Spatial organization in cyclic Lotka-Volterra systems

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 average size, left-angle l(t)right-angle ∼tα, where α=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, left-angle L(t)right-angle ∼tβ, with β=1 and 2/3 for N=3 and 4, respectively. For N≥5, the system quickly reaches a frozen state with noninteracting 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 relevant extensions are also analyzed. copyright 1996 The American Physical Society