  Population dynamics in systems composed of cyclically competing species has
been of increasing interest recently. Here, we investigate a system with four
or more species. Using mean field theory, we study in detail the trajectories
in configuration space of the population fractions. We discover a variety of
orbits, shaped like saddles, spirals, and straight lines. Many of their
properties are found explicitly. Most remarkably, we identify a collective
variable which evolves simply as an exponential: $\mathcal{Q}% \propto
e^{\lambda t}$, where $\lambda$ is a function of the reaction rates. It
provides information on the state of the system for late times (as well as for
$t\rightarrow -\infty $). We discuss implications of these results for the
evolution of a finite, stochastic system. A generalization to an arbitrary
number of cyclically competing species yields valuable insights into universal
properties of such systems.
