Summary

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 species-area 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 (shoot-incidence) or to the existence of a rooted plant in the area (rooted-incidence). For the species-area 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 species-area 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, species-area 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 species-area 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 cross-section of a stalk or trunk at the ground) of this species in this area (point-incidence).

·      For the sampling scheme and its empiric species-area 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 species-area functions, depending on the sampling scheme, is being used as an opportunity to draw up a compelling terminological system: So the species-area functions - in the broader sense - are being subdivided into species-accumulation curves (for nested areas) and true species-area curves (for separate areas).

·      Because of the non-uniform 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 species-area 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 Poisson-distribution, 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) species-area 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 non-dominant species) fulfil the criteria. These are exactly the areas having the so-called „visual homogeneity“ (Dengler 2006). The terms „quasi-homogeneity” (Rauschert 1969) resp. „quasi-vegetation 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 so-called power law of the species-area 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 Germany (maps 1:25.000) are a little bit different in their sizes. What is the effect on their species numbers?

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 non-homogeneity have on the species-area functions?

·      How can a species-area function, basing on the point-incidence, be applied to rooted-incidence or shoot-incidence?

The null models (conceptional models) for the species-area 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 species-area 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 D-function 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 species-area curves and frequency distribution of the species being found can be produced: From the frequency distribution conclusions can be drawn as to the accompanying species-area 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, standard-lines procedures by van der Maarel) and open questions, for example those of the asymptotic species-area functions, of the so-called S-shaped 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 t-values to characterize plant communities. The t-method 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 t-function, by means of which the increase of the species-area functions can be studied. From that point of view it seems to be justified to call this the t-approach or the t-theory 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 species-area-data.

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 point-presence (with relationship to rooted- or shoot-presence),

definition of nested / disjunct areas,

system of sample designs,

accumulation curves / true species-area curves,

systematic of calculating the average of species-area data,

definition of species richness, species density (Kap. 11.1).

 

Homogeneity and vegetation equality (Kap. 6)

Homogeneity: definition using concept of grid-presence, random distribution, independence,

vegetation equality: concept of grid-presence.

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

Coleman-model Ia

 

Ugland-model

 IIa,

Hurlbert-model IIb

are equal:

rarefaction-model.

nested areas

Blackman-model Ib

 

approximation

Kylin-model I‘

--

Relationships between descriptive and conceptional models (Kap. 8.7)

Mutual complementing, verification,

investigation of the inner structures of the species-area 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 species-area functions and distribution of frequences.

 

Applications

Simulation as addition to the empiric investigation of vegetation (Kap. 7.9, 8.5.2),

t-, Emra-test to homogeneity resp. to inhomogeneity (Kap. 7.7, 8.7.1),

application to floristic mapping (Kap. 7.11),

asymptotic and S-shaped species-area functions (Kap. 9.1),

minimal area using t-concept (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 t-concept (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: 0344-8002.

Zu beziehen über: AG Geobotanik in Schl.Holst. u. Hamburg, Ökologie-Zentrum, Olshausenstr. 75, D 24098 Kiel. Preis 7,50 Euro + Versandkosten

 

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