  We study the evolution of a system of $N$ interacting species which mimics
the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5
species, spatial inhomogeneities develop spontaneously in initially homogeneous
systems. The arising spatial patterns form a mosaic of single-species domains
with algebraically growing size, $\ell(t)\sim t^\alpha$, where $\alpha=3/4$
(1/2) and 1/3 for N=3 with sequential (parallel) dynamics and N=4,
respectively. The domain distribution also exhibits a self-similar spatial
structure which is characterized by an additional length scale, ${\cal
L}(t)\sim t^\beta$, with $\beta=1$ and 2/3 for N=3 and 4, respectively. For
$N\geq 5$, the system quickly reaches a frozen state with non interacting
neighboring species. We investigate the time distribution of the number of
mutations of a site using scaling arguments as well as an exact solution for
N=3. Some possible extensions of the system are analyzed.
