The Concept of Evenness/Unevenness: Less Evenness or More Unevenness?

While evenness is understood to be maximal if all types (species, genotypes, alleles, etc.) are represented equally (via abundance, biomass, area, etc.), its opposite, maximal unevenness, either remains conceptually in the dark or is conceived as the type distribution that minimizes the applied evenness index. The latter approach, however, frequently leads to conceptual inconsistency due to the fact that the minimizing distribution is not specifiable or is monomorphic. The state of monomorphism, however, is indeterminate in terms of its evenness/unevenness characteristics. Indeed, the semantic indeterminacy also shows up in the observation that monomorphism represents a state of pronounced discontinuity for the established evenness indices. This serious conceptual inconsistency is latent in the widely held idea that evenness is an independent component of diversity. As a consequence, the established evenness indices largely appear as indicators of relative polymorphism rather than as indicators of evenness. In order to arrive at consistent measures of evenness/unevenness, it seems indispensable to determine which states are of maximal unevenness and then to assess the position of a given type distribution between states of maximal evenness and maximal unevenness. Since semantically, unevenness implies inequality among type representations, its maximum is reached if all type representations are equally different. For given number of types, this situation is realized if type representations, when ranked in descending order, show equal differences between adjacent types. We term such distributions "stepladders" as opposed to "plateaus" for uniform distributions. Two approaches to new evenness measures are proposed that reflect different perspectives on the positioning of type distributions between the closest stepladders and the closest plateaus. Their two extremes indicate states of complete evenness and complete unevenness, and the midpoint is postulated to represent the turning point between prevailing evenness and prevailing unevenness. The measures are graphically illustrated by evenness surfaces plotted above frequency simplices for three types, and by transects through evenness surfaces for more types. The approach can be generalized to include variable differences between types (as required in analyses of functional evenness) by simply replacing types with pairs of different types. Pairs, as the new types, can be represented by their abundances, for example, and these can be modified in various ways by the differences between the two types that form the pair. Pair representations thus consist of both the difference between the paired types and their frequency. Omission of pair frequencies leads to conceptual ambiguity. Given this specification of pair representations, their evenness/unevenness can be evaluated using the same indices developed for simple types. Pair evenness then turns out to quantify dispersion evenness.

Severe limitations of the FEve metric of functional evenness and some alternative metrics.

The metric of functional evenness FEve is an example of how approaches to conceptualizing and measuring functional variability may go astray. This index has several critical conceptual and practical drawbacks: Different values of the FEve index for the same community can be obtained if the species have unequal species abundances; this result is highly likely if most of the traits are categorical.Very minor differences in even one pairwise distance can result in very different values of FEve.FEve uses only a fraction of the information contained in the matrix of species distances. Counterintuitively, this can cause very similar FEve scores for communities with substantially different patterns of species dispersal in trait space.FEve is a valid metric only if all species have exactly the same abundances. However, the meaning of FEve in such an instance is unclear as the purpose of the metric is to measure the variability of abundances in trait space. We recommend not using the FEve metric in studies of functional variability. Given the wide usage of FEve index over the last decade, the validity of the conclusions based on those estimates is in question. Instead, we suggest three alternative metrics that combine variability in species distances in trait space with abundance in various ways. More broadly, we recommend that researchers think about which community properties (e.g., trait distances of a focus species to the nearest neighbor or all other species, variability of pairwise interactions between species) they want to measure and pick from among the appropriate metrics.

Factorization of joint metacommunity diversity into its marginal components: an alternative to the partitioning of trait diversity.

Diversity in metacommunities is traditionally viewed to consist of the diversity within communities ([Formula: see text]) that is complemented by the differences between communities ([Formula: see text]) so as to result in the total diversity ([Formula: see text]) of the metacommunity. This perception of the partitioning of diversity, where [Formula: see text] is a function of [Formula: see text] and [Formula: see text] (usually [Formula: see text] with all components specified as effective numbers), has several drawbacks, among which are (1) [Formula: see text] is an average that can be taken over communities in many ways, (2) complete differentiation among communities cannot always be uniquely inferred from [Formula: see text] and [Formula: see text], (3) different interpretations of [Formula: see text] as effective number of communities (e.g., distinct or monomorphic) are possible, depending on the choice of ideal situations to which the respective effective numbers refer, and (4) associations between types (species, genotypes, etc.) and community affiliations of individuals are not explicitly covered by [Formula: see text] and [Formula: see text]. Item (4) deserves special regard when quantifying metacommunity diversity. It is argued that this requires consideration of the joint distribution of type-community combinations together with its diversity (joint diversity) and its constituent components: type and community affiliation. The quantification of both components can be affected by their association as realized in the joint distribution. It is shown that under this perception, the joint diversity can be factorized into a leading and an associated component, where the first characterizes the minimum number of communities required to obtain the observed joint diversity given the observed type distribution, and the second specifies the effective number of types represented in the minimally required number of communities. Multiplication of the two yields the joint diversity. Interchanging the roles of community and type, one arrives at the dual factorization with leading minimum number of types and associated effective number of communities. The two dual factorizations are unambiguously defined for all measures of diversity and can be used, for example, to indicate structural characteristics of metacommunities, such as type differentiation among communities and associated type polymorphism. The information gain of the factorization approach is pointed out in comparison with the classical and more recent modified approaches to partitioning total type diversity into diversity within and between communities. The use of factorization in analyses of latent community subdivision is indicated.

Effects of reproductive resource allocation and pollen density on fertilization success in plants.

BACKGROUND: Declining resources due to climate change may endanger the persistence of populations by reducing fecundity and thus population fitness via effects on gamete production. The optimal mode of generative reproduction allocates the limited resources to ovule and pollen production in proportions that maximize the number of fertilized ovules in the population. In order to locate this optimum and derive reproduction modes that compensate for declined resources to maintain reproductive success, a model of gamete production, pollen dispersal, and ovule fertilization is developed. Specification of opportunities for compensation is given priority over specification of physiological or evolutionary mechanisms of adaptation. Thus model parameters summarize gametic production resources, resource investment per gamete, resource allocation as proportion of resources invested in ovules, and pollen density as size of the pollen dispersal range and proportion of pollen retained within the range. Retained pollen disperses randomly, and an ovule is fertilized if at least one pollen settles on its surface. The outcome is the expected number of fertilized ovules. RESULTS: Maximization of fertilization success is found to require the investment of more gametic production resources in ovules than in pollen, irrespective of the parameter values. Resource decline can be compensated by adjusting the resource allocation if the maximum expected number of fertilized ovules after the decline is not less than the expected number the population experienced before the decline. Compensation is also possible under some conditions by increasing the pollen density, either by raising a low pollen retention or by shrinking the dispersal range. CONCLUSION: Fertilization success in populations affected by resource decline may be maintainable by adjustment of the sexual allocation of gametic production resources or by increasing pollen density. The results have implications for insect pollination, sexual allocation bias, management measures, and metapopulation fragmentation.

Effective numbers in the partitioning of biological diversity.

Admissible measures of diversity allow specification of the number of types (species, alleles, etc.) that are "effectively" involved in producing the diversity (the "diversity effective number", also referred to as "true diversity") of a community or population. In metacommunities, effective numbers additionally serve in partitioning the total diversity (symbolized by γ) into one component summarizing the diversity within communities (symbolized by α) and an independent component summarizing the differences between communities (symbolized by ß). There is growing consensus that the ß-component should be treated in terms of an effective number of "distinct" communities in the metacommunity. Yet, the notion of distinctness is shown in the present paper to remain conceptually ambiguous at least with respect to the diversity within the "distinct" communities. To overcome this ambiguity and to provide the means for designing further desirable effective numbers, a new approach is taken that involves a generalized concept of effective number. The approach relies on first specifying the distributional characteristics of partitioning diversity among communities (among which are differentiation, where the same types tend to occur in the same communities, and apportionment, where different types tend to occur in different communities), then developing the indices which measure these characteristics, and finally inferring the effective numbers from these indices. MAJOR RESULTS: (1) The ß-component reflects apportionment characteristics of metacommunity structure and is quantified by the "apportionment effective number" of communities (number of effectively monomorphic communities). Since differentiation between communities arises only as a side effect of apportionment, the common interpretation of the ß-component in terms of differentiation is unwarranted. (2) Multiplicative as well as additive methods of partitioning the total type diversity (γ) involve apportionment effective numbers of communities that are based on different apportionment indices. (3) "Differentiation effective numbers" of communities exist but do not conform with the classical concept of partitioning total type diversity into components within and between communities. (4) Differentiation characteristics are measured as effective numbers of distinct types (rather than communities) from the dual perspective, in which the roles of type and community membership are exchanged. This is relevant e.g. in studies of endemism and competitive exclusion. (5) For Shannon-Wiener diversity, all of the differentiation and apportionment effective numbers are equal, with the exception of those representing additive partitioning. (6) Under either perspective, that is dual or non-dual, measures of compositional differentiation (as originally suggested for the assessment of ß-diversity) do not figure in the partitioning of total diversity into components, since they do not build on the intrinsic concept of diversity.

Classifying measures of biological variation.

Biological variation is commonly measured at two basic levels: variation within individual communities, and the distribution of variation over communities or within a metacommunity. We develop a classification for the measurement of biological variation on both levels: Within communities into the categories of dispersion and diversity, and within metacommunities into the categories of compositional differentiation and partitioning of variation. There are essentially two approaches to characterizing the distribution of trait variation over communities in that individuals with the same trait state or type tend to occur in the same community (describes differentiation tendencies), and individuals with different types tend to occur in different communities (describes apportionment tendencies). Both approaches can be viewed from the dual perspectives of trait variation distributed over communities (CT perspective) and community membership distributed over trait states (TC perspective). This classification covers most of the relevant descriptors (qualified measures) of biological variation, as is demonstrated with the help of major families of descriptors. Moreover, the classification is shown to open ways to develop new descriptors that meet current needs. Yet the classification also reveals the misclassification of some prominent and widely applied descriptors: Dispersion is often misclassified as diversity, particularly in cases where dispersion descriptor allow for the computation of effective numbers; the descriptor GST of population genetics is commonly misclassified as compositional differentiation and confused with partitioning-oriented differentiation, whereas it actually measures partitioning-oriented apportionment; descriptors of ß-diversity are ambiguous about the differentiation effects they are supposed to represent and therefore require conceptual reconsideration.

The analysis of association between traits when differences between trait States matter.

Because of their elementary significance in almost all fields of science, measures of association between two variables or traits are abundant and multiform. One aspect of association that is of considerable interest, especially in population genetics and ecology, seems to be widely ignored. This aspect concerns association between complex traits that show variable and arbitrarily defined state differences. Among such traits are genetic characters controlled by many and potentially polyploid loci, species characteristics, and environmental variables, all of which may be mutually and asymmetrically associated. A concept of directed association of one trait with another is developed here that relies solely on difference measures between the states of a trait. Associations are considered at three levels: between individual states of two variables, between an individual state of one variable and the totality of the other variable, and between two variables. Relations to known concepts of association are identified. In particular, measures at the latter two levels turn out to be interpretable as measures of differentiation. Examples are given for areas of application (search for functional relationships, distribution of variation over populations, genomic associations, spatiogenetic structure).

Distribution of variation over populations.

Understanding the significance of the distribution of genetic or phenotypic variation over populations is one of the central concerns of population genetic and ecological research. The import of the research decisively depends on the measures that are applied to assess the amount of variation residing within and between populations. Common approaches can be classified under two perspectives: differentiation and apportionment. While the former focuses on differences (distances) in trait distribution between populations, the latter considers the division of the overall trait variation among populations. Particularly when multiple populations are studied, the apportionment perspective is usually given preference (via F(ST)/G(ST) indices), even though the other perspective is also relevant. The differences between the two perspectives as well as their joint conceptual basis can be exposed by referring them to the association between trait states and population affiliations. It is demonstrated that the two directions, association of population affiliation with trait state and of trait state with population affiliation, reflect the differentiation and the apportionment perspective, respectively. When combining both perspectives and applying the suggested measure of association, new and efficient methods of analysis result, as is outlined for population genetic processes. In conclusion, the association approach to an analysis of the distribution of trait variation over populations resolves problems that are frequently encountered with the apportionment perspective and its commonly applied measures in both population genetics and ecology, suggesting new and more comprehensive methods of analysis that include patterns of differentiation and apportionment.

Relational diversity.

In biology, the measurement of diversity traditionally focusses on reporting number of unambiguously distinguishable types, thus referring to qualitative (discontinuously varying) traits. Inclusion of frequencies or other weights has produced a large variety of diversity indices. Quantitative (continuously varying) traits do not readily fit into this perspective. In fact, in the context of quantitative traits, the concept of diversity is not always clearly distinguished from the (statistical) notion of dispersion. In many cases the ambiguity even extends to qualitative traits. This is at variance with the broad spectrum of diversity issues ranging, e.g., from ecological and genetic aspects of diversity to functional, structural, systematic, or evolutionary (including phylogenetic) aspects. In view of the urgent need for a more consistent perspective, it is called to attention that all of these aspects, whether of qualitative or quantitative nature, can be gathered under the common roof of binary relations (for qualitative traits two objects are related, for example, if they share the same trait state). A comprehensive concept of (relational) diversity can be developed in two steps: (1) determine the number of unrelated pairs of objects among all admissible pairs as a measure of implicit (relative) diversity, (2) invoke the concept of effective number to transform the implicit measure of diversity into an explicit (absolute) measure. The transformation operates by equating the observed implicit diversity to the implicit diversity obtained for the ideal model of an equivalence relation with classes of equal size. The number of these classes specifies the effective number as an explicit measure of diversity. The wealth of problems that can be treated from this unified perspective is briefly addressed by classifying and interpreting established diversity indices in the light of relational diversity. Desirable applications to the above-mentioned aspects are specified with the help of types of relations such as order, hierarchical, and tree relations. Corresponding biological issues including taxonomic community diversity, mating system, food web, sociological, cladistic and phylogenetic, or hypercycle diversity are suggested for future consideration.

Measuring differentiation among populations at different levels of genetic integration.

BACKGROUND: Most genetic studies of population differentiation are based on gene-pool frequencies. Population differences for gene associations that show up as deviations from Hardy-Weinberg proportions (homologous association) or gametic disequilibria (non-homologous association) are disregarded. Thus little is known about patterns of population differentiation at higher levels of genetic integration nor the causal forces. RESULTS: To fill this gap, a conceptual approach to the description and analysis of patterns of genetic differentiation at arbitrary levels of genetic integration (single or multiple loci, varying degrees of ploidy) is introduced. Measurement of differentiation is based on the measure Delta of genetic distance between populations, which is in turn based on an elementary genic difference between individuals at any given level of genetic integration. It is proven that Delta does not decrease when the level of genetic integration is increased, with equality if the gene associations at the higher level follow the same function in both populations (e.g. equal inbreeding coefficients, no association between loci). The pattern of differentiation is described using the matrix of pairwise genetic distances Delta and the differentiation snail based on the symmetric population differentiation DeltaSD. A measure of covariation compares patterns between levels. To show the significance of the observed differentiation among possible gene associations, a special permutation analysis is proposed. Applying this approach to published genetic data on oak, the differentiation is found to increase considerably from lower to higher levels of integration, revealing variation in the forms of gene association among populations. CONCLUSION: This new approach to the analysis of genetic differentiation among populations demonstrates that the consideration of gene associations within populations adds a new quality to studies on population differentiation that is overlooked when viewing only gene-pools.

Detecting local establishment strategies of wild cherry (Prunus avium L.).

BACKGROUND: P. avium, a pioneer tree species that colonizes early forest successional stages, is assumed to require an effective strategy allowing stably repeatable rounds of local establishment, dispersal and local extinction. Consequently, the early replacement of cherry by climax tree species makes the establishment of several local generations very unlikely, especially in central European continuous cover forests. This has to be seen in connection with the mixed reproduction system involving asexual reproduction as a complementary adaptational strategy. Tests of the local establishment of wild cherry must therefore consider the possibility of first generation establishment via seedling recruitment potentially followed by an asexual generation (root suckering). Successful establishment can therefore be determined only among adult individuals with the option of detecting vegetative reproduction at these stages. To test the implied suggestion about local establishment strategies of wild cherry, nuclear microsatellites were used to analyse patterns of asexual propagation among adult stages that have been subjected to one of two major types of forest management. These management types, the historical "coppice with standards system" (CWS) and the "high forest system" (HFS), can be reasonably assumed to have affected the reproduction system of P. avium. RESULTS: Clear differences were found in the reproduction pattern between two stands representing the two forest management types: 1) Clonal propagation is observed in both management systems, but with a distinctly higher frequency in the CWS. Hence, sexual recruitment as a first local generation is followed by a second asexual generation in both, whereas in the CWS there is evidence for an additional clonal generation. 2) The estimation of amounts of clonal reproduction critically depends on the assumptions about multilocus gene associations. This is revealed by the application of newly developed methods of quantifying gene associations. 3) Haplotype diversities are higher in the CWS and found to be associated with a large degree of heterozygosity for the second largest clonal group. 4) Seed set was sparse over the last eight years of observation in the CWS stand. CONCLUSION: This study provides useful guidelines for more comprehensive investigations, particularly on the interrelationships between degrees of cloning and capacity of sexual reproduction, amounts of multilocus gene associations, effects of heterozygosity on cloning success, and sustainability of different forest management types.

The isolation principle of clustering: structural characteristics and implementation.

The isolation principle rests on defining internal and external differentiation for each subset of at least two objects. Subsets with larger external than internal differentiation form isolated groups in the sense that they are internally cohesive and externally isolated. Objects that do not belong to any isolated group are termed solitary. The collection of all isolated groups and solitary objects forms a hierarchical (encaptic) structure. This ubiquitous characteristic of biological organization provides the motivation to identify universally applicable practical methods for the detection of such structure, to distinguish primary types of structure, to quantify their distinctiveness, and to simplify interpretation of structural aspects. A method implementing the isolation principle (by generating all isolated groups and solitary objects) is proven to be specified by single-linkage clustering. Basically, the absence of structure can be stated if no isolated groups exist, the condition for which is provided. Structures that allow for classifications in the sense of complete partitioning into disjoint isolated groups are characterized, and measures of distinctiveness of classification are developed. Among other primary types of structure, chaining (complete nesting) and ties (isolated groups without internal structure) are considered in more detail. Some biological examples for the interpretation of structure resulting from application of the isolation principle are outlined.

Spatiogenetic characteristics of beech stands with different degrees of autochthony.

BACKGROUND: Autochthony in forest tree stands is characterized by a number of criteria, among which the range over which stands act as a population has been suggested to play a central role. Therefore, measures are needed for the delineation of populations or the detection of subpopulation structure. It is argued here that methods of population delineation must be based on the combined consideration of spatial distances and genetic differences between adult individuals. Conventional approaches and a set of newly developed methods are applied to seven isozyme loci in four beech stands which are distinguished by different types of forest management based on natural regeneration. RESULTS: Permutation analyses show that correlations between spatial distances and genetic differences vary only little in the studied beech stands. In view of the popularity of this and related descriptors of spatiogenetic covariation, this result came as a surprise. The newly developed methods lead to a different conclusion. Significant spatiogenetic structure is indicated in all stands when considering the mean and variance of spatiogenetic separation, where separation is measured by the smallest spatiogenetic difference of an individual from all others. Spatiogenetic difference is measured here by a combination of the spatial distances and genetic differences between individuals. This descriptor indicates the existence of spatiogenetic clusters in the beech stands. In order to arrive at an explicit representation of cluster structure as a representation of subpopulation structure, two types of cluster structure (primary and alpha-isolated) are distinguished, both of which reflect desirable characteristics of subpopulation structure. Particularly in the alpha-isolated structure, the proportion of individuals organized in clusters, the effective size, and the effective number of clusters clearly distinguish and consistently rank the four stands with respect to their types of forest management and the associated criteria of autochthony. CONCLUSION: The surprisingly high correspondence between our descriptors of spatiogenetic structure and forest management types confirms the appropriateness of the applied measure of cluster isolation and of the criterion for the choice of the level alpha of cluster isolation. The two types of cluster structure and their characteristic descriptors are thus suggested to be promising tools for the detection of subpopulation structure. To include the effects of long-distance gene flow, the presented methods can be extended as outlined to larger spatial scales in order to detect higher order population structure.