Spatial point pattern analysis of available and exploited resources

A patchy spatial distribution of resources underpins many models of population regulation and species coexistence, so ecologists require methods to analyse spatially‐explicit data of resource distribution and use. We describe a method for analysing maps of resources and testing hypotheses about species' distributions and selectivity. The method uses point pattern analysis based on the L‐function, the linearised form of Ripley's K‐function. Monte Carlo permutations are used for statistical tests. We estimate the difference between observed and expected values of L(t), an approach with several advantages: 1) The results are easy to interpret ecologically. 2) It obviates the need for edge correction, which has largely precluded the use of L‐functions where plot boundaries are “real”. Including edge corrections may lead to erroneous conclusions if the underlying assumptions are invalid. 3) The null expectation can take many forms, we illustrate two models: complete spatial randomness (to describe the spatial pattern of resources in the landscape) and the underlying pattern of resource patches in the landscape (akin to a neutral landscape approach). The second null is particularly useful to test whether spatial patterns of exploited resource points simply reflect the spatial patterns of all resource points. We tested this method using over 100 simulated point patterns encompassing a range of patterns that might occur in ecological systems, and some very extreme patterns. The approach is generally robust, but Type II decision errors might arise where spatial patterns are weak and when trying to detect a clumped pattern of exploited points against a clumped pattern of all points. An empirical example of an intertidal lichen growing on barnacle shells illustrates how this technique might be used to test hypotheses about dispersal mechanisms. This approach can increase the value of survey data, by permitting quantification of natural resource patch distribution in the landscape as well as patterns of resource use by species.     

Ripley''s K(d)函数分析种群空间分布格局的边缘校正

Ripley’sK(d)函数是分析种群空间分布格局最常用的方法,边缘校正是此方法的关键问题。传统的边缘校正包括3种情形：(i)圆完全包含在样地内;(ii)圆与样地的一条边相交;(iii)圆与样地的两条边相交,并有2个或3个交点。实际上,还存在第(iv)种情形,即圆与样地两条边相交有4个交点的情形。本文从数学上证明,传统边缘校正中,当边缘校正属于第(iv)种情形时,仍按第(iii)种情形进行校正,必然导致权重估计偏大,L(d)值偏高,从而影响结果分析。采用我国东北长白山天然云冷杉林样地资料验证了此结论。结果表明,天然云冷杉林中,3个优势种群林木分布格局特点是：鱼鳞云杉和臭冷杉的分布格局曲线非常相近,而与椴木的差异较大。原因是鱼鳞云杉和臭冷杉具有相似的生态学特性。椴木在小尺度显著聚集,因为椴木常在林隙更新。相比较而言,建群种云冷杉分布的均匀程度要比椴木高。     

An analysis of the vegetation pattern in a semi-arid Eucalyptus populnea woodland in north-west New South Wales

Shrub invasion of Eucalyptus populnea woodland is of concern to the pastoral industry. As part of an ecosystem study, different analyses were used to identify spatial pattern and associations in the tree and shrub species, with the aim of identifying what factors were most influential on the ecosystem. The results suggested that there were no strong edaphic or topographic influences on the distribution of plants. The principal influence on the arrangement of trees and shrubs was the large Eucalyptus trees, which were randomly arranged, and the canopy of which covered 15% of the area. It is concluded that any disturbance of the large Eucalyptus trees would have intense effects on the whole ecosystem. Qualitative and quantitative measurements of the plant species in sample quadrats were subjected to a variety of pattern analyses including association analysis (DIVINF), correlation analysis, principal components analysis and several classification programs including POLYDIV and MULCLAS. A criticism of such analyses is that positive results are both inevitable and not subject to statistical proof, whilst their strength is that they can simplify complicated sets of data. In this case the patterns suggested by the results were visually apparent and no additional insights were achieved. The dangers inherent in relying on a single pattern analysis were revealed when the groups of quadrats produced by the different analyses were mapped. Even when the groupings derived from different analyses were based on a similar species content, their spatial arrangement was dissimilar.     

Rings, circles, and null-models for point pattern analysis in ecology

A large number of methods for the analysis of point pattern data have been developed in a wide range of scientific fields. First-order statistics describe large-scale variation in the intensity of points in a study region, whereas second-order characteristics are summary statistics of all point-to-point distances in a mapped area and offer the potential for detecting both different types and scales of patterns. Second-order analysis based on Ripley's K-function is increasingly used in ecology to characterize spatial patterns and to develop hypothesis on underlying processes; however, the full range of available methods has seldomly been applied by ecologists. The aim of this paper is to provide guidance to ecologists with limited experience in second-order analysis to help in the choice of appropriate methods and to point to practical difficulties and pitfalls. We review (1) methods for analytical and numerical implementation of two complementary second-order statistics, Ripley's K and the O-ring statistic, (2) methods for edge correction, (3) methods to account for first-order effects (i.e. heterogeneity) of univariate patterns, and (4) a variety of useful standard and non-standard null models for univariate and bivariate patterns. For illustrative purpose, we analyze examples that deal with non-homogeneous univariate point patterns. We demonstrate that large-scale heterogeneity of a point-pattern biases Ripley's K-function at smaller scales. This bias is difficult to detect without explicitly testing for homogeneity, but we show that it can be removed when applying methods that account for first-order effects. We synthesize our review in a number of step-by-step recommendations that guide the reader through the selection of appropriate methods and we provide a software program that implements most of the methods reviewed and developed here.     

植物种群空间分布的点格局分析

植物种群在群落中的分布格局与空间尺度有着密切关系,传统的样方取样及其格局分析方法,只能分析一种尺度下的格局。本文引入一种新的格局分析方法--点格局分析,其是以种群空间分布的坐标点图为基础,通过本文对美国密西根州克林顿县栎林3个优势种格局分析,它有3个明显的优点：1）能够分析各种尺度下的种群格局和种间关系,结果清楚,直观;2）所描述的结果更符合实际,尤其是对群落结构的描述;3）它有利于定点观察,研究时间与种群格局的关系,本文分析的3个种集群特征明显,但随尺度的变化有不同的分布趋势,3个种间的关系也是如此。     

On explicit formulas of edge effect correction for Ripley's K-function

Abstract. The analysis of spatial pattern in plant ecology usually implies the solution of some edge effect problems. We present in this paper some explicit formulas of edge effect correction that should enable plant ecologists to analyse a wider range of real field data. We consider the local correcting factor of edge effect for Ripley's K-function, that can also be used for other statistics of spatial analysis based on the counting of neighbours within a given distance. For both circular and rectangular study areas, we provide a review of explicit formulas and an extension of these formulas for long and narrow plots. In the case of irregular-shaped study plots, we propose a generalization of the method that computes edge effect correction by excluding triangular surfaces from a simple (rectangular or circular) initial shape. An example in forest ecology, where the soil characteristics determine a study plot of complex shape, illustrates how this edge effect correction can be effective in avoiding misinterpretations.     

The spatial and temporal distribution of the grain aphid Sitobion avenae in winter wheat

The spatial and temporal distribution of the grain aphid Sitobion avenae F. (Homoptera: Aphididae) was studied within a field of winter wheat during the summer of 1996. Sampling was done using four nested grids comprising 133 locations. Analysis by Taylor's power law gave results typical for insect populations. Analysis by SADIE (Spatial Analysis by Distance Indices) showed spatial pattern due to edge effects and sampling scale, and positive but mild spatial association, although spatial patterns were ephemeral. Reasons for these findings and the implications for integrated crop management are discussed.     

A test of the central–marginal hypothesis using population genetics and ecological niche modelling in an endemic salamander (Ambystoma barbouri)

The central–marginal hypothesis (CMH) predicts that population size, genetic diversity and genetic connectivity are highest at the core and decrease near the edges of species' geographic distributions. We provide a test of the CMH using three replicated core‐to‐edge transects that encompass nearly the entire geographic range of the endemic streamside salamander (Ambystoma barbouri). We confirmed that the mapped core of the distribution was the most suitable habitat using ecological niche modelling (ENM) and via genetic estimates of effective population sizes. As predicted by the CMH, we found statistical support for decreased genetic diversity, effective population size and genetic connectivity from core to edge in western and northern transects, yet not along a southern transect. Based on our niche model, habitat suitability is lower towards the southern range edge, presumably leading to conflicting core‐to‐edge genetic patterns. These results suggest that multiple processes may influence a species' distribution based on the heterogeneity of habitat across a species' range and that replicated sampling may be needed to accurately test the CMH. Our work also emphasizes the importance of identifying the geographic range core with methods other than using the Euclidean centre on a map, which may help to explain discrepancies among other empirical tests of the CMH. Assessing core‐to‐edge population genetic patterns across an entire species' range accompanied with ENM can inform our general understanding of the mechanisms leading to species' geographic range limits.     

长白落叶松人工林天然更新幼苗分布格局及其研究方法的比较

长白落叶松是东北东部地区主要造林树种之一．该文研究了55a生长白落叶松人工林天然更新幼苗的空间分布格局．测定了样地内（30m×18m）所有天然更新幼苗的坐标、地径和苗高．在幼苗坐标点图的基础上,进行了23个取样尺度（离散尺度）下的相邻格子法的空间格局分析：采用Poisson分布、负二项分布和正二项分布拟合幼苗的离散分布,方差均值比、Morisita指数和最近邻体法确定了空间分布格局,C-指数、Green指数和负二项参数K值度量了它们的聚集强度．利用更新幼苗的坐标点图,采用地统计学中自々半方差分析方法和空间点格局方法（采用Ripley二次分析法）研究了尺度连续变化条件下更新幼苗空间分布格局和聚集强度的变化并确定了其聚集规模,利用最近邻体法的计算方法分析了更新幼苗的自然定植特点．结果表明：1）更新幼苗所有离散尺度下（样方面积从1m^2到18m^2）均呈聚集分布．连续尺度下,在0—10．5m的半径区间内呈聚集分布,在10．5—12．3m的半径区间内呈随机分布．2）离散尺度下（样方面积从1m^2到18m^2）,聚集强度随尺度的增加而增加．连续尺度下,在0．10．5m的半径区间内,聚集强度随尺度的增加表现出先增加后降低的趋势,在半径（尺度）为4．2m处聚集强度最大．3）此次调查的样地范围内天然更新幼苗呈现聚集状态生长的平均最大聚集斑块（聚集规模）的半径为4．79m．4）天然更新幼苗之间的平均距离为0．30-0．42m,更新幼苗生长在与母树相距2．99—3．26m之外的空间．5）相邻格子法（传统的样方方法）和空间点格局分析法对格局的判别和格局强度的确定基本一致,但是空间点格局分析法能够反映出尺度连续变化条件下格局和格局强度的变化．它与最近邻体法和地统计学方法等以种群分布的坐标点图为数据源的     

Point pattern analysis of different age-classes of Larix principis-rupprechtii in Luya Mountain Reserve, Shanxi Province, China

Larix principis-rupprechtii forest is an important vegetation formation and has a large distribution area in Luya Mountain Reserve, China. Spatial pattern analysis on individual trees in different age-classes of Larix principis-rupprechtii was made in this paper. Here, we employed the technique of point pattern analysis, which could analyze patterns under all scales along a gradient. It was based on spatial mapped points of individual distribution. The results of this study showed that the densities of the five age-classes varied in the order: age-class 3 > age-class 4 > age-class 5 > age-class 2 > age-class 1. Although age-classes 1 and 2 have much fewer individuals than other three age-classes do, the population was stable at present. However, it would be necessary to take some measures for improving population regeneration for a long-time view. The individuals of all age-classes focused on clumping distribution in space; however, their distribution pattern varied with the change of scale. This mainly depended on biological features of Larix principis-rupprechtii and forest environments, but it also meant that the scale was an important factor in controlling spatial distribution pattern of tree individuals. The feature of clumping distribution became more significant with the increase of age. The relationships between individuals in different age-classes were almost all significantly correlated with each other. These associations became more significant within the older age-classes. This suggested that the individuals of different age-classes were interdistributed, by which the population could get benefits in resource utilization. The technique of point pattern analysis is effective and easy to be used in species pattern study. Its results are more closer to the reality, especially for community structure. Translated from Acta Ecologica Sinica, 2004, 24(1): 35–40 [译自: 生态学报]     

An evaluation of the state of spatial point pattern analysis in ecology

Over the last two decades spatial point pattern analysis (SPPA) has become increasingly popular in ecological research. To direct future work in this area we review studies using SPPA techniques in ecology and related disciplines. We first summarize the key elements of SPPA in ecology (i.e. data types, summary statistics and their estimation, null models, comparison of data and models, and consideration of heterogeneity); second, we review how ecologists have used these key elements; and finally, we identify practical difficulties that are still commonly encountered and point to new methods that allow current key questions in ecology to be effectively addressed. Our review of 308 articles published over the period 1992–2012 reveals that a standard canon of SPPA techniques in ecology has been largely identified and that most of the earlier technical issues that occupied ecologists, such as edge correction, have been solved. However, the majority of studies underused the methodological potential offered by modern SPPA. More advanced techniques of SPPA offer the potential to address a variety of highly relevant ecological questions. For example, inhomogeneous summary statistics can quantify the impact of heterogeneous environments, mark correlation functions can include trait and phylogenetic information in the analysis of multivariate spatial patterns, and more refined point process models can be used to realistically characterize the structure of a wide range of patterns. Additionally, recent advances in fitting spatially‐explicit simulation models of community dynamics to point pattern summary statistics hold the promise for solving the longstanding problem of linking pattern to process. All these newer developments allow ecologists to keep up with the increasing availability of spatial data sets provided by newer technologies, which allow point patterns and environmental variables to be mapped over large spatial extents at increasingly higher image resolutions.     

Characterizing phase‐related differences in behaviour of Schistocerca gregaria with spatial distribution analysis

The behaviour of locusts has been studied extensively using two approaches: (1) analysing a single individual's response to a group stimulus or (2) using group conditions to look at aggregation patterns. The second approach has, in contrast with the first one, not been improved in terms of statistical analyses since the 1960s. In the present study, we propose a spatial statistics approach of point‐pattern analysis to improve the group‐based assessment of behavioural phase characterization. This diagnostic tool was developed and tested in the laboratory with comparative analysis of solitarious (isolation‐reared) and gregarious (crowd‐reared) desert locusts, Schistocerca gregaria (Forskål) (Orthoptera: Acrididae). The spatial distribution patterns of 10 either solitarious or gregarious third‐instar hoppers were characterized with nearest neighbour distance measurements in a circular arena. The temporal sequence of spatial disposition of locusts was recorded with a digital camera taking photographs at regular intervals. The approach of point‐pattern analysis focused on the spatial distribution of observed events and allowed us to make inferences about the underlying process that generated them. The results confirmed that our diagnostic tool could identify that crowd‐reared hoppers tended to aggregate more to conspecifics than isolation‐reared ones. We could also verify that isolation‐reared hoppers were less active than crowd‐reared ones, but this was only true at the beginning of the experiments. The spatial statistics approach proposed in the present study could help with observations of phase‐related differences in the behaviour of locusts.     

Can isotropy vs. anisotropy in the spatial association of plant species reveal physical vs. biotic facilitation?

In dryland ecosystems and other harsh environments, a large part of the vegetation is often clustered, appearing as ‘islands’. If ‘independent’ species, usually colonizers, can be distinguished from species which are ‘dependent’ on the presence of the colonizing species for successful establishment and/or persistence, the type of spatial pattern of the association ‐ isotropic (spatially symmetric) or anisotropic (spatially asymmetric) ‐ can give information on the underlying environmental factors driving the process of association. Modified spatial pattern analysis based on Ripley's K‐function can be applied to bivariate clustered patterns by cardinal direction in order to detect possible anisotropy in the pattern of association. The method was applied to mapped distribution patterns of two types of semi‐arid shrubland in southeastern Spain. In shrubland of Retama sphaerocarpa, low shrubs of Artemisia barrelieri were significantly clustered under the canopy of the Retama shrubs in all four cardinal directions, suggesting an isotropic facilitation effect. In low shrubland dominated by Anthyllis cytisoides and Artemisia barrelieri, Anthyllis shrubs occurred more frequently than expected on the eastern side (and downslope) of an Artemisia shrub. The possible environmental factors driving the two association patterns are discussed and recommendations for further applications of the analytical method are given.     

A test for spatial relationships between neighbouring plants in plots of heterogeneous plant density

Abstract. Maps of plant individuals in (x, y) coordinates (i.e. point patterns) are currently analysed through statistical methods assuming a homogeneous distribution of points, and thus a constant density within the study area. Such an assumption is seldom met at the scale of a field plot whilst delineating less heterogeneous subplots is not always easy or pertinent. In this paper we advocate local tests carried out in quadrats partitioning the plot and having a size objectively determined via a trade‐off between squared bias and variance. In each quadrat, the observed pattern of points is tested against complete spatial randomness (CSR) through a classical Monte‐Carlo approach and one of the usual statistics. Local tests yield maps of p‐values that are amenable to diversified subsequent analyses, such as computation of a variogram or comparison with co‐variates. Another possibility uses the frequency distribution of p‐values to test the whole point pattern against the null hypothesis of an inhomogeneous Poisson process. The method was demonstrated by considering computer‐generated inhomoge‐neous point patterns as well as maps of woody individuals in banded vegetation (tiger bush) in semi‐arid West Africa. Local tests proved able to properly depict spatial relationships between neighbours in spite of heterogeneity/clustering at larger scales. The method is also relevant to investigate interaction between density and spatial pattern in the presence of resource gradients.     

红松存在/不存在数据的多尺度空间分布格局

红松属小兴安岭地区地带性植被优势种,该地区也是其分布的北缘。在景观尺度上开展红松的分布格局研究有利于进一步了解红松分布机理、未来迁移过程等问题,对其经营和保护有重要意义。将景观指数法与点格局分析法结合,设定8个空间尺度,利用红松存在/不存在数据,通过计算各空间尺度上红松聚集程度和景观指数,分析小兴安岭地区红松种群在多尺度上的分布格局。研究结果表明,小尺度上红松聚集分布明显,随机分布区多处于其聚集分布区的边缘,均匀分布区则散布在其聚集分布区内。景观指数研究表明,通过景观指数可判断红松聚集分布格局趋势,而不能判断均匀分布、随机分布格局趋势,因为它们在多尺度下景观指数波动大,不能用景观指数来描述分布格局。研究得出如下结论:1)红松主要分布在其分布区的核心区域内,在分布区边缘和过渡带上呈随机分布,2)存在/不存在数据能够用来分析种群的多尺度空间分布格局,3)空间尺度的变化会引起树种分布格局的变化,随机分布随尺度增加,边缘化程度加强,4)单一尺度上,景观格局指数不能完全描述种群分布格局;而在多尺度上,变化趋势稳定的景观指数表明聚集分布存在,而波动剧烈的景观指数常与随机分布和均匀分布联系在一起,5)地形因子中,红松对坡度和海拔两个因子变化敏感。     

Spatial pattern of downed logs and wood‐decaying fungi in an old‐growth Picea abies forest

Abstract. Since many wood‐living forest species are influenced by the dynamics of coarse woody debris (CWD), information about the spatial pattern of CWD under natural conditions is essential to understand species distributions. In this study we examined the spatial pattern of downed logs and wood‐decaying fungi in an old‐growth boreal Picea abies forest in northwestern Sweden that is governed by gap‐phase dynamics. The spatial pattern of wood‐decaying fungi was studied to draw conclusions about species dispersal abilities. A total of 684 logs with a diameter > 10 cm were mapped and analysed with Ripley's K‐function. The distribution of all logs taken together displayed a significant aggregated pattern up to 45 m. The different decay stages also deviated from random expectations. Fairly fresh logs and logs in the middle decay stage were clumped up to about 25 and 35 m respectively, and late decayed logs aggregated up to 95 m. Logs with diameters from 10–29 cm were aggregated up to 25 m, whereas logs ≥30 cm diameter were randomly distributed. The result suggests that gap‐dynamics do have an impact on the spatial pattern of the CWD, creating fine‐scale clumping. The random distribution of large logs may result from the slightly regular spacing of large living trees. The spatial patterns of 16 species (n > 20) of wood‐decaying fungi were analysed with Ripley's K‐function. Three patterns were aggregated, for Gloeophyllum sepiarium, Coniophora olivacea and Vesiculomyces citrinus. These results indicate that the distribution of most species at the stand level is generally not influenced by dispersal limitations.     

Structural heterogeneity and spatial autocorrelation in a natural mature Pinus sylvestris dominated forest

The tree species composition, vertical stratification and patterns of spatial autocorrelation at the tree and quadrate (25 × 25 m) scales were studied in a natural mature PinuS sylvestris dominated forest in eastern Finland. For the analyses we mapped the locations and dimensions of trees taller than 10 m in a 9 ha (300 × 300 m) area, and within this area we mapped all trees taller than 0.3 m on a core plot of 4 ha (200 × 200 m). The overall tree size distribution was bimodal. the dominant layer and the understory forming the peak frequencies. Pinus sylvestris dominated the main canopy, together with scattered Betula pendula and Picea abies. Alnus incana, Populus tremula, Salix caprea, Sorbus aucuparia and Juniperus communis occurred only in the under- and middlestories. Autocorrelation analysis (semivarianee) of tree size variation revealed spatial patterns, which were strongly dependent on the size of trees included in the analysis. When all living trees, including the understory regeneration, were taken into account, the autocorrelation pattern ranged up to 35 m inter-tree distances, reflecting the spatial scale of understory regeneration patches. Competitive interaction among middle- and upperstory trees (height>10 m) had contrasting effects on autocorrelation pattern depending on spatial scale. At the fine scale, dominant trees suppressed their smaller close neighbors (asymmetric competition), which was shown as increased tree size variation at small inter-tree distances (<2 m). At slightly larger inter-tree distances, specifically among large trees of similar size, competition was more symmetrical, which resulted in decreased tree size variation at these inter-tree distances (3–4 m). This effect was seen most clearly in the dominant trees, there being a clear autocorrelation pattern in tree size up to inter-tree distances of ～4 m. At the quadrate scale (25 × 25 m) the analysis revealed high local variation in structural characteristics such as tree height diversity (THD), tree species diversity (H) and autocorrelation of tree height. The analysis suggests that naturally developed P. sylvestris forests exhibit complex small-scale patterns of structural heterogeneity and spatial autocorrelation in tree size. These patterns may be important for stand-scale habitat diversity and can have aggregated effects on ecosystem dynamics at larger spatial scales though their influence on the spread of disturbance and regeneration after disturbance.     

A new non‐parametric method for analyzing replicated point patterns in ecology

Most ecological studies that involve point pattern analyses are based on a single plot, which prevent the separation of the effects of various processes that could act simultaneously, as well as limiting the conclusions that can be extracted from these studies. However, considering the spatial distribution of individuals in several plots as replicates of the same process could help to differentiate its specific effects from those of other confounding processes. Thus, we introduce a new method for analyzing spatial point patterns that are replicated according to a two‐factorial design. By summarizing the spatial patterns as K‐functions, the proposed method computes the average K‐functions for each level of the two factors (i.e. predictors) and for each combination of levels, before estimating the sum of squared deviations from the overall mean K‐function. Inferences of the strength of the relationship between the predictors, their interaction, and the spatial structure are made based on a non‐parametric bootstrap procedure, which considers the dependency among spatial scales. We illustrate the proposed approach based on an analysis of the effects of altitude (with two levels: low and high) and slope (with two levels: flat and steep slopes) on the spatial pattern of Croton wagneri, a dominant shrub in an Andean dry scrubland. Our method detected a significant effect of the interaction between slope and altitude, which could not have been detected using current point pattern analysis methodology. The prevalence of single‐plot analysis in ecological studies may be due to a lack of familiarity with appropriate methods for replicated point patterns, as well as the greater complexity of these methods and the absence of appropriate software. Our approach can be applied to a significant number of ecological questions while maintaining a simple, understandable, and easily reportable methodological framework.     

沙地樟子松天然林南缘分布区林木竞争、空间格局及其更新特征

以呼伦贝尔沙地樟子松地理分布南缘天然林为研究对象,采用Hegyi竞争指数与点格局等分析方法,研究了沙地樟子松天然林的种群结构、种内竞争、空间格局及幼树更新等特征。结果表明:1)所调查区域天然沙地樟子松纯林处于中幼龄阶段,属增长型种群;2)其竞争指数与对象木胸径服从幂函数关系CI=242.24D~(-1.12)(R~2=0.91);3)幼树在小尺度上呈聚集分布,中树与大树在中大尺度上呈随机分布;幼树与中树在小尺度上呈正相关性,中树与大树在小尺度呈负相关性;4)竞争指数与更新幼树和存活更新幼树的密度均呈显著正相关性。在林分管理中需要充分考虑林木竞争、空间格局以及种群更新的关系,本研究可为沙地樟子松天然林的经营管理与保护提供重要的科学依据。