Spatial pattern model of herbaceous plant mass at species level

Ecological Informatics 24 (2014) 124–131

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Spatial pattern model of herbaceous plant mass at species level Jun Chen a,⁎, Masae Shiyomi b a b

College of Animal Science and Technology, Northwest A&F University, Yangling, Shaanxi 712100, China The Open University of Japan, Ibaraki Study Center, Bunkyo 2, Mito 310-0056, Japan

a r t i c l e

i n f o

Article history: Received 14 November 2013 Accepted 6 August 2014 Available online 13 August 2014 Keywords: Exponential distribution Gamma distribution Herbaceous plant community Random spatial pattern Spatial heterogeneity

a b s t r a c t Individual plant species distribute according to their own spatial pattern in a community. In this study, we proposed an index for measuring the spatial heterogeneity in mass (dry weight) of individual plant species. First, we showed that the frequency distributions for mass of individual plant species per quadrat in a plant community are expressed using the gamma distribution with two parameters of λ (mean) and p. The parameter p is a measure indicating the level of spatial heterogeneity of plant mass as follows: (1) when p = 1, the plant mass per quadrat has a random pattern; (2) when p N 1, the plant mass has a spatial pattern with a lower heterogeneity than would be expected in the random pattern; and (3) when p b 1, the plant mass has a spatial pattern with a higher heterogeneity than would be expected in the random pattern. The p value for a given species can easily be calculated by the following equation if we use the moment method: (mean plant mass among quadrats)2 / (variance of plant mass among quadrats). The scatter diagram of (λ, p) for individual plant species, exhibits the spatial characteristics of each species in the community. We illustrated two examples of the (λ, p) diagram from data for individual species composing actual communities in a semi-natural grassland and a weedy grassland. Frequency distributions for the plant mass of individual species per quadrat followed the gamma distribution, and indi vidual species exhibited an inherent level of spatial heterogeneity. © 2014 Published by Elsevier B.V.

1. Introduction Small-scale spatial patterns in vegetation are formed by various external factors, such as small-scale spatial variations in nutrient concentrations, water conditions, soil salinity, soil pH, grazing by animals, and plant diseases (e.g., Gokalp et al., 2010; Mallants et al., 1996; Yasuda et al., 2003). The development of spatial patterns is also related to internal factors in vegetation, such as species-to-species interactions (e.g., Kull and Zobel, 1991; van der Maarel and Sykes, 1993; van der Maarel et al., 1995; Zobel et al., 2000). Silvertown and Charlesworth (2001) reviewed the biological mechanisms leading to plant heterogeneity and classified the factors causing spatial patterns into the following six types: (1) niche separation processes, (2) spatial segregation processes, (3) recruitment limitation processes, (4) pest-pressure processes, (5) storage effect processes and (6) density independent processes. Small-scale spatial patterns of vegetation are, in general, measured by determining the frequency of occurrence, density of individuals, plant cover and biomass (Bonham, 2013). Through these measures, the vegetation structure, such as species richness and species composition, can be evaluated numerically (Shiyomi et al., 2010). We divide the research methods of spatial pattern formation in a community into two categories: the first involves the frequency distributions of plant abundance such as the frequency of occurrence and ⁎ Corresponding author. Tel.: +86 186 8185 1233. E-mail address: [email protected] (J. Chen).

http://dx.doi.org/10.1016/j.ecoinf.2014.08.001 1574-9541/© 2014 Published by Elsevier B.V.

density per unit ground area (e.g., Greig-Smith, 1964; Harte et al., 2005; Pielou, 1977; Shiyomi, 1981), and the second uses various measures of spatial series, such as fractal analysis and semivariance analysis (e.g., Manurer, 1994; Palmer, 1988; Roe et al., 2011; Schabenberger and Gotway, 2005). In this study, we refer to the first of the two methods, and focus on the frequency distribution of the mass (dry weight) of individual plant species per unit ground area. To determine the spatial pattern of vegetation, the criterion that expresses the standard pattern of spatial distribution must be elucidated. In many cases, the standard is “random spatial pattern”; for example, in a classic index by David and Moore (1954), I = variance/mean − 1 = 0 indicates a random pattern in ‘count’ data, such as the number of aphids per plant, and in the index of Morisita (1962), Iδ = 1 indicates a random pattern in count data of individuals. Since the spatial pattern of a single plant species in a community is influenced by ‘continuous’ variables such as cover and plant mass (weight), these continuous variables also have to have criteria to determine the pattern. In this paper, we propose a criterion to determine the spatial pattern of mass for a plant species and accordingly consider a model of the spatial pattern. Shiyomi et al. (1983) attempted to fit a frequency model of total plant mass constructing a plant community per unit ground area using a gamma distribution. The frequency distributions of the mass of pooled plant species per unit ground area were recorded under various stocking rates, and were fitted to the gamma distribution. Chen et al. (2008) reported that the gamma distribution that described the

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frequency distribution of plant mass per unit ground area could be derived from the beta distribution that described the frequency distribution of plant cover per unit ground area. In this paper, we derive and use the gamma distribution to measure the spatial heterogeneity in mass of individual plant species. We first define ‘a random spatial pattern for mass of a single plant species’, and show that the spatial pattern of plant mass is expressed using a parameter contained in the gamma distribution. Second, we determine whether plant mass of a herbaceous species population is spatially distributed according to a pattern more or less heterogeneous than the random pattern. Next we determine whether the gamma distribution and its parameters adequately describe the spatial heterogeneity in the mass of a herbaceous plant species making up a community, based on two examples from a semi-natural grassland and a weedy grassland.

The n + 1 short segments are formed on the line segment (Fig. 1a). The small segment, y, intercepted by the adjacent two random points follows the power distribution according to the theory of order statistics (e.g., Zwillinger and Kokoska 2000). That is, y is a new random length following the power distribution. The density function of y, f(y), is expressed by the following form:

2. Model of spatial pattern for mass of a single plant species

in which the short segment of y with a random length approaches asymptotically a random distribution according to the exponential distribution (Eq. (3)) (Fig. 1b), and the mean and variance of the right side of Eq. (3) approach: θ(n + 1)−1 → λ and nθ2 (n + 1)−2 (n + 2)−1 → λ2, respectively. Therefore, in Eq. (3), [mean]2 / [variance] = 1. We define the following three situations in spatial pattern of y: (1) if [mean]2 / [variance] = 1, y randomly distributes spatially based on the above definition; (2) if [mean]2 / [variance] N 1 (i.e., the standard deviation is relatively small compared to the mean), y distributes with a lower spatial heterogeneity than would be expected in a random case (the larger [mean]2 / [variance], the lower the heterogeneity is) and; (3) if [mean]2 / [variance] b 1 (i.e., the standard deviation is relatively large compared to the mean), y distributes with a higher

2.1. Random pattern in mass of a plant species For simplicity, we consider a line segment with a given length of θ (N0). Then, suppose that the line segment is cut at n random points, each of whose length from the origin is x1, x2, …, xn (0 ≤ xi ≤ θ; i = 1, 2, …, n), where x follows a uniform distribution and is generally referred to as a random length (Fig. 1a shows an example for n = 3). The density function of x, f(x), is expressed as: −1

f ðxÞ ¼ θ for 0 ≤ x ≤ θ ¼ 0; otherwise:

ð1Þ

n−1 −1

θ f ðyÞ ¼ nðθ − yÞ ¼ 0; otherwise:

for 0 ≤ y ≤ θ

ð2Þ

The mean and variance of Eq. (2) are expressed as θ(n + 1)−1 and nθ2 (n + 1)−2(n + 2)−1, respectively. Now, under the condition of θn−1 = λ: constant, for θ → ∞ f ðyÞ ¼ nðθ − yÞ

n−1 −1

θ

−y=λ

→e

–1

λ ; for y ≥ 0; λ N 0;

ð3Þ

Fig. 1. The random spatial pattern of plant mass. (a) Three lengths, x1, x2 and x3, generated according to the uniform distribution from the origin on a line segment with finite length of θ (Eq. (1)). Four small segments with lengths of y1, y2, y3 and y4, in that order from the origin follow a power distribution (Eq. (2)). (b) An extension of the line segment of length θ to a half-line with an infinite length. Short line segments of y1, y2, … (in order from the origin) follow an exponential distribution. (c) When the same plant mass, h, lies at any point of the half-line, the area of each small rectangle with a given height h on the half-line, i.e. h × yi, represents the plant mass on a small segment of yi, and follows the exponential distribution, which is random according to the exponential distribution. (d) Let us exchange the height and width of each rectangle in Fig. 1c, that is, suppose that yi × h = zi indicates (the plant mass density) × (the quadrat size on the ith quadrat). Then, each zi follows an exponential distribution.

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heterogeneity than would be expected in a random case (the smaller [mean]2 / [variance], the higher the heterogeneity is). We put hereafter: 2

½mean =½variance ¼ p;

ð4Þ

and use p as an index of the degree of spatial heterogeneity of plant mass per unit ground area. If the same plant mass, h, of a plant species lies at any point on the half-line shown in Fig. 1b, the area of each small rectangle with a constant height h on the half-line (Fig. 1c), i.e., the amount of h × yi, must follow the exponential distribution as well as yi. Furthermore, even if we exchange the width and height of each rectangle in Fig. 1c, the area of the rectangle, yi × h = zi, follows the same exponential distribution, where zi expresses the plant mass on the i-th quadrat with a constant size h (Fig. 1d). 2.2. A frequency distribution model for mass of a plant species per unit ground area We assumed the following situation: (1) the mass of a single plant species per unit ground area changes with time according to the logistic assumption under a condition of continuous animal grazing; (2) the carrying capacity of the logistic equation determined by physical, chemical environments and biological interactions operating among plant species stochastically varies according to a normal distribution; and (3) plant growth is balanced by defoliation. The frequency distribution of dry mass of a single plant species per unit ground area follows a gamma distribution under the above assumptions (Appendix A). The density function of the gamma distribution of z (the mass of a given plant species per unit ground area) is expressed by: f ðzÞ ¼

zp−1 pp −pz=λ e ; z ≥ 0; p N 0; λ N 0; Γ ðpÞλp

ð5Þ

where λ and p are the mean and the index expressing heterogeneity level of plant mass per unit ground area, respectively. If these assumptions are not much different from the actual growth process, the spatial pattern of species would be approximated by Eq. (5). The exponential distribution (Eq. (3)) is a special case of Eq. (5) for p = 1. Fig. 2 shows gamma distributions (density functions) with a mean λ = 5 for various p's. For p b 1 (the plant mass distributes with a higher

heterogeneity than would be expected in a random case), the density function forms an L-shaped curve with a long, relatively high right tail; for p = 1, the plant mass distributes randomly among quadrats and the density function forms an L-shaped curve; and for p N 1, the plant mass distributes with a lower heterogeneity than would be expected in a random case and the density function forms a one-peak curve with a long, relatively low right tail. For a large p, the density function approaches the normal distribution, and for p → ∞, the density exists at only one point of x = 5 because the variance vanishes, which means that the plant mass is distributed completely evenly over the area. 2.3. Estimation of parameters of gamma distribution Let the n plant mass data be z1, z2, …, zn for one species, and let the sample mean, sample variance and sample value of p be expressed ^ and p ^ σ ^ , respectively. The values of λ ^ were obtained by ^ 2 and p by λ, the following two methods: ^ is given as the arithmetic (1) The moment (ME) estimate, where λ ^ 2 =σ ^¼λ ^ 2. mean of z1, z2, …, zn and p (2) The maximum likelihood estimate (MLE), where the logarithm of likelihood of the gamma density function, l, is given by:   n
 X ^ p ^ p ^ ¼ −n ln Γ ðp ^Þ þ pn ln ^: ^−1Þ l λ; ln zi −np þ ðp ^ λ i¼1 ^ and p ^ are estimated using the following two equations, Then, λ n ^Þ ^ X ∂l Γ 0 ðp p ln zi ¼ 0 ¼ −n þ n ln þ ^Þ ^ ^ Γ ðp ∂p λ i¼1 n X ∂l ^þ zi ¼ 0; ¼ −nλ ^ ∂λ i¼1

^Þ is the gamma function of p ^ and Γ′( p ^ ) is dΓðp ^Þd=dp ^ . The where Γ ðp ^ and p ^, σ ^ λ 2 and σ ^ p 2, are respectively estimated by: sample variances of λ ^ 2 n−1 ^ λ2 ¼ λ σ and ^ p2 ¼ σ

" ∞ X i¼0

#−1 n n − : ^ ^ þ iÞ2 p ðp

ð6Þ

These variances are used for statistical tests, and for the confidence ^ and p ^ using the standard normal distribution for a large n. intervals for λ 3. Field examples 3.1. Materials and methods

Fig. 2. Gamma distributions with various heterogeneity values, p, when the mean, λ, is 5 (see text). For p b 1, the density function forms an L-shaped curve with a long, relatively high right tail; for p = 1, the density function forms an L-shaped curve; and for p N 1, the density function forms a one-peak curve with a long, relatively low right tail. For a large p, the density function approaches the normal distribution, and for p → ∞, the density exists at only one point of x = 5.

We provided two sets of survey data as examples to be analyzed using the above model. The first survey was conducted at a grazed semi-natural grassland of the National Institute of Livestock and Grassland Science located in Nasushiobara, Japan. We clipped plants at ground level in 100 quadrats measuring 10 × 10 cm positioned at 0.5m intervals on a 50-m line in August 2005. We removed the dead plant materials, divided the live plants according to species, and then oven-dried and weighed the plant material. The second example was a vegetation survey conducted at a weedy grassland located on the Ibaraki University campus, Mito, Japan. A 1 × 1-m square was divided into 100 quadrats measuring 10 × 10 cm and left without treatment following hand-weeding in May 2006. In October at the end of the growing season, plants in each quadrat were

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clipped at the ground level. The clipped plants were separated according to species, and plant materials were oven-dried and weighed. 3.2. Results We found 23 different species in the survey of semi-natural grass^ and level of spatial heterogeland. The per-quadrat mean plant mass, λ, ^, of the superior 10 species in terms of the number of occurrences neity, p are shown in Table 1 (hereafter omitting the symbol ^ on λ and p). In this survey, Zoysia japonica showed a much higher λ and p than the other species. Only Z. japonica displayed p N 1; i.e., only this species exhibited a lower spatial heterogeneity than would be expected for a random pattern. The values of p for Agrostis clavata, Carex nervata, and Paspalum thunbergii were all close to unity (Ac, Cn, and Pt, respectively; Fig. 3a), indicating a nearly random pattern. All other species with low mass exhibited values of p much smaller than unity, indicating a high level of spatial heterogeneity. Fourteen additional species, with very small amounts of plant mass, exhibited high heterogeneity (additional plant species in Fig. 3a). Most of the intraclass correlation coefficients in plant mass between two adjacent cells were between −0.2 and 0.2. However, the following four species showed relatively high intraclass correlations: Z. japonica (0.434), Hydrocotyle sibthorpioides (0.687), Liriope minor (0.466), and Potentilla freyniana (0.294) (not shown in tables and figures). All of these species with a high intraclass correlation can grow and reproduce through both stolons and seeds. The high positive intraclass correlation may cause underestimation in the variance of plant mass and heterogeneity (p) in these species. Six species (Ixeris dentata, Z. japonica, Andropogon virginicus, Anthoxanthum odoratum, H. sibthorpioides, and L. minor) with sufficient occurrences (N26) for the goodness-of-fit test, fit the gamma distribution (Fig. 4). All species except for Z. japonica showed L-shaped gamma distributions because p b 1. Note that Z. japonica, which had a large plant mass, was ubiquitous, while other species with lower masses were unevenly distributed spatially (see “Occurrence” in Table 1). Seventeen species were identified in the sampled square of the weedy grassland, and we estimated the mean plant mass (λ) and heterogeneity (p) of each individual species (Table 1). Digitaria adscendens (Da) was comparatively ubiquitous and showed high mass, and Setaria viridis (Sv) had relatively high mass (Table 1, Fig. 3b). All species exhibited a value of p less than unity, indicating a tendency for more heterogeneous spatial patterns than would be expected in a random

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case. Of these species, 10 exhibited very low p values (b0.07) and very low plant mass [λ b 0.06 g (0.01 m2)−1; Fig. 3b]. We calculated the intraclass correlation coefficients in aboveground plant mass between two adjacent cells. The following four species showed relatively high correlations: S. viridis (0.226), Commelina communis (0.855), Equisetum arvense (0.506), and Erigeron philadelphicus (0.286). Using the MLE to estimate the parameters, we evaluated the frequency distributions of plant mass for D. adscendens, E. philadelphicus, and S. viridis, which had sufficient occurrences for the goodness-of-fit test (Fig. 5). All of these species formed L-shaped curves with values of p less than unity, and all cases fit well with the gamma distribution (Table 1). Spatial variation in species-to-species interactions within a community caused spatial variations in the carrying capacity of the logistic equation, as well as spatial variations in parameters such as soil fertility, microclimate, and grazing intensity. Actually, the following species combinations showed relatively high correlations in the semi-natural grassland [C. nervata/P. thunbergii (0.254), C. nervata/A. virginicus (0.257), A. odoratum/H. sibthorpioides (0.231), H. sibthorpioides/ A. clavata (0.285), and P. freyniana/A. clavata (0.500)] and in the weedy grassland [D. adscendens/S. viridis (0.205) and E. philadelphicus/ Cyperus microiria (0.240)]. If samples are collected under using random sampling, the confidence limits for λ and p are estimated as follows. As an example, we show confidence limits for Z. japonica, since the intraclass correlation (0.434) was higher than zero, meaning that the variance (0.493; Table 1) used in the calculation of confidence would be somewhat plimits ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ 1:390  1:96  0:493=100 ¼ 1:25−1:53g underestimated, as: λ ^ (Table 1), 1.96 is the value of the (0.01 m2)− 1 where 1.390 = λ cumulative normal distribution for the probability = 0.95, 100 is the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ 4:474  1:96  0:3871 ¼ 3:24−5:70; number of quadrats, and p ^; and 0.3871 = σ ^ p^ 2 calculated based on Eq. (6). where 4.474 = p 4. Discussion 4.1. Model The spatial patterns of biological populations and communities undergo spatio-temporal changes due to the environmental conditions such as physical, chemical and biological factors both outside and inside the population and community. Studies of these spatial patterns are key

Table 1 Species names, plant type, and parameter values estimated using the maximum likelihood method in the semi-natural grassland and weedy grassland. Species name, family name and plant type

Mean λ, DM g(0.01 m2)−1

Variance

p

Occurrencea

χ2 test

a. Semi-natural grassland Zoysia japonica Steud,; Poaceae, a shortgrass with clonal reproduction, perennial Paspalum thunbergii Kunth.; Poaceae with medium height, horizontal growth, perennial Andropogon virginicus L.; bunch type Poaceae with medium height, tufted type, perennial Liriope minor (Maxim.) Makino; Liliaceae with clonal reproduction, short height, perennial Anthoxanthum odoratum L.; Poaceae with short height, tufted type, perennial Hydrocotyle sibthorpioides Lam.; Umbelliferae with clonal reproduction, perennial Ixeris dentate (Thunb.) Makino; Compositae, anemochory perennial, small stump and short height Polygala japonica Houtt.; Polygalaceae, perennial, very small stump and short height Agrostis clavata Trin.; Poaceae with short height, untufted type, perennial Carex nervata Fr. et Sav.; Cyperaceae with short height, perennial

1.390 0.067 0.061 0.051 0.032 0.029 0.009 0.003 0.003 0.002

0.493 0.057 0.030 0.004 0.007 0.002 0.000 0.000 0.000 0.000

4.474 0.208 0.304 0.594 0.324 0.391 0.639 0.687 0.837 0.909

100 12 26 65 44 63 66 19 20 10

nsb – ns ns ns ns ns –c – –

b. Weedy grassland Digitaria adscendens (H.B.K.) Henr.; Poaceae with tall height, tufted type, annual Setaria viridis (L.) Beauv.; Poaceae with tall height; a weed of cultivated field, tufted type, annual Equisetum arvense L.; Equisetales; summer-green perennial, rhizoa reproduction Cayratia japonica (Thunb.) Gagnep.; Vitaceae, bindweed with a large volume, perennial Cyperus microiria Steud.; Cyperaceae with short height, annual Erigeron philadelphicus L.; Compositae, overwintering with small rosette, flowering in spring

2.466 0.458 0.082 0.067 0.014 0.011

5.605 0.795 0.044 0.063 0.001 0.000

0.602 0.359 0.479 0.215 0.413 0.605

89 41 22 15 46 70

ns ns – – – ns

a b c

The number of occurrences of each species per 100 quadrats. Not statistically significant for the goodness-of-fit test to the gamma distribution. Not tested because of insufficient number of occurrences for the χ2 test.

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Fig. 3. Observed examples of (λ, p) plots at the semi-natural grassland (a) and the weedy grassland (b). a-1. Observations from the semi-natural grassland. a-2. Same as Fig. 3a-1, but enlarged around the origin. b-1. Observations from the weedy grassland. b-2. Same as Fig. 3b-1, but enlarged around the origin. MLE and ME means estimated values based on the maximum likelihood and moment, respectively. Plant species with low mass (λ) and low values of p around the origin are omitted. The line of p = 1 indicates a random spatial pattern. Species names: Aa: Acalypha australis L., Ac: Agrostis clavata Trin., Ao: Anthoxanthum odoratum L., Ap: Artemisia princeps Pampan., Av: Andropogon virginicus L., Cc: Commelina communis L., Cj: Cayratia japonica (Thumb.) Gagnep., Cm: Cyperus microiria Steud., Da: Digitaria adscendens (H.B.K.) Henr., Ea: Equisetum arvense L., Ep: Erigeron philadelphicus L., Hs: Hydrocotyle sibthorpioides Lam., Id: Ixeris dentata (Thumb.) Nakai, Lj: Lysimachia japonica Thumb., Lm: Liriope minor (Maxim.) Makino, Oc: Oxalis corniculata L., Pc: Pleioblastus chino (Franch. et Savat.) Makino, Pj, Polygala japonica Houtt., Pt: Paspalum thunbergii Kunth, Sv: Setaria viridis (L.) Beauv., Vt: Vickie tetrasperma (L.) Schreb., Zj: Zoysia japonica Steud. (Editorial Committee of Dictionary of Terminology in Grassland Science, 2000).

to elucidating the organization and mechanisms operating in the biological population and community. In 1940s, research in this area began with studies examining the spatial patterns formed by individual biological organisms. In those studies of the occurrence of individuals, Poisson and negative binomial distributions were often applied as models in field studies of biology and agriculture (Bliss and Fisher, 1953; David and Moore, 1954; Feller, 1943; Fisher, 1941; Greig-Smith, 1964; Williams, 1964). On the other hand, the frequency of individuals having a specific attribute, such as a contagious disease within a family, is described using binomial distribution or beta-binomial distribution (Hughes and Madden, 1993; Kemp and Kemp, 1956a,b; Madden and Hughes, 1995; Shiyomi and Takai, 1979; Skellam, 1948). Similar to ‘discrete’ quantities such as the number of individuals and frequency of disease-carrying individuals in a family, biomass and cover, which are ‘continuous’ quantities, also form a spatial pattern within a population and community. Studies on the modeling of continuous quantities have delayed compared to those of discrete quantities. Various 'methods' of analyzing continuous quantities were published in the 1970s (e.g., Pielou, 1964, 1977), but explicit ‘models’ of continuous quantities are lacking.

In this study, we determined the random biomass pattern, which is a natural tool for judging the level of spatial heterogeneity of an individual species, that is, p = [mean of plant mass per quadrat]2 / [variance of plant mass among quadrats] = 1. The gamma distributions constructed based on several assumptions (Appendix A) statistically fitted well with the actual observed frequency distributions of the mass of individual plant species (Figs. 4 and 5), under conditions that the number of occurrence is sufficient to estimate the MLE and to statistically test the goodness-of-fit. Therefore, the assumptions contained in modeling a gamma distribution (Appendix A) are not far from the truth observed in nature. In field surveys of the spatial patterns of organisms, the quadrat size used is a subject of debate (e.g., Bonham, 2013; Fortin, 1999). Parameter values, such as the mean number of plants, the number of species, biomass per measurement unit and the species dispersion level, are affected by quadrat size. Harte et al. (2005) proposed a survey method to estimate community parameters for different quadrat sizes by analyzing small-scale species richness. Although a fixed quadrat size was used in the examples in this paper, extension to surveys with different quadrat-sizes is feasible by connecting adjacent quadrats.

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Fig. 4. Gamma distribution fit to the observed frequency distribution of plant mass data for the six species, with sufficient occurrences for goodness-of-fit test, identified in the semi-natural grassland. For the parameter values estimated and χ2 tests, see Table 1.

4.2. Use of index p to measure spatial heterogeneity in plant mass The value of p seems to be an effective index of the spatial heterogeneity of individual species in field surveys of actual grasslands, because the analysis of spatial patterns using p shows good agreement with our visual judgment. For example, in A. clavata (Ac), Polygala japonica [λ = 0.002 ± 0.001 g (0.01 m2)−1 (mean ± standard error), p = 0.073 ± 0.019], and I. dentata (Id), the fact that their individuals are small in

size and solitary in dispersion (i.e., grew without bunching) caused their lower heterogeneity, and these species showed relatively high values of p (Fig. 3a). In contrast, since A. odoratum (Ao) and H. sibthorpioides (Hs) generally form large clusters, they showed relatively low values of p (Fig. 3a). These apparent characteristics of species agreed with our judgment based on the index. The species behavior in the weedy grassland based on the (λ, p) plot (Fig. 3b) is also consistent with our visual judgment. It is, for example, visually confirmed that

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low frequency of occurrence and a low mass, distribute with high heterogeneity (i.e., low p). This indicates that the mass and frequency of occurrence of individual plant species relate strongly to the spatial pattern of that species. These facts indicate that heterogeneity is determined by both the intrinsic characteristics of individual species, such as reproductive type and life form, and the plant mass and frequency of occurrence of the species. A similar phenomenon was found in the relationships between the number of occurrences and the spatial heterogeneity value of individual species in a grassland described based on the beta-binomial distribution (Shiyomi et al., 2000) and the power law (Yasuda et al., 2003). These species-to-species plots of value of p against plant mass λ (as in Fig. 3) will aid ecological understanding of individual species' characteristics in plant communities. Analyzing the spatial pattern of the mass of individual plant species will not only contribute to our understanding of community characteristics, but also assist in the control and management of various biological processes such as succession, vegetation disturbance, biological diversity conservation, and in the diagnosis and prediction of vegetative succession. For example, spatial heterogeneity in plant mass is caused by patchy succession processes and local disturbances such as animal grazing, fire, salinization, invasion by alien species and intra-/interspecific competition (Martens et al., 2000). Spatial heterogeneity in plant mass will also affect biomass yield in grasslands (Day et al., 2003). Clarification of the cause-and-effect relationships between plants and the environment in these community processes, and how to manage this spatial heterogeneity, will contribute to grassland studies. Appendix A. Gamma distribution as the frequency distribution of per-quadrat mass of a species Suppose that the mass of a given plant species per unit area at time t is w = w(t), the growth rate of the species (dw) during an infinitesimal period of time (dt) is given as dw/dt. Let a, r′ and k′ be the defoliation rate of plant mass per unit area by senescence and grazing (for example, in a pasture) during dt, the growth coefficient and the carrying capacity of plant mass in the environment, respectively. We assume that all three coefficients, a, r′ and k′, for the plant species are constant. The soil fertility, water content of soil, micro-climate, grazing intensity and the numbers and densities of surrounding plant species affect the carrying capacity. The following logistic equation containing defoliation rate a (Shinozaki, 1976) is obtained: dw=dt ¼ r0wð1 –w=k0Þ–a:

ðA1Þ

Eq. (A1) is transformed to the following generalffi logistic form by the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi re-parameterization such that: c ¼ ð1−4a=r0=k0Þ; r ¼ cr0; k ¼ ck0 and z ¼ w–kð1–cÞ=ð2cÞ: dz=dt ¼ rzð1–z=kÞ: Next, we transform from time unit t to new time unit τ = rt/k, then the logistic equation simplifies: Fig. 5. Gamma distribution fit to the observed frequency distribution of plant mass data for three species, with sufficient occurrences for goodness-of-fit test, identified in the weedy grassland. For the parameter values estimated and χ2 test, see Table 1.

Cayratia japonica, which forms large patches covering undergrowth, exhibits high heterogeneity compared with E. philadelphicus, which disperses with pappus and produces small, solitary seedlings at random sites before winter. Actually, E. philadelphicus (Ep) showed relatively high values of p, whereas C. japonica (Cj) showed low values of p (Fig. 3b). We showed empirically that ubiquitous plant species with a large plant mass, such as Z. japonica (Zj) in Fig. 3a, distribute with low spatial heterogeneity (i.e., high value of p), whereas most plant species with

dz=dτ ¼ zðk–zÞ; z ¼ zðτÞ:

ðA2Þ

In actual grasslands, the carrying capacity, k, will stochastically vary among small-scale areas, because the soil fertility, water content of soil, microclimate, grazing, and the numbers and densities of surrounding plant species, such as inter- and intra-specific interactions, are different among small-scale areas. Therefore, in Eq. (A2), we make k = k0 + Λ (τ) where k0 is constant, and suppose that Λ (τ) randomly changes according to the normal distribution N(0, σ2), depending on new time unit τ. Then, Eq. (A1) is written as follows: dz=dτ ¼ zðk0 –zÞ þ zΛ ðτÞ:

ðA3Þ

J. Chen, M. Shiyomi / Ecological Informatics 24 (2014) 124–131

Here, we obtain the mean M and variance V of plant mass z dependent on time τ by applying Stratonovich's integral calculation (Zwillinger and Kokoska, 2000); i.e.,
 2 M ðz; τÞ ¼ z k0 –z þ σ =2 ; 2 2

V ðz; τÞ ¼ σ z :

ðA4Þ

By substituting Eq. (A4) for Kolmogorov's forward equation (Bharuchia-Reid, 1960; Boswell and Dennis, 1979), we obtain:   i ∂  ∂f ðz; τÞ 1 ∂2 h 2 2 1 2 f ð z; τ Þ ; σ z f ð z; τ Þ − −z þ ¼ z k σ 0 2 ∂z2 2 ∂τ ∂z

ðA5Þ

where f (z, τ) is the density function of plant mass per unit ground area at time τ. If we assume the plant mass is in an equilibrium state, Eq. (A5) is written as follows:   i 1dh 2 2 1 2 σ z f ðzÞ ¼ z k0 −z þ σ f ðzÞ; 2 dz 2

ðA6Þ

where f(z) is the density function of plant mass per unit ground area at the equilibrium state. Then, we obtain the explicit density function of f(z) of plant mass per unit ground area (z) from Eq. (A6) under the ∞ condition ∫ f ðzÞdz ¼ 1: 0

p−1 p

f ðzÞdz ¼

z p −pz=λ e dz; z ≥ 0; p N 0; λ N 0 ðgamma distributionÞ: Γ ðpÞλp

The equilibrium state (as shown in Eq. (A6)) would be attained when either the growth stage reaches maximum biomass in non-grazed grasslands (like the present weedy grassland), or when continuous grazing is carried out in a grassland (like the present semi-natural grassland), such that the conditions almost maintain a constant plant mass. References Bharuchia-Reid, A.T., 1960. Element of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York, pp. 167–234. Bliss, C.L., Fisher, R.A., 1953. Fitting the negative binomial distribution to biological data and a note on efficient fitting of the negative binomial. Biometrics 9, 176–200. Bonham, C.D., 2013. Measurements for Terrestrial Vegetation, 2nd edition. John Wiley & Sons, New York, pp. 10–13, (20–42, 186–188). Boswell, M.T., Dennis, B.C., 1979. Ecological contributions involving statistical distributions. In: Ord, J.K., Patil, G.P., Taillie, C. (Eds.), Statistical Distributions in Ecological Work. International Co-operative Publishing House, Fairland, pp. 196–260. Chen, J., Shiyomi, M., Hori, Y., Yamamura, Y., 2008. Frequency distribution models for spatial patterns of vegetation abundance. Ecol. Model. 211, 403–410. David, F.N., Moore, P.G., 1954. Notes on contagious distribution in plant populations. Ann. Bot. Lond. New Ser. 18, 47–53. Day, K.J., Hutchings, M.J., John, E.A., 2003. The effects of spatial pattern of nutrient supply on yield, structure and mortality in plant populations. J. Ecol. 91, 541–553. Editorial Committee of Grassland Science Dictionary (Japan), 2000. Dictionary of Grassland Science. Yokendo Publishers, Tokyo. Feller, W., 1943. On a general class of contagious distribution. Ann. Math. Stat. 14, 389–400. Fisher, R.A., 1941. The negative binomial distribution. Ann. Eugen. Lond. 11, 182–187. Fortin, M.J., 1999. Effects of quadrat size and data measurement on detection of boundaries. J. Veg. Sci. 10, 43–50.

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Gokalp, Z., Basaran, M., Uzun, O., Serin, Y., 2010. Spatial analysis of some physical soil properties in a saline and alkaline grassland soil of Kayseri, Turkey. Afr. J. Agric. Res. 5, 1127–1137. Greig-Smith, P., 1964. Quantitative Plant Ecology, 3rd edition. Butterworths, London, pp. 54–104, (171–289). Harte, J., Conlisk, E., Ostling, A., Green, J.L., Adam, B., Smith, A.B., 2005. A theory of spatial structure in ecological communities at multiple spatial scale. Ecol. Monogr. 75, 179–197. Hughes, G., Madden, L.P., 1993. Using the beta-binomial distribution to discreet-aggregated patterns of disease incidence. Phytopathology 83, 759–763. Kemp, C.D., Kemp, A.W., 1956a. Generalized hypergeometric distribution. J. R. Stat. Soc. B18, 202–211. Kemp, C.D., Kemp, A.W., 1956b. The analysis of point quadrat data. Aust. J. Bot. 4, 167–174. Kull, K., Zobel, M., 1991. High species richness in an Estonian wooded meadow. J. Veg. Sci. 2, 711–714. Madden, R.H., Hughes, G., 1995. Plant disease incidence: distribution, heterogeneity, and temporal analysis. Annu. Rev. Phytopathol. 33, 529–564. Mallants, D., Mohanty, B., Jacques, D., Jan, F., 1996. Spatial variability of hydraulic properties in a multi-layered soil profile. Soil Sci. 161, 167–181. Manurer, B.A., 1994. Geographical Population Analysis, Tools for the Analysis of Biodiversity. Blackwell Scientific Publications, London. Martens, S.N., Breshears, D.D., Meyer, C.W., 2000. Spatial distributions of understory light along the grassland/forest continuum: effects of cover, height, and spatial pattern of tree canopies. Ecol. Model. 126, 79–93. Morisita, M., 1962. Iδ-index, a measure of dispersion of individuals. Res. Popul. Ecol. 4, 1–7. Palmer, L.W., 1988. Fractal geometry: a tool for describing spatial patterns of plant communities. Plant Ecol. 75, 91–102. Pielou, E.C., 1964. The spatial pattern of two-phase patchworks of vegetation. Biometrics 20, 156–167. Pielou, E.C., 1977. Mathematical Ecology. Wiley-Interscience, NY, pp. 148–152, (118– 266). Roe, C.M., Parker, G.C., Korsten, A.C., Lister, C.J., Weatherall, S.B., Lodge, R.H.E.L., Wilson, J.B., 2011. Small-scale spatial autocorrelation in plant communities: the effects of spatial grain and measure of abundance, with a improved sampling scheme. J. Veg. Sci. 23, 471–482. Schabenberger, O., Gotway, C.A., 2005. Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC, Boca Raton, pp. 133–214. Shinozaki, K., 1976. Effect of removal on a growing system. Physiol. Ecol. Jpn. 17, 101–107. Shiyomi, M., 1981. Mathematical ecology of spatial pattern of biological populations. Bull. Natl. Inst. Agric. Sci. Ser. A 27, 1–29. Shiyomi, M., Takai, A., 1979. The spatial pattern of infected or infested plants and negative hyper-geometric series. Jpn. J. Appl. Entomol. Zool. 23, 224–229. Shiyomi, M., Akiyama, T., Takahashi, S., 1983. A spatial pattern model of plant mass in grazing pasture 1. J. Jpn. Grassl. Sci. 28, 372–381. Shiyomi, M., Takahashi, S., Yoshimura, J., 2000. A measure for spatial heterogeneity of a grassland vegetation based on the beta-binomial distribution. J. Veg. Sci. 11, 627–632. Shiyomi, M., Chen, J., Yasuda, T., 2010. Spatial heterogeneity in species richness and species composition. Grassl. Sci. 56, 153–159. Silvertown, J., Charlesworth, D., 2001. Introduction to Plant Population Biology, 4th edition. Blackwell Science, London, pp. 223–239. Skellam, J.G., 1948. A probability distribution derived from the binomial distribution by regarding the probability of success as variable between sets of trials. J. R. Stat. Soc. Ser. B 10, 257–261. van der Maarel, E., Sykes, M.T., 1993. Small-scale species turnover in a limestone grassland: the carousel model and some comments on the niche concept. J. Veg. Sci. 4, 179–188. van der Maarel, E., Noest, V., Palmer, M.W., 1995. Variation in species richness on small grassland quadrats: niche structure or small-scale plant mobility? J. Veg. Sci. 6, 741–752. Williams, C.B., 1964. Patterns in the Balance of Nature and Related Problems in Quantitative Ecology. Academic Press, London, pp. 192–268. Yasuda, T., Shiyomi, M., Takahashi, S., 2003. Differences in spatial heterogeneity at the species and community levels in semi-natural grasslands under different grazing intensities. Grassl. Sci. 49, 101–108. Zobel, M., Otsus, M., Liira, J., Moora, M., Möls, T., 2000. Is small-scale species richness limited by seed availability or microsite availability? Ecology 81, 3274–3282. Zwillinger, D., Kokoska, S., 2000. Standard Probability and Statistics, Tables and Formulae. Chapman & Hall/CRC, Boca Raton, 61-62, 462-463.