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European Summer School on Soil Survey Pedometrics

tools for the analysis of soil typological maps

and georeferenced soil databases

- J.J. Ibáñez (CCMA)
- Centro de Ciencias Mediambientales
- Consejo Superior de Investigaciones Científicas

Madrid (Spain)

In Memory

- Fridland (Russian Pedologist)
- Hole and Campbell (American Pedologists)
- Soil Pattern Analysis (Soil Maps)

Statistical Tools in Pedometrics

- Typological Analysis of Pedotaxa and soilscapes

This Dissertation - Continuum approach Geostatistics of single soil

properties - Hybrid Methods Typological Analysis

Geostatistics (into each soil polygon) - Other Methods Fractals, Non Lineal Dynamics,

Fussy Sets, Neuronal Networks, Numerical

Taxonomies

Premises Against the opinion of some

pedometrisians, without formation of the state of

the art in biological taxonomies, biodiversity

and conservation biology, we could consider soil

types or pedotaxa in similar fashion to biotaxa

(including ecosystems and habitats). There are

any epistemologic arguments against this

perspective. The continuum dilemma and the

spatial delimitation of natural bodies affect,

both to pedologic and biologic entities. We

recognize that, as is the case of ecosystems,

habitats and some species, the boundaries of

pedotaxa are fuzzy. Furthermore, the hierarchies

of biological taxonomies and the delimitation /

characterization of many biotaxa are also, at

least, fuzzy and arbitraries. We recommend to

these incredulous pedometrisians to consult the

abundant and suitable biological literature.

Introduction

- Conceptual parallelisms
- We need only count
- 1. The different classes of objects
- 2. The number of individuals or coverage in each

classes - Biotaxa Pedotaxa
- Ecosystems Soilscapes
- Ecoregions Soil Regions
- Biomes Pedomes
- Ecosphere Pedosphere

Introduction

- Data source
- 1. Soil Surveys (Maps)
- 2. Geographical Soil Databases (Inventories)
- 3. Soil Information Systems

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The Taxonomic Vision of the Pedometrisians

Slatter Philosophy

From Soil Maps to Soil Resources Map

- Measuring pedosphere surface (non trivial)
- Measuring pedosphere volume (non trivial)
- Pedosphere structure (typological approach)
- Pedosphere properties (continuum appr.)
- Measuring pedosphere vulnerability (risks)
- Measuring pedosphere state (deterioration)

Introduction

- Index and models of diversity have proved to be

valuable in ecology. By using them, ecologists

have studied the structure and organization of

ecological entities at different scales (from

plots to biomes). - These methodologies have been applied to the

study of abiotic structures of landscapes on very

few occasions, despite their potential.

Introduction

- At the same time, this may be one of the ways to

explore, quantify and compare the complexity

abiotic landscapes in different areas and

environments. - This is a type of spatial pattern analysis

Information Lost (Map Generalization and Scales

The concept of diversity Huston (1995) "The

concept of diversity has two primary components,

and two unavoidable value judgments. The

primary components are statistical properties

that are common to any mixture of different

objects, whether the objects are balls of

different colors, segments of DNA that code for

different proteins, species or higher taxonomic

levels, or soil types or habitat patches on a

landscape. .Each of these groups of items has

two fundamental properties 1. the number of

different types of objects (e.g., species, soil

types) in the mixture or sample and 2. The

relative number or amount of each different type

of object. The value judgements are .whether

the selected classes are different enough to be

considered separate types of objects and

.whether the objects in a particular class are

similar enough to be considered the same type.

On these distinctions hangs the quantification

of biological diversity".

Preliminaries on Diversity Analysis From a

methodological point of view, the different ways

of measuring diversity may be grouped into three

categories - indices of richness number of

categories (e.g. biological species, communities,

pedotaxa, soilscapes, etc.) known to occur in a

defined sampling area - indices based on

proportional abundance of categories not only

the number but also their relative abundance (in

our case, the relative area occupied by each

pedotaxa) are taken into account - models of the

distribution of abundance of categories these

provide the most complete description but also

the least abridged. Hollow curves Measuring

conectance and pedocomplexity Species-area

relationships (Spars) Species-time relationships

(Divergent Pedogenesis) Species-energy

relationships Nested subset analysis Area

Selection Methods etc. .

The Hollow Curves

Hollow curves Everywhere

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Ranked-Abundance Plot of Island Areas in Km2

Hollow curve

Ranked-Abundance List (area in Km2) of

taxonomic pedotaxa Hollow curve

Pedotaxa

Area(Km2)

Ic

7303,65

Lc

5061,71

Rc

3267,15

Ie

2589,8

RO

1503,48

Jc

982,91

Be

751,21

Bc

654,07

Bk

556,63

Id

415,35

Lv

332,2

To

301,8

I

211,25

E

186,26

Bd

134,07

Lo

121,31

U

113,22

Vc

98,51

IRO

72,24

Bh

68,52

Oe

29,33

Gc

28,83

Re

19,5

Richness indexes

- For many scientists the concept of diversity is

restricted to an inventory of the number of

different objects ("richness") present in a

certain ecosystem, landscape, survey or taxonomic

scheme. - It is possible to distinguish between
- "numerical richness"
- "species density".

- Numerical richness is the ratio between the

number of different soil taxa found and that of

the soil profiles analyzed.

In soil surveys

- Taxa density refers to the number of soil taxa

per sampling area

Indices based on the proportional abundance of

objects. (Diversity sensu stricto)

- Is the most frequent way of estimating diversity.

- Diversity s.s. may be divided into two elements.
- "richness"
- "evenness"
- The concept referring to the relative abundance

of each object. - This distinction is logical, since for two

different plots of land with the same area and

identical richness, the most diverse will be that

where the different types occupy equal area and

are equally probable therefore.

A summary of the characteristics and performance

of a range of diversity statistics

Diversity concept (sensu stricto) (2)

- Diversity indexes most used in ecology come from

the Theory of Information. The most popular is

the - Shannon Index (Shannon and Weaver, 1949)
- Diversity is thus equated with the amount of

uncertainty that exists regarding the species

(objects) of an individual selected at random

from a population. The more species there are,

and the more even their representation, the

greater the uncertainty and hence the greater the

diversity. - Information content, which is a measure of

uncertainty, is a reasonable measure of diversity

Diversity concept (sensu stricto) (3)

- The mathematical expression of the Shannon's

Index is - where
- H' is the negative entropy "negentropy" or

diversity of the population - pi is the proportion of individuals found in this

ith taxa. - In fact, the true value of pi is unknown, and it

is estimated by ni/N, where ni is the number of

individuals of the object considered, and N the

total number of individuals collected (it may

also be the percentage of surface area occupied

by this ith object). - The value of H' is the sum of the proportions of

the individual objects multiplied by the negative

logarithm of the proportion. - Information is maximum when the probabilities

(proportional abundance) of all objects (S) are

equal. It is then equal to the ln S. Information

is 0 if there is only one possibility (one

object), i.e. diversity is 0.

Example of a diversity analysis on the

composition of a hypothetical soilscape

diversity statistics and abundance distribution

model (best fit by chi-squared test). N Number

of observations S Richness H' Negentropy

index E Evenness Hmax Maximum negentropy.

Diversity concept (sensu stricto) (4)

- The value of the Shannon Index is usually between

1.5 and 4.5. - In nature it has not been possible to detect

values larger than 5 bits although it is

mathematically possible ( why?). - Although the Shannon index considers evenness, it

is also possible to measur it independently. - For certain conditions of richness (S), maximum

possible negentropy Hmax occurs in situations

where all objects are equiprobable - H' Hmax lnS (2)
- The relation between negentropy observed and

maximum negentropy may be used, therefore, as a

measure of evenness E - E H/Hmax H/ lnS (3)
- The E index can take any value between 0 and 1,

where 1 represents the situation in which all

species or objects are equiprobable (e.g. when

they occupy the same area) and tend to 0 when

there is a highly non-uniform distribution of

relative abundance (i.e. where one object

dominates over all others).

Some limitations of diversity analysis

- Estimates of diversity can be made on information

from several scales or different intensity

sampling. For example, as a consequence of map

generalization, considerable information is lost

(e.g., taxa with scarce spatial representation

are not usually considered). Thus, the results

obtained varies according to the quality and

quantity of the available information . - The figures obtained for richness and entropy

indices are determined by the maximum number of

taxa which the classification used allows.

Why Drainage Basins

- According to the Horton Laws size-frequency

distributions of drainage basins are conform to

power laws. - They are also fractal structures

The fractal nature of river networks and the

Horton Laws (2)

- For many purposes and increase of the order (or

rank) of a basin could be understood like an

increase of complexity. - Many branching systems follow Horton Laws. For

example - Mammalian circulatory and respiratory systems
- Plant vessel-bundle vascular system
- Plant radical systems
- In these disciplines the Strahler classification

also has been used with profusion.

Horton Laws are an example of space-filling

fractals. This is a natural and common structure

for ensuring a very fast and economic flow (and

sometimes exportation) of matter.

In space filling fractals energy dissipated to

distribute resources is minimized.

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The fractal nature of river networks and the

Horton Laws (3)

- Horton Law of Stream number
- Nk RB(s-k) (1)

Where Nk the number of streams of order K Rs

the branching ratio (a constant characteristic of

the basin) S the order of the basin

Horton Law of Stream Length Lk RL(k-1)L1

(2)

Where Lk average lengh of Kth order stream RL

stream lengh ratio, a constant for any given basin

Horton Law Stream Slopes S'k Rs (1-k) S'1 (3)

Where S'k is the average slope of the Kth

order streams Rs (1-k) is the stream slope

ratio ( a constant)

Where A'K is the average area drained by each

Kth order stream RA(k-1) the stream area ratio

Schumm Law of Stream Area A'K RA(k-1)A'1 (4)

Here Ak is defined to include not only the area

drained directly by the particular stream but

also that drained by all its tributaires of lower

order

The fractal nature of river networks and the

Horton Laws (4)

- The fact that the variables Nk, L'k, S'K and A'K

are all related to stream order by either a

direct or an inverse geometric series suggest

that they are also related to each other by

simple power laws (Allometric Growth Law). For

example, by eliminating K between equations 2 and

4, we find - L's/L'1 (A'K/A1)n (5)
- Where n ln RL / ln RA (6)
- Hack Law of Mainstream Lengh-Basin Area
- L CAn' or empirically L 1.4A2/3 (7)
- Where
- L mainestream lengh
- C a constant with experimental figure ? 1.4 and

n' ? 0.6 - Therefore the river open system grows

allometrically according to the general power law - Y axb

Pedodiversity at scale 1 20.000 (Drainage basins)

Pedodiversity and Diversity of Plant Communities

in the Iberian Peninsula according to Basins Rank

Pedodiversity and Diversity of Plant Communities

in the Iberian Peninsula according the to size of

the basins

Diversity, drainage basins in geomorphology and

pedology (1)

- Fluvial patterns are filling fractal trees (see

the nature of Horton Laws) - The evolution of fluvial systems (increase of

fractal dimension of fluvial courses) induces an

increase in complexity of pedogeomorphological

landscapes and biological ones.. - Fluvial incision (chronosequences) also increases

negentropy and richness of geomorphological and

soil landscapes.

Diversity, drainage basins in geomorphology and

pedology (2)

- In high structured drainage basins, diversity and

richness of soils and geomorphologic units

increases from top (mainly erosive) to bottom

(mainly depositional). - There are close connections between the land's

fractal dimension and its geomorphologic and

pedological negentropy (Shannon diversity Index). - Are Drainage basins and their soil and

geomorphologic cover "dissipative systems" ?. - Increase of soilscapes, geomorphologic landscapes

and plant landscapes diversities, in

correspondence with drainage basins

hierarchisation, may be scale invariant.

Pedodiversity at Global Level

Pedodiversity at Continental Level

Aegean Archipelago Correlation between two

hierarchical levels (FAO)

Pedodiversity and Richness in the Aegean

Archipelago

Pedodiversity in plots of different soilscapes

(Terrace chronosequence in the Henares River

Edafodiversidad a pequeñas Escalas Cronosecuencia

del Río Henares

Genetic Pedodiversity

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Diversity statistics for observed horizon

sequences and their absolute and relative

frequency for the Abal soil series Belgium

(original data from Van Orshoven et al., 1991).

Measuring Complexity and Connectance Graphos

Measuring Complexity two approaches

Pedocomplexity and Connectance (Henares River

dataset)

Beta (Differentiation) Diversity

Example of the application of how inventory

diversity and differentiation diversity concepts

could be applied in 1250000 scale natural region

soil map (the hypothetical are covered by the

soil map is 1 Millon ha)

Cascade hierarchy of diversities

Beta Diversity Analysis Pedodiversity at Global

Level

Beta Diversity Analysis Pedodiversity at Global

Level

Abundance Distribution Models

Abundance Distribution Models

- Abundance distribution models (e.g., geometric

series, logseries, lognormal distributions, gamma

distributions, power laws, broken stick models,

etc.) are statistical tools which ecologists have

applied for decades with the object of analyzing

the intrinsic regularities of various ecological

entities. - It would be interesting to apply these techniques

in pedology with the object of detecting the

similarities and differences between the entities

of both natural resources. - The same is true respect other abiotic resources

such as, geology, geomorphology, climatology, and

so on. - The two former will give approximately straight

lines when log abundance is plotted against rank

whilst the latter shows up as a straight plot

when abundance is plotted against log rank

Rank-Abundance Plots

Rank-Abundance Plots

Abundance Distribution Models Geometric Series

Summary of the most popular abundance

distribution models in ecology Taking into

account the underlying fragmentation mechanisms

of each model, the broken-stick model is the

statistical realistic expression of an

intrinsically uniform distribution and in the

opposite the geometric and log series models the

statistical expression of an uneven distribution.

The log normal model is in most respects

intermediate between broken-stick and geometric

or log series extreme models.

Geometric series model (1) This model (Motomura

1932) is based in the assumption that one entity

is subdivided recursively following a fixed rule

for a given fraction k. If the entity is

idealized by the unit segment, the first fragment

measures k, the next one is a fraction of what

remains, then it measures k(1-k) and so on. It is

obtained the series k, k(1-k), k(1-k)2,. ,

k(1-k)i-1. It is a geometric series for the first

under the hypothesis that the abundance of the

taxa are proportional to the numbers of the above

series, they correspond to the ranked-abundance

list as proportions. For a community of NT

individuals, NT Ck k(1-k)i-1 is the size of the

i-th most abundant taxa, where Ck is a constant

which value is being ST the total number of

taxa, in order to ensure that

Logseries model (1) Suppose that the mechanism of

the geometric series model stems from the fact

that the taxa arrive at successive uniform time

intervals and proceed to preempt a fraction of

the remaining niche before the arrival of the

next, then, the randomization of the time

intervals leads to a log series distribution

(Fisher 1943). According to the log series

distribution the frequency of species with N

individuals is for N1,2, where

and are constants. Then, it is a

discrete distribution.

Geometric vs. Logarithmic series

Abundance Distribution Models Lognormal Series

Log normal model (1) It is a mixture of

geometric and broken stick models. There are

recursivity and randomization. The segment of

unit length is broken, at a randomly chosen

point, into two parts. One part is chosen

randomly and independently of its length, and is

broken into two parts chosen one point of its

points randomly. One of the three resulting parts

is chosen randomly and independently of its

length, and is broken into two parts chosen one

point of its points randomly. And so on. Then,

the distribution of the lengths of all the parts

tends to log normal form.

Log normal model (2) If we now assume that the

species in a multy-speciose community have

divided up some limiting resource among

themselves in this manner, and that the abundance

of each taxa is proportional to its share of the

resource, then the taxa-abundance distribution is

log normal. This fragmentation process has been

applied previously in soil sciences in connection

with aggregates and particle-size distribution

(Hatch 1933 Kolmogorov 1941, among others). In

this case, the density function is where

and are the random variable mean and

standard deviation of It is also used the

parameter ? to characterize the distribution.

Abundance Distribution Models Broken Stick model

Summary of the most popular abundance

distribution models in ecology Broken-stick

model In this case the unit segment is broken,

randomly and at once, among the number of

observed taxa for taxa, points are located

randomly in the unit interval so that the

predicted sizes are proportional to the sizes of

the obtained subintervals. As before, it may be

computed the abundance distribution. In this

case, the density function is a negative

exponential with parameter for

Diversidad Area en el tiempo y diferentes

geoambientes

Distribution Abundance Models

- Geométric Series
- Logarithmic Series
- Lognormal Distr.
- Broken Stick Model
- Others
- Power Law
- Fractals

- ? Small samples
- ? Environmental Perturbations
- ? Larger non-perturbed Communities
- ? SPARs and SPITs
- ?Ecological Dynamics
- ? Evolutive Dynamics
- ?General System Dyn.
- ? Fit Problems

Observed and expected values of the cumulative

distribution functions of pedotaxa area

Pedotaxa Area

Observed

Expected

Log-Normal

Logarithmic

Geometric

Broken Stick

20

1

1,727

8,987

0,199

0,423

29

2

2,511

9,894

1,641

0,610

30

3

2,594

9,977

1,773

0,631

69

4

5,255

12,025

5,006

1,426

73

5

5,478

12,164

5,224

1,505

99

6

6,774

12,915

6,407

2,017

114

7

7,420

13,262

6,955

2,307

122

8

7,740

13,429

7,218

2,460

135

9

8,227

13,677

7,611

2,706

187

10

9,863

14,476

8,875

3,662

212

11

10,513

14,783

9,362

4,105

302

12

12,372

15,643

10,736

5,618

333

13

12,885

15,879

11,115

6,110

416

14

14,041

16,414

11,979

7,362

557

15

15,505

17,108

13,112

9,278

655

16

16,281

17,489

13,741

10,470

752

17

16,916

17,810

14,277

11,548

983

18

18,072

18,424

15,317

13,756

1504

19

19,665

19,363

16,967

17,298

2590

20

21,242

20,474

19,077

20,917

3268

21

21,762

20,907

19,979

21,889

5062

22

22,519

21,631

21,678

22,790

7304

23

22,962

22,127

23,101

22,974

Distribution functions of the models and

observed values of pedotaxa area

Ranked-Abundance List of the models

and of observed values

(Logarithmic scale)

Diversity and Area For many conservation

biologists, one of the most important empirical

observations in ecology is that larger areas

contain more species than smaller areas. For

understanding how communities are organized or

what the consequences of reserve design are for

the maintenance of biodiversity, any relationship

is more informative. Yet an understanding of

what determines species-area curves remains

elusive at the date.

F(N) Total number of taxa with size

in excess of N (km2) Species or Peodotaxa

accumulation curves (Aegean Isles

The theory of island biogeography The theory of

island biogeography (MacArthur 1960, 1965,

MacArthur and Wilson 1963, 1967) has been for

decades the keystone in biodiversity analysis on

islands, as well in conservation biology. This

theory rest on the assumption that the number of

species residing in a habitat is the result of an

equilibrium between immigration and extinction.

A goal of the theory is the explanation of the

dependence of species richness upon such factors

as area and the proximity and magnitude of

sources of immigrants.

The Predictions of the theory of island

biogeography 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 (i) The species

numbers increase with the area according to a

power law. Much interest has centred on the value

of p, the slope of the ln S versus ln A

regression. the pedorichness-area relationships

conform to a power law whose exponent is around

the value of 0.25. An impressive body of field

data has been accumulated to support the validity

of the equations and assumptions of the theory.

Observed values often fall in the range 0.2-0.4

Habitat heterogeneity However the mechanism

leading SPARs are at best obscure. There are

currently other alternatives, which differ from

that of island biogeography. One of them

focussed on habitat-heterogeneity a larger

area encompasses more diverse habitats, which

support more species. Thus, habitat

heterogeneity approach would argue that the

number of species reflects the range of habitats

included in the area.

Richness-area curves (1) The dependencies of the

number of species (or the number of other

different taxonomic categories) on the area

(referring in particular to islands and other

analogous habitats of patchy spatial occurrence),

that are explored in ecological literature, may

be classified into two kind of models. Those

that could be approximated by a linear relation

between log S and log A (log S p log Aq for

certain constants p and q ) and those which

approximation is a linear relation between S and

log A (S a log Ab for certain constants a and

b ). First relation is a power function with

exponent p and prefactor ( ). Second

relation is simply a logarithmic function of S in

terms of A (or an exponential function of A in

terms of S ). The first will be called the

power model and the latter the logarithmic model.

Richness-area curves (3) The log normal and the

broken-stick models yield power curves. For the

log normal model the exponent is related with

parameter by the expression (May 1975)

(eq. 1) For the

broken-stick model . Let us remark

that for the lognormal model, the smallest

exponent is obtained for ,

that is to say, for the so called canonical

lognormal model. The geometric and log series

model yield logarithmic expressions for the

richness-area curves. It holds (May 1975) that

(eq. 2) Summarizing, while a

logarithmic expression directly results from the

log series distribution of species-abundance

combined with the assumption that population size

scales linearly with habitat area, the canonical

log normal distribution with the same assumption

leads to a power expression.

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Pedodiversity-Area Relationships

- Continents, Drainage, Isles, Basins at Different

Scales and Plots

Power laws fit of pedotaxa of the Aegean

Archipelago (1.000.000)

Power Law Fits of Pedotaxa and Vascular Plants In

Brithish Isles

Drainage Basins at 11.000 Scale in Spain

pedodiversity-area relationships and

biodiversity-area relationships

Spars for Drainage basins at the scale 120.000

Pedosphere Power Laws and different Scales From

14M to 1 500m

Result of power law fits of pedotaxa-area

relationships for the six tested datasets

Nested Subset Analysis

Nested Subsets and taxa-range size distribution

- A common pattern of taxa distribution has been

termed nested subsets by ecologists - Species present in any particular biological

assemblage tend also to be present in larger

assemblages. - The same is true for the so called species-range

size distribution - Locally abundant species tend to be widespread,

whereas locally rare species tend to be narrowly

distributed

Nested Subset Matrix

Non-Perfect Nested Matrix

Results of the nested subset analysis for the

Aegean Archipelago

Results of Nested Calculator for the six tested

datasets

Hollow curve and power law for pedo-taxa range

size distribution in Aegean Archipelago

Pedotaxa number of presences in the more of 120

analyzed islands

Conclusions The regularities detected in the

pedological entities tested are surprisingly

similar to those in ecological literature. Both,

diversity-area relationships, abundance

distribution models, nested subsets, taxa

range-size distributions appear to follow the

same trend in ecology and pedology. Thus, for

example, our results suggest that the biological

assumptions underlying the Theory of Island

biogeography could not be the cause of the last

relation and the generalized value of the 0.25

for the exponent. Thus, in contrast with the

generalized ecologist opinion, neither power laws

for SPARs, Nested subsets, fractals structures,

and log-normal distributions for DEMs etc, do not

appears to be a property of the structure of the

communities and the spatial distribution of the

species respectively.

Other Typological Pedometrics Methods

- Will be analysed in the dissertation on

Biodiversity Loss

Power Laws and Fractals

Complementarity Methods for the Design of

Networks of Natural Soil Reserves

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The End Fin

Why Landmasses

- According to Mandelbrot (among others), the

size-frequency distribution of landmasses fit to

a power law - size-frequency distribution of islands in a given

archipelago is an example for small landmasses - That is a problem of fragmentation

Why Drainage Basins

- The surface fragmentation of landmasses in

drainage basins is an example of fractal

structures - The channels are also filling fractal trees

Background (1) Williamson (1981) found an

almost perfect linear relationship between the

logarithm of the number of geological types and

the logarithm of the mapped area (a power law)

using a geological map. He concluded that

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 (see also Rafe et al.

1985).

Background (2) Cody (1983) observed that bird

diversity on islands in the Gulf of California

increase in a stepwise fashion as island area

increased. He suggested that this was controlled

by the geomorphology of drainage basins. On

different-sized islands larger islands could

support higher-order rank channels and therefore

a greater range of riparian habitats than smaller

islands (see also Ibáñez et al. 1990 and 1994).