Random Landscape Research Articles

Line Creep in Paper Peeling

We have studied experimentally the dynamics of the separation of a sheet of paper into two halves in a peeling configuration. The experimental setup consists of a peeling device, where a fracture front is driven along the plane of the paper, with a constant force. The theoretical picture is how an elastic line interacts with a random landscape of fracture toughness. We compare the results with theoretical simulations in several aspects. One recent finding concerns the autocorrelation function of the average front position. The data from the experiments produces so-called cusps or singularities in the correlation function, as predicted by the functional renormalization group theory for elastic lines. Comparisons with simulations with either a short range or a long range elastic kernel demonstrate that the latter agrees with the experimental observations, as expected.

Organic glasses: cluster structure of the random energy landscape

AbstractAn appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for “deep” clusters where value of the field on each site is much greater than the rms disorder. We also demonstrate significant modification of the properties of energetic disorder in organic glasses at the vicinity of the electrode.

Organic glasses: cluster structure of the random energy landscape

An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for "deep" clusters where value of the field on each site is much greater than the rms disorder. We also demonstrate significant modification of the properties of energetic disorder in organic glasses at the vicinity of the electrode.

Effect of electric field on diffusion in disordered materials

An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for “deep” clusters where value of the field on each site is much greater than the rms disorder. We also demonstrate significant modification of the properties of energetic disorder in organic glasses at the vicinity of the electrode.

Some physical applications of random hierarchical matrices

The investigation of spectral properties of random block-hierarchical matrices as applied to dynamic and structural characteristics of complex hierarchical systems with disorder is proposed for the first time. Peculiarities of dynamics on random ultrametric energy landscapes are discussed and the statistical properties of scale-free and polyscale (depending on the topological characteristics under investigation) random hierarchical networks (graphs) obtained by multiple mapping are considered.

Declines in moose population density at Isle Royle National Park, MI, USA and accompanied changes in landscape patterns

Ungulate herbivores create patterns of forage availability, plant species composition, and soil fertility as they range across large landscapes and consume large quantities of plant material. Over time, herbivore populations fluctuate, producing great potential for spatio-temporal landscape dynamics. In this study, we extend the spatial and temporal extent of a long-term investigation of the relationship of landscape patterns to moose foraging behavior at Isle Royale National Park, MI. We examined how patterns of browse availability and consumption, plant basal area, and soil fertility changed during a recent decline in the moose population. We used geostatistics to examine changes in the nature of spatial patterns in two valleys over 18 years and across short-range and long-range distance scales. Landscape patterns of available and consumed browse changed from either repeated patches or randomly distributed patches in 1988–1992 to random point distributions by 2007 after a recent record high peak followed by a rapid decline in the moose population. Patterns of available and consumed browse became decoupled during the moose population low, which is in contrast to coupled patterns during the earlier high moose population. Distributions of plant basal area and soil nitrogen availability also switched from repeated patches to randomly distributed patches in one valley and to random point distributions in the other valley. Rapid declines in moose population density may release vegetation and soil fertility from browsing pressure and in turn create random landscape patterns.

Driven particle in a random landscape: Disorder correlator, avalanche distribution, and extreme value statistics of records

We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .

Exact sampling of corrugated surfaces

We discuss an algorithm for the exact sampling of vectors satisfying a set of pairwise difference inequalities. Applications include the exact samplingof skew Young Tableaux, of configurations in the Bead Model, and of corrugated surfaceson a graph, that is random landscapes in which at each vertex corresponds a localmaximum or minimum. As an example, we numerically evaluate with high precision thenumber of corrugated surfaces on the square lattice. After an extrapolation to thethermodynamic limit, controlled by an exact formula, we put into evidence a discrepancywith previous numerical results.

Phase transition in a random minima model: mean field theory and exact solution on theBethe lattice

We consider the number and distribution of minima in random landscapes defined on non-Euclideanlattices. Using an ensemble where random landscapes are reweighted by a fugacity factorz for each minimum that they contain, we construct first a ‘two-box’mean field theory. This exhibits an ordering phase transition atzc = 2 above which one box contains an extensive number of minima. Theonset of order is governed by an unusual order parameter exponentβ = 1, motivating us to study the same model on the Bethe lattice.Here we find from an exact solution that for any connectivityμ+1>2 there is an ordering transition with a conventional mean field order parameter exponentβ = 1/2, but with the region where this behaviour is observable shrinking in size as1/μ in the mean fieldlimit of large μ. We show that the behaviour in the transition region can also be understood directlywithin a mean field approach, by making the assignment of minima ‘soft’. Finally wedemonstrate, in the simplest mean field case, how the analysis can be generalized toinclude both maxima and minima. In this case an additional first-order phasetransition appears, to a landscape in which essentially all sites are either minima ormaxima.

Relevant Source Area of Pollen in patchy cultural landscapes and signals of anthropogenic landscape disturbance in the pollen record: A simulation approach

Using the HUMPOL simulation computer model we explored the effects of various factors characteristic of mosaic cultural landscapes on the Relevant Source Area of Pollen (RSAP sensu Sugita) of small lakes (50 m radius), and the representation of NAP and human-impact pollen indicators in small lakes. The following aspects were tested: 1) effect of variations in the spatial distribution of taxa/patches and species diversity on RSAP, 2) effect of changes in the proportion of landscape openness and size of openings on the RSAP and the pollen signal, and 3) the value of low-abundance anthropogenic indicator species to detect openness in the landscape. Moreover, we examined a methodological aspect of the MOSAIC computer program, i.e. the variability in RSAP due to inherent variations between replicates of random landscape simulation scripts using the program MOSAIC. We found differences in the RSAP estimates between runs using replicates of the same landscape created in the program MOSAIC. Such differences need to be taken into account when interpreting results involving the use of MOSAIC replicates. The simulations suggest that, if the model assumptions are valid, the RSAP is sensitive to vegetation structure in terms of patch size, and taxa/patch diversity and spatial distribution, whatever the properties of the taxa modelled. The longer the distance from each sampling site to get a sufficient cover of all taxa within the landscape, the larger the RSAP. If the spatial distribution of patches is very uneven and/or some taxa/patches are characterised by low frequencies in the randomized landscape, this distance will become longer. Further analysis of the results and new modelling experiments are necessary to provide better means for testing the presented hypothesis on the effect of spatial vegetation/landscape evenness on RSAP. The simulations indicate that, in hypothetical mosaic tree/herb landscapes including common NW European taxa of traditional cultural landscapes, RSAP estimates are relatively robust, i.e. will seldom be smaller than 1000 m and larger than 2500 m in radius in the case of 50 m radius lakes. NAP percentage is not a reliable “measure” of landscape openness. A NAP percentage value may represent a large range of openness percentage cover depending on the size and spatial distribution of the openness patches. Common pollen indicators of human impact and landscape openness will be represented in a pollen assemblage from small lakes when the landscape is relatively open (≥ 30%) and a minimum of 300–1000 grains is counted. Findings of pollen indicators of human impact may not be as reliable indicators of land use within the RSAP as has sometimes been assumed, especially when human activity is small scale. Moreover, it is important to realize that a small number of grains may indicate relatively large overall openness cover in the landscape, especially if the open areas consist of few large clearings rather than many small clearings.

Slow relaxation, dynamic transitions, and extreme value statistics in disordered systems

We show that the dynamics of simple disordered models, like the directed trap model and the random energy model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape.

Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelatedrandom fitness landscape in the limit of infinite genome size, which implies that eachmutation generates a new fitness value drawn from a probability distributiong(w). This is the finite population version of Kingman’s house of cards model (Kingman1978 J. Appl. Probab. 15 1). In contrast to Kingman’s work, the focus here is on unbounded distributionsg(w) which lead to an indefinite growth of the population fitness. The model is solvedanalytically in the limit of infinite population size and simulated numerically for finiteN. When the genome-wide mutation probabilityU is small, the long-time behavior of the model reduces to a point process of fixationevents, which is referred to as a diluted record process (DRP). The DRP is similarto the standard record process except that a new record candidate (a numberthat exceeds all previous entries in the sequence) is accepted only with a certainprobability that depends on the values of the current record and the candidate.We develop a systematic analytic approximation scheme for the DRP. At finiteU the fitness frequency distribution of the population decomposes into astationary part due to mutations and a traveling wave component due toselection, which is shown to imply a reduction of the mean fitness by a factor of1−U compared to the limit.

Anomalous mobility of dislocation kink solitons in disordered solid solutions

Dislocation kink solitons in disordered solid solutions provide an example of quasiparticles showing anomalous kinetics-i.e., the nonlinear dependence of the displacement x on the time t , x approximately t(delta) (delta<1) . To describe the dynamic phase transition from the ordinary linear to anomalous regime, the dynamics of a quasiparticle in an energy landscape that performs a correlated random walk on the energy scale was theoretically studied. The phase diagram was characterized by the calculated temperature dependence of the threshold driving force F(th) below which the average velocity of quasiparticles vanishes. The exponent delta of the kinetic equation for the anomalous phase, x approximately t(delta), was determined by simple statistical arguments using the concepts of the "optimal fluctuation method." The dependence of the threshold driving force F(th) on the concentration of solute atoms and statistical properties of a random energy landscape relevant to disordered solid solutions was calculated. The correlations between steps of the random potential were shown to modify the concentration dependence of F(th), thereby providing a qualitative explanation of experimental data on the dislocation pinning in solid solutions.

Scaling laws of marine predator search behaviour

Many free-ranging predators have to make foraging decisions with little, if any, knowledge of present resource distribution and availability. The optimal search strategy they should use to maximize encounter rates with prey in heterogeneous natural environments remains a largely unresolved issue in ecology. Lévy walks are specialized random walks giving rise to fractal movement trajectories that may represent an optimal solution for searching complex landscapes. However, the adaptive significance of this putative strategy in response to natural prey distributions remains untested. Here we analyse over a million movement displacements recorded from animal-attached electronic tags to show that diverse marine predators-sharks, bony fishes, sea turtles and penguins-exhibit Lévy-walk-like behaviour close to a theoretical optimum. Prey density distributions also display Lévy-like fractal patterns, suggesting response movements by predators to prey distributions. Simulations show that predators have higher encounter rates when adopting Lévy-type foraging in natural-like prey fields compared with purely random landscapes. This is consistent with the hypothesis that observed search patterns are adapted to observed statistical patterns of the landscape. This may explain why Lévy-like behaviour seems to be widespread among diverse organisms, from microbes to humans, as a 'rule' that evolved in response to patchy resource distributions.

Appraisal of random and systematic land cover transitions for regional water balance and revegetation strategies

This study describes the use of landscape transition analysis as a means for effective basin management. Land cover transitions from 1995 to 2002 were analyzed using a cross-tabulation matrix for an important economic zone in south-west Victoria, Australia. Specifically, the matrix was used to determine whether the transitions were random or systematic. Random landscape transitions occur when a land cover replaces other land covers in proportion to their availability. Systematic landscape transitions occur when there are deviations from random patterns, and land use types ‘target’ other land use types for replacement. The analysis was conducted with 11 land cover categories and showed that dryland pastures have been systematically losing area to dryland crops and blue gum ( Eucalyptus globulus) plantations. Dryland crops have systematically expanded in the north-east of the catchment, an area where increasing in-stream salinization has occurred concurrently with this transition. The systematic expansion of the blue gum plantations has been predominantly in the south-west of the catchment and has the potential to reduce stream flows and groundwater recharge in an already water-stressed region, as blue gums use more water than the dryland pastures they are replacing. All other transitions were largely random. These findings have implications for land use planning in the study area for regional water balance and revegetation strategies. Landscape transition analysis is a cost-effective means of contributing to the management of water resources at a regional scale, and is highly recommended for future basin planning.

Multiscale Habitat Use by Wolverines in British Columbia, Canada

ABSTRACT Wolverine (Gulo gulo) distribution in British Columbia, Canada, includes multiple‐use lands where human use and resource extraction may influence habitat selection. We evaluated seasonal habitat use by resident adult wolverines using radiotelemetry locations from 2 multiple‐use landscapes in British Columbia. Food, predation risk, and human disturbance hypotheses were considered in logistic regression analyses of used and random landscapes. Male wolverine habitat associations were most supported by the food hypothesis in both summer and winter. Moose (Alces alces) winter ranges, valley bottom forests, and avalanche terrain were positively associated with winter male wolverine use. Habitat use by male wolverines in winter was also negatively associated with helicopter skiing areas in the Columbia Mountains. Habitat associations of females were more complex; combinations of variables supporting food, predation risk, or human disturbance hypotheses were included in most supported models from both summer and winter in both study areas. Females were associated with alpine and avalanche environments where hoary marmot (Marmota caligata) and Columbia ground squirrel (Spermophilus columbianus) prey are found in summer. Roaded and recently logged areas were negatively associated with female wolverines in summer. In the Columbia Mountains, where winter recreation was widespread, females were negatively associated with helicopter and backcountry skiing. Moose winter ranges within rugged landscapes were positively associated with females during winter. Our analysis suggests wolverines were negatively responding to human disturbance within occupied habitat. The population consequences of these functional habitat relationships will require additional focused research. Our spatially explicit models can be used to support conservation planning for resource extraction and tourism industries operating in landscapes occupied by wolverines.

Phase transition in a random NK landscape model

An analysis for the phase transition in a random NK landscape model, NK ( n , k , z ) , is given. This model is motivated from population genetics and the solubility problem for the model is equivalent to a random ( k + 1 ) -SAT problem. Gao and Culberson [Y. Gao, J. Culberson, An analysis of phase transition in NK landscapes, Journal of Artificial Intelligence Research 17 (2002) 309–332] showed that a random instance generated by NK ( n , 2 , z ) with z > z 0 = 27 − 7 5 4 is asymptotically insoluble. Based on empirical results, they conjectured that the phase transition occurs around the value z = z 0 . We prove that an instance generated by NK ( n , 2 , z ) with z < z 0 is soluble with positive probability by providing a polynomial time algorithm. Using branching process arguments, we prove again that an instance generated by NK ( n , 2 , z ) with z > z 0 is asymptotically insoluble. The results show the phase transition around z = z 0 for NK ( n , 2 , z ) . In the course of the analysis, we introduce a generalized random 2-SAT formula, which is of self interest, and show its phase transition phenomenon.

Exploring landscape structure effect on termite territory size using a model approach

The foraging territory of the Formosan subterranean termite, Coptotermes formosanus Shiraki, was simulated by using a lattice model in order to study how landscape structure affects the foraging territory. Three kinds of landscape were generated on lattice space: ideal, random and fractal landscape. Each lattice cell had a value ranging from 0.0 to 1.0, interpreted as transition probability, P trans, which represents spatially distributed property of the landscapes. The heterogeneity of the fractal landscape was characterized by a parameter, H, controlling aggregation of lattice cells with higher value of P trans. Higher H values corresponded to higher aggregation levels. The model made use of minimized local rules based on empirical data that determines the development of the foraging territory. Additionally, seasonal cycle (summer and winter season), and obstacles which hinder the growth of the territory were incorporated in the model as environmental variables. Territory size was largest in the ideal landscape while it was larger in the random landscape than in the fractal landscape. As obstacle density increased, the territory size decreased. In the fractal landscape, the territory size increased, decreased, and increased again as H increased.

How to define a patch: a spatial model for hierarchically delineating organism-specific habitat patches

Landscape analysis and delineation of habitat patches should take into account organism-specific behavioral and perceptual responses to landscape structure because different organisms perceive and respond to landscape features over different ranges of spatial scales. The commonly used methods for delineating habitat based on rules of contiguity do not account for organism-specific responses to landscape patch structure and have undesirable properties, such as being dependent on the scale of base map used for analysis. This paper presents an improved patch delineation algorithm, “PatchMorph,” which can delineate patches across a range of spatial scales based on three organism-specific thresholds: (1) land cover density threshold, (2) habitat gap maximum thickness (gap threshold), and (3) habitat patch minimum thickness (spur threshold). This algorithm was tested on an “idealized” landscape with landscape gaps and spurs of known size, and delineated patches as expected. It was then applied to delineate patches from a neutral random fractal landscape, which showed that as the input gap and spur thickness thresholds were increased, the number of patches decreased from 59 (low thresholds) patches to 1 (high thresholds). The algorithm was then applied to model western yellow-billed cuckoo (Coccyzus americanus occidentalis) nesting habitat patches based on spur and gap thresholds specific to this organism. Both these analyses showed that fewer patches were delineated by PatchMorph than by rules of contiguity, and those patches were larger, had smoother edges, and had fewer gaps within the patches. This algorithm has many applications beyond those presented in this paper, including habitat suitability analysis, spatially explicit population modeling, and habitat connectivity analysis.

Density of stationary points in a high dimensional random energy landscape and the onset of glassy behavior

The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of size L = R √N corresponds to the onset of exponential in N growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures, a simple variational upper bound on the true free energy of the R = ∞ version of the problem has been constructed and it has been shown that this approximation can recover the position of the whole de-Almeida-Thouless line.