Scaling laws and simulation results for the self-organized critical forest-fire model

Ergodicity breaking and restoration in models of heat transport with microscopic reversibility

The behavior of lattice models in which time reversibility is enforced at the level of trajectories (microscopic reversibility) is studied analytically. Conditions for ergodicity breaking are explored, and a few examples of systems characterized by an additional conserved quantity besides energy are presented. All the systems are characterized by ergodicity restoration when put in contact with a thermal bath, except for specific choices of the interactions between the atoms in the system and the bath. The study shows that the additional conserved quantities return to play a role in nonequilibrium conditions. The similarities with the behavior of some mesoscale systems, in which the transition rates satisfy detailed balance but not microscopic reversibility, are discussed.

Family-size variability grows with collapse rate in a birth-death-catastrophe model

Forest-fire and avalanche models support the notion that frequent catastrophes prevent the growth of very large populations and as such, prevent rare large-scale catastrophes. We show that this notion is not universal. A new model class leads to a paradigm shift in the influence of catastrophes on the family-size distribution of subpopulations. We study a simple population dynamics model where individuals, as well as a whole family, may die with a constant probability, accompanied by a logistic population growth model. We compute the characteristics of the family-size distribution in steady state and the phase diagram of the steady-state distribution and show that the family and catastrophe size variances increase with the catastrophe frequency, which is the opposite of common intuition. Frequent catastrophes are balanced by a larger net-growth rate in surviving families, leading to the exponential growth of these families. When the catastrophe rate is further increased, a second phase transition to extinction occurs when the rate of new family creations is lower than their destruction rate by catastrophes.

Forest-fire model with natural fire resistance

Observations suggest that contemporary wildfire suppression practices in the United States have contributed to conditions that facilitate large, destructive fires. We introduce a forest-fire model with natural fire resistance that supports this theory. Fire resistance is defined with respect to the size and shape of clusters; the model yields power-law frequency-size distributions of model fires that are consistent with field observations in the United States, Canada, and Australia.

Age distribution of trees in stationary forest system

A statistical theory for the age distribution of spatially dominant trees in a stationary forest system is developed. The result depends whether or not mortality is spatially correlated, as well as whether or not the stand boundaries are pre-determined. In the case of spatially non-correlated mortality, the tree age distribution is an exponential with survival rate as the base. In the case of spatially correlated mortality within a stand with pre-determined boundaries, the age distribution within the stand is an exponential with natural base. For a small stand, the median life span of the stand is inversely proportional to the number of trees (n); the median life span in relation to stand closure time is inversely proportional to nln(n). For a large stand, the stand life does not extend to the closure time. The behaviour of a forest system without fixed stand boundaries depends on the dimensionality of the system. In the case of a one-dimensional system, the longevity distribution is exponential, most of the trees however having the same longevity. Consequently, the probability density of tree age is constant. However, the probability mass of size of catastrophe destroying a particular tree is evenly distributed. This is due to trees being rapidly born on empty areas in the beginning of the life cycle, and clusters rapidly growing into larger ones close to the end of tree life.

Phase diagram and critical behavior of a forest-fire model in a gradient of immunity

The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl [Physica A 199, 183 (1993)], in which each site of a lattice can be in three possible states: occupied by a tree, empty, or occupied by a burning tree (fire). The trees grow at empty sites with probability p, healthy trees catch fire from adjacent burning trees with probability (1-g), where g is the immunity, and a burning tree becomes an empty site spontaneously. In this paper we study the FFMIT by means of the recently proposed gradient method (GM), considering the immunity as a uniform gradient along the horizontal axis of the lattice. The GM allows the simultaneous treatment of both the active and the inactive phases of the model in the same simulation. In this way, the study of a single-valued interface gives the critical point of the active-absorbing transition, whereas the study of a multivalued interface brings the percolation threshold into the active phase. Therefore we present a complete phase diagram for the FFMIT, for all range of p, where, besides the usual active-absorbing transition of the model, we locate a transition between the active percolating and the active nonpercolating phases. The average location and the width of both interfaces, as well as the absorbing and percolating cluster densities, obey a scaling behavior that is governed by the exponent α=1/(1+ν), where ν is the suitable correlation length exponent (ν(⊥) for the directed percolation transition and ν for the standard percolation transition). We also show that the GM allows us to calculate the critical exponents associated with both the order parameter of the absorbing transition and the number of particles in the multivalued interface. Besides, we show that by using the gradient method, the collapse in a single curve of cluster densities obtained for samples of different side is a very sensitive method in order to obtain the critical points and the percolation thresholds.

Models, data and mechanisms: quantifying wildfire regimes

The quantification of wildfire regimes, especially the relationship between the frequency with which events occur and their size, is of particular interest to both ecologists and wildfire managers. Recent studies in cellular automata (CA) and the fractal nature of the frequency-area relationship they produce has led some authors to ask whether the power-law frequency-area statistics seen in the CA might also be present in empirical wildfire data. Here, we outline the history of the debate regarding the statistical wildfire frequency-area models suggested by the CA and their confrontation with empirical data. In particular, the extent to which the utility of these approaches is dependent on being placed in the context of self-organized criticality (SOC) is examined. We also consider some of the other heavy-tailed statistical distributions used to describe these data. Taking a broadly ecological perspective we suggest that this debate needs to take more interest in the mechanisms underlying the observed power-law (or other) statistics. From this perspective, future studies utilizing the techniques associated with CA and statistical physics will be better able to contribute to the understanding of ecological processes and systems.

Lattice geometry, gap formation and scale invariance in forests

The geometry of the lattice used in ecological modeling is important because of the local nature of ecological interactions. The latter can generate complex behavior such as criticality (scale-invariance). In this work, we implement two slightly different forest disturbance models on three lattices, each with square, triangular and hexagonal symmetry, in order to study the effect of geometry. We calculate the density distribution of gaps in a forest and find bumps in the distribution at sizes that depend on lattice geometry. Similar bumps were observed in real data but remained unexplainable. We suggest that these bumps provide information about the geometry and scale of ecological interactions. We also found an effect of geometry on the conditions under which criticality appears in model forests. These conditions appear to be more biologically realistic, and also linked to the likelihood of local disturbance propagation. The scaling exponent of the gap-size distribution, however, was found to be independent of both model and geometry, a hallmark of universality.

Competitive coexistence in a dynamic landscape

This paper investigates the effect of a dynamic landscape on the persistence of many interacting species. We develop a multi-species community model with an evolving landscape in which the creation and destruction of habitat are dynamic and local in space. Species interactions are also local involving hierarchical competitive trade-offs. We show that dynamic landscapes can reverse the trend of increasing species richness with higher fragmentation observed in static landscapes. The increase in the species-area exponent from a homogeneous to a fragmented landscape does not occur when dynamics are turned on. Thus, temporal aspects of the processes that generate and destroy habitat appear dominant relative to spatial characteristics. We also demonstrate, however, that temporal and spatial aspects interact to influence the persistence time of individual species, and therefore, rank-abundance curves. Specifically, persistence in the model increases in habitats with faster local turnover because of the presence of dynamic corridors.

Investigation of the forest-fire model on a small-world network

It is shown that the forest-fire model of Bak et al. run on a square lattice network with additional long-range interactions in the spirit of a small-world network results in a scale-free system reminiscent of self-organized criticality without recourse to fine tuning. As the number of these long-range interactions is increased, the cluster size distribution exponent is found to decrease in magnitude as the small-world regime is entered, indicating a change in its universality class. It is suggested that such a model could have applicability in the study of disease spreading in human populations.

Fractal dynamics of electric discharges in a thundercloud

We have investigated the fractal dynamics of intracloud microdischarges responsible for the formation of a so-called drainage system of electric charge transport inside a cloud volume. Microdischarges are related to the nonlinear stage of multiflow instability development, which leads to the generation of a small-scale intracloud electric structure. The latter is modeled by using a two-dimensional lattice of finite-state automata. The results of numerical simulations show that the developed drainage system belongs to the percolation-cluster family. We then point out the parameter region relevant to the proposed model, in which the thundercloud exhibits behavior corresponding to a regime of self-organized criticality. The initial development and statistical properties of dynamic conductive clusters are investigated, and a kinetic equation is introduced, which permits us to find state probabilities of electric cells and to estimate macroscopic parameters of the system.

Broken scaling in the forest-fire model

We investigate the scaling behavior of the cluster size distribution in the Drossel-Schwabl forest-fire model (DS-FFM) by means of large scale numerical simulations, partly on (massively) parallel machines. It turns out that simple scaling is clearly violated, as already pointed out by Grassberger [P. Grassberger, J. Phys. A 26, 2081 (1993)], but largely ignored in the literature. Most surprisingly, the statistics do not seem to be described by a universal scaling function, and the scale of the physically relevant region seems to be a constant. Our results strongly suggest that the DS-FFM is not critical in the sense of being free of characteristic scales.

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 us 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.

Forest-fire models as a bridge between different paradigms in self-organized criticality

We turn the stochastic critical forest-fire model introduced by Drossel and Schwabl [Phys. Rev. Lett. 69, 1629 (1992)] into a completely deterministic threshold model. This model has many features in common with sandpile and earthquake models of self-organized criticality. Our deterministic forest-fire model exhibits in detail the same macroscopic statistical properties as the original Drossel-Schwabl model. We use the deterministic model to elaborate on the relation between forest-fire, sandpile, and earthquake models.

Corrections to scaling in the forest-fire model

We present a systematic study of corrections to scaling in the self-organized critical forest-fire model. The analysis of the steady-state condition for the density of trees allows us to pinpoint the presence of these corrections, which take the form of subdominant exponents modifying the standard finite-size scaling form. Applying an extended version of the moment analysis technique, we find the scaling region of the model and compute nontrivial corrections to scaling.

Forest fires: An example of self-organized critical behavior

Despite the many complexities concerning their initiation and propagation, forest fires exhibit power-law frequency-area statistics over many orders of magnitude. A simple forest fire model, which is an example of self-organized criticality, exhibits similar behavior. One practical implication of this result is that the frequency-area distribution of small and medium fires can be used to quantify the risk of large fires, as is routinely done for earthquakes.

On the critical behaviour of simple epidemics

We show how ideas and models which were originally introduced to gain an understanding of critical phenomena can be used to interpret the dynamics of epidemics of communicable disease in real populations. Specifically, we present an analysis of the dynamics of disease outbreaks for three common communicable infections from a small isolated island population. The strongly fluctuating nature of the temporal incidence of disease is captured by the model, and comparisons between exponents calculated from the data and from simulations are made. A forest-fire model with sparks is used to classify the observed scaling dynamics of the epidemics and provides a unified picture of the epidemiology which conventional epidemiological analysis is unable to reproduce. This study suggests that power-law scaling can emerge in natural systems when they are driven on widely separated time-scales, in accordance with recent analytic renormalization group calculations.

A scaling analysis of measles epidemics in a small population

We present a detailed analysis of the pattern of measles outbreaks in the small isolated community of the Faroe Islands. Measles outbreaks in this population are characterized by frequent fade-out of infection resulting in long intervals when the disease is absent from the islands. Using an analysis of the distribution of epidemic sizes and epidemic durations we propose that the dynamical structure observed in the measles case returns reflects the existence of an underlying scaling mechanism. Consequently the dynamics are not as purely stochastic as is usually thought for epidemiological systems of this sort. We use a lattice-based epidemic model to provide a theoretical estimate of the scaling exponents and show that a conventional compartmental SEIR model is unable to reproduce this result. The methods discussed in this paper are general and represent a novel way to consider the dynamics of any other communicable disease where there is frequent fade-out in the case returns.

Power laws governing epidemics in isolated populations

Temporal changes in the incidence of measles virus infection within large urban communities in the developed world have been the focus of much discussion in the context of the identification and analysis of nonlinear and chaotic patterns in biological time series. In contrast, the measles records for small isolated island populations are highly irregular, because of frequent fade-outs of infection, and traditional analysis does not yield useful insight. Here we use measurements of the distribution of epidemic sizes and duration to show that regularities in the dynamics of such systems do become apparent. Specifically, these biological systems are characterized by well-defined power laws in a manner reminiscent of other nonlinear, spatially extended dynamical systems in the physical sciences. We further show that the observed power-law exponents are well described by a simple lattice-based model which reflects the social interaction between individual hosts.