Boundary Spatiotemporal Correlations in a Self-Organized Critical Model of Punctuated Equilibrium

TL;DR

This paper investigates boundary effects in a 1D self-organized critical model of biological evolution, revealing how boundary conditions influence avalanche size distributions and correlations, with exact calculations and numerical verifications.

Contribution

It provides exact analytical results for boundary avalanche exponents and correlations in a self-organized critical model, highlighting boundary-bulk universality and effects of boundary conditions.

Findings

Avalanche size distribution exponent at the boundary is 7/4.

Bulk-like behavior with exponent 3/2 is restored under specific boundary conditions.

Boundary avalanches share the same fractal dimensions as bulk avalanches.

Abstract

In a semi-infinite geometry, a 1D, M-component model of biological evolution realizes microscopically an inhomogeneous branching process for $M \to \infty$. This implies in particular a size distribution exponent $\tau'=7/4$ for avalanches starting at a free end of the evolutionary chain. A bulk--like behavior with $\tau'=3/2$ is restored if `conservative' boundary conditions strictly fix to its critical, bulk value the average number of species directly involved in an evolutionary avalanche by the mutating species located at the chain end. A two-site correlation function exponent ${\tau_R}'=4$ is also calculated exactly in the `dissipative' case, when one of the points is at the border. These results, together with accurate numerical determinations of the time recurrence exponent $\tau_{first}'$, show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as in the bulk, and their distribution exponents obey the basic scaling laws holding there.