Exact results for spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

Fitness fluctuations in the Bak-Sneppen model

We study the one-dimensional Bak-Sneppen model for the evolution of species in an ecosystem. Of particular interest are the temporal fluctuations in fitness variables. We numerically compute the power spectral density and apply the finite-size scaling method to get data collapse. A clear signature of 1/f^{α} noise with α≈1.2 (long-time correlations) emerges for both local and global (or average) fitness noises. The limiting value of the spectral exponent, 0 or 2, corresponds to no interaction or a random neighbor version of the model, respectively. The local power spectra are spatially uncorrelated and also show an additional scaling, ∼1/L, in the frequency regime L^{-λ}≪f≪1/2, where L is the linear extent of the system.

A general mathematical method for predicting spatio-temporal correlations emerging from agent-based models

Agent-based models are used to study complex phenomena in many fields of science. While simulating agent-based models is often straightforward, predicting their behaviour mathematically has remained a key challenge. Recently developed mathematical methods allow the prediction of the emerging spatial patterns for a general class of agent-based models, whereas the prediction of spatio-temporal pattern has been thus far achieved only for special cases. We present a general and mathematically rigorous methodology that allows deriving the spatio-temporal correlation structure for a general class of individual-based models. To do so, we define an auxiliary model, in which each agent type of the primary model expands to three types, called the original, the past and the new agents. In this way, the auxiliary model keeps track of both the initial and current state of the primary model, and hence the spatio-temporal correlations of the primary model can be derived from the spatial correlations of the auxiliary model. We illustrate the agreement between analytical predictions and agent-based simulations using two example models from theoretical ecology. In particular, we show that the methodology is able to correctly predict the dynamical behaviour of a host–parasite model that shows spatially localized oscillations.

Bak-Sneppen model: Local equilibrium and critical value

The Bak-Sneppen (BS) model is a very simple model that exhibits all the richness of self-organized criticality theory. At the thermodynamic limit, the BS model converges to a situation where all particles have a fitness that is uniformly distributed between a critical value p_{c} and 1. The p_{c} value is unknown, as are the variables that influence and determine this value. Here we study the BS model in the case in which the lowest fitness particle interacts with an arbitrary even number of m nearest neighbors. We show that p_{c} verifies a simple local equilibrium relation. Based on this relation, we can determine bounds for p_{c} of the BS model and exact results for some BS-like models. Finally, we show how transformations of the original BS model can be done without altering the model's complex dynamics.

Scaling and organization of electroencephalographic background activity and alpha rhythm in healthy young adults

The coexistence of the broad-band fluctuation and alpha rhythm of the brain dynamics is studied based on the zero-crossing property of the local electroencephalographic (EEG) recording in eyes closed and eyes open. A two-component zero-crossing scenario, consisting of a broad-band fractal and narrow-band rhythm components, is assumed. Scaling is found in the power law distribution p(tau) approximately tau(-nu) of the crossing time interval tau of the broad-band fluctuation. In alpha dominant brain state, the alpha rhythm interval L also exhibits scaling in the form of power law distribution: p(L) approximately L(phi). Our main result is the relationship nu + phi approximately 3 that characterizes the organization of these two prominent features of the brain dynamics. The possible role of self-organized criticality of punctuated equilibrium in this organization is discussed.

Devil's-staircase-like behavior of the range of random time series with record-breaking fluctuations

We propose insight into the analysis of the record-breaking fluctuations in random time series, which permits to distinguish between the self-organized criticality and the record dynamics (RD) scenarios of system evolution, using a finite time series realization. Performed analysis of the time series associated with the historical prices of different commodities has shown that the evolution of commodity markets is controlled by the record-breaking fluctuations as it is outlined by the RD. Furthermore, we found that the sizes of record-breaking fluctuations follow a fat-tailed distribution and the devil's-staircase-like records of price ranges are multiaffine and persistent, nevertheless, the high moments (q> q(C) >2) of their q-order height-height correlation functions behave logarithmically.

Rapid self-organized criticality: Fractal evolution in extreme environments

We introduce the phenomenon of rapid self-organized criticality (RSOC) and show that, like some models of self-organized criticality (SOC), RSOC generates scale-invariant event distributions and 1/f noise. Unlike SOC, however, RSOC persists despite more than an order of magnitude variation in driving rate and displays extremely thick and dynamic branching geometry. Starting with an initial set of parameter values, we perform two numerical experiments in which nonequilibrium RSOC systems are tuned towards their critical points. The approach to the critical state is tracked using average branching rates, which must equal 1 if systems are genuinely critical.

Extremal dynamics and the approach to the critical state: experiments on a three dimensional pile of rice

The evolution of the growth of a ricepile is studied in three dimensions. With time, the pile approaches a critical state with a certain slope. Assuming extremal dynamics in the evolution of the pile, the way the critical state is approached is dictated by the scaling properties of the critical state itself. Experimentally, we determine the envelope of the maximal slope, which is a measure for the distance from the critical state, as well as the growth of the average avalanche size with time. These quantities obey power-law scaling, where the experimental exponents are in good agreement with those obtained from an earlier determination of the critical state properties and extremal dynamics. Furthermore, we discuss the influence of the transient state on the avalanche size distribution, which may have applications in the prevention of large avalanches in natural systems.

dc = 4 is the upper critical dimension for the Bak-Sneppen model

Numerical results are presented indicating d(c) = 4 as the upper critical dimension for the Bak-Sneppen evolution model. This finding agrees with previous theoretical arguments, but contradicts a recent Letter [Phys. Rev. Lett. 80, 5746 (1998)] that placed d(c) as high as d = 8. In particular, we find that avalanches are compact for all dimensions d< or =4 and are fractal for d>4. Under those conditions, scaling arguments predict a d(c) = 4, where hyperscaling relations hold for d< or =4. Other properties of avalanches, studied for 1< or =d< or =6, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

In a semi-infinite geometry, a one-dimensional, M-component model of biological evolution realizes microscopically an inhomogeneous branching process for M-->infinity. This implies a size distribution exponent tau(')=7/4 for avalanches starting at a free, "dissipative" end of the evolutionary chain. A bulklike behavior with tau(')=3/2 is restored by "conservative" boundary conditions. These are such as to 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. Together with accurate numerical determinations of the time recurrence exponent tau(')(first), these results show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as those in the bulk, and their distribution exponents obey the basic scaling laws holding there.

The role of extreme events in the impacts of selective tropical forestry on erosion during harvesting and recovery phases at Danum Valley, Sabah

Ten years' hydrological investigations at Danum have provided strong evidence of the effects of extremes of drought, as in the April 1992 El Niño southern oscillation event, and flood, as in January 1996. The 1.5 km2 undisturbed forest control catchment experienced a complete drying out of the stream for the whole 1.5 km of defined channel above the gauging station in 1992, but concentrated surface flow along every declivity from within a few metres of the catchment divide after the exceptional rains of 19 January 1996. Under these natural conditions, erosion is episodic. Sediment is discharged in pulses caused by storm events, collapse of debris dams and occasional landslips. Disturbance by logging accentuates this irregular regime. In the first few months following disturbance, a wave of sediment is moved by each storm, but over subsequent years, rare events scour sediment from bare areas, gullies and channel deposits. The spatial distribution of sediment sources changes with time after logging, as bare areas on slopes are revegetated and small gullies are filled with debris. Extreme storm events, as in January 1996, cause logging roads to collapse, with landslides leading to surges of sediment into channels, reactivating the pulsed sediment delivery by every storm that happened immediately after logging. These effects are not dampened out with increasing catchment scale. Even the 721 km2 Sungai Segama has a sediment yield regime dominated by extreme events, the sediment yield in that single day on 19 January 1996 exceeding the annual sediment load in several previous years. In a large disturbed catchment, such road failures and logging-activity-induced mass movements increase the mud and silt in floodwaters affecting settlements downstream. Management systems require long-term sediment reduction strategies. This implies careful road design and good water movement regulation and erosion control throughout the logging process.