Spatial organization in cyclic Lotka-Volterra systems,
L. Frachebourg, P.L. Krapivsky, and E. Ben-Naim
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.
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