For about one century the
mathematical laws of the relationship between the number of species and the
size of areas have been researched on plant communities, on the vegetation of
larger areas, and on animal stocks. In doing so, power functions (Arrhenius 1920a,b,
1921, Preston 1962a,b), more or
less modified logarithm functions (Archibald
1949, Williams 1995, Schmitt 2001), and asymptotic functions
(Archibald 1949, Williams 1995, Schmitt 2001) are used as descriptive models for the approximation of the data.
Arrhenius
(1921), making use of the probability calculus, already pointed out a conceptional model, which opens up the
relationship between the number of species and the size of area. Similar models
were developed by Kylin (1926), Blackman (1935), Sanders (1968), Hurlbert
(1971), Coleman (1981), Bammert (1992), Williams (1995) and Ugland
et al. 2003. The relationship between these models was mostly unclear
till now.
Generally
speaking, a consistent distinction between descriptive and conceptional models
according to Wissel (1992a), which
would make it possible to point out the respective methodical strengths and weaknesses, is not made.
To
record the number of species depending on the size of area, mostly nested areas
or separate (i.e. non overlapping) areas („islands“)
are chosen. Only since Pielou
(1975) described the problem, particular attention has been paid to the problem
that these cases have to be distinguished with regard to content and, if
necessary, to terminology, too. However, until recently there have been
irritations regarding these requirements which are even intensified by the
fact that for both described cases there exist several sampling schemes, which
again have different effects on the speciesarea relationship. There are also
differences concerning the question under what circumstances a plant can be
regarded as belonging to an area: Here you can refer to the vertical projection
of above ground parts of a shoot (shootincidence) or to the existence of a
rooted plant in the area (rootedincidence). For the speciesarea data the
incidence definition is of great importance, at least for relatively small
areas. An additional problem arises from the fact that the conditions, under
which the models are being developed and used, are not always described in a
sufficiently clear manner. Thus, for example, the regression calculation is
sometimes used in a dubious way or an unjustified extrapolation beyond the
field of validity is made. The important question, which common qualities the
areas should have to design a fitting speciesarea function, is not asked in
all cases, so that the results are often not comparable. Homogeneity is
described by many authors as a desirable quality of the area being researched,
but after decades of attempts at definitions meanwhile the opinion prevails
that there is no acceptable and satisfying solution and that therefore
homogeneity has to be dealt with intuitively.
The
difficulties in researching the connection between the number of species and the
size of area do not exist, because not enough valuable knowledge has been
assembled but because the approaches, the test objects, and the aims are so
diverse. As a result, there are large gaps in the definition basis, the general
idea and the knowledge of the connections. Therefore the objective of this
Ph.D. thesis is the attempt to fill such gaps and to formulate a theory to the
overall complex. Abb. 78
shows in an overall view, by what partial steps this objective can be reached.
I
propose to call the scientific investigation of the number of species speciometry, and to call the
investigation of the dependence of the number of species on the size of area geospeciometry. Geospeciometry
requires a clear definition basis. This is difficult as an entire system of
terms has to be created, which is logical, consistent and complete (for the
purpose being planned). It has to integrate all previous approaches as far as
possible and make them comparable to each other. To begin with, individual,
incidence, sampling scheme, speciesarea curve and species richness are key terms:
· The term ”individual”
for plants is often problematical. Following the concept of Keys & Harper (1974) I use the term ”ramet” instead. Plants then appear as monoramets or
polyramets.
· For the modelling of speciesarea
functions a special definition of incidence is indispensible: I regard a
species as being present in an area, if there exists at least one point of presence
(= centre of the crosssection of a stalk or trunk at the ground) of this
species in this area (pointincidence).
· For the sampling scheme and its
empiric speciesarea curves I propose a strictly dichotomic system of
alternatives, in this version containing and unambiguously naming
7 models, but also being expandable, if necessary. Then problems with
forming the average can be discussed without reference to this. But it is
important to devise a system which unites as many procedures used in
geospeciometry as possible and thus provides the basis for a comparative
consideration.
· The discussion about the different
models of speciesarea functions, depending on the sampling scheme, is being
used as an opportunity to draw up a compelling terminological system: So the
speciesarea functions  in the broader sense  are being subdivided into speciesaccumulation
curves (for nested areas) and true speciesarea curves (for separate areas).
· Because of the nonuniform use of
the term ”species richness” in the literature, a clarification
setting is necessary, in which I am referring to Hurlbert (1971) and Magurran
(2004). In a pragmatic approach to species richness  in the broader sense  a distinction is being made between species
richness  in the narrower sense  (referring to individuals) and species density (referring to areas).
To study the
complete complex of the speciesarea relationships, the problem of homogeneity has to be dealt with.
Especially for plant formations, for which naturally an assignment to individuals
is often difficult, the previous approaches with methods of probability theory
are not sufficient. These approaches, however, make up the raw material, from
which a full concept of homogeneity can be formed. In the definition of a grid homogeneity, being presented in this Ph.D. thesis,
examples of the literature are being used and joined together and the dispersion
model, widely used since Svedberg
(1922), is being integrated, too:
· To avoid the difficulty of
individual reference I refer to areas, which are covered by a grid. The plant
species are then being represented by their incidences in the area (Blackman 1935 can be regarded as a
predecessor).
· Otherwise I follow Bammert (1992) who divides homogeneity
up into two components (intraspecific and interspecific). Apart from that it is
necessary to describe in precise mathematical terms what is often casually
called „distribution at random“ and „independent“.
· Distributing the incidences
„randomly“ in a grid, a hypergeometric distribution is
being produced. As a consequence, the Poissondistribution, being used by those
advocating the theory of the dispersion models, and also the binomial
distribution present themselves as more or less good approximations.
To draw up
a (true) speciesarea curve of islands, a common characteristic of the islands
is needed too, on which a model can be constructed. Requiring a homogenous vegetation on the whole area of all islands
would not be appropriate. I am introducing the term ”vegetation equality” instead, which, in
analogy to the homogeneity, shows an intraspecific and an interspecific
component. As a sample for the equality of vegetation two islands can serve,
which have the same plant communities with equal share of area.
Homogeneity
and vegetation equality are categorial quantities and theoretical constructs.
The problem of the existence of homogeneity or vegetation equality, however,
can practically be checked for real formations, too. For that purpose some
tests to be used on a sample area have been developed. It can be shown, that in
the sense of the definition there exist homogeneous areas resp. areas with
equal vegetation in nature, although these are rare exceptions. Much more
frequent are those cases, in which nearly all species and sets of species
(namely all except a few nondominant species) fulfil the criteria. These are exactly
the areas having the socalled „visual homogeneity“ (Dengler 2006). The terms
„quasihomogeneity” (Rauschert
1969) resp. „quasivegetation equality” refer to such cases. From this it
becomes clear that there are natural grades of homogeneity, although so far
there are no satisfactory solutions for the recording of such intermediate
levels. Furthermore by the definition it is clearly to be seen, that
homogeneity and vegetation equality are qualities with regard to a particular
grid size.
The variety of models and methods is a strength of speciometry. But this becomes a weakness, if
no instruments for comparing and arranging data are available. Mathematical
tools of this kind are now being provided in this Ph.D. thesis. There exist
functions, by which the increase of particular base functions can be described:
The parameters t (tau) and D (delta) are being defined as
follows depending on the area size A
(S(A)= species number in dependence on the
area size):
and .
The t and D terminology is being presented for the first
time in this Ph.D. thesis. The conception based on it is being connected to
the findings of some authors, that by doubling the area size, the power
functions produce constant factors and the logarithmic functions produce
constant summands of the species numbers.
The many
descriptive models of the species area curves, being used until now, are standing
next to each other without any closer relationship and without any basis of comparison.
It is an aim of this Ph.D. thesis to provide a summary theory to this problem.
Essential tools for that are the parameters t and D. In addition six axioms have been
formulated, which give the mathematical qualities of species area functions
being desired and considered to be important. In all descriptive models known
until now (power functions, logarithmic functions, and asymptotic functions),
the fulfilment of these axioms has been checked and the properties of the t and D functions have been analysed. In that way an exact view of the increasing qualities of the
potential modelling functions has been developed. It became apparent
that the power functions, the logarithmic functions, and the asymptotic
functions can be put into an order on the basis of their increase and that –
according to the current findings – the set of these function models seems to
be sufficient for the approximation of all species area data, under the
condition that there are at least approximatly homogeneous areas resp. areas
with equal vegetation.
The
qualities of all descriptive modelling functions can now be summarized in just
one function form (”universal form”)
,
in which t(A), with values between 1 and 2, decreases
strictly monotonously or stays constant with increasing area size. This
approach is a generalization of the socalled power law of the
speciesarea relationship, which is based on a constant t and permits the connection to the entire scientific experience,
being found in the literature. The approach presented here, however, is more
versatile and more flexible, especially if it can be assessed, in which order
of magnitude t is varying. From this a lot of
application possibilities can be derived in the field of floristic mapping, for
example the following questions could be answered:
· Which part of flora has been
recorded, if the area can be examined just partly?
· What effect does an alteration of the
grid field size have on the species number?
· The Messtischblätter of
Assuming,
that the species area data follow a power function, questions like the
following ones can be answered, too, by corresponding ideas of the model:
· What effect does nonhomogeneity
have on the speciesarea functions?
· How can a speciesarea function,
basing on the pointincidence, be applied to rootedincidence or
shootincidence?
The null
models (conceptional models) for the speciesarea relationship, being
described in the literature, partly stand next to each other without an inner
connection, partly there has not been discussed, which mathematical kind of
function they are to be assigned to. Until now the relationship with the
descriptive modelling functions is unknown. Finding answers to these open
questions is the purpose of this Ph.D. thesis.
As a
starting point of my own derivation of the speciesarea functions, which is
based on probability theory and assumes homogeneity, two sorts of ballot box
tests are being used: with and without repetition. Taking into consideration
the different sample designs you get four functions, namely the Coleman
function, the Blackman function (being named by me), the Ugland function (being
named by me), and the Hurlbert = rarefaction function. Their relationships to
each other, their mathematical classification, their qualities, and their
relationship to the descriptive models can be studied in that way. It could
even be shown that the Ugland function and the rarefaction function are
identical. It can be shown as well, how the differing approaches of Kylin (1926), Bammert (1992), and Williams
(1995) relate to the four basic models.
By means
of the t and Dfunction the descriptive and the conceptional
species area functions can be linked together. Basing on the conceptional
models, the effect of different frequency distributions on the shape of the
species area curve can be studied more precisely by simulation. In addition by
comparing the conceptional models with the empirical data, new scientific
findings for the homogeneity problem are being produced, and a criterion of
homogeneity is being yielded (“Emra” test).
Furthermore
procedures are being presented, by which a bijection between speciesarea
curves and frequency distribution of the species being found can be produced:
From the frequency distribution conclusions can be drawn as to the accompanying
speciesarea functions and vice versa. The alteration of frequency
distribution by increasing the area size and the relationship with the
frequency law of Raunkiaer can be studied, too.
The
leading principle with the descriptive models as well as with the conceptional
models is the gradual adaptation of the conditions being found under ideal
conditions (homogeneous areas resp. areas with equal vegetation) to actual
conditions, characteristic of nature.
Starting
out from this basis certain methods can be analyzed more precisely (e.g.
Hobohm‘s a diversity, standardlines
procedures by van der Maarel) and open questions, for example those of the
asymptotic speciesarea functions, of the socalled Sshaped curves, and of the extrapolation of
species numbers, can be arranged into a general view. Taking up the topic of diversity I suggest to
use a vector of species density and tvalues to characterize plant
communities. The tmethod even allows a relation to
the problem of the minimal area size
of a plant community.
Finally in a general survey those elements are
presented, which can be regarded as essential elements of theory of geospeciometry. Actually all common methods can
successfully be summarized either under the umbrella of the descriptive models
or under that of the conceptional models. Establishing some connections between
both models as a total system their individual legitimacy can be emphasized.
The most important and also actually new instrument for the theory of the
geospeciometry is the tfunction, by means of which the increase of
the speciesarea functions can be studied. From that point of view it seems to
be justified to call this the tapproach or the ttheory of
geospeciometry.
This Ph.D. thesis is expected to prepare the
ground for a theoretical speciometry.
At the same time the practical
orientation is being looked for. Important computation methods are being
carried out for data of my own or for those available from the literature.
The
basic elements of geospeciometric theory, which are explained in this Ph.D.
thesis, mostly are in accordance with the latest scientific findings. They are
compatible with all essential methods, being used until now, and in many
respects they even provide a belated confirmation of procedures, being used
more intuitively until now, for example the assumption of the power function
for speciesareadata.
Finally
it is possible to show, that speciometry can be applied to topics of completely
different fields, e.g. to the number of different words in dependence on the
length of the text.

Theory of geospeciometry 

Assumptions (Kap. 5) Concept of ramets (as alternative to the
concept of individuals), definition of pointpresence (with
relationship to rooted or shootpresence), definition of nested / disjunct areas, system of sample designs, accumulation curves / true speciesarea
curves, systematic of calculating the average of
speciesarea data, definition of species richness, species density (Kap. 11.1). 
Homogeneity and vegetation equality (Kap. 6) Homogeneity: definition using concept of
gridpresence, random distribution, independence, vegetation equality: concept of
gridpresence. 

Parameter of increase t, D (Kap. 7.1, 7.3) Definition and application, characteristic of increase of power, logarithm
and asymptotic functions. 

descriptive models (Kap. 7) Axioms 1–6, functions of modelling: power, logarithm and asymptotic functions
universal form, degree of increase. 
Conceptional models (Kap. 8) 


with repetiton 
without repetiton 

islands 
Colemanmodel Ia 
Uglandmodel IIa, Hurlbertmodel IIb are equal: rarefactionmodel. 

nested areas 
Blackmanmodel Ib 

approximation 
Kylinmodel I‘ 
 

Relationships between descriptive and conceptional models (Kap. 8.7) Mutual complementing, verification, investigation of the inner structures of the
speciesarea functions and the frequences, the effect of extreme frequences. 

Frequence and distribution of frequences (Kap. 10) Isomorphism of probability of presences and
frequences, relationship between speciesarea functions
and distribution of frequences. 
Applications Simulation as addition
to the empiric investigation of vegetation (Kap. 7.9, 8.5.2), t, Emratest to homogeneity resp. to inhomogeneity (Kap. 7.7, 8.7.1), application to floristic mapping (Kap. 7.11), asymptotic and Sshaped speciesarea functions (Kap. 9.1), minimal area using tconcept (Kap. 9.2), investigation of existing procedures, e.g. procedure of standard lines of van der
Maarel (Kap. 11.3.2), index of diversity using tconcept (Kap. 11.3.3). 
Abb. 78: Overall
view of those parts of this thesis PhD., which are
important for theory of geospeciometry.
Christensen, E. (2007): Eine Theorie zur Beziehung zwischen Artenzahl und Flächengröße. – Mitt. AG Geobot. Schl.Holst. Hamb. 64, 296 S., Kiel., ISSN: 03448002.
Zu beziehen über: AG Geobotanik in Schl.Holst. u. Hamburg, ÖkologieZentrum, Olshausenstr. 75, D 24098 Kiel. Preis 7,50 Euro + Versandkosten
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