The geometry of percolation fronts in two-dimensional lattices with spatially varying densities

Percolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x, has received less attention. Previous studies with long-range spatial variations in p(x) have only investigated cases where p has a finite, non-zero gradient at the critical point p_c. Here we extend the theory to two-dimensional cases in which the gradient can change from zero to infinity. We present scaling laws for the width and length of the hull (i.e. the boundary of the spanning cluster). We show that the scaling exponents for the width and the length depend on the shape of p(x), but they always have a constant ratio 4/3 so that the hull's fractal dimension D=7/4 is invariant. On this basis, we derive and verify numerically an asymptotic expression for the probability h(x) that a site at a given distance x from p_c is on the hull.Comment: 13 pages, 7 figures, to appear in New Journal of Physic

Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry

A change in the environmental conditions across space—for example, altitude or latitude—can cause significant changes in the density of a vegetation type and, consequently, in spatial connectivity. We use spatially explicit simulations to study the transition from connected to fragmented vegetation. A static (gradient percolation) model is compared to dynamic (gradient contact process) models. Connectivity is characterized from the perspective of various species that use this vegetation type for habitat and differ in dispersal or migration range, that is, “step length” across the landscape. The boundary of connected vegetation delineated by a particular step length is termed the “ hull edge.” We found that for every step length and for every gradient, the hull edge is a fractal with dimension 7/4. The result is the same for different spatial models, suggesting that there are universal laws in ecotone geometry. To demonstrate that the model is applicable to real data, a hull edge of fractal dimension 7/4 is shown on a satellite image of a piñon‐juniper woodland on a hillside. We propose to use the hull edge to define the boundary of a vegetation type unambiguously. This offers a new tool for detecting a shift of the boundary due to a climate change

Five main phases of landscape degradation revealed by a dynamic mesoscale model analysing the splitting, shrinking, and disappearing of habitat patches

The ecological consequences of habitat loss and fragmentation have been intensively studied on a broad, landscape-wide scale, but have less been investigated on the finer scale of individual habitat patches, especially when considering dynamic turnovers in the habitability of sites. We study changes to individual patches from the perspective of the inhabitant organisms requiring a minimum area for survival. With patches given by contiguous assemblages of discrete habitat sites, the removal of a single site necessarily causes one of the following three elementary local events in the affected patch: splitting into two or more pieces, shrinkage without splitting, or complete disappearance. We investigate the probabilities of these events and the effective size of the habitat removed by them from the population's living area as the habitat landscape gradually transitions from pristine to totally destroyed. On this basis, we report the following findings. First, we distinguish four transitions delimiting five main phases of landscape degradation: (1) when there is only a little habitat loss, the most frequent event is the shrinkage of the spanning patch; (2) with more habitat loss, splitting becomes significant; (3) splitting peaks; (4) the remaining patches shrink; and (5) finally, they gradually disappear. Second, organisms that require large patches are especially sensitive to phase 3. This phase emerges at a value of habitat loss that is well above the percolation threshold. Third, the effective habitat loss caused by the removal of a single habitat site can be several times higher than the actual habitat loss. For organisms requiring only small patches, this amplification of losses is highest during phase 4 of the landscape degradation, whereas for organisms requiring large patches, it peaks during phase 3

The geometry of percolation fronts in two-dimensional lattices with spatially varying densities

10.1088/1367-2630/14/10/103019NEW JOURNAL OF PHYSICS141

From virtual plants to real communities: A review of modelling clonal growth

International audienceClonal plants grow by the production of semi- autonomous modules (ramets), and form complex branching structures which may provide communication/ resource flow channels between the units. These characteristic features have made clonal plants a challenging subject for spatial modelling. We review the advance of ideas and new directions in theoretical research since the last review (Oborny and Cain, 1997). We place clonal growth models into a general framework of spatial population dynamic models, comparing individual ramets of a clone with individuals in a non-clonal population. We discuss three specificities of clonal spreading: (1) ramets can be physiologically integrated through the network of branching structures; (2) formation of new ramets occurs by the growth of these branching structures which can be directional, following architectural rules; and (3) formation of new ramets can be adjusted to the environment by phenotypic plasticity. We review methods by which these traits have been implemented into models. We summarize model predictions, for the spatial structure and fitness of clonal plants, and link these predictions with existing empirical data. Emphasis is given to the contributions that theoretical studies could provide for experimental studies in the field. We emphasize the following recent major developments: (i) a much better understanding of emergent consequences of various clonal growth rules over broad spatial and temporal scales has been reached. (ii) Links have been found to other complex systems. For example, a key problem of integration vs. splitting of connecting structures has been shown to be closely related to a problem in percolation theory. (iii) Interactions between physiological integration, architectural growth and plastic responses have been demonstrated; research on these interactions has generally shown a large degree of contingency in the effects of these traits. Finally, we outline some areas for future research

Consensus time in a voter model with concealed and publicly expressed opinions

10.1088/1742-5468/aac14aJOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT2018

Agent-based neutral competition in two-community networks

Competition between alternative states is an essential process in social and biological networks. Neutral competition can be represented by an unbiased random drift process in which the states of vertices (e.g., opinions, genotypes, or species) in a network are updated by repeatedly selecting two connected vertices. One of these vertices copies the state of the selected neighbor. Such updates are repeated until all vertices are in the same "consensus" state. There is no unique rule for selecting the vertex pair to be updated. Real-world processes comprise three limiting factors that can influence the selected edge and the direction of spread: (1) the rate at which a vertex sends a state to its neighbors, (2) the rate at which a state is received by a neighbor, and (3) the rate at which a state can be exchanged through a connecting edge. We investigate how these three limitations influence neutral competition in networks with two communities generated by a stochastic block model. By using Monte Carlo simulations, we show how the community structure and update rule determine the states' success probabilities and the time until a consensus is reached. We present a heterogeneous mean-field theory that agrees well with the Monte Carlo simulations. The effectiveness of the heterogeneous mean-field theory implies that quantitative predictions about the consensus are possible even if empirical data (e.g., from ecological fieldwork or observations of social interactions) do not allow a complete reconstruction of all edges in the network.Comment: 13 pages, 4 figures, to be published in Physical Review

Transition from Connected to Fragmented Vegetation across an Environmental Gradient: Scaling Laws in Ecotone Geometry

10.1086/599292AMERICAN NATURALIST1741E23-E3

Changes in the Gradient Percolation Transition Caused by an Allee Effect

The establishment and spreading of biological populations depends crucially on population growth at low densities. The Allee effect is a problem in those populations where the per-capita growth rate at low densities is reduced. We examine stochastic spatial models in which the reproduction rate changes across a gradient g so that the population undergoes a 2D-percolation transition. Without the Allee effect, the transition is continuous and the width w of the hull scales as in conventional (i.e., uncorrelated) gradient percolation, proportional to g^(-0.57). However, with a strong Allee effect the transition is first order and w is proportional to g^(-0.26).Comment: 4 pages, 4 figures, Phys. Rev. Lett., in print, supplementary material available at http://www2.imperial.ac.uk/~mgastner/publications/suppl_PRL_Jan11.pd