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Introduction

The distribution of vegetation across environmental gradients has received considerable attention since the beginnings of vegetation science, for example, in gradient analysis (Whittaker and Niering 1965). Interest in this topic increased further after reports of significant shifts in vegetation boundaries (e.g., tree lines) due to recent climate changes (Allen and Breshears 1998; Danby and Hik 2007). Detecting boundary shifts has been suggested for biomonitoring climate change (Kimball and Weihrauch 2000; but see Zeng and Malanson 2006 and references therein).

Because the geometries of most natural vegetation boundaries are complicated, it is not self‐evident what to measure, that is, which feature of a boundary could be sufficiently general to enable comparisons between years and/or geographic regions. Observing the vegetation along a gradient, for example, from lower to higher altitudes across an alpine tree line, often reveals a complex pattern of patches. Inside the forest zone, a relatively high cover of forest is interspersed with gaps of various sizes. The average gap size typically increases toward the edge, where small gaps become large enough to coalesce. At that point, the forest becomes fragmented. Farther uphill, the sizes of fragments decrease and the forest vegetation vanishes; see the map in Zeng and Malanson (2006). Apart from alpine tree lines, the same general pattern has also been documented at the edges of a gallery forest (map in Loehle et al. 1996) and of piñon‐juniper woodlands (map in Milne et al. 1996). Several studies have emphasized that natural boundaries are not straight lines but gradual (“soft”; Forman 1995), with a transition zone (ecotone) of considerable spatial extent.

In this article, we study the geometry of this transition zone with spatially explicit numerical simulations. Our main objective is to model the effect of smooth changes in environmental conditions. Moving along a gradient, the density of a particular vegetation type (e.g., forest) decreases. A decrease in density inevitably leads to fragmentation. An abrupt change from the connected to the fragmented state occurs even along a smooth gradient. This phenomenon is studied here with gradient percolation models. The simplest case is the gradient random map (GRM). There, the probability that a site will be occupied by a certain vegetation type changes continuously through space, without correlations between the states of different sites. This model is static, in the sense that it does not account for the dynamics of site occupancy. As a step toward more realism, we also investigate the gradient contact process (GCP), in which local colonization and extinction events are modeled explicitly and a dynamic boundary is investigated. We assume that the colonization distance is limited, so that the states of adjacent sites are correlated.

In both the GRM and the GCP, space is represented by a square lattice, and only two states of the lattice sites, vacant and occupied, are distinguished. We analyze the spatial pattern of occupied sites from the perspective of a species that moves (i.e., migrates or disperses) across this patchy landscape. We assume that two occupied sites are connected and hence belong to the same vegetation patch if and only if the species can move from one site to the other without stepping on a vacant site in between. Crossing a vacant area is possible, but residence (i.e., ending the step there) is not permitted. Various step lengths are tested, and the environmental gradient is also varied.

This organism‐centered description of the landscape allows us to analyze connectivity on various spatial scales. Both empirical and modeling studies have emphasized that the same landscape can be connected or fragmented, depending on the movement range of the actual species (Forman 1995; Ims 1995; With and Crist 1995; With et al. 1997; Wiens 1997; Solé and Bascompte 2006). For example, an area with wetland patches in North Carolina was found to be connected for mink (Mustela vison; ) but fragmented for prothonotary warblers (Prothonotaria citrea; dispersal range mostly <12 km; Bunn et al. 2000). By changing the step length in the simulations, we can test whether species with different dispersal ranges experience different geometries.

For each step length, we determine which sites are connected and hence belong to the same patch (i.e., percolation cluster). The largest patch is generally the one that connects the highest number of sites in the region of low density to the region of highest density. If the gradient points along the horizontal x‐direction from sparse to full vegetation cover, this patch will span from the bottom to the top of the lattice (fig. 1), a property that can be used in delineating the edge. The probability of two distinct large patches in the densest region is negligible in a sufficiently large lattice. Therefore, we call the largest patch the “connected patch.” All the other patches we call “fragments.” Finally, we delineate the connected patch’s hull edge (Milne et al. 1996), which is a set of the farthest points a species can reach without experiencing habitat fragmentation.

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Figure 1: A, Examples of percolation neighborhoods. Suppose the gray site in the center is occupied. If any other site within the highlighted neighborhood is occupied, it belongs to the same patch as the gray site when the maximum step length s is 1 (top), 2 (middle), or 4 (bottom). B, Definition of the hull edge for . The hull edge can be delineated using a left‐turning biased walk, which finds the leftmost path within the connected patch (for details, see “Determination of the Hull Edge by a Biased Walk”). Numbers and crosses mark the steps along the path. C, Hull edge for . Now the walk can skip over some vacant sites along the way. Relative to B, the boundary shifts to the left, and fewer fragments remain. Distinct fragments are labeled with different letters.

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We show that the hull edge has a fractal structure and that its fractal dimension depends on neither the gradient nor the step length of the species. The fractal dimension is, furthermore, the same in the GRM and the GCP and is thus insensitive to correlations introduced by a limited colonization distance. Because of this robust fractal property, we propose that the hull edge is a good candidate for locating and monitoring vegetation boundaries and comparing them across years, geographic regions, or biomes.

Percolation through a Landscape with an Environmental Gradient

A frequently used method for studying the connectivity structure of a landscape is percolation modeling. The main focus of percolation theory is spatial spreading in a heterogeneous medium consisting of occupied and vacant sites (Stauffer and Aharony 1994). In a typical ecological setting, an occupied site is one containing a particular habitat type (i.e., suitable for the particular organism). The significance of percolation theory for ecology was recognized in the late 1980s (Gardner et al. 1987). Since then, a number of papers have investigated habitat maps with simulations and field data (Gustafson and Parker 1992; Milne 1992; Plotnick and Gardner 1993; Loehle et al. 1996; With 1997; Li 2002; He and Hubbell 2003; Solé et al. 2005). Effects of percolation on population dynamics have also been investigated (Andrén 1994; With and Crist 1995; Bascompte and Solé 1996; Oborny et al. 2007).

A commonly used neutral model in ecology relevant to percolation is the so‐called random map. In a random map, the lattice sites are occupied or vacant independently of each other. Conventionally, the map is uniform, in the sense that every site has the same probability p of being occupied (i.e., no gradient is assumed). After the set of occupied sites is determined, it is partitioned into patches, depending on the maximum step length s. All the sites belonging to the same patch can be connected by a walk that moves from one occupied site to another, taking steps no longer than s. For sites in distinct patches, no such path exists. In this article, we study patches generated for s equal to 1, 2, and 4 lattice spacings (fig. 1A).

Although the construction of uniform random maps (URMs) is simple, they have several intriguing properties. When the occupancy probability p is changed continuously from 0 to 1, the landscape suddenly changes from a fragmented to a connected state, that is, there is a critical value p_c at which the random map undergoes a phase transition. As p increases and approaches p_c, the average patch size S of a randomly chosen occupied site increases very rapidly. In an infinite lattice near p_c, S follows the scaling law , with and hence diverges at p_c. Increasing p further, the probability P that a randomly chosen occupied site belongs to an infinitely large connected patch becomes larger than 0. The value of P follows a scaling law , with . With the connected patch excluded, the average size of the fragmented patches above p_c decreases following the same scaling law as that below p_c, .

While the existence of a sharp transition from fragmented to connected landscapes is already remarkable, it is even more surprising that the critical exponents β and γ are universal; that is, they are independent of the geometry of the lattice and the step length s. A square lattice has the same critical exponents as triangular, hexagonal, or even more complex lattices. We can even work in continuous space without referring to any underlying lattice (Meester and Roy 1996). The only factor determining the values of these exponents is the dimension of the system (Stauffer and Aharony 1994), which in landscape ecology is typically . Therefore, random maps possess some robust features that are insensitive to the local details of the model. (For ecological implications, see Oborny et al. 2007.)

The gradient random map (GRM) is a straightforward generalization of the URM for modeling spatial distributions along environmental gradients. We assume that the occupancy probability p(x) changes with the spatial position x along a gradient. The simplest assumption is a linear function, where g_p is a constant gradient (fig. 2A). The GRM was first introduced in statistical physics to model the diffusion of particles in a solid (Sapoval et al. 1985). In ecology, Milne et al. (1996) applied the GRM for studying the ecotone between grassland and piñon‐juniper woodlands in New Mexico. They interpreted the change from fragmented to connected woodland cover as the percolating phase transition of the GRM.

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Figure 2: Comparison of the three gradient models investigated in this article. A, Gradient random map (GRM). B, Gradient contact process with linear change in the colonization rate (GCP_c). C, Gradient contact process with linear change in the extinction rate (GCP_e). The highlighted curves in A–C are the hull edges for step length . Occupied sites (gray) are less aggregated on the random map A than in the contact processes B and C. D–F, Overall population density n and hull density h, averaged over 1,000 independent realizations, for the GRM (D), the GCP_c (E), and the GCP_e (F). Upper horizontal axis: local values of p, c, or e. Lower horizontal axis: spatial position x.

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To determine the precise position where the connected‐to‐fragmented transition occurs, it is necessary to identify the boundary, or hull edge, of the connected patch. The hull edge in the simplest case, , is defined as the set of all sites in the connected patch that are adjacent to exterior vacant sites (fig. 1B). These are vacant sites connected via nearest or next‐nearest neighbors to the largest contiguous vacant area around the connected patch. Metaphorically, the hull edge is the “coastline” where the connected patch is in contact with the surrounding “ocean” of vacant sites. Interior vacant “lakes” are disregarded. The hull edge can be delineated by a left‐turning biased walk along the coastline (see “Determination of the Hull Edge by a Biased Walk”). This method can also be generalized to define the hull edge for step lengths (fig. 1C).

A common feature between our work and the study by Milne et al. (1996) is the use of various step lengths ( and in Milne et al.; , 2, and 4 in our study). The results of Milne et al. clearly show a transition from a connected to a fragmented state with the decrease of woodland cover p. We develop this approach further by investigating the geometry of the hull edge for the GRM as well as other landscape models.

A Dynamic Model of Occupancy: The Contact Process

The occupation by a particular type of vegetation of a real landscape is autocorrelated, at least on some spatial scales, unlike vegetation on a random map. Several studies develop a null model by introducing a scale‐free, fractal structure (e.g., Palmer 1988; Milne 1990; Plotnick et al. 1993; With et al. 1997; With and King 1999; Olff and Ritchie 2002). An alternative method is to aggregate sites on one or more spatial scales (see a review in Keitt 2000). Most approaches are static, in the sense that the generated pattern does not change over time, but some dynamic methods have also been proposed. The simplest dynamic process that creates aggregation by repeated colonization and extinction events is the so‐called contact process.

Originally introduced to model contagion in epidemics (Harris 1974), the contact process has found applications in many other fields as a simple model of spatial spreading. In the context of ecology, the contact process is the simplest spatially explicit implementation of the logistic model of population growth and also of the Levins model of metapopulation growth (Levins 1969; see Dytham 2000 for a review of spatially explicit Levins models). Accordingly, several studies have used the contact process to investigate the spreading and persistence of (meta)populations in space (Barkham and Hance 1982; Crawley and May 1987; Anderson and May 1991; Durrett and Levin 1994; Levin and Durrett 1996; Holmes 1997; Franc 2004; Solé and Bascompte 2006; see Oborny et al. 2007 for a review).

In the contact process, two states of the sites are distinguished: occupied and vacant. An occupied site can colonize vacant sites in the neighborhood. The basic contact process, as introduced by Harris (1974), operates on a square lattice. The rates of colonization c and extinction e are the same in every site. Let us call this basic model without any gradient the uniform contact process (UCP). The rules described in figure 3 generate a straightforward, spatially explicit version of a logistic, or Levins, process. Although not analytically solvable, efficient numerical simulations are possible (see “Contact Process Algorithm”).

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Figure 3: Updating rules in the contact process. A, If a site is vacant, then we count the number of occupied neighbors k in its four‐site neighborhood. During a small time interval dt, the focal site becomes occupied with a probability . In other words, c is the rate at which one occupied site attempts to colonize any of its four neighboring sites. This neighborhood‐dependent rule expresses local density dependence of the colonization process. B, Occupied sites become vacant (i.e., extinction occurs) with a probability . The extinction process is assumed to be independent of the neighbors.

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To our knowledge, we are the first to apply the contact process as a model of broad‐scale vegetation dynamics on the landscape level. Since the contact process assumes spreading through neighborhood contacts, the only vegetation types that are suitable for this model are those in which the proximity of existing patches is important for the local community assembly, that is, in which long “jumps” are unlikely. Within this constraint, we are free to choose the spatial extent of a lattice site (i.e., the lattice constant) and the size of the neighborhood. Holland et al. (2007) have pointed out that some features of spatially explicit models can depend on the geometry of the neighborhood. However, Lennon et al. (1997) varied the neighborhood size in the uniform contact process (UCP), where c and e are the same for all sites, and concluded that several important features, such as the overall ratio of occupied to vacant sites, are rather insensitive to the neighborhood definition. Therefore, the arbitrary choice of a four‐site neighborhood should not make the results too specific: they are likely to be valid, at least qualitatively, for a broader set of models. For the sake of simplicity, we consider only colonization from a four‐site neighborhood (fig. 3; note that the organism that is assumed to move between the vegetation patches can step outside this neighborhood if ).

Because we are interested in vegetation transition zones, we modify the UCP by introducing an environmental gradient. In a gradient contact process (GCP), the colonization rate c and/or the extinction rate e changes with the spatial position x. For example, c(x) may vary linearly while e(x) is constant, or vice versa, where g_c and g_e are constant gradients. We refer to the model of equation (2) as GCP_c and that of equation (3) as GCP_e. Examples are shown in figures 2B and 2C, respectively.

Both rules have been applied in metapopulation ecology for the study of range limits of species (Holt and Keitt 2000). Lennon et al. (1997) applied somewhat more complex rules for colonization, testing various neighborhood sizes and shapes. Both studies showed a phenomenon of fundamental importance: a sharp boundary may emerge even across a smooth environmental gradient. A small change in c or e can cause occupancy to decline sharply in the vicinity of the range limit. In spite of this sharpness, the boundary is not frozen but fluctuates over time (fig. 5 of Lennon et al. 1997). The standard deviation of occupancy peaks around the region where the mean shows rapid decline (Holt and Keitt 2000; Antonovics et al. 2006). Results from Lennon et al. (1997) suggest that the shape of the gradient is relatively unimportant. Changing e(x) from a linear to an exponential function in equation (3) caused quantitative but not qualitative changes in the pattern of occupancy.

Our work is closely related to these studies, but we do not focus on the position x along the gradient where the vegetation vanishes. Instead, we examine the hull edge, that is, the boundary between connected and fragmented vegetation, where occupancy is still rather high. Our goal is to show that, although different models (GRM, GCP_c, GCP_e) may differ in their dynamics, spatial autocorrelations, and visual appearance (fig. 2), they have several scaling laws in common.

Numerical Simulations

We perform numerical simulations to study the GRM, GCP_c, and GCP_e models (according to eqq. [1], [2], and [3], respectively). To construct the GRM, we randomly fill all sites with the predetermined probabilities p(x). To produce the GCP patterns, we let the contact process run from an initial state until equilibration (see “Removing Transients and Finite‐Size Effects in the GCP” for details). Then we identify the connected patch and delineate the hull edge for step lengths , 2, and 4. We now present several variables to characterize the patterns.

Overall density profile. The overall density profile n(x) is the number of occupied sites in column x divided by L, the number of sites in that column. (In our simulations, we use .) Clearly, n(x) increases monotonically from unfavorable to favorable conditions. In the GRM, this change is linear (fig. 2D), according to equation (1). In the GCP_c and GCP_e, the emergence of the density profile is more complex. It results from dynamic colonization‐extinction processes and shows a rather sharp increase from to positive values of n (fig. 2E, 2F). The threshold where the transition from zero to positive density occurs is at in the GCP_c and in the GCP_e. Therefore, the rate of colonization must exceed the rate of extinction by ≈65% to enable survival. This is approximately the same value observed in the corresponding uniform contact process (Marro and Dickmann 1999).

Hull density profile. To locate the transition between connected and fragmented vegetation, we calculate the hull density h(x), the fraction of sites in column x belonging to the hull edge. Despite different overall density profiles n(x), h(x) is in all three models bell shaped and symmetric about the maximum (fig. 2D–2F, similar to the GRM results of Milne et al. 1996). There are several tasks in which a well‐defined place along the gradient has to be singled out as the “borderline” of the actual vegetation type. We propose to use the mean x‐coordinate for this purpose. It is located at rather high densities. For , for example, (GRM) or (GCP_c and GCP_e). The reason for the shift is the emergence of spatial autocorrelations in the GCPs. (Similar shifts have been observed in other correlated lattices by Mendelson [1997].) A shorter step size shifts the borderline toward a higher density. For example, in the GRM, increases from 0.09 to 0.59 as s decreases from 4 to 1. It is notable that the same value, , was found by Milne and colleagues in a similar GRM at . The same authors confirmed this theoretical prediction, using empirical data from a piñon‐juniper woodland edge in New Mexico (Milne et al. 1996).

Width and length of the hull. We define the width of the hull w as the standard deviation of the x‐coordinates around the mean . The length of the hull u is defined as the number of lattice sites visited by the left‐turning walk along the edge. We present the formal definitions of these variables and the detailed results in “Width and Length of the Hull” in the appendix. Here we summarize the main results. Larger gradients, that is, steeper slopes, or longer step lengths s lead to smaller w and u. We find that both variables scale as power laws, and , where g stands for the gradient in the respective vegetation model (g_p, g_c, or g_e). Interestingly, the exponents depend on neither s nor the particular model, although occupied sites in the GCPs are aggregated, unlike in the GRM (see “Aggregation of the Occupied Sites in the GCPs”). The result suggests that the GCP’s autocorrelations do not have any significant effect on these fundamental geometric features of the hull edge.

Fractal dimension. We calculate the fractal dimension D_f of the hull edge with the equipaced polygon method (Batty and Longley 1994; a detailed description of this method is given in “Fractal Dimension and the Equipaced Polygon Method”). The results (fig. 4) suggest the validity of scaling laws , where is the average distance between k steps in the left‐turning walk. Two different scaling regimes can be distinguished. For small k, the value of D_f fits (least squares fit for ); at larger values, there is a crossover to . This indicates that the hull edge is a fractal at small length scales and becomes similar to a simple straight line when viewed in coarser resolution. It is remarkable that the fractal dimension is approximately , and this is the same at every value of step length s and in every model (GRM, GCP_c, and GCP_e). The value has been conjectured in earlier theoretical studies for the particular case of and GRM (Sapoval et al. 1985). Recent mathematical work supports this conjecture (Smirnov and Werner 2001). Our results lead to the surprising conclusion that this value is not specific to the GRM or , and we may expect universal scaling behavior at the hull edge, independent of the exact interactions between occupied and vacant sites.

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Figure 4: Fractal dimension of the hull edge. A, To determine the fractal dimension, we calculate the average of ( ), the distances between k steps in the left‐turning walk for a fixed number k. In A, . B–D, The average distance versus k on double‐logarithmic axes. For small k, scales as the four‐sevenths power of k, indicating a fractal dimension . For large k, there is a crossover to . The gradients used for the figure are . The specific choice of the gradients influences the crossover positions between the two different scaling regimes, but the exponents remain the same for other gradients. (Each data point is the average over 10 realizations. The error bars are smaller than the symbols.)

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Application to Empirical Data

As an example of the hull edge of an empirical vegetation transition, we investigate the boundary of a piñon‐juniper woodland on a slope of the Sandia Mountains in central New Mexico (following Milne et al. [1996], who had described the hull edge of the same vegetation type in this region). We used a high‐resolution Google Earth image (fig. 5A) to obtain a binary matrix ( ; fig. 5B), in which woody and nonwoody vegetation are distinguished. The largest (connected) woodland patch was identified, and its hull edge was outlined and analyzed by the method described for the simulated data. (See “Background Information about Figure 5A, 5B” for a detailed description of the site and of the image analysis.)

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Figure 5: Edge of a piñon‐juniper woodland along a slope in the Sandia Mountains (Cibola National Forest, New Mexico). A, Satellite image. B, Processed binary image. One pixel is around . The hull edge for is marked in black. C, Overall density n and hull density h as functions of position in the north‐south direction. D, Estimation of the fractal dimension with step lengths , 2, or 3. The notation is the same as in figure 4.

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The results show remarkable similarities to the simulations. The overall density profile is half‐bell shaped and suggests a rather steep slope, where n increases from low ( ) to high densities ( ) over less than 200 m (fig. 5C). The density of occupancy at the borderline, , is approximately 0.48 at and 0.60 at . The latter is in agreement with the result of Milne et al. (1996) and with the theoretical value of the percolation threshold in uniform random maps at (0.592746; Stauffer and Aharony 1994).

The fractal geometry of the hull in the Sandia Mountains example also shows similarities to our models. Over more than one order of magnitude, one can plausibly fit a power law, (fig. 5D). Thus, at length scales of around 10–100 m, we confirm the fractal dimension that we found in our simulations. There is also a clear indication of the predicted crossover to linear scaling at large step lengths.

Discussion

Applicability of the Model and Further Questions

In order to observe an edge pattern emerging from colonization‐extinction events, a sufficiently large area is needed relative to the distance of spreading. For example, if the vegetation can spread maximally 1 m per year under ideal conditions and the area over which the environment changes is 500 m, causing a decline from high to zero vegetation cover, then the available system size is 500 sites (lattice cells) in the x direction. This can be sufficiently large for our approach. In the transversal (y) direction, the lattice should be large enough for the estimation of the fractal dimension. The size and orientation of the lattice have to be chosen such that the hull edge spans from to the maximum y‐coordinate. As a rough guideline, the length should be at least 200 sites. We do not recommend testing the predictions on small (e.g., ) lattices, because the geometric features of the hull are unlikely to be observable.

Our models are based on several simplifying assumptions. As we have pointed out, the values of the critical exponents in the scaling laws, including the fractal dimension, are expected to be robust against many kinds of modifications in the model. We list some potential factors that may change the values of the exponents or cause the scaling laws to break down.

Nonlinear changes in the environment. The environmental conditions were assumed to change linearly (eqq. [1]–[3]). This assumption rarely holds in reality, but if the change in the environmental conditions is smooth, a linear approximation may be applicable. Because the same scaling exponents are found for a linear density profile (GRM) and for two nonlinear density profiles (GCP_c and GCP_e), we believe that linearity in the environment or in the population’s response is not mandatory.

Colonization distance. In our model, we assume that the vegetation spreads only to the nearest sites. The model can be readily extended by assuming that farther sites can be colonized as well, as long as the colonized neighborhood is finite or the probability of colonization decreases sufficiently fast with distance. This assumption is realistic because at least some component species are likely to be dispersal limited. If, however, self‐assembly of the vegetation can take place at an arbitrary distance from existing vegetation patches, the GCP is not applicable. A modified GCP, where colonization at a randomly chosen site is attempted with rate q, interpolates between the GCP_e for and a GRM for and .

Inhomogeneities. Another concern about the applicability of the model is the occurrence of an inhomogeneity in the environment in addition to the gradient already present. This is likely to occur in almost all natural landscapes; for example, depth and fertility of the soil may vary in space. According to investigations in the UCP, many properties of the contact process remain unchanged by background heterogeneity if the heterogeneous pattern is fine‐grained in space and can fluctuate randomly over time, even if the frequency is very low (Szabó et al. 2002). But so‐called quenched disorder (i.e., a static heterogeneity) influences the behavior of the system fundamentally, altering, for example, the time dependence of occupancy (Dickman and Moreira 1998; Szabó et al. 2002). We do not know of any study investigating the GCP with a heterogeneous background. From the existing studies on the UCP we hypothesize that heterogeneity in the environmental background would not influence the model’s predictions unless the sizes of vegetation patches are smaller than the correlation length of the heterogeneity.

Nonlinear density dependence. In both the UCP and the GCP, the probability of colonizing a vacant site is proportional to the number of occupied sites in its neighborhood (fig. 3). This assumption appears to be realistic: the higher the cover of that vegetation type in the neighborhood, the higher, proportionally, is the probability of occupying the vacant site. Nevertheless, this assumption does not necessarily hold in all situations. Nonlinear density dependence may fundamentally change the behavior of the system, destroying the scaling relations completely or introducing new values for the exponents (see a review in Ódor 2004). This area is largely unknown, especially for gradients.

Memory effects. Feedback from the state of vegetation (empty vs. occupied) to the state of environment is another promising field of research. For example, a vacant site that has been occupied before can be better colonized because of soil formation, or autotoxicity by allelopathic compounds may increase the probability of extinction after some time of occupancy. Such memory effects may have consequences for the dynamics and the scaling laws.

Multiple vegetation types. Our model is a “single‐species” approach, in the sense that only one vegetation type is considered and interactions between vegetation types are disregarded. This simplification may be judicious where the vegetation type under study meets bare ground (e.g., mud at a lakeshore) or dominates strongly over the alternative vegetation type. For example, an altitudinal or latitudinal tree line may be assumed (at least, in some situations) to spread until the living conditions are suitable for the trees. Predictive models about the distribution of forest vegetation often disregard the effect of the alternative, nonforest vegetation. In many cases, however, the formation of the edge may be better approximated by a “two‐species” model: a struggle zone is formed by an interaction between two vegetation types. Since the hull edge is at rather high densities (fig. 2), it is possible that the struggle zone is farther away from the hull edge, so that the existence of an alternative vegetation type does not influence predictions about scaling laws. But this condition must be checked in every two‐species case. The end of the struggle zone (i.e., the farthest point at which the alternative vegetation type occurs) must not reach up to the hull edge; otherwise, both species have to be modeled simultaneously.

Conclusion

In summary, we describe and analyze two neutral models: a simple one with no spatial correlation (GRM) and a more complex one with spatial correlation (GCP). Both models are neutral, in the sense that either no interaction between the sites is assumed (in the GRM) or the interactions are assumed to be the most parsimonious (in the GCP). Both models predict a fractal dimension for the hull edge and additional scaling laws for its length and width. In spite of the simplicity of these models, the scaling laws appear to be independent of the local details of the dynamics. The meaning of “local” depends on the characteristic length and time scales (correlation length and relaxation time). An example of a woodland edge in the Sandia Mountains, New Mexico, shows the applicability of the model to field data.