A fractal approach for detecting spatial hierarchy and structure on mussel beds
检测贻贝床空间层次和结构的分形方法
Abstract Within beds of blue mussel (Mytilus edulis L.), individuals are aggregated into small patches, which in turn are incorporated into bigger patches, revealing a complex hierarchy of spatial structure. The present study was done to find the different scales of variation in the distribution of mussel biomass, and to describe the spatial heterogeneity on these scales. The three approaches compared for this purpose were fractal analysis, spatial autocorrelation and hierarchical (or nested) analysis of variances (ANOVA). The complexity (i.e. patchiness) of mussel aggregations was described with fractal dimension, calculated with the semivariogram method. Three intertidal mussel beds were studied on the west coast of Sweden. The distribution of wet biomass was studied along transects up to 128 m. The average biomasses of blue mussels on the three mussel beds were 1825plusmn;210, 179plusmn;21 and 576 plusmn;66 g per 0.1m2, respectively, and the fractal dimensions of the mussel distribution were 1.726plusmn;0.010,1.842plusmn;0.014 and1.939plusmn;0.029 on transects 1-3, respectively. Distributions of mussels revealed multiscaling behaviour. The fractal dimension significantly changed twice on different scales on the first bed (thus showing three scaling regions), the second and third beds revealed two and three scaling regions, respectively. High fractal dimension was followed by significant spatial autocorrelation on smaller scales. The fractal analysis detects the multiple scaling regions of spatial variance even when the spatial structure may not be distinguished significantly by conventional statistical inference. The study shows that the fractal analysis, the spatial autocorrelation analysis and the hierarchical ANOVA give complementary information about the spatial variability in mussel populations.
摘要 在紫贻贝的贻贝床上,个体聚合成小块,进而形成更大的斑块,揭示了空间结构的复杂层次。本研究的目的是找到不同规模的贻贝生物量的分布变化，并描述这些规模上的空间异质性。为此目的比较的三种方法是分形分析、 空间自相关分析和方差的分层 （或嵌套）分析 (ANOVA)。贻贝聚集的复杂性用分形维数来描述，用半方差法来计算。在瑞典西海岸，对三个潮间带贻贝床进行了研究。研究了湿生物质的分布，沿断面可达128米。在三个贻贝床上，紫贻贝的平均生物量分别为1825plusmn;210、179 plusmn;21和576 plusmn;66克每0.1平方米。在1-3断面上，贻贝分布的分形维数分别为1.726plusmn;0.010,1.842plusmn;0.014和1.939plusmn;0.029。贻贝的分布显揭示了多规模行为。在第一张贻贝床的不同规模上，分形维数显着改变了两次（从而显示出三个缩放区域），第二和第三张贻贝床分别显示两个和三个缩放区域。在较小的斑块上，高分形维数紧跟着显著的空间自相关。分形分析检测多规模区域的空间方差，甚至当空间结构可能无法通过传统的统计推断显著区分时。研究表明，分形分析、空间自相关分析和分层方差分析提供关于贻贝种群空间变异的互补信息。
Introduction
Estimation of scales of variation and the degrees of spatial heterogeneity is a necessary step to describe the distribution of natural populations, and for building a proper sampling design for later studies. Description of the variability and predictability of the environment requires reference to the particular range of scales that is relevant to the organisms or processes being examined (Levin 1992). In the present study we used three different approaches to address this problem: the commonly used techniques of spatial autocorrelation (Legendre and Fortin 1989) and hierarchical ANOVA (Underwood 1997), and the concept and tools of fractals, introduced by Mandelbrot (1977). A fractal is a complex geometrical shape, constructed of smaller copies of itself, and an ideal mathematical fractal has the same structure (is self-similar) on an infinite range of scales (Mandelbrot 1982). In contrast to simple geometric objects, fractals possess non-integer dimension. Geometric objects in Euclidean geometry are described using integer dimensions (0 for a point, 1 for a line, 2 for a plane and 3 for a volume), while fractal dimension does not need to be an integer. It may take any value between the bounding integer topological dimensions, and increases with the increase in complexity of a geometric object. For selfsimilar mathematical fractals, the fractal dimension (D) is calculated as:
D =/ ⑴
for an irregular or fragmented geometric object that can be subdivided into N similar parts, each of which is a copy of the whole, reduced k times. Strict or ideal fractal shapes are not found in nature (although mathematical self-similarity may be visualised when viewing for example snowflakes and ferns down to a certain scale), which is why calculation of the fractal dimension of natural patterns and shapes is based on statistical selfsimilarity (classical examples are clouds and coastlines) and is done over a range of scales (see e.g. Cox and Wang 1993; Hastings and Sugihara l993).
引言
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