A power law model for analyzing spatial patterns of vegetation abundance in terms of cover, biomass, density, and occurrence: derivation of a common rule

The cover, biomass, density, and frequencies of occurrence of individual species are important measures for characterizing plant communities. Examination of the statistical properties of vegetation data obtained through a quadrat survey provides insight into the spatial structure of plant communities and plant species diversity. In community analysis, we assume that x is theoretical variance calculated under the assumption that the abundance of a species follows a random distribution among quadrats (which we can obtain easily), and y is quadrat-to-quadrat variance calculated based on the observed data for a species. If the relationship y = Axᵇ or logy = logA + blogx holds for two measurable traits of the plant community, x and y, the relationship is referred to as the power law, where A and b are constants. Then, an ideal index for measuring the spatial heterogeneity (δ) of the vegetation variable for each species was defined as δ = logy – logx. If δ = 0, the spatial heterogeneity of the species is random; if δ > 0, the heterogeneity exceeds a random pattern; and if δ < 0, the heterogeneity is less than a random pattern. In previous works in 2001 and 2014, we demonstrated that using the power law based on occurrence and density is a valuable approach to vegetation surveys and analyses. In this paper, the power law is mathematically extended and practical evidence is given for its application to cover and biomass of each species within a plant community. All measurements of cover, biomass, density, and frequency of occurrences were fitted to the power law, and plots of δ against the abundance values of these variables for each species in the community well described abundance and spatial pattern of each species in the environment. The power law is applicable not only to small-scale quadrats such as 50 × 50 cm but also to large-scale vegetation maps obtained using remote sensing and aerial photographic techniques.