193 research outputs found
Crossover from Percolation to Self-Organized Criticality
We include immunity against fire as a new parameter into the self-organized
critical forest-fire model. When the immunity assumes a critical value,
clusters of burnt trees are identical to percolation clusters of random bond
percolation. As long as the immunity is below its critical value, the
asymptotic critical exponents are those of the original self-organized critical
model, i.e. the system performs a crossover from percolation to self-organized
criticality. We present a scaling theory and computer simulation results.Comment: 4 pages Revtex, two figures included, to be published in PR
Scaling laws and simulation results for the self--organized critical forest--fire model
We discuss the properties of a self--organized critical forest--fire model
which has been introduced recently. We derive scaling laws and define critical
exponents. The values of these critical exponents are determined by computer
simulations in 1 to 8 dimensions. The simulations suggest a critical dimension
above which the critical exponents assume their mean--field values.
Changing the lattice symmetry and allowing trees to be immune against fire, we
show that the critical exponents are universal.Comment: 12 pages, postscript uuencoded, figures included, to appear in Phys.
Rev.
Solitons in the one-dimensional forest fire model
Fires in the one-dimensional Bak-Chen-Tang forest fire model propagate as
solitons, resembling shocks in Burgers turbulence. The branching of solitons,
creating new fires, is balanced by the pair-wise annihilation of oppositely
moving solitons. Two distinct, diverging length scales appear in the limit
where the growth rate of trees, , vanishes. The width of the solitons, ,
diverges as a power law, , while the average distance between solitons
diverges much faster as .Comment: 4 pages with 2 figures include
Renormalization group approach to the critical behavior of the forest fire model
We introduce a Renormalization scheme for the one and two dimensional
Forest-Fire models in order to characterize the nature of the critical state
and its scale invariant dynamics. We show the existence of a relevant scaling
field associated with a repulsive fixed point. This model is therefore critical
in the usual sense because the control parameter has to be tuned to its
critical value in order to get criticality. It turns out that this is not just
the condition for a time scale separation. The critical exponents are computed
analytically and we obtain , and ,
respectively for the one and two dimensional case, in very good agreement with
numerical simulations.Comment: 4 pages, 3 uuencoded Postcript figure
The self-organized critical forest-fire model on large scales
We discuss the scaling behavior of the self-organized critical forest-fire
model on large length scales. As indicated in earlier publications, the
forest-fire model does not show conventional critical scaling, but has two
qualitatively different types of fires that superimpose to give the effective
exponents typically measured in simulations. We show that this explains not
only why the exponent characterizing the fire-size distribution changes with
increasing correlation length, but allows also to predict its asymptotic value.
We support our arguments by computer simulations of a coarse-grained model, by
scaling arguments and by analyzing states that are created artificially by
superimposing the two types of fires.Comment: 26 pages, 7 figure
Phase Transitions in a Forest-Fire Model
We investigate a forest-fire model with the density of empty sites as control
parameter. The model exhibits three phases, separated by one first-order phase
transition and one 'mixed' phase transition which shows critical behavior on
only one side and hysteresis. The critical behavior is found to be that of the
self-organized critical forest-fire model [B. Drossel and F. Schwabl, Phys.
Rev. Lett. 69, 1629 (1992)], whereas in the adjacent phase one finds the spiral
waves of the Bak et al. forest-fire model [P. Bak, K. Chen and C. Tang, Phys.
Lett. A 147, 297 (1990)]. In the third phase one observes clustering of trees
with the fire burning at the edges of the clusters. The relation between the
density distribution in the spiral state and the percolation threshold is
explained and the implications for stationary states with spiral waves in
arbitrary excitable systems are discussed. Furthermore, we comment on the
possibility of mapping self-organized critical systems onto 'ordinary' critical
systems.Comment: 30 pages RevTeX, 9 PostScript figures (Figs. 1,2,4 are of reduced
quality), to appear in Phys. Rev.
Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model
We present the analytic solution of the self-organized critical (SOC)
forest-fire model in one dimension proving SOC in systems without conservation
laws by analytic means. Under the condition that the system is in the steady
state and very close to the critical point, we calculate the probability that a
string of neighboring sites is occupied by a given configuration of trees.
The critical exponent describing the size distribution of forest clusters is
exactly and does not change under certain changes of the model
rules. Computer simulations confirm the analytic results.Comment: 12 pages REVTEX, 2 figures upon request, dro/93/
Some Properties of the Speciation Model for Food-Web Structure - Mechanisms for Degree Distributions and Intervality
We present a mathematical analysis of the speciation model for food-web
structure, which had in previous work been shown to yield a good description of
empirical data of food-web topology. The degree distributions of the network
are derived. Properties of the speciation model are compared to those of other
models that successfully describe empirical data. It is argued that the
speciation model unifies the underlying ideas of previous theories. In
particular, it offers a mechanistic explanation for the success of the niche
model of Williams and Martinez and the frequent observation of intervality in
empirical food webs.Comment: 23 pages, 6 figures, minor rewrite
Critical Kauffman networks under deterministic asynchronous update
We investigate the influence of a deterministic but non-synchronous update on
Random Boolean Networks, with a focus on critical networks. Knowing that
``relevant components'' determine the number and length of attractors, we focus
on such relevant components and calculate how the length and number of
attractors on these components are modified by delays at one or more nodes. The
main findings are that attractors decrease in number when there are more
delays, and that periods may become very long when delays are not integer
multiples of the basic update step.Comment: 8 pages, 3 figures, submitted to a journa
An explanatory model for food-web structure and evolution
Food webs are networks describing who is eating whom in an ecological
community. By now it is clear that many aspects of food-web structure are
reproducible across diverse habitats, yet little is known about the driving
force behind this structure. Evolutionary and population dynamical mechanisms
have been considered. We propose a model for the evolutionary dynamics of
food-web topology and show that it accurately reproduces observed food-web
characteristic in the steady state. It is based on the observation that most
consumers are larger than their resource species and the hypothesis that
speciation and extinction rates decrease with increasing body mass. Results
give strong support to the evolutionary hypothesis.Comment: 16 pages, 3 figure
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