We study the evolution of a system of {ital N} interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with {ital N}{lt}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, {l_angle}l({ital t}){r_angle}{approximately}{ital t}{sup {alpha}}, where {alpha}=3/4 (1/2) and 1/3 for {ital N}=3 with sequential (parallel) dynamics and {ital N}=4, respectively. The domain distribution also exhibits a self-similar spatial structure which is characterized by an additional length scale, {l_angle}{ital L}({ital t}){r_angle}{approximately}{ital t}{sup {beta}}, with {beta}=1 and 2/3 for {ital N}=3 and 4, respectively. For {ital N}{ge}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 {ital N}=3. Some relevant extensions are also analyzed. {copyright} {ital 1996 The American Physical Society.}
[1]
Vito Volterra,et al.
Leçons sur la théorie mathématique de la lutte pour la vie
,
1931
.
[2]
N. Rashevsky,et al.
Mathematical biology
,
1961,
Connecticut medicine.
[3]
P. A. P. Moran,et al.
An introduction to probability theory
,
1968
.
[4]
M. Bramson.
Convergence of solutions of the Kolmogorov equation to travelling waves
,
1983
.
[5]
Josef Hofbauer,et al.
The theory of evolution and dynamical systems
,
1988
.
[6]
Sidney Redner,et al.
Kinetics and Spatial Organization of Competitive Reactions
,
1994
.
[7]
George Sugihara,et al.
Fractals in science
,
1995
.