and
logistic growth, populations with randomized and biased migration, populations
competing for a resource, coexisting species, and populations with phase-space
interactions. The method provides an alternative to Monte Carlo simulations and
may be useful in the fast analysis of biological data and/or removal of
deterministic trends.
