Spatial organization in cyclic Lotka-Volterra systems

TL;DR

This paper investigates the spatial patterns and dynamics of cyclic Lotka-Volterra systems with a small number of species on a lattice, revealing how spatial inhomogeneities and domain structures evolve over time.

Contribution

It provides a detailed analysis of spatial pattern formation and domain growth in cyclic Lotka-Volterra models for N<5, including exact solutions and scaling laws.

Findings

Spatial inhomogeneities develop spontaneously for N<5.

Domain size grows algebraically with time, with exponents depending on N and dynamics.

For N≥5, the system quickly reaches a frozen state.

Abstract

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.