Conventional and Multifractal Approaches to Pedodiversity Analysis at Planetary Level

Pedodiversity and Biodiversity Area-Relationships at Planetary Level: What is the Connection?

F. Nachtergeale(1)J.J. Ibáñez(2,,*) , A. Brú (2), F. San José(3), J. Caniego(3)

(1)TechnicalOfficerLand Classification, Land and Plant Nutrient Management Service, FAO HQ Rome, Italy. .

(2) Centro de Ciencias Medioambientales, Consejo Superior de Investigaciones Científicas, Madrid, Spain.

(3) Dpto. de Matemática Aplicada a la Ing. Agronómica, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, Madrid, Spain.

*Corresponding author.

E-mail addresses:; ; (J.J. Ibáñez), (J. Caniego), (F. San José).

Fax number: (34)91 564 08 00 Teleph. number (34)91 745 25 00

Abstract

The study of the diversity of natural non-biological resources is in its infancy. Only in the field of the pedology diversity analysis begin to be considered as an interesting pedometrics tool. At the date, several studies show that biodiversity and pedodiversity seems follows the same patterns. One of them concern to the diversity-area relationships. This study examines pedodiversity and biodiversity-area relationships at global level using countries as geometric support. FAO soil datasets and IUCN-WCMC biological datasets has been used in this analysis. The results obtained show that biodiversity of some biological target groups (mammals, total birds, breeding birds, reptiles, amphibians, vascular plants, and the whole of vertebrates plus vascular plants) and pedodiversity are conform to power laws. Furthermore both diversity-area relationships at strongly correlated at global level. Thus, when a country is very diverse in a given taxa, also trend to be diverse in other taxa, including soils. Punctual divergences of power laws are explained in terms of environmental determinants. The results obtained seem corroborate habitat heterogeneity hypothesis but not the Theory of Island Biogeography. Implications of the results on management of natural resources are outlined. Because the Weibull distribution seems fit well to many biological and pedological datasets, a deeper study of the relations between this distribution and power laws are required.

Keywords: Pedodiversity-area relationships, biodiversity-area relationships, biodiversity-pedodiversity relation, statistical distribution models, power laws, Weibull distributions.

1. Introduction

Currently pedodiversity analysis began to be a growing industry (Table 1) (e.g. Ibáñez et al. 1990, 1995a, 1998, 2004a, 2004b; Phillips 1999, 2001, 2004; Guo et al. 2003a, 2003b; Amundson et al. 2003), and has been recognized as a novel pedometric tool (McBratney et al. 2000), as biodiversity analysis was decades ago. In similar way to biodiversity, pedodiversity is a frame type of spatial pattern analysis.This new topic has put forward some important problems from the theoretical and practical point of view. The regularities detected in pedodiversityanalysishave surprisingly parallelisms in biodiversity one (Johnson and Simberloff, 1974; Ibáñez et al.1998, 2004a). What are the reasons for these similarities? To which extend soils maybe the best single predictor of biotaxa richness? What are the implications for environmental management and assessment?

Table 1. Relevance of Biodiversity Analysis

Because it’sparamount importance in conservation biology, species-area relationships (SARs) studies is one of the topics of major interest in ecology (Rosenzweig 1995; Ibáñez et al. 2003). Thus, for example, SARs is used as a predictor of the number of species in a given area in order to select sites and design networks of biological reserves(Rosenzweig 1995; Whittaker 1998). These concepts have been applied to soils (Ibáñez et al. 1995a, 2004a and the references therein), considering the relative extension of each pedotaxon within a given region, at very disparate scales. Pedologists also begin to consider that the design of networks of soil reserves could be a very interesting way to preserve pedodiversity and belowground biodiversity (McBratney 1992; Ibáñez et al. 2003; Amundson et al. 2003).

Nowadays, ecologists search a biological explanation of the regularities detected using SARs. Two rival frameshave been proposed: The called Theory of Island Biogeography (MacArthur and Wilson, 1963, 1967) and the termed habitat heterogeneity hypothesis (Williamson 1981).

It has been found that biotic and abiotic resources follow similar patterns for islands (Ibáñez et al. 2004a, 2004b), drainage basins (Ibáñez et al. 2003, 2004b), European countries (Ibáñez at al. 1995b) United Estates of America (Guo et al 2003a) and plots in river terraces (Saldaña and Ibáñez 2004). In particular, the pedorichness-area relationship conform to the well establish pattern in ecology, the power-law model (May 1975). Moreover, the exponent of the power law, at least in archipelagoes coincides with the theory (Preston, 1960; MacArthur and Wilson 1963;1967; May 1975, 1981) and observations reported in the ecological literature (Rosenzweig,1995). Power laws, has been considered the sign of fractal structures(Mandelbrot, 1983), but not all of the datasets that fit to power laws are fractals (Schroeder 1991).

FAO databases include the necessary information in order to estimate the pedotaxa-area relationships by countries at world wide levels and coarse scales. In addition IUCN-WCMC database list the number of certain target taxa for the most of world countries. Thus, a joint analysis of both databases permits an analysis of how pedodiversity and biodiversity area relationships are correlated at global level using countries as a geometric support.

The main shortcoming of these databases consists in the use of countries as the basic units. Several environments, bioclimates, world geological regions, etc. are included in many countries (mainly in the larger ones). Thus it is possible carried on a finer analysis of the pedodiversity if natural units’ instead administrative boundaries were available, or using a scaling grid cell method and GIS tools. However that is not the case in biodiversity analysis. However also is true that the most inventories at global level also made use of these administrative boundaries. So this criticism did not only concern to pedology and biology. In contrast by the same reasons it is possible compare and analyses assemblage compositions of pedotaxa and biotaxa by countries at global level.Ibáñez et al. (1995b) show as in pedotaxa-area relationships in Europe, by countries is also conform to a power law with a exponent that is into the range obtained in biodiversity and pedodiversity literature for natural units.

The “Theory of Island Biogeography” (McArthur and Wilson 1963 1967) assumes that the number of species in a given habitat or island is the result of a dynamic equilibrium between immigration and extinction. The driving forces of this equilibrium are the proximity and magnitude of sources of immigrants. Two predictions of the “Theory of Island Biogeography”, among others, are: (i) the most widespread distribution abundance patterns of the species assemblages into island (as in mainland) communities is the canonical lognormal distributions (McArthur and Wilson 1965, 1967; May 1975), and (ii) the species’ numbers increase with the area according to a power law (Preston 1960; MacArthur and Wilson 1967; May 1975; Rosenzweig 1995 a huge of studies support the validity of the equations and assumptions of the theory (O’Neill 2000).

However, how we can see above there are, at least other line of reasoning, which differ from that of island biogeography: “habitat heterogeneity”. The habitat heterogeneity hypothesis claims that the number of species reflects the range of habitats, being the latter correlated with the area size (Williamson 1981; Cody 1983; Rafe et al. 1985). Ibáñez et al. (1990; 1995, 1998, 2003, 2004a; 2004b) shows as pedodiversity analysis corroborate the habitat heterogeneity hypothesis against the Theory of Island Biogeography. Because pedodiversity-area relationships follow a power law, with the same exponent value in islands, appealing to biological mechanisms it is not necessary. Thus, Williamson (1981) claim:“Should this pattern of environmental variation be found to be common, then at least some species-area variation found in plants could be ascribed reasonably directly to the environment in which they are growing, while that of animals might be ascribed either to the environment, or to the plants, or to both”. The same must be true for soils (Ibáñez et al 2000, 2001, 2003., 2001; Phillips 2001). In a similar way, Johnson and Simberloff (1974) observed in the British Isles that the number of soil types was the best predictor of the richness of vascular plants. They suggest that the importance of soils in determining island species number shows that the area could be a surrogate index of habitat (soils types) heterogeneity.

In this paper we show at world wide level, using countries as geometrical supports: (i) how biodiversity and pedodiversity are related; (ii) the fits of species-area relationships and some biological target groups to different statistical distribution models; (iii) the fits of pedotaxa-area relationship to different statistical distribution models; (iv) search mathematical similitude and differences between both spatial patterns; and (v) testing biological conjectures about species-area relationships (Theory of Island Biogeographyversus Habitat Heterogeneity hypothesis).

Material and Methods

Material

In order to get our objectives we hade use of the following datasets:

•Pedotaxa assemblages of the pedosphere by countries (all world countries). This dataset is the FAO Soil Database (unpublished) that has been kindly provided by Freddy Nachtergaele. Pedotaxa corresponds to the 2º level FAO units (nº = 131, plus 4 miscellany units)using the classification of 1974 (FAO 1974). The geometric support are administrative units, mainly countries, but also a few group of segregated regions due to social or bellic conflicts (e.g. Gaza, Golan, Kashmir), getting 209 units. Raw data in Km2 was transformed in percentages dividing the area covered by a given pedotaxon by the total country area.

•Species number of biological target taxa for 164 countries, using IUCN-WCMC database (Groombridge and Jenkins 2002). This database only present the results for countries largest than 5.000 Km2. The analysed taxonomic groups are the following: mammals, total birds, breeding birds, reptiles, amphibians, vascular plants, and the whole of vertebrates plus vascular plants.

Methods

Biologists analyze the relationship between the number of species and the area making use of two kinds of models: those that can be approximated by a linear relation between and , i.e. where and are constants (May 1975), and those which approximate to a linear relation between and , i.e. where and are constants (May 1975, 1981). Ibáñez et al. (2004a) emphasize that the first relation is a power function with exponent and prefactor () and the second is simply a logarithmic function of in terms of or alternatively an exponential function of in terms of . The former are termed will the power model and the latter the logarithmic model. Commonly biological and pedological datasets fit well to the power law and logarithmic models. However due to sampling deficiencies, it is common that biological and pedological inventories, as other many datasets fit well to power laws and to lognormal distributions (Korwin 1992). Furthermore, it has been demonstrated that the same dataset fit well to another statistical distribution models (Magurran 1988; Korwin 1992). For this reason, our datasets fit to more than forty different distribution models were performed using CurveExpert v.3.2™ software (; Regressions were calculated and correlation coefficients used to estimate the degree of each fit.A correlation matrix between all the variables considered was carried out using the STATISTICA v.6.0TM software (Statsoft Ltd

Results and Discussion

Table 2 shows the fit of the biological and pedological datasets to target statistical distribution models, as well as the best fit among the forty models tested using CurveExpert software. It is clear that the best fit of many biological taxa is the Weibull distribution (Weibull 1951) whereas pedotaxa and the whole of all vertebrates and plants (as indicator of global diversity) are conforming first to a power law. Brú and Ibáñez (2004)show that many datasets fit well simultaneously to the Weibull distribution and the power law. Furthermore is intriguing that this fact appear in several hundreds of web sites concerning with fractal topics (Brú and Ibáñez 2004). Furthermore, a comparative mathematical analysis between pedological and biological classifications, show that both mental constructs get better fits to the Weibull that to the lognormal and power law ones, against the results on this topic published in previous papers (Brú andIbáñez 2004; Ibáñez andRuiz-Ramos 2005). The question is that very scarce numbers of researches test the Weibull distribution in comparison with rival and most popular statistical distribution models.

Table 2. Testing the fit of biological and pedological datasets to different statistical distribution models using the number of pedotaxa and biotaxa by country

Although, the number of quotations on Weibull distributions and its interrelations with power laws, fractals and multifractals in the www and reputed international journals papers is huge (Table 3). Furthermore the number of accumulated quotations has been growth very fast, being conforming to a gaussian model (R = 0.9998), to a Power law (R = 0.999) andto a Weibull distributions (R = 0.996)themselves (Figure 1).It is noticeable that this increase start at the beginning of the seventies, just when the citations on fractals, power laws, and other abundance distribution models also growth with a similart pattern (unpublished data).This values are clear indicators of that a problem exists to discern between the power laws, Weibull distributions and lognormal distributions, among others.

Table 3. WWW Search Results on Statistical Distributions Models, Fractals and Multifractals with Special Emphasis on the Weibull distributions

(*) Nº. of Web pages and publications relate with the topic involved Science Direct, since the year 1823 (searching along all the text)

In our opinion, the study of the relationships between the latter distributions, fractals and multifractal deserve more attention. Thus, the problem to discern the best fit between different abundance distribution models is huge (Magurran 1988; Korwin 1992).

It is noticeable and intriguing that once again pedotaxa, as a non-biological resource follows the same pattern that the biological ones, as Ibáñez et al. (2004a and 2004b) shown in Aegean, British and Canary archipelagoes, among other sites.The same is true between biological and pedological classifications (Ibáñez andRuiz-Ramos 2005).

Table 4 shows the fit to a power law of taxa-area relationships to the tested pedological and biological datasets. The best fit to correspond to the pedotaxa at 2nd FAO level (FAO 1974). It is also noticeable that the value of the exponent obtained for pedotaxa is into the range of the values empirical obtained by biologists for islands (they consider that the observed values often fall in the range 0.2-0.4 are in agreement with the theory) (May 1975) and the near of the predicted value of 0.25 obtained mathematically by McArthur and Wilson (1963) and (May 1975). Ibáñez et al. (1995b) obtain exactly the same value exponent (0.21) and similar constant (1.12 for the European Union countries and 2.38 at world wide level), but they use the 1st FAO level istead the 2nd one. It is clear that all datasets get the exponent values obtained by the ecologists.

Table 4.Testing the fit of biological and pedological datasets to a power law using the number of pedotaxa and biotaxa by country

(*) where S = Number of species or pedotaxa; A = area, and R = coefficient of correlation.

Table 5 shows the correlation matrix between all the variables analyzed (nº taxa by country) for world countries that have this information (145 = 75% of the population). The most interesting information is that all the variables are significantly correlated among them. Thus, as in the case of the British (Johnson and Simberloff 1974) and Canary Islands (Ibáñez et al. 2004b), biodiversity of target groups and pedodiversity are correlated. Furthermore, against the interest that ecologist has on biodiversity area-relationships, pedodiversity is the variable that shows a strongest correlation with the area. This fact could be of interest in conservation biology. Because many countries have not completed biodiversity inventories, it is permissible to use pedodiversity as a surrogate indicator of aboveground biodiversity. If in the future soil biologists show that there is a positive and strong correlation between the number of belowground biological species and the number of pedotaxa, it will be also possible use pedotaxa richness as surrogate indicator of soil biodiversity. We must also have into account that, at the date there is not a single site where scientists are compiling an inventory of the whole biodiversity. Thus, the information on soil biodiversity is very poor.