Diversification and diversity partitioning - BioOne

Paleobiology, 40(2), 2014, pp. 162–176 DOI: 10.1666/13041

Diversification and diversity partitioning Michael Hautmann

Abstract.—Model calculations predict that pathways of alpha- and beta-diversity in diversifying ecosystems notably differ depending on the relative role of competition, predation, positive effects of species’ interactions, and environmental parameters. Four scenarios are discussed, in which alpha- and beta-diversity are modeled as a function of increasing gamma-diversity. The graphic illustration of this approach is herein called a-b-c plot, in which the x-axis indicates increasing diversification rather than absolute time. In purely environmentally controlled systems, beta-diversity maintains near-maximum values throughout the diversification interval, whereas mean alpha-diversity increases linearly, with a slope being reciprocal to beta-diversity. A second scenario is based on the assumption that increasing richness will have predominantly positive effects on the addition of further species; here, alpha- and beta-diversity increase simultaneously (though not necessarily at the same rates) and without reaching a predictable upper limit. In ecosystems that are characterized by low competition between species, mean alpha-diversity asymptotically approaches a saturation level, whereas the increase in betadiversity accelerates until alpha-diversity stagnates, and then continues to rise linearly. If competition is high, addition of species first increases beta-diversity until no further habitat contraction is possible, followed by a period in which alpha-diversity increase through adaptive divergence becomes the principal drive of diversification. Because there is a continuous transition between the late stage of the low-competition model and the early stage of the high-competition scenario, both can be combined in a single model of diversity partitioning under the premise of a diversity-dependent increase of competition. This summary model predicts three phases of diversity accumulation: (1) a niche overlap phase, (2) a habitat contraction phase, and (3) a niche differentiation phase. The models herein discussed provide a potential tool to assess the question which factors primary controlled the diversification of life over geological times. ¨ Michael Hautmann. Paläontologisches Institut und Museum, Karl Schmid-Strasse 4, CH-8006 Zurich, Switzerland. E-mail: [email protected] Accepted: 4 October 2013 Published online: 17 January 2014

Introduction Which factors primarily control the diversification of life? The Darwinian answer to this question is competition, as illustrated in Darwin’s wedge metaphor, which suggests an upper limit of the number of the world’s species (Gould 1985). With some modern additions from island biogeography and population dynamics, this view also underlies Sepkoski’s (1984) influential interpretation of the Phanerozoic fossil record by means of coupled logistic equations. However, many ecologists emphasize that species also provide resources for other species, thus outweighing negative effects of competition (e.g., Whittaker 1977), or that supposedly negative interactions such as competition and predation actually drive adaptive evolution, rather than depressing diversification (see Benton and Emerson 2007). Finally, any significant role of biotic interactions in shaping long-term trends of Ó 2014 The Paleontological Society. All rights reserved.

biodiversity has been disputed given the great influence of environmental factors on diversity and diversity trends (Gould 1985; Barnoski 2001). Advocates of different concepts have sought support for their views from comparison of mathematical models with global compilations of the Phanerozoic fossil record (e.g., Sepkoski 1984; Benton 2001; Lane and Benton 2003; Stanley 2007). However, inadequacies of the fossil record and the practical impossibility of a taxonomically coherent census of all taxa that ever lived on Earth, including the lack of global data on the species level, open each of these attempts to criticism. Sample-standardization techniques (Alroy et al. 2008; Alroy 2010) are a promising approach for reducing bias due to differences in the sampling effort and the quality of the fossil record, but other problems such as lack of taxonomic coherence 0094-8373/14/4002-0002/$1.00

163

DIVERSIFICATION AND DIVERSITY PARTITIONING

and poor stratigraphic resolution of the data still remain. There are, however, alternatives to the analysis of global patterns in the attempt to identify the principal controls of biodiversification. Biodiversity is organized at different levels: (1) within a community, (2) between communities, (3) within a landscape, (4) between landscapes, and (5) within a biogeographic region (Whittaker 1977). This paper explores how different potential drivers of biodiversity change affect diversity trends at the first two of these levels. For this purpose, I describe four scenarios mathematically, considering situations of environmentally controlled habitat boundaries, positive effects of species’ interactions, low competition, and high competition. Application of these models to paleodata requires only the analysis of a limited set of adjacent habitats through a critical time interval, which is a feasible task provided that a sequence of comparable habitats can be indentified from geological data. Concepts of Diversity Partitioning In a series of influential papers, Whittaker (1960, 1972, 1975, 1977) introduced a hierarchical concept of biodiversity, which recognizes five different levels at which biodiversity is organized. The lowest of these levels is the within-habitat level, in which alpha-diversity indicates the degree of niche differentiation in a given community. Alpha-diversity is most commonly expressed as the number of species occurring in the community under consideration, although there are alternative approaches that also take species’ abundances into account (e.g., Shannon’s H, effective richness [Olszewski 2010]). Beta-diversity is defined as between-habitat diversity, that is, the degree by which communities of adjacent habitats differ in their species composition. Gammadiversity in its original definition is the product of alpha- and beta-diversity and thus the overall sum of the taxa that inhibit all habitats of a given landscape (‘‘landscape diversity’’). Analogous to beta-diversity, delta-diversity has been defined as the degree of taxonomic differentiation between landscapes, whereas epsilon-diversity indicates the actual diversity

in a given region or biogeographic unit. Because of the difficulty of defining a landscape versus a region, delta- and epsilon diversity have seldom been used by subsequent authors. In current practice, alpha- and gamma diversity are usually understood as diversity at the finest and at the largest scale of observation, respectively (e.g., Sepkoski 1988; Holland 2010). Both are inventory diversities; i.e., they can be expressed by the number of taxa that they comprise. In contrast, betadiversity is a differentiation diversity that measures the taxonomic differences between the units (e.g., communities) that constitute the level of alpha-diversity (Whittaker 1977: Table 1). Beta-diversity is usually expressed as the ratio of gamma-diversity to mean alphadiversity; thus gamma-diversity is the product of alpha- and beta-diversity (multiplicative diversity partitioning). In this formulation, beta-diversity is a dimensionless factor that is equivalent to the number of taxonomically completely different communities that are necessary to explain the observed diversity at the gamma level (Olszewski 2010). However, beta-diversity can also be measured additively as the diversity missing from the mean alpha-diversity to account for the total inventory diversity (Lande 1996). The advantage in this additive diversity partitioning (ADP) is that beta-diversity can be expressed by the same unit as inventory diversity (e.g., richness, entropy); thus ADP is increasingly used in field studies on diversity partitioning (Patzkowsky and Holland 2007; Heim 2009). A drawback, however, is that ADP produces counterintuitive results in certain situations, e.g., in the situation of homogeneous ecosystems with discrete habitat boundaries described below. In the following models, alpha-diversity (a) is measured as the number of species (S) in a given community (i.e., richness): a ¼ S:

ð1Þ

An alternative measure of alpha-diversity is effective richness (Seff; Jost 2006), which is defined as Euler’s number (e) to the power of the Shannon index (H). Effective richness is

164

MICHAEL HAUTMANN

equivalent to the number of taxa that would result in the same entropy value if all were equally abundant and thus translates nonintuitive entropy units to more easily understood richness units (Olszewski 2010). Alt h o u g h e ff e c t i v e r i c h n e s s h a s s o m e interesting properties (see Olszewski 2010), it is not included in the following models because it introduces a quantitative aspect that requires different reasoning than the richness-based approach used herein. Moreover, it has the problematic aspect that effective richness of a single community (alpha-diversity) may exceed effective richness of the system (gamma-diversity). Beta-diversity is herein understood as the ratio of the total number of species (St) in a set of adjacent communities, divided by the average number of species per community ¯ (¼ mean alpha-diversity; a): b¼

St a¯

ð2Þ

In this definition, beta-diversity represents the factor by which overall diversity in a set of adjacent habitats exceeds the mean alphadiversity of its constituting habitats. Gamma-diversity (c) indicates the overall number of species in a ‘‘landscape’’ or paleogeographic region and is identical to St in equation (2) if all habitats of the region are considered. In this situation, beta- and gamma-diversity are linked by c ¼ a¯ *b:

ð3Þ

As an alternative to the multiplicative approach in equations (2) and (3), gammadiversity can also be calculated additively as the sum of alpha-and beta-diversity: c ¼ a¯ þ b:

ð4Þ

In this formulation, beta-diversity has the same unit as alpha-diversity and represents the difference between regional diversity and mean alpha-diversity (Olszewski 2010). The models presented below can be formulated for both a multiplicative and an additive approach, but as discussed below, the multiplicative approach reflects processes better than does the additive approach.

Changing diversity is usually modeled against time, as in the standard models of global diversification (e.g., Sepkoski 1984; Benton 2001; Brayard et al. 2009; Solé et al. 2010). When modeling diversity trends on the different levels of alpha- and beta-diversity simultaneously, however, an a priori theory on how each depends on the other is required. An alternative method used herein is modeling alpha- and beta-diversity as a function of gamma-diversity. This approach corresponds to a hypothetical experiment in which species are added to a closed ecosystem and the resulting changes in alpha- and beta-diversity are recorded. The graphic illustration of this method is herein called a-b-c plot. In constructing such plots, functions for either alphaor beta-diversity are derived from theoretical considerations, and the complementary component is simply calculated from equation (3) in the multiplicative or equation (4) in the additive formulation. Because diversification by definition means increasing overall biodiversity in time, increasing gamma-diversity defines a time’s arrow, although with a variable scale in most cases. Apart from the advantage for the modeling procedure, a-b-c plots are also more practical for analyses of collected diversity data. Timedependent diversification models might produce curves that differ only in nuances, at least during certain intervals (e.g., during the later phase in the logistic and hyperbolic models in Sol e´ et al. 2010). Given the uncertainty of radiometric age data and the high variability of sedimentation rates, it is currently impossible to estimate time-differences between stratigraphically successive samples with the precision required for testing such models. In contrast to any kind of geological time measurement, gamma-diversity is a parameter of the paleontological samples themselves, and thus it is directly available and avoids introduction of extrinsic errors. The relation between local (alpha) and regional (gamma) diversity was previously addressed less formally by Ricklefs (1987). Figure 1 in Ricklefs (1987) illustrates two scenarios, one called the ‘‘enrichment model,’’ in which local diversity always represents a


165

more or less constant fraction of the regional geology and continues to increase with regional diversity. In contrast, the ‘‘saturation model’’ describes a situation in which local community diversity decouples from regional diversity as soon as an upper limit (‘‘saturation’’) is reached, which is set by interactions between species within the community. Regional diversity increase beyond the point of local saturation is limited to ‘‘increased geographical turnover,’’ i.e., to an increase of betadiversity. The premises of these two models are similar to the low- and high-competition models discussed below, although I suggest herein that a continuous transition between the two is possible. Patzkowsky and Holland (2003) applied Ricklefs’ (1987) models to paleodata from the Ordovician, which supported his enrichment model, essentially corresponding to the early phase of the lowcompetition model presented below. Concepts of Communities and Community Boundaries Probably the most widely accepted definition of a community is that of Mills (1969), which states that ‘‘community means a group of organisms occurring in a particular environment, presumably interacting with each other and with the environment, and separable by means of ecological survey from other groups.’’ The vagueness of this definition results from a general disagreement about the question whether communities are discrete units that are tied together by biotic interactions or rather reflect local assemblages of taxa that co-occur due to a variety of local factors and gradually pass into each other if the environmental gradient is not too high (e.g., Elliot and Gray 2009: Fig. 3.13). Apart from the nature of the environmental gradient, the prevalence of either of these two extremes depends of course also on the type of ecosystem under consideration. Highly coevolved or habitat-structuring communities undoubtedly have sharp boundaries with adjacent communities (e.g., reefs and adjacent level-bottom communities [e.g., Boucot 1981]). In contrast, gradual transitions prevail in terrestrial plant communities that are chiefly dependent on abiotic resources (e.g., Whit-

FIGURE 1. Plot illustrating alpha- and beta-diversity measurements in a transect of gradually changing communities (A), discrete communities (B), and an intermediate situation (C). In all cases, gamma-diversity is 7, beta-diversity is 3 (the inverse of 1/3, which is the fraction of the transect that is occupied by each species, as indicated by horizontal lines), and mean alpha-diversity is 7/3 ¼ 2.33. In A, diversity at equally spaced sample points (stippled lines) is indicated, averaging 21/9 ¼ 2.33 in accordance with mean alpha-diversity of the system. In the discrete case (B), the same value results as the mean of the three communities. In the intermediate case illustrated in C, samples cluster in two communities, each containing four species. This overestimates the actual alpha-diversity of 2.33, measured as the mean of equally spaced sample points indicated by stippled lines.

taker 1960). Paleoecological studies have traditionally adhered to the concept of discrete communities (Fursich 1977; Aberhan 1992; ¨ Hofmann et al. 2013a,b); this prevalence,

166

MICHAEL HAUTMANN

however, stems at least partly from the widespread use of cluster analyses for the identification of paleocommunities, because cluster analysis is set up to create discrete clusters, even from a continuous distribution of data (Bambach and Bennington 1996: p. 146). Additionally, paleontological data are typically sampled at large spatial and temporal scales, which hamper recognition of gradual transitions. Nevertheless, if biotic interaction occurs it is clear that the distribution of a species along an environmental gradient cannot be completely independent from the distribution of other species. The resulting covariance between occurrences of species constitutes communities as statistical entities that are conceptually useful even in absence of discrete boundaries. The concept of diversity partitioning can be applied to both a discrete and a gradually interchanging distribution of communities, but different approaches are needed to calculate alpha- and beta-diversity for these two cases. In the discrete-entity situation, mean alpha-diversity is simply the mean richness of all communities, and the ratio of gammadiversity to mean alpha-diversity gives betadiversity (eq. 3). For the gradual situation, I propose to measure beta-diversity as the inverse of the average fraction that each species occupies on the transect: b¼

1 ; tf

ð5Þ

where tf ¼ transect fraction occupied by species. It is easily seen that b in this formulation gives the number of taxonomically completely different communities that sum up to the observed diversity at the gamma level and is thus equivalent to the definition of betadiversity in the situation of discrete communities (Fig. 1). Having determined beta-diversity, mean alpha-diversity can be calculated from equation (3). Alternatively, mean alphadiversity can also be obtained from the mean of equally spaced point diversities along the transect, provided that the sampling distance is small enough to catch all transitions. Equation (5) helps to understand the examples

illustrated in Figures 2–5, but it is also applicable to field surveys. Figure 1A,B illustrates measurements of alpha- and beta-diversity for gradual and discrete community boundaries and the equivalence of alpha- and beta-diversity values in these two situations. Figure 1C refers to an intermediate state that is probably the most realistic situation for most ecosystems. It demonstrates that the practice of merging ‘‘recurrent assemblages’’ (i.e., samples that are statistically similar enough to form a discrete cluster) into a single paleocommunity leads to an overestimation of alpha-diversity and a corresponding underestimation of betadiversity. This procedure is equivalent to an artificial extension of species ranges to all sample points that constitute the paleocommunity. It may be argued that the absence of a taxon in certain samples of the cluster is likely to be due to its low abundance or to preservation problems, but some bias will still remain if homogeneity within the paleocommunity cannot be demonstrated. Pragmatic solutions to minimize this bias include (1) using trophic nucleus diversity, which is likely to be representative for all samples, and (2) measuring alpha-diversity in terms of entropy (Shannon’s H, effective richness), which minimizes the contribution of rare taxa that occur only in few samples. Alternatively, point diversities (rather than statistically identified groups of samples) might be used, but this procedure requires regularly spaced sampling points, a precondition that is hardly feasible in paleontologic field surveys. Models of Alpha- and Beta-Diversity Changes as a Function of Gamma-Diversity The starting condition for the following models is a situation where ecosystems were largely vacated at the beginning of the studied time interval. At the local scale, this condition is given, e.g., in newly formed volcanic islands (the Galapagos Islands being a prominent example), whereas the Cambrian–Ordovician radiation of metazoans and intervals that followed global mass extinction events provide such a situation at the global scale. The models are mostly set up from a gradual viewpoint, because this is the more general


167

FIGURE 2. Diversity partitioning in homogeneous, environmentally separated communities. A, B, Example illustrating the case of three environmentally separated communities. Horizontal lines in A indicate the distribution of species along the transect; numbers refer to alpha-, beta-, and gamma-diversity for each stage. Gamma diversity increases from 9 to 18 solely by doubling alpha-diversity. C, Generalized a-b-c plot for multiplicative diversity partitioning showing constancy of beta-diversity and linearly increasing alpha-diversity. D, Same as B for additive diversity partitioning (ADP). ADP indicates a steady increase in beta-diversity that exceeds the increase in mean alpha-diversity. This reflects the effect of habitat separation on the actual numbers of species in a region, but it fails to explain the underlying process, which is solely within-habitat diversification.

situation. An exception is the case where strong environmental boundaries between habitats create discrete community boundaries, or if biotic interactions cause co-occurrences of species in more or less discrete units. A requirement for the application of the models is that stratigraphically sequential sets of samples comprise the same segment of the environmental gradient. All models represent end-cases, in which just one factor is considered as the principal control on biodiversification. Diversity trends in actual ecosystems will be controlled by a mixture of these factors,

but owing to the large differences in the predicted pattern, it is likely that the prevalent factor will essentially shape the observed pattern and thus can be identified in an a-bc plot. Homogeneous Ecosystems with Discrete, Environmentally Controlled Habitat Boundaries Rationale.—The basic assumption in this scenario is that species are not allowed to cross environmentally imposed boundaries between habitats. Although this appears

168

MICHAEL HAUTMANN

unrealistic in most ecosystems, such boundaries may actually exist, e.g., the Nile River and the adjacent Sahara desert. In a less rigorous scenario, an approximation of this situation occurs if negative environmental effects in adjacent habitats exceed negative effects of interspecific competition within the primary habitat for a given set of species. Example.—Three adjacent habitats are inhabited by communities with three species each (S ¼ 3). In each habitat, alpha-diversity doubles by immigration or evolutionary processes. According to equation (3), gammadiversity doubles as well (from 9 to 18), whereas beta-diversity holds its value, which is equal to the number of habitats (three; Fig. 2A,B). Modeling.—During diversification in a set of adjacent, ecologically undersaturated habitats, beta-diversity will stay constant throughout time. Taking equation (3) from above, betadiversity is simply equal to the number of habitats present (n), i.e., the maximum possible value of beta-diversity. Conversely, alphadiversity increases linearly with increasing gamma-diversity, with the slope being the reciprocal of number of habitats (n ¼ betadiversity; Fig. 2C): b¼n

ð6Þ

a¯ ¼ c=n:

ð7Þ

and

Discussion.—The basic nature of this scenario invites a comparison between the multiplicative and the additive approach of diversity partitioning. According to equation (4), betadiversity in the additive approach is given as b ¼ c a¯ ¼ c nc ¼ cð1 n1Þ. This means that the additive approach indicates a linear increase of beta-diversity (Fig. 2D), which is even larger than the increase of mean alphadiversity, because n1 , 1 n1 for n . 2 (the minimum condition for diversity partitioning). This does not adequately reflect the underlying biological process, which is solely within-habitat diversification. This example highlights that the choice of either a multiplicative or an additive approach is not arbitrary. Additive diversity partitioning is useful if it is

intended to compare absolute diversity values at different hierarchic levels, whereas the multiplicative approach should be chosen if the biological processes that drive diversity partitioning are sought. Ecosystems Dominated by Positive Feedbacks between Species Rationale.—In his concluding paper on the evolution of species diversity in land communities, Whittaker (1977: p. 24) assumed that diversity evolves as a self-augmenting process, in which ‘‘the evolution of species diversity provides resources that make possible the addition of further species to the community.’’ This process increases alphadiversity without a ‘‘clearly definable, effectively predictable stopping point for diversity increase in evolutionary time.’’ Whittaker also assumed a concurrent increase of beta-diversity, because new species preferentially interject themselves in between the abundance maxima of existing species and thereby narrow their habitats. This assumption introduces a competition component in the Whittaker’s argument that appears somewhat contradictory to his view of within-habitat evolution, but the net result of a concurrent increase in alpha- and beta-diversity may also result from positive effects of species interaction alone. Figure 3A illustrates a situation in which local diversity (not abundance) maxima narrow mean habitat width over evolutionary time because diversity maxima are predicted to have an attractive effect under the premise that the resource aspect of species diversity determines the overall pattern. Modeling.—Because gamma-diversity is the product of mean alpha-diversity and betadiversity (eq. 3), the mathematical expression of a scenario in which both alpha- and betadiversity contribute to increasing gammadiversity is a

aðcÞ ¼ cc

ð8Þ

and b

bðcÞ ¼ cc with a þ b ¼ c.

ð9Þ


169

Factors a and b (and thus c) are not necessarily constant through time; thus equations (9) and (10) describe a general trend around which the values of alpha- and betadiversity are allowed to fluctuate. In the most simple case where alpha- and beta-diversity contribute equally to the increase of gamma diversity, a ¼ b ¼ 1; i.e., alpha- and betadiversity are each the square root of gamma diversity. However, dominance of either alpha- or beta-diversity increase appears possible, leading to divergent curves between the two (Fig. 3C). Discussion.—Models assuming positive feedbacks between species predict a hyperbolic increase of overall diversity in time (Solé et al. 2010), which in practice is probably difficult to distinguish from an exponential pattern. The diversity-partitioning-based model illustrated in Figure 3 is independent of timing and makes the prediction that alpha- and betadiversity as a function of gamma-diversity increase in phase, with a decreasing rate but without a predictable upper limit. Ecosystems with Low Competition Rationale.—Probably the most dominant view among paleontologists is that competition has a limiting effect on regional and global species diversity. The dominance of this assumption is for the most part due to the elegant interpretation of the Phanerozoic fossil record by means of coupled logistic equations derived from population dynamics and island biogeography (Sepkoski 1978, 1979, 1981, 1984), which require an upper boundary of the global species’ carrying capacity. However, there has also been intense criticism of this model, summarized by Benton (2001) and Stanley (2007). This criticism focused on the interpretation of the global Phanerozoic diversity curve presented by Sepkoski (1984, 1998), which may or may not be biased by the taxonomic level (families and genera rather

FIGURE 3. Diversity partitioning in systems that are controlled by positive effects of species interaction. A, B, Example illustrating the attracting effect of points of maximum diversity (stippled lines). Horizontal lines in A indicate the distribution of species along the transect;

numbers refer to alpha-, beta-, and gamma-diversity for each stage. C, Generalized a-b-c plot according to equations (8) and (9). Line in the middle is the square root function representing the case where alpha- and betadiversity increase at the same rate. Indication of alphaand beta-diversity is arbitrary and can be interchanged.

170

MICHAEL HAUTMANN

than species), changes in the quality of the fossil record, or general flaws in compiling taxa without in-depth revision by experts. My aim here is not to reassess this debate, but it can be shown that diversity partitioning allows an alternative approach to the subject. In this endeavor, I first make a distinction between low- and high-competition ecosystems, which are predicted to differ in pathways of alpha- and beta-diversity when diversification proceeds. Although both models may sufficiently describe diversification for most types of ecosystems, it is also possible to combine them into a comprehensive scenario of diversity partitioning under the regime of successively increasing competition that is useful at a larger scale of observation. Low competition can have several different causes, yet the following three cases are probably most common: 1. Species only slightly detract from each other’s resources; i.e., pairwise competition coefficients in the Lotka-Volterra equation (e.g., Pianka 1974: p. 2141, eq. 1) are low. Species with low pairwise competition coefficients may coexist over long (even geological) time intervals, provided that the carrying capacities for both species are similar (Sepkoski 1996). 2. Abundances are kept well below the carrying capacities, e.g., because of intensive predation (Stanley 2008) or unstable environmental conditions. 3. The number of competing species is low, e.g., on newly formed islands or soon after a mass extinction event. Because low competition allows a high degree of niche overlap (e.g., Pianka 1974), the addition of species will not lead to significant habitat contraction of existing communities under these premises, and diversity will increase chiefly at the community FIGURE 4. Diversity partitioning in a low-competition system, as described in the main text. A, B, Example showing random addition of species with broadly overlapping occurrences along the transect. Habitat width slightly decreases as more species are added. Horizontal lines in A indicate the distribution of species along the transect; numbers refer to alpha-, beta-, and gamma-

diversity for each stage. C, Generalized a-b-c plot according to equations (10), (12), and (13), showing declining increase of mean alpha-diversity as the saturation level (amax) is approached, contrasted with accelerated increase of beta-diversity that finally passes into a linear slope. Setting a0 ¼ b0 is for clearer illustration only.


(alpha-diversity) level (stages a and b in Fig. 4A,B). As diversification proceeds, however, a basic prediction of the Lotka-Volterra equation is that competition will increase by the sheer increase of competing species, even if these have low mutual competition coefficients. This increase in competition forces species into their ecological optimum, thereby contracting their habitat width and thus increasing beta-diversity of the system (stage c in Fig. 4A–B). As a consequence, the role of betadiversity progressively increases during diversification, whereas the increase in alphadiversity levels out when communities approach a state of saturation. Modeling.—Because the increase in alphadiversity progressively declines as alpha-diversity approaches its saturation level, the mathematical description of the scenario outlined above is a limited growth function, herein written as aðcÞ ¼ amax þ ða0 amax Þekðc0 cÞ ;

ð10Þ

where a0 ¼ initial mean alpha-diversity, amax ¼ maximum mean alpha-diversity, and c0 ¼ initial gamma-diversity. The constant k must be constrained to account for the model assumption that the initial increase of gamma-diversity is solely due to increasing mean alpha-diversity, which implies that the initial increase of alpha-diversity is the reciprocal of the initial beta-diversity (b0). da 1 ðc0 Þ ¼ : dc b0

ð11Þ

Derivation of (10), inserting conditions for c0 (11) and solving for k gives k¼

1 : b0 ða0 amax Þ

ð12Þ

Inserting (10) in equation (3) and solving for b gives: bðcÞ ¼

c amax þ ða0 amax Þekðc0 cÞ

:

ð13Þ

As shown in the Figure 4C, the increase in beta-diversity slightly accelerates until alphadiversity approaches its maximum. When alpha-diversity stagnates at its maximum,

171

beta-diversity passes into a straight line with the slope 1/amax. Discussion.—Marine soft-bottom communities combine low mutual competition with high rates of predation that keep the species’ abundances well below their carrying capacities (Stanley 2008), and they are therefore an appropriate example for testing the predictions for diversity partitioning in a lowcompetition setting. Previous records from the Ordovician (Patzkowsky and Holland 2003) and the Early Triassic (Hofmann et al. 2013a,b) are indeed in accordance with the model described above, with linearly increasing alpha-diversity and virtually constant beta-diversity in the early phase of the Paleozoic and Mesozoic diversification of marine life, respectively. Hofmann et al. (2013a,b) have already predicted that alphadiversity levels out at a certain saturation point, but neither their study nor that of Patzkowsky and Holland (2003) captured later stages of diversification. The model thus still needs to be tested on the basis of data from a longer interval of diversification. Ecosystems with High Competition Rationale.—In contrast to the low-competition scenario, high competition occurs if mutual competition coefficients are high (e.g., due to limited resources), abundances are near the carrying capacity (e.g., due to low predation), and/or the number of competing species is high. Consequently, a high-competition situation may be innate to certain ecosystems, or it may follow the low-competition situation if diversity exceeds a certain threshold. As in the case of the low-competition model, the theoretical framework for high-competition systems follows the general consensus about community dynamics described by the Lotka-Volterra equation. A possible exception from this prerequisite has recently been reported by Olszewski (2012), who found on the basis of a consumerresource model of competition that highly competitive systems can persist in non-equilibrium states over geological time scales. Evaluating the effects of this particular case on diversity partitioning will be an important future task.

172

MICHAEL HAUTMANN

In a high-competition setting, competitive exclusion creates sharp boundaries between the occurrences of ecologically similar species. Addition of new species, either by immigration or by evolutionary processes, is expected to occur by wedging in between existing species at their habitat boundaries, thereby reducing their habitat width. Thus, the addition of species initially increases beta-diversity only, without affecting mean alpha-diversity (stages a and b in Fig. 5A,B). However, the habitat area of any community has a minimum size imposed by the need for space and resources of its members (e.g., the range in which animals usually move, or the areadependent primary production necessary to maintain a minimum population size), which sets a limit to possible habitat contraction. This creates a situation in which the roles of alphaand beta-diversity are reversed, relative to the case of low-competition systems: The increase in beta-diversity declines as the system diversifies and finally levels out when no further habitat contraction is possible. Conversely, alpha-diversity begins to increase and becomes the sole agent of increasing gammadiversity beyond the point of maximum habitat contraction (stage c in Fig. 5A,B). The late stage of diversification in a high-competition setting thus shows a similar pattern as the early stage in a low-competition scenario, both of which are characterized by increasing mean alpha-diversity and stagnant beta-diversity. This is not contradictory, because different processes of increasing alpha-diversity underlie the models. In the low-competition case, mean alpha-diversity increases because this situation allows a high degree of niche overlap. In a high-competition setting, mean alpha-diversity increases by niche differentiation within habitats, because competition

FIGURE 5. Diversity partitioning in a high-competition system, as described in the main text. A, B, Example illustrating that insertion of new species at contact points of existing species leads to habitat contraction and a corresponding increase in beta-diversity (stages a and b). Subsequent increase in niche differentiation (stage c) results in increasing alpha-diversity. Horizontal lines in

A indicate the distribution of species along the transect; numbers refer to alpha-, beta-, and gamma-diversity for each stage. C, Generalized a-b-c plot according to equations (14) to (16), showing declining growth rates of beta-diversity contrasted with accelerated increase of alpha-diversity. Initial values for alpha- and beta-diversity are identical to terminal values in Figure 5, allowing construction of a composite model shown in Figure 6.


excludes coexistence and habitats cannot be further contracted. Modeling.—As in the low-competition case, the situation can be described by a limited growth function, in which the increase of betadiversity (rather than alpha-diversity as in the case of low competition) progressively declines as beta-diversity approaches its saturation level. Thus, bðcÞ ¼ bmax þ ðb0 bmax Þekðc0 cÞ

ð14Þ

and aðcÞ ¼

c bmax þ ðb0 bmax Þekðc0 cÞ

:

ð15Þ

In a gradual scenario as depicted in Figure 5A,B, the initial increase in alpha-diversity is zero; thus, analogous to equation (12), k is constrained by k¼

1 : a0 ðb0 bmax Þ

ð16Þ

Alternatively, a scenario of discrete community boundaries is possible as well, caused by a combination of strong biotic interactions (e.g., due to mutualism or consumer-resource interactions) on the one hand and competitive exclusion on the other. Such interactiondetermined communities may respond collectively to habitat contraction of one of their species, which reduces diversity at the points where new species are added and thus decreases mean alpha-diversity in the early phase. In this case k is larger than in the gradual scenario: k.

1 : a0 ðb0 bmax Þ

ð17Þ

Discussion.—A high-competition setting requires a minimum number of interacting species, but it is still possible that this scenario occurs relatively early in the process of diversification provided that regional resources are limited. Alternatively, it may represent a late stage in the process of diversification that succeeds a low-competition situation as soon as a critical number of species has been reached. Diversification curves of the lowand high-competition models smoothly pass into each other and can be combined into a

173

single diversification model that covers the full range of increasing competition during regional diversification. Summarized Model of DiversificationDriven Increase in Competition The predictions for the early phase of the high-competition model are identical to the predictions for the late phase of the lowcompetition model (i.e., stagnant mean alphadiversity and linearly increasing beta-diversity), and both models can be linked conceptually by assuming that the highest state of competition in the first model corresponds to the initial state of competition in the second. Thus, a combined model can be constructed, in which the conditions of the equilibrium state in low-competition model are taken as starting conditions of the high-competition model. Figure 6 shows a combined competition-based diversification model that predicts three different phases herein referred to as (1) niche overlap phase, (2) habitat contraction phase, and (3) niche differentiation phase. Niche Overlap Phase.—As long as competition between species is insignificant, the addition of species has no effect on the habitat width of existing species, nor does it not add a stimulus for adaptive divergence. Initially, alpha-diversity is thus built up without a significant effect on beta-diversity. This situation gradually changes as more and more species are added, because the competitive pressure on a given species results from resource detraction by the sum of all competing species. Competition will thus start to affect the system if the number of competitors is sufficiently high, even if pairwise competition is low. Habitat Contraction Phase.—As competition increases, the habitat width of species decreases, which makes beta-diversity the principal driver of diversification. The degree of maximum habitat contraction depends on a variety of factors, including species traits and the nature of the environmental gradient, and it appears possible that in some ecological settings (e.g., marine soft-bottom communities) the habitat contraction phase represents the final stage of diversity partitioning that is realized.

174

MICHAEL HAUTMANN

FIGURE 6. Composite model illustrating pathways of alpha- and beta-diversity under the premise of a diversitydependent increase of competition. In the niche overlap phase (green), competition is weak, allowing the addition of species without affecting the distribution of other species. The first phase gradually passes into the habitat contraction phase (yellow), in which increasing competition forces species into their ecological optima, thereby narrowing their habitat width. Species are added preferentially at habitat boundaries of existing species during this phase. If no further habitat contraction is possible, gamma-diversity may still rise by adaptive divergence between species, which essentially increases mean alpha-diversity (niche differentiation phase; red). (For the references to color in this figure legend, the reader is referred to the web version of this article.) See text for further details.

Niche Differentiation Phase.—If no further habitat contraction is possible to accommodate additional species, diversity may still increase by adaptive divergence, leading to increased niche differentiation in a community. No upper limit of this process can be predicted (Whittaker 1977), and thus the niche differentiation phase represents the final stage of diversification in this model. Discussion.—It is noteworthy that the processes that underlie these three phases differ significantly in the time scales at which they operate. In the niche overlap phase, the effect of newly added species on diversity partitioning is immediate in the sense that there is no effect apart from the addition itself. In contrast, the process that underlies the habitat contraction phase is ecological in nature, and can be expected to act at a time scale of several tens or hundreds of generations. Finally, evolutionary processes drive the niche differentiation phase, which may vary from some thousands to some hundred thousands or millions of years. Although the different time scales of the processes have no direct bearings on the overall duration during which they dominate the system, their increasing length co-determines the succession of the three phases; e.g., the habitat contraction phase

necessarily precedes the niche differentiation phase because habitat contraction allows accommodation of species much more quickly than evolutionary changes that lead to adaptive divergence. The summarized model of diversity partitioning in competition-controlled systems makes explicit predictions about trends in alpha- and beta-diversity that allow us to test the prevalence or non-prevalence of competition in diversifying biota. Unfortunately, comprehensive data on diversity partitioning over time intervals that are long enough to capture the critical transitions between the predicted phases do not yet exist. Incipient data from the early phases of the Paleozoic and Mesozoic diversifications of marine level-bottom communities are in accordance with the predictions for the initial niche overlap phase (see above), but data from later phases will be required to finally establish a dominant role for competition in shaping diversity patterns over evolutionary times. Summary Do biotic interactions have a positive or negative effect on biodiversification, or is biodiversity primary controlled by environmental factors? These key questions in evolu-


tionary paleobiology have previously been debated largely on the basis of global diversification curves, which however have left sufficient wiggle room for being quoted as evidence for each of these views (see Benton and Emerson 2007 for a review). An alternative approach to these questions is to study changes in diversity partitioning in sets of adjacent habitats, which avoids some of the major problems associated with global diversity curves. Four principal models have been presented herein, which predict that pathways of alpha- and beta-diversity during biotic diversification notably differ depending on which factor governs this process. A strong environmental gradient between ecosystems will maintain constantly high beta-diversity, limiting any change in gamma-diversity to within-habitat diversification. In the case of prevalently positive effects of species interactions on diversity growth, both alpha-and beta-diversity are predicted to increase simultaneously. Low interspecific competition and a moderate environmental gradient combine to a situation where mean alpha-diversity initially increases linearly but reaches virtual stasis in later stages of diversification. Conversely, the increase of beta-diversity accelerates with increasing gamma-diversity until a linear increase is reached as the increase in alpha-diversity stagnates. High interspecific competition first promotes habitat contraction and later adaptive divergence, making first beta-diversity and then alpha-diversity the principal driver of diversification. The lowand high-completion scenarios can be combined into a single model that predicts three phases of diversity partitioning in a competition-controlled setting: (1) niche overlap phase, (2) habitat contraction phase, and (3) niche differentiation phase. Collecting comprehensive paleorecord data that allow for the identification of dominant patterns in diversity partitioning will be an important future task in the endeavor to test diversification theories in general. Acknowledgments F. Menzel (Mainz) provided useful comments on competition and speciation in Recent ecosystems. I thank my wife, Stefanie, for

175

introducing me to MATLAB. Constructive criticism by reviewers T. D. Olszewski and M. E. Patzkowsky and handling editor W. Kiessling significantly improved the manuscript. Support by the Swiss National Science Foundation (grant 200021-121774) is gratefully acknowledged. Literature Cited Aberhan, M. 1992. Palokologie und zeitliche Verbreitung ben¨ thischer Faunengemeinschaften im Unterjura von Chile. Beringeria 5:3–174. Aberhan, M., and T. K. Baumiller. 2003. Selective extinction among Early Jurassic bivalves: a consequence of anoxia. Geology 31:1077–1080. Alroy, J. 2010. Geographical, environmental and intrinsic biotic controls on Phanerozoic marine diversification. Palaeontology 53:1211–1235. Alroy, J., M. Aberhan, D. J. Bottjer, M. Foote, F. T. Fursich, P. J. ¨ Harries, A. J. W. Hendy, S. M. Holland, L. C. Ivany, W. Kiessling, M. A. Kosnik, C. R. Marshall, A. J. McGowan, A. I. Miller, T. D. Olszewski, M. E. Patzkowsky, S. E. Peters, L. Villier, P. J. Wagner, N. Bonuso, P. S. Borkow, B. Brenneis, M. E. Clapham, L. M. Fall, C. A. Ferguson, V. L. Hanson, A. Z. Krug, K. M. Layou, E. H. Leckey, S. Nurnberg, C. M. Powers, J. A. Sessa, C. Simpson, A. ¨ ´ Tomaˇsovych, and C. C. Visaggi. 2008. Phanerozoic trends in the global diversity of marine invertebrates. Science 321:97–100. Bambach, R. K., and J. B. Bennington. 1996. Do communities evolve? A major question in evolutionary paleoecology. Pp. 123– 160 in Jablonski et al. 1996. Barnosky, A. D. 2001. Distinguishing the effects of the Red Queen and Court Jester on Miocene mammal evolution in the northern Rocky Mountains. Journal of Vertebrate Paleontology 21:172– 185. Benton, M. J. 2001. Biodiversity on land and in the sea. Geological Journal 36:211–230. Benton, M. J., and B. C. Emerson. 2007. How did life become so diverse? The dynamics of diversification according to the fossil record and molecular phylogenetics. Palaeontology 50:23–40. Boucot, A. J. 1981. Principles of benthic marine paleoecology. Academic Press, New York. Brayard, A., G. Escarguel, H. Bucher, C. Monnet, T. Bruhwiler, N. ¨ Goudemand, T. Galfetti, and J. Guex. 2009. Good genes and good luck: ammonoid diversity and the end-Permian mass extinction. Science 325:1118–1121. Elliot, M., and J. S. Gray. 2009. Ecology of marine sediments. Oxford University Press, Oxford. Fursich, F. T. 1977. Corallian (Upper Jurassic) marine benthic ¨ associations from England and Normandy. Palaeontology 20:337–385. Gould, S. J. 1985. The paradox of the first tier: an agenda for paleobiology. Paleobiology 11:2–12. Heim, N. A. 2009. Stability of regional brachiopod diversity structure across the Mississippian/Pennsylvanian boundary. Paleobiology 35:393–412. Hofmann, R., M. Hautmann, and H. Bucher. 2013a. A new paleoecological look at the Dinwoody Formation (Lower Triassic, Western U.S.): intrinsic versus extrinsic controls on ecosystem recovery after the end-Permian mass extinction. Journal of Paleontology 87:854–880. Hofmann, R., M. Hautmann, M. Wasmer, and H. Bucher. 2013b. Palaeoecology of the Virgin Formation (Utah, USA) and its

176

MICHAEL HAUTMANN

implications for the Early Triassic recovery. Acta Palaeontologica Polonica 58:149–173. Holland, S. M. 2010. Additive diversity partitioning in palaeobiology: revisiting Sepkoski’s question. Palaeontology 53:1237– 1254. Jablonski, J., D. H. Erwin, and J. H. Lipps, eds. 1996. Evolutionary paleobiology: essays in honor of James W. Valentine. University of Chicago Press, Chicago. Jost, L. 2006. Entropy and diversity. Oikos 113:363–375. Lande, R. 1996. Statistics and partitioning of species diversity, and similarity among multiple communities. Oikos 76:5–13. Lane, A., and M. J. Benton. 2003. Taxonomic level as a determinant of the shape of the Phanerozoic marine biodiversity curve. American Naturalist 162:265–276. Mills, E. L. 1969. The community concept in marine zoology, with comments on continua and instability in some marine communities: a review. Journal of the Fisheries Research Board of Canada 26:1415–1428. Olszewski, T. D. 2010. Diversity partitioning using Shannon’s entropy and its relationship to rarefaction. In J. Alroy and G. Hunt, eds. Quantitative methods in paleobiology. Paleontological Society Papers 16:95–116. ———. 2012. Persistence of high diversity in nonequilibrium ecological communities: Implications for modern and fossil ecosystems. Proceedings of the Royal Society of London B 279:230–236. Patzkowsky, M. E., and S. M. Holland. 2003. Lack of community saturation at the beginning of the Paleozoic plateau: the dominance of regional over local processes. Paleobiology 29:545–560. ———. 2007. Diversity partitioning of a Late Ordovician marine biotic invasion: controls on diversity in regional ecosystems. Paleobiology 33:295–309. Pianka, E. R. 1974. Niche overlap and diffuse competition. Proceedings of National Academy of Sciences USA 71:2141– 2145.

Ricklefs, R. E. 1987. Community diversity: relative roles of local and regional processes. Science 235:167–171. Sepkoski, J. J., Jr. 1978. A kinetic model of Phanerozoic taxonomic diversity. I. Analysis of marine orders. Paleobiology 4:223–251. ———. 1979. A kinetic model of Phanerozoic taxonomic diversity. II. Early Phanerozoic families and multiple equilibria. Paleobiology 5:222–251. ———. 1981. A factor analytic description of the Phanerozoic marine fossil record. Paleobiology 7:36–53. ———. 1984. A kinetic model of Phanerozoic taxonomic diversity. III. Post-Paleozoic families and mass extinctions. Paleobiology 10:246–267. ———. 1988. Alpha, beta, or gamma: where does all the diversity go? Paleobiology 14:221–234. ———. 1996. Competition in macroevolution: the double wedge revisited. Pp. 211–255 in Jablonski et al. 1996. ———. 1997. Biodiversity: past, present, and future. Journal of Paleontology 71:533–539. ———. 1998. Rates of speciation in the fossil record. Philosophical Transactions of the Royal Society London B 353:315–326. ˜ J. M. Montoya, and D. H. Erwin. 2010. Solé, R. V., J. Saldana, Simple model of recovery dynamics after mass extinction. Journal of Theoretical Biology 267:193–200. Stanley, S. M. 2007. An analysis of the history of marine animal diversity. Paleobiology Memoirs No. 4. Paleobiology 33(Suppl. to No. 4). ———. 2008. Predation defeats competition on the seafloor. Paleobiology 34:1–21. Whittaker, R. H. 1960. Vegetation of the Siskiyou Mountains, Oregon and California. Ecological Monographs 30:279–338. ———. 1972. Evolution and measurements of species diversity. Taxon 21:213–251. ———. 1975. Communities and ecosystems, 2nd ed. Macmillan, New York. ———. 1977. Evolution of species diversity in land communities. Evolutionary Biology 10:1–67.