Plant size and spatial pattern in a natural population of Myosotis micrantha

Abstract. We examined spatial distributions and plant sizes along a transect through a natural population of a winter annual, Myosotis micrantha. A size hierarchy existed, as indicated by high values of Gini coefficients of inequality for plant mass and correlated measures. Plants with no immediate conspecific neighbors were larger than plants with one or more near neighbors, suggesting that competition from near neighbors depressed plant size. However, there was strong positive spatial autocorrelation in plant size: large plants were associated with large neighbors and small ones with small neighbors. Plant size was also positively correlated with the combined biomass of near neighbors. The population formed a two-phase mosaic of patches of relatively large plants alternating with patches of smaller plants. The data suggest that individual plants compete with conspecifics, but the effects of competition are symmetrical. The most likely explanations for this spatially structured size hierarchy are variation in plant density, patchy distribution of resources, or a combination of the two.     

Multiple site dissimilarity quantifies compositional heterogeneity among several sites,while average pairwise dissimilarity may be misleading

Several measures of multiple site dissimilarity have been proposed to quantify the overall heterogeneity in assemblage composition among any number of sites. It is also a common practice to quantify such overall heterogeneity by averaging pairwise dissimilarities between all pairs of sites in the pool. However, pairwise dissimilarities do not account for patterns of co‐occurrence among more than two sites. In consequence, the average of pairwise dissimilarities may not accurately reflect the overall compositional heterogeneity within a pool of more than two sites. Here I use several idealized examples to illustrate why pairwise dissimilarity measures fail to properly quantify overall heterogeneity. Thereafter, the effect of this potential problem in empirical patterns is exemplified with data of world amphibians. In conclusion, when the attribute of interest is the overall heterogeneity in a pool of sites (i.e. beta diversity) or its turnover or nestedness components, only multiple site dissimilarity measures are recommended.     

Spind: a package for computing spatially corrected accuracy measures

Using an appropriate accuracy measure is essential for assessing prediction accuracy in species distribution modelling. Therefore, model evaluation as an analytical uncertainty is a challenging problem. Although a variety of accuracy measures for the assessment of prediction errors in presence/absence models is available, there is a lack of spatial accuracy measures, i.e. measures that are sensitive to the spatial arrangement of the predictions. We present ‘spind’, a new software package (based on the R software program) that provides spatial performance measures for grid‐based models. These accuracy measures are generalized, spatially corrected versions of the classical ones, thus enabling comparisons between them. Our method for evaluation consists of the following steps: 1) incorporate additional autocorrelation until spatial autocorrelation in predictions and actuals is balanced, 2) cross‐classify predictions and adjusted actuals in a 4 × 4 contingency table, 3) use a refined weighting pattern for errors, and 4) calculate weighted Kappa, sensitivity, specificity and subsequently ROC, AUC, TSS to get spatially corrected indices. To illustrate the impact of our spatial method we present an example of simulated data as well as an example of presence/absence data of the plant species Dianthus carthusianorum across Germany. Our analysis includes a statistic for the comparison of spatial and classical (non‐spatial) indices. We find that our spatial indices tend to result in higher values than classical ones. These differences are statistically significant at medium and high autocorrelation levels. We conclude that these spatial accuracy measures may contribute to evaluate prediction errors in presence/absence models, especially in case of medium or high degree of similarity of adjacent data, i.e. aggregated (clumped) or continuous species distributions.     

A Computer Programme Package for the Statistical Analysis of Spatial Point Processes in a Square

A computer programme for the statistical analysis of point data in a square is described. Several tests for randomness of the distribution of points are possible. The most comprehensive of these are comparisons of the empirical distributions of the inter-point and closest neighbour distances with their respective expected distributions under complete randomness, and tests based on Ripley' L function; using these, significant aggregation or regularity can be identified. It is also possible to calculate statistics of properties (“attributes”) associated with each spatial point, as well as to compare statistics for sub-areas of the experimental square. Several measures of spatial autocorrelation are available, amongst them correlograms and variograms. The programme can also find the tesselation of the study area and correlate tile properties with the point attributes. The procedures are illustrated by references to the spatial distribution and mound heights of Trinevitermes trinervoides on a study area in South Africa. Although the programme was developed specifically for application in entomology, it could be used to analyse data from many other disciplines.     

Multilocus estimation of genetic structure within populations

Spatial structure of genetic variation within populations is well measured by statistics based on the distribution of pairs of individual genotypes, and various such statistics have been widely used in experimental studies. However, the problem of uncharacterized correlations among statistics for different alleles has limited the applications of multiallelic, multilocus summary measures, since these had unknown sampling distributions. Usually multiple alleles and/or multiple loci are required in order to precisely measure spatial structures, and to provide precise indirect estimates of the amount of dispersal in samples of reasonable size. This article examines the correlations among pair-wise statistics, including Moran I-statistics and various measures of conditional kinship, for different alleles of a locus. First the correlations are mathematically derived for random spatial distributions, which allow averages over alleles and loci to be used as more powerful yet exact test statistics for the null hypothesis. Then extensive computer simulations are conducted to examine the correlations among values for different alleles under isolation by distance processes. For loci with more than three alleles, the results show that the correlations are remarkably and perhaps surprisingly small, establishing the principle that then alleles behave as nearly independent realizations of space-time stochastic processes. The results also show that the correlations are largely robust with respect to the degree of spatial structure, and they can be used in a straightforward manner to form confidence intervals for averages. The results allow a precise connection between observations in experimental studies and levels of dispersal in theoretical models.     

放牧和刈割条件下草山草坡群落空间异质性分析

采用变异矩分析和分形方法,研究了草山草坡群落在放牧和刈割条件下的空间异质性及空间自相关性,结果表明,群落空间格局有尺度依赖性,刈割条件下空物异质性及空间相关性弱,多样性梯度即β多样性小,放牧消除地形引起的样地差异,因而使空间异质性简单化。     

Markov Random Field Modeling of the Spatial Distribution of Proteins on Cell Membranes

Cell membranes display a range of receptors that bind ligands and activate signaling pathways. Signaling is characterized by dramatic changes in membrane molecular topography, including the co-clustering of receptors with signaling molecules and the segregation of other signaling molecules away from receptors. Electron microscopy of immunogold-labeled membranes is a critical technique to generate topographical information at the 5–10 nm resolution needed to understand how signaling complexes assemble and function. However, due to experimental limitations, only two molecular species can usually be labeled at a time. A formidable challenge is to integrate experimental data across multiple experiments where there are from 10 to 100 different proteins and lipids of interest and only the positions of two species can be observed simultaneously. As a solution, we propose the use of Markov random field (MRF) modeling to reconstruct the distribution of multiple cell membrane constituents from pair-wise data sets. MRFs are a powerful mathematical formalism for modeling correlations between states associated with neighboring sites in spatial lattices. The presence or absence of a protein of a specific type at a point on the cell membrane is a state. Since only two protein types can be observed, i.e., those bound to particles, and the rest cannot be observed, the problem is one of deducing the conditional distribution of a MRF with unobservable (hidden) states. Here, we develop a multiscale MRF model and use mathematical programming techniques to infer the conditional distribution of a MRF for proteins of three types from observations showing the spatial relationships between only two types. Application to synthesized data shows that the spatial distributions of three proteins can be reliably estimated. Application to experimental data provides the first maps of the spatial relationship between groups of three different signaling molecules. The work is an important step toward a more complete understanding of membrane spatial organization and dynamics during signaling.     

Non-stationarity and local approaches to modelling the distributions of wildlife

Despite a growing interest in species distribution modelling, relatively little attention has been paid to spatial autocorrelation and non-stationarity. Both spatial autocorrelation (the tendency for adjacent locations to be more similar than distant ones) and non-stationarity (the variation in modelled relationships over space) are likely to be common properties of ecological systems. This paper focuses on non-stationarity and uses two local techniques, geographically weighted regression (GWR) and varying coefficient modelling (VCM), to assess its impact on model predictions. We extend two published studies, one on the presence–absence of calandra larks in Spain and the other on bird species richness in Britain, to compare GWR and VCM with the more usual global generalized linear modelling (GLM) and generalized additive modelling (GAM). For the calandra lark data, GWR and VCM produced better-fitting models than GLM or GAM. VCM in particular gave significantly reduced spatial autocorrelation in the model residuals. GWR showed that individual predictors became stationary at different spatial scales, indicating that distributions are influenced by ecological processes operating over multiple scales. VCM was able to predict occurrence accurately on independent data from the same geographical area as the training data but not beyond, whereas the GAM produced good results on all areas. Individual predictions from the local methods often differed substantially from the global models. For the species richness data, VCM and GWR produced far better predictions than ordinary regression. Our analyses suggest that modellers interpolating data to produce maps for practical actions (e.g. conservation) should consider local methods, whereas they should not be used for extrapolation to new areas. We argue that local methods are complementary to global methods, revealing details of habitat associations and data properties which global methods average out and miss.     

Does landscape composition alter the spatiotemporal distribution of the pine processionary moth in a pine plantation forest?

Landscape composition and physiognomy affect community structure and species distribution across space and time. The pine processionary moth (PPM) (Thaumetopoea pityocampa Den. & Schiff., Lepidoptera, Notodontidae) is a common pine defoliator throughout southern Europe and Mediterranean countries. We surveyed the spatiotemporal distribution of the PPM in a pine plantation forest in southwestern France and used the density of the winter nests as a proxy for population density. The study spanned 4 years (2005–2008) and showed a high temporal variability in nest density. We found a strong edge effect with nest densities at stand edges more than twice as large as within-stand densities. At the landscape scale, the spatial distribution of the moth exhibited a significant spatial autocorrelation in 3 out of 4 years of our study. The spatial scales of the autocorrelation ranged from ca. 2 km to more than 22 km. We found a positive correlation between spatial distributions corresponding to certain sampling years, but the relationship was not systematic. Landscape configuration appeared to be an important driver of the PPM spatial pattern. Bivariate Moran’s I correlograms showed that patch richness density as well as the percentage of local landscape covered by various land uses were correlated with population density. The study showed that accounting for landscape characteristics may be important in order to understand forest insect pest distribution, even in cases where the host species is abundant and homogeneously distributed throughout the study area, e.g., pure plantation forests.     

Competing populations on fragmented landscapes with spatially structured heterogeneities: improved landscape generation and mixed dispersal strategies

Interactions between two species competing for space were studied using stochastic spatially explicit lattice-based simulations as well as pair approximations. The two species differed only in their dispersal strategies, which were characterized by the proportion of reproductive effort allocated to long-distance (far) dispersal versus short-distance (near) dispersal to adjacent sites. All population dynamics took place on landscapes with spatially clustered distributions of suitable habitat, described by two parameters specifying the amount and the local spatial autocorrelation of suitable habitat. Whereas previous results indicated that coexistence between pure near and far dispersers was very rare, taking place over only a very small region of the landscape parameter space, when mixed strategies are allowed, multiple strategies can coexist over a much wider variety of landscapes. On such spatially structured landscapes, the populations can partition the habitat according to local conditions, with one species using pure near dispersal to exploit large contiguous patches of suitable habitat, and another species using mixed dispersal to colonize isolated smaller patches (via far dispersal) and then rapidly exploit those patches (via near dispersal). An improved mean-field approximation which incorporates the spatially clustered habitat distribution is developed for modeling a single species on these landscapes, along with an improved Monte Carlo algorithm for generating spatially clustered habitat distributions.      

Spatial autocorrelation and dispersal in mistletoes: field and simulation results

The infections of two species of mistletoes in Baja California, Mexico were investigated for spatial patterns of abundance, and for an effect of the dispersal patterns of mistletoe seeds on these spatial patterns. Mistletoe distributions were mapped and the dispersal of mistletoe seeds was observed. Most mistletoes seeds were dispersed locally to the parent tree or to nearby trees. While mistletoe distributions were highly clumped at the level of the individual tree, no spatial pattern was found above the scale of the individual tree. Infected trees were no more clumped than the overall host population, and infected trees had no more mistletoes on nearby surrounding trees than did uninfected trees. Trees showed no spatial autocorrelation in the number of mistletoes they supported. Simulations using a spatially explicit simulation model with local dispersal and stochasticity in seed dispersal, host mortality, and mistletoe mortality were used to interpret the field results. Simulation results suggest that dispersal patterns affect the overall level of variance in the number of mistletoes per tree but do not lead to spatial patterns in abundance above the scale of the tree. Thus, both simulation and field systems give the surprising result that local dispersal does not lead to spatial autocorrelation in the numbers of mistletoes per tree.Abbreviations AI = Arroyo Inspiracion - VSR = Valle San Rafael     

Multiple regression on distance matrices: a multivariate spatial analysis tool

I explore the use of multiple regression on distance matrices (MRM), an extension of partial Mantel analysis, in spatial analysis of ecological data. MRM involves a multiple regression of a response matrix on any number of explanatory matrices, where each matrix contains distances or similarities (in terms of ecological, spatial, or other attributes) between all pair-wise combinations of n objects (sample units); tests of statistical significance are performed by permutation. The method is flexible in terms of the types of data that may be analyzed (counts, presence–absence, continuous, categorical) and the shapes of response curves. MRM offers several advantages over traditional partial Mantel analysis: (1) separating environmental distances into distinct distance matrices allows inferences to be made at the level of individual variables; (2) nonparametric or nonlinear multiple regression methods may be employed; and (3) spatial autocorrelation may be quantified and tested at different spatial scales using a series of lag matrices, each representing a geographic distance class. The MRM lag matrices model may be parameterized to yield very similar inferences regarding spatial autocorrelation as the Mantel correlogram. Unlike the correlogram, however, the lag matrices model may also include environmental distance matrices, so that spatial patterns in species abundance distances (community similarity) may be quantified while controlling for the environmental similarity between sites. Examples of spatial analyses with MRM are presented.     

Colonization rules and spatial distributions in ecology

Because they are intuitive and mathematically straight-forward, colonization rules are often used to model spatial patterns in ecology. Colonization rules assign individuals to categories according to the locations of previous colonists. In this note, a compact introduction to colonization rules in ecology is presented with implications for autocorrelation and spatial distributions. I use the colonization rule approach to unify a diverse set of spatial and species diversity analyses, exploring future extensions to incorporate greater realism.     

Evaluating,partitioning, and mapping the spatial autocorrelation component in ecological niche modeling: a new approach based on environmentally equidistant records

Most species data display spatial autocorrelation that can affect ecological niche models (ENMs) accuracy‐statistics, affecting its ability to infer geographic distributions. Here we evaluate whether the spatial autocorrelation underlying species data affects accuracy‐statistics and map the uncertainties due to spatial autocorrelation effects on species range predictions under past and future climate models. As an example, ENMs were fitted to Qualea grandiflora (Vochysiaceae), a widely distributed plant from Brazilian Cerrado. We corrected for spatial autocorrelation in ENMs by selecting sampling sites equidistant in geographical (GEO) and environmental (ENV) spaces. Distributions were modelled using 13 ENMs evaluated by two accuracy‐statistics (TSS and AUC), which were compared with uncorrected ENMs. Null models and the similarity statistics I were used to evaluate the effects of spatial autocorrelation. Moreover, we applied a hierarchical ANOVA to partition and map the uncertainties from the time (across last glacial maximum, pre‐insustrial, and 2080 time periods) and methodological components (ENMs and autocorrelation corrections). The GEO and ENV models had the highest accuracy‐statistics values, although only the ENV model had values higher than expected by chance alone for most of the 13 ENMs. Uncertainties from time component were higher in the core region of the Brazilian Cerrado where Q. grandiflora occurs, whereas methodological components presented higher uncertainties in the extreme northern and southern regions of South America (i.e. outside of Brazilian Cerrado). Our findings show that accounting for autocorrelation in environmental space is more efficient than doing so in geographical space. Methodological uncertainties were concentrated in outside the core region of Q. grandiflora's habitat. Conversely, uncertainty due to time component in the Brazilian Cerrado reveals that ENMs were able to capture climate change effects on Q. grandiflora distributions.     

Multi-scale variation in spatial heterogeneity for microbial community structure in an eastern Virginia agricultural field

To better understand the distribution of soil microbial communities at multiple spatial scales, a survey was conducted to examine the spatial organization of community structure in a wheat field in eastern Virginia (USA). Nearly 200 soil samples were collected at a variety of separation distances ranging from 2.5 cm to 11 m. Whole-community DNA was extracted from each sample, and community structure was compared using amplified fragment length polymorphism (AFLP) DNA fingerprinting. Relative similarity was calculated between each pair of samples and compared using geostatistical variogram analysis to study autocorrelation as a function of separation distance. Spatial autocorrelation was found at scales ranging from 30 cm to more than 6 m, depending on the sampling extent considered. In some locations, up to four different correlation length scales were detected. The presence of nested scales of variability suggests that the environmental factors regulating the development of the communities in this soil may operate at different scales. Kriging was used to generate maps of the spatial organization of communities across the plot, and the results demonstrated that bacterial distributions can be highly structured, even within a habitat that appears relatively homogeneous at the plot and field scale. Different subsets of the microbial community were distributed differently across the plot, and this is thought to be due to the variable response of individual populations to spatial heterogeneity associated with soil properties.     

Do vegetation patch spatial patterns disrupt the spatial organization of plant species?

The mechanisms that structure plant diversity and generate long-range correlated spatial patterns have important implications for the conservation of fragmented landscapes. The ability to disperse and persist influences a plant species’ capacity for spatial organization, which can play a critical role in structuring plant diversity in metacommunities. This study examined the spatial patterns of species diversity within a network of patches in Cabo de Gata Natural Park, southeastern Spain. The objectives were to understand how the spatial heterogeneity of species composition (beta diversity) varies in a structured landscape, and how the long-range spatial autocorrelation of plant species is affected by the spatial configuration of patches.The mechanisms underlying the spatial distribution of plants acted at two scales. Between patches, spatial variation in species distributions was greater than that expected based on spatial randomization, which indicated that movement among patches was restricted. Within patches, diffusion processes reduced spatial variability in species distributions, and the effect was more prominent in large patches. Small patch size negatively influenced the long-range spatial autocorrelation of characteristic species, whereas inter-patch distance had a stronger effect on species frequency than it had on the disruption of spatial organized patterns.The long-range spatial autocorrelation was evaluated based on the dispersal abilities of the species. Among the 106 species evaluated, 39% of the woody species, 17% of the forbs, and 12% of the grasses exhibited disrupted long-range spatial autocorrelation where patches were small. The species that are more vulnerable to the effects of fragmentation tended to be those that have restricted dispersal, such as those that have short-range dispersal (atelechoric), e.g., Phlomis purpurea, Cistus albidus, Teucrium pseudochamaepytis, Brachypodium retusum, and the ballistic species, Genista spartioides. Helianthemum almeriense is another vulnerable species that has actively restricted dispersal (antitelechory), which is common in arid regions. Wind dispersers such as Launaea lanifera were less vulnerable to the effects of fragmentation. Long-distance dispersers whose persistence depends on facilitative interactions with other individuals, e.g., allogamous species such as Thymus hyemalis, Ballota hirsuta, and Anthyllis cytisoides, exhibit disrupted long-range spatial autocorrelation when patch size is reduced.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.     

Combining spatial and phylogenetic eigenvector filtering in trait analysis

Aim To analyse the effects of simultaneously using spatial and phylogenetic information in removing spatial autocorrelation of residuals within a multiple regression framework of trait analysis. Location Switzerland, Europe. Methods We used an eigenvector filtering approach to analyse the relationship between spatial distribution of a trait (flowering phenology) and environmental covariates in a multiple regression framework. Eigenvector filters were calculated from ordinations of distance matrices. Distance matrices were either based on pure spatial information, pure phylogenetic information or spatially structured phylogenetic information. In the multiple regression, those filters were selected which best reduced Moran's I coefficient of residual autocorrelation. These were added as covariates to a regression model of environmental variables explaining trait distribution. Results The simultaneous provision of spatial and phylogenetic information was effectively able to remove residual autocorrelation in the analysis. Adding phylogenetic information was superior to adding purely spatial information. Applying filters showed altered results, i.e. different environmental predictors were seen to be significant. Nevertheless, mean annual temperature and calcareous substrate remained the most important predictors to explain the onset of flowering in Switzerland; namely, the warmer the temperature and the more calcareous the substrate, the earlier the onset of flowering. A sequential approach, i.e. first removing the phylogenetic signal from traits and then applying a spatial analysis, did not provide more information or yield less autocorrelation than simple or purely spatial models. Main conclusions The combination of spatial and spatio‐phylogenetic information is recommended in the analysis of trait distribution data in a multiple regression framework. This approach is an efficient means for reducing residual autocorrelation and for testing the robustness of results, including the indication of incomplete parameterizations, and can facilitate ecological interpretation.     

Finding spatial regularity in mosaic landscapes: two methods integrated

Two methods were employed to find spatial regularity in a complicated mountain landscape of Beijing, China on the basis of functional and structural affinities. The first approach applied Affinity Analysis based on species composition to landscape. The mosaic diversity of the landscape was 3.5298>3, which means the study landscape is complex and controlled by multiple environmental gradients. These landscape types were divided into 3 parts according to the mean affinity values of 0.2143 and 0.7857 (0.5±1 SD). Modal sites are the central types of the landscape, which include a zonal broad-leaved forest of the region and a conifer plantation replacing the former. Outliers are found in the highest altitude and the lowest, both have few species in common with the above two modal types. The remaining landscape types are intermediate sites, which are transitional between modals and outliers, broadly distributed throughout mountain environments. Neighbor types have more species in common than those more widely separated, which probably distributed adjacently in space or in similar quality habitat. The other method employed is the new TWINSPAN analysis by substituting spatial neighboring data of landscape types for species composition data. It clearly divided the landscape types into three groups, i.e., subalpine, middle and low mountain groups, which were correlated with altitude, as well as influenced by human disturbance. The new TWINSPAN classification method is more reliable in finding spatial gradient of patchy landscapes than affinity analysis; however, affinity analysis is useful in finding species diversity pattern and the importance of landscape types in a region. Integrating advantages of the two methods could supply complete and reliable information on how landscape types are distributed in space, which environmental gradient dominates the spatial distribution of the landscape types, as well as where important and unusual types are located.     

A geostatistical analysis of small-scale spatial variability in bacterial abundance and community structure in salt marsh creek bank sediments

Small-scale variations in bacterial abundance and community structure were examined in salt marsh sediments from Virginia's eastern shore. Samples were collected at 5 cm intervals (horizontally) along a 50 cm elevation gradient, over a 215 cm horizontal transect. For each sample, bacterial abundance was determined using acridine orange direct counts and community structure was analyzed using randomly amplified polymorphic DNA fingerprinting of whole-community DNA extracts. A geostatistical analysis was used to determine the degree of spatial autocorrelation among the samples, for each variable and each direction (horizontal and vertical). The proportion of variance in bacterial abundance that could be accounted for by the spatial model was quite high (vertical: 60%, horizontal: 73%); significant autocorrelation was found among samples separated by 25 cm in the vertical direction and up to 115 cm horizontally. In contrast, most of the variability in community structure was not accounted for by simply considering the spatial separation of samples (vertical: 11%, horizontal: 22%), and must reflect variability from other parameters (e.g., variation at other spatial scales, experimental error, or environmental heterogeneity). Microbial community patch size based upon overall similarity in community structure varied between 17 cm (vertical) and 35 cm (horizontal). Overall, variability due to horizontal position (distance from the creek bank) was much smaller than that due to vertical position (elevation) for both community properties assayed. This suggests that processes more correlated with elevation (e.g., drainage and redox potential) vary at a smaller scale (therefore producing smaller patch sizes) than processes controlled by distance from the creek bank.