Self-organized Criticality and Complexity in Historical Landscape Patterns

J. Bolliger
Department of Forest Ecology and Management, University of Wisconsin, Madison WI 53706

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA

 D. J. Mladenoff
Department of Forest Ecology and Management, University of Wisconsin, Madison WI 53706

(Accepted 29 August 2002)


Self-organization describes the evolution process of complex structures where systems emerge spontaneously, driven internally by variations of the system itself.  Self-organization to the critical state is manifested by scale-free behavior across many orders of magnitude (Bak et al. 1987, Sole et al. 1999).  Spatial scale-free behavior implies fractal properties and is quantified by the fractal dimension.  Temporal scale-free behavior is evident in power spectra of fluctuations that obey power laws.  Self-organized criticality is a universal phenomenon that likely produces some of the fractals and power laws observed in nature.

We investigated the historical landscape of southern Wisconsin (USA) (60,000 km2) for self-organization and complexity.  The landscape is patterned into prairies, savannas, and open and closed forests, using data from the United States General Land Office Surveys that were conducted during the 19th century, at a time prior to Euro-American settlement.

We applied a two-dimensional cellular automaton model with one adjustable parameter.  Model evolution replaces a cell that dies at random times by a cell chosen randomly from within a circular radius r, where r typically takes values between 1 (local) and 10 units (regional).  Cluster probability is used to measure the degree of organization.  The model landscape self-organizes to a realistic critical state if neighborhoods of intermediate size (r = 3) are chosen, indicating that (a) no particular time or space scale for the clusters is singled out, i.e. the spatial dependence is fractal, and temporal fluctuations in the cluster probability exhibit power laws; (b) a simple model suffices to replicate the landscape pattern resulting from complex spatial and temporal interactions.

Measures of comparison between the observed and the simulated landscape show good agreement: fractal dimensions for simulated (1.6) and observed landscapes (1.64), cluster probabilities for simulated (32.3%) and observed (32.6%) landscapes, and algorithmic complexity for simulated (6792 bytes) and observed (6205 bytes) landscapes.  The results are robust towards variation of initial and boundary conditions as well as perturbations.

Ref: J. Bolliger, J. C. Sprott, and D. J. Mladenoff, OIKOS 100, 541-553 (2003).

The complete paper is available in PDF format.