SOC in a population model with global control

TL;DR

This paper analyzes a population model controlled by global removal, revealing its close relation to directed percolation and demonstrating that critical phenomena and extinction dynamics follow DP scaling laws.

Contribution

It establishes the connection between the population control model and directed percolation, including scaling laws and critical exponents, and explores implications for extinction and applications.

Findings

Model exhibits anomalous scaling laws in large populations.

Critical exponents relate to those of directed percolation.

Family extinction follows directed percolation scaling laws.

Abstract

We study a plant population model introduced recently by J. Wallinga [OIKOS {\bf 74}, 377 (1995)]. It is similar to the contact process (`simple epidemic', `directed percolation'), but instead of using an infection or recovery rate as control parameter, the population size is controlled directly and globally by removing excess plants. We show that the model is very closely related to directed percolation (DP). Anomalous scaling laws appear in the limit of large populations, small densities, and long times. These laws, associated critical exponents, and even some non-universal parameters, can be related to those of DP. As in invasion percolation and in other models where the r\^oles of control and order parameters are interchanged, the critical value $p_c$ of the wetting probability $p$ is obtained in the scaling limit as singular point in the distribution of infection rates. We show that a mean field type approximation leads to a model studied by Y.C. Zhang et al. [J. Stat. Phys. {\bf 58}, 849 (1990)]. Finally, we verify the claim of Wallinga that family extinction in a marginally surviving population is governed by DP scaling laws, and speculate on applications to human mitochondrial DNA.