Simple Cellular Automaton Model Research Articles

Self-organized criticality: Does it have anything to do with criticality and is it useful?

Abstract. Three aspects of complexity are fractals, chaos, and self-organized criticality. There are many examples of the applicability of fractals in solid-earth geophysics, such as earthquakes and landforms. Chaos is widely accepted as being applicable to a variety of geophysical phenomena, for instance, tectonics and mantle convection. Several simple cellular-automata models have been said to exhibit self-organized criticality. Examples include the sandpile, forest fire and slider-blocks models. It is believed that these are directly applicable to landslides, actual forest fires, and earthquakes, respectively. The slider-block model has been shown to clearly exhibit deterministic chaos and fractal behaviour. The concept of self-similar cascades can explain self-organized critical behaviour. This approach also illustrates the similarities and differences with critical phenomena through association with the site-percolation and diffusion-limited aggregation models.

A microscopic model for packet transport in the Internet

A microscopic description of packet transport in the Internet by using a simple cellular automaton model is presented. A generalised exclusion process is introduced which allows to study travel times of the particles (‘data packets’) along a fixed path in the network. Computer simulations reveal the appearance of a free flow and a jammed phase separated by a (critical) transition regime. The power spectra are compared with empirical data for the round trip time (RTT) obtained from measurements in the Internet. We find that the model is able to reproduce the characteristic statistical behaviour in agreement with the empirical data for both phases (free flow and congested). The ocurrence of different phases is therefore due to jamming properties and are not related to the structure of the network. Moreover the model shows, as observed in reality, critical behaviour (1/ f-noise) for paths with critical load.

Towards a realistic microscopic description of highway traffic

Simple cellular automata models are able to reproduce the basic properties of highway traffic. The comparison with empirical data for microscopic quantities requires a more detailed description of the elementary dynamics. Based on existing cellular automata models, we propose an improved discrete model incorporating anticipation effects, reduced acceleration capabilities and an enhanced interaction horizon for braking. The modified model is able to reproduce the three phases (free-flow, synchronized, and stop-and-go) observed in real traffic. Furthermore we find a good agreement with detailed empirical single-vehicle data in all phases.

Cellular-automata models applied to natural hazards

The concept of self-organized criticality evolved from studies of three simple cellular automata models: the forest-fire, slider-block, and sandpile models. Each model is associated with natural hazards which have frequency-size statistics that are well approximated by power law distributions. These distributions have important implications for probabilistic hazard assessments.

Models of earthquake faults with long-range stress transfer

Seismologists, and more recently computational and condensed-matter physicists, have made extensive use of computer modeling to investigate the physics of earthquakes. The authors describe a simple cellular automaton model that explores the possible relationship between Gutenberg-Richter scaling law and critical phenomena. The presence of scale invariance coupled with the ubiquity of this law in diverse geological conditions indicates that the large-scale statistical properties of earthquakes may be insensitive to environmental details. In this case, relatively simple models will properly capture the essential physics responsible for these behaviors.

Dynamics of a shallow fluidized bed.

The results of the experimental study of the dynamics of a shallow fluidized bed are reported. The behavior of granular material is controlled by the interplay of two factors--levitation due to the upward airflow, and sliding back due to gravity. Near the threshold of instability, the system shows critical behavior with remarkably long transient dynamics. The experimental observations are compared with a simple cellular automata model.

RIPPLES VERSUS GIANT DUNES IN A SALTATION-AVALANCHE MODEL

A simple stochastic cellular automaton model is proposed for ripples and dunes formation. Saltation and avalanches are the unique ingredients of the model. The dynamics of ripple formation is found to be logarithmic in time while the ripple merging as well as the dune growth is faster and occurs by bursts. The ripple state turns out to be metastable.

Circular Kinks on the Surface of Granular Material Rotated in a Tilted Spinning Bucket

We find that circular kinks form on the surface of granular material when the axis of rotation is tilted more than the angle of internal friction of the material. Radius of the kinks is measured as a function of the spinning speed and the tilting angle. Stability consideration of the surface results in an explanation that the kink is a boundary between the inner unstable and outer stable regions. A simple cellular automata model also displays kinks at the stability boundary.

Self-Organized Criticality Applied to Natural Hazards

The concept of self-organizedcriticality evolved from studies of three simplecellular-automata models: the sand-pile, slider-block,and forest-fire models. In each case, there is asteady “input” and the “loss” is associated with afractal (power-law) distribution of “avalanches.” Each of the three models can be associated with animportant natural hazard: the sand-pile model withlandslides, the slider-block model with earthquakes,and the forest-fire model with forest fires. We showthat each of the three natural hazards havefrequency-size statistics that are well approximatedby power-law distributions. The model behaviorsuggests that the recurrence interval for a severeevent can be estimated by extrapolating the observedfrequency-size distribution of small and mediumevents. For example, the recurrence interval for amagnitude seven earthquake can be obtained directlyfrom the observed frequency of occurrence of magnitudefour earthquakes. This concept leads to thedefinition of a seismic intensity factor. Both globaland regional maps of this seismic intensity factor aregiven. In addition, the behavior of the modelssuggests that the risk of occurrence of large eventscan be substantially reduced if small events areencouraged. For example, if small forest fires areallowed to burn, the risk of a large forest fire issubstantially reduced.

Self-organized criticality in a block lattice model of the brittle crust

An earthquake model is introduced, in which the brittle crust is treated as a two-dimensional system of many blocks divided by faults, and the mechanical behavior of the faults is described by the Burridge-Knopoff stick-slip model. The coherent system naturally evolves into a self-organized critical state. Some universal scaling laws of seismicity, such as the Gutenberg-Richter law with the b value in agreement with the observational result and the fractal feature of fault patterns, are reproduced. Some ambiguity in simple cellular automata models is also solved.

A Simple Three-States Cellular Automaton for Modelling Excitable Media

Wave propagation in excitable media provides an important example of spatiotemporal self-organization. The Belousov–Zhabotinsky (BZ) reaction and the impulse propagation along nerve axons are two well-known examples of this phenomenon. Excitable media have been modelled by continuous partial differential equations and by discrete cellular automata. Here we describe a simple three-states cellular automaton model based on the properties of excitation and recovery that are essential to excitable media. Our model is able to reproduce the dynamics of patterns observed in excitable media.

Metastable states in a microscopic model of traffic flow

It is a well known fact that metastable states of very high throughput and hysteresis effects exist in traffic flow, which the simple cellular automaton model of traffic flow and its continuous generalization fail to reproduce. It is shown that the model can be generalized to give a one-parametric family of models, a part of which reproduces the metastable states and the hysteresis. The models that have that property and those that do not that are separated by a transition that can be clearly identified.

Banded vegetation: survival under drought and grazing pressure based on a simple cellular automaton model

In western NSW, Australia, banded vegetation communities in grassland and shrubland remain widespread after up to 150 years of livestock grazing and a number of severe droughts. A simple model based on cellular automata is employed to investigate the robustness of such communities. It is shown that grazing pressure, simulated by the loss of plants at random within the landscape, has little effect on survival of the banded structure, unless pressure is extreme. Drought, associated with the systematic loss of plants in the driest locations, results in more extensive band fragmentation. The model suggests that these seemingly sensitive runoff–runon systems are in fact quite robust, and may well have changed little morphologically since European settlement. Species composition and plant age structure may, however, have been affected.

Banded vegetation: development under uniform rainfall from a simple cellular automaton model

Banded vegetation communities are known from semi-arid and arid landscapes in many parts of the world, in grasslands, shrublands, and woodlands. The origin of the distinctive patterning has been the subject of speculation, a common view being that banding evolves through the decline of more complete vegetation cover because of climatic deterioration or through grazing disturbance. A simple model based on cellular automata is employed to test the hypothesis that plausible mechanisms of water partitioning in spatially unstructured plant communities can bring about the development of banding. It is shown that without any climatic change or external disturbance, strongly developed banding can emerge from an initially random distribution of plants. Physical processes underlying the water partitioning, some of which remain unresearched, are discussed, and management implications noted.

Effects of the spatial and temporal ordering of events on the behaviour of a simple cellular automaton

Using a simple cellular automaton model, I demonstrate that the timing mechanism for events on different sites within the model can have a large impact on macroscopic model predictions. This implies that modellers must consider how to iterate their models so as to produce timing of events which match the biology of the problem rather than allowing a potentially unbiological timing (and hence an unrealistic model) to be imposed unintentionally by computational expedience. I discuss how to achieve this and suggest that model presentation should always describe the timing mechanism fully in order to allow others to repeat published simulation experiments.

Percolation Times in Two-Dimensional Models For Excitable Media

The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $Z^2$ is initially assigned one of the states 0, 1 or 2. At each tick of a discrete--time clock, the configuration changes according to the following synchronous rule: changes $1\to 2$ and $2\to 0$ are automatic, while an $x$ in state 0 may either stay in the same state or change to 1, the latter possibility occurring iff there is at least one representative of state 1 in the local neighborhood of $x$. Starting from a product measure with just 1's and 0's such dynamics quickly die out (turn into 0's), but not before 1's manage to form infinite connected sets. A very precise description of this ``transient percolation'' phenomenon can be obtained when the neighborhood of $x$ consists of 8 nearest points, the case first investigated by S. Fraser and R. Kapral. In addition, first percolation times for related monotone models are addressed.

Are rainforests self-organized in a critical state?

The spatial distribution of low-canopy gaps in the Barro Colorado Island rainforest (Panama) is shown to exhibit fractal properties. A simple cellular automata model (the “Forest Game”) was constructed in order to simulate the gap dynamics of such forests as well as the observed macroscopic spatial regularities. Generalized fractal dimensions are studied as a function of several relevant parameters. The observed and simulated fractal behaviour is shown to be related to self-similar dynamics of biomass. This result is interpreted as related to the emergence of a class of “self-organized critical state”.

Bacterial population autowave patterns: spontaneous symmetry bursting

Abstract Bacteria are known to form autowave patterns (population waves) like those formed by propagating nerve impulses, phase transitions, concentration waves in the Belousov-Zhabotinsky reaction, etc. The formation of bacterial waves is due to the ability of bacteria to drift (through chemotaxis) into the regions with higher attractant concentration. As a result, in contrast to other types of autowaves, bacterial population waves have not only a diffusion component of a bacterial flow but a chemotaxis flow as well. We present the experimental results of the study of spontaneous symmetry loss of bacterial autowave patterns. We show that this phenomenon can be simulated with a simple cellular automata model, and symmetry bursting depends on the parameters characterizing chemotactic sensitivity and motility of the cells forming the population wave. In the experiments in vivo we show that the distortion of a bacterial wave shape can be initiated by bacterial density fluctuations in the parent population that the bacterial waves flake off from.

Bacterial population autowave patterns: spontaneous symmetry bursting

Bacteria are known to form autowave patterns (population waves) like those formed by propagating nerve impulses, phase transitions, concentration waves in the Belousov-Zhabotinsky reaction, etc. The formation of bacterial waves is due to the ability of bacteria to drift (through chemotaxis) into the regions with higher attractant concentration. As a result, in contrast to other types of autowaves, bacterial population waves have not only a diffusion component of a bacterial flow but a chemotaxis flow as well. We present the experimental results of the study of spontaneous symmetry loss of bacterial autowave patterns. We show that this phenomenon can be simulated with a simple cellular automata model, and symmetry bursting depends on the parameters characterizing chemotactic sensitivity and motility of the cells forming the population wave. In the experiments in vivo we show that the distortion of a bacterial wave shape can be initiated by bacterial density fluctuations in the parent population that the bacterial waves flake off from.

Dynamical and critical behavior of a simple discrete model of the cellular immune system.

A simple cellular automata model has been constructed to investigate the interactions between the two T-helper subset cell types (${\mathrm{T}}_{\mathrm{H}}$1 and ${\mathrm{T}}_{\mathrm{H}}$2) in a lymph node during chronic parasitic infection. The model exhibits behavior similar to a phase transition as a function of the antigenic burden placed on the host. At low antigen density the behavior of the model resembles that of a ``paramagnetic'' phase in which both T-helper cell subset cells can coexist. Above a threshold antigen density then one or other of the ${\mathrm{T}}_{\mathrm{H}}$ subset cells becomes dominant and forms a single, connected, infinite cluster (equivalent to a ``ferromagnetic'' phase). Much of the phenomenological behavior of the model is seen to be in good agreement with that observed in animal models of parasitic infection.