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Ecology Letters, (2005) 8: 846–856 doi: 10.1111/j.1461-0248.2005.00795.x LETTER Interpreting the ‘selection effect’ of biodiversity on ecosystem function Jeremy W. Fox Department of Biological Sciences, University of Calgary, 2500 University Dr. NW, Calgary AB T2N 1N4, Canada Correspondence: E-mail: [email protected] Abstract Experimental ecosystems often function differently than expected under the null hypothesis that intra- and interspecific interactions are identical. Recent theory attributes this to the ‘selection effect’ (dominance by species with particular traits), and the ‘complementarity effect’ (niche differentiation and/or facilitative interactions). Using the Price Equation, I show that the ‘selection effect’ only partially reflects dominance by species with particular traits at the expense of other species, and therefore is only partially analogous to natural selection. I then derive a new, tripartite partition of the difference between observed and expected ecosystem function. The ‘dominance effect’ is analogous to natural selection. ‘Trait-independent complementarity’ occurs when species function better than expected, independent of their traits and not at the expense of other species. ‘Trait-dependent complementarity’ occurs when species with particular traits function better than expected, but not at the expense of other species. I illustrate the application of this new partition using experimental data. Keywords Additive partition, biodiversity, complementarity effect, ecosystem function, natural selection, Price Equation, selection effect, tripartite partition. Ecology Letters (2005) 8: 846–856 INTRODUCTION Many ecosystem-level properties and functions (e.g. total biomass, primary productivity) comprise the aggregate functional contributions of individual species. Because species differ in their ecological attributes, the aggregate functioning of a diverse ecosystem may differ from what would be expected under the null hypothesis that all species are identical. Species loss might therefore be expected to alter ecosystem function, a possibility of serious concern given current, historically high rates of biodiversity loss (Loreau et al. 2001). To predict the likely effects of biodiversity loss on ecosystem function, it would be helpful to understand how interspecific differences in ecological attributes affect aggregate ecosystem functioning. Recent theory (Loreau & Hector 2001) suggests that two effects can cause a diverse mixture of species to function differently than would be expected based on the functional performance of individual species growing in monoculture. The ‘selection effect’ is interpreted as reflecting dominance of the mixture by species that function at high levels in monoculture, while the ‘complementarity effect’ is interpreted as reflecting niche differences and/or facilitative 2005 Blackwell Publishing Ltd/CNRS interactions among species. Loreau & Hector (2001) derive an additive partition of these two effects, under the null hypothesis that intra- and interspecific interactions are identical. Under the null hypothesis, individuals of any given species function equally well in monoculture and mixture. The additive partition has been widely used to interpret the effect of experimental manipulations of plant diversity on total plant biomass or primary productivity (e.g. Loreau & Hector 2001; Fridley 2002; Hector et al. 2002; van Ruijven & Berendse 2003; Hooper & Dukes 2004; Hooper et al. 2005; Spehn et al. 2005). Interestingly, Loreau & Hector (2001) suggest that their additive partition is partially analogous to the Price Equation, which is used in evolutionary biology to partition the causes of evolutionary change (Price 1970, 1972, 1995; Frank 1995, 1997). Specifically, the ‘selection effect’ in the additive partition seems analogous to natural selection in evolution. For instance, natural selection favouring individuals with large body size will tend to increase the mean body size in the next generation, other things being equal. Similarly, a ‘selection effect’ favouring dominance of a mixture by species with high monoculture functioning will tend to increase mixture functioning over that expected under the null hypothesis, Interpreting the ‘selection’ effect 847 other things being equal. The standard interpretation of the ‘selection effect’ is that it captures the extent to which species with high monoculture yields dominate a mixture at the expense of other species, an interpretation motivated by the analogy with natural selection in the Price Equation (Loreau & Hector 2001). Numerous authors interpret the ‘selection effect’ in this manner (Loreau & Hector 2001; Sala 2001; Fridley 2002; Hector et al. 2002; van Ruijven & Berendse 2003; Hooper et al. 2005; Spehn et al. 2005, but see Petchey 2003; Hooper & Dukes 2004). It is this analogy between the ‘selection effect’ and natural selection that seems to make the additive partition partially analogous to the Price Equation; Loreau & Hector (2001) do not suggest any evolutionary analogue to the ‘complementarity effect’. Here I clarify the relationship between the ‘selection effect’ and natural selection in the Price Equation. I demonstrate that the ‘selection effect’ in the additive partition actually combines selection sensu Price (1970, 1972, 1995) with other processes that have no evolutionary analogue. I go on to argue that the fundamental insight of Loreau & Hector (2001) – that there is an analogy between dominance of a mixture by species with particular traits, and natural selection favouring individuals with particular traits – is valid, but needs to be developed more precisely. Drawing on the treatment of natural selection in the Price Equation, I derive a new tripartite partitioning of the ways in which the aggregate function of a mixture of species can deviate from that expected under the null hypothesis that all individuals of a given species function equally well in mixture or monoculture. The tripartite partition comprises the ‘dominance effect’, the ‘trait-dependent complementarity effect’, and the ‘trait-independent complementarity effect’. Dominance is equivalent to selection sensu Price (1970, 1972, 1995), and occurs when species with particular traits dominate at the expense of others. Trait-dependent complementarity occurs when growth in mixture rather than monoculture increases the functioning of species with particular traits, but not at the expense of other species. Trait-independent complementarity occurs when growth in mixture rather than monoculture increases the functioning of species, independent of their traits and not at the expense of other species. Neither form of complementarity has (or is intended to have) any evolutionary analogue, or any counterpart in the Price Equation. The ‘selection effect’ of Loreau & Hector (2001) combines the dominance and trait-dependent complementarity effects. I illustrate the application of this new tripartite partition using data from the BIODEPTH experiment (Spehn et al. 2005), and show that the tripartite partition provides new ecological insight into these data. THE PRICE EQUATION I begin by discussing the Price Equation, since an understanding of the Price Equation is required for an understanding of both the additive partition and my new tripartite partition. While neither the additive partition or my new tripartite partition is (or is intended to be) an instance of the complete Price Equation, one of the terms in each partition is intended to correspond to the natural selection term in the Price Equation. Several derivations of the Price Equation are available (e.g. Price 1972; Frank 1995, 1997), but these derivations differ in their details and often discuss crucial points only briefly. For ease of reference, and for the sake of fixing terminology and notation, I first sketch the derivation the Price Equation, emphasizing a crucial point not emphasized in most derivations. In general, the Price Equation partitions the difference between two ‘corresponding’ populations in the weighted mean of some property of the ‘objects’ comprising the populations (Price 1995). In order to calculate these weighted means, objects are categorized into ‘types’ according to their property values, with all objects of a given type assumed to have the same property value (the average value of objects of that type). The weight assigned to each type of object is some measure of the frequency or amount of objects of that type in each population. Populations ‘correspond’ when the objects comprising the populations have some kind of 1 : 1 relationship with one another. The generality and the power of the Price Equation arise from the flexibility with which ‘correspondence’ may be defined. The definition of correspondence is crucial because the way in which ‘correspondence’ is defined partially dictates how objects are categorized into types (Price 1995). For instance, in evolutionary biology, the two populations are the parental and offspring populations, with offspring ‘corresponding’ to their parent(s) (Price 1995; Frank 1997). As an illustration, consider an asexual species with uniparental inheritance (the more complex book-keeping required to account for biparental inheritance is irrelevant here; see Frank 1997). If the property of interest is some aspect of the phenotype of individual organisms (the ‘objects’), then the frequency of parents of phenotype (type) i is simply the relative abundance of parents of phenotype i. However, the nature of the correspondence between parents and offspring dictates that the frequency of offspring of type i be defined as the relative abundance of offspring of parents of phenotype i, not the relative abundance of offspring of phenotype i. That is, each offspring is indexed by the phenotype of its parent, not its own phenotype. For instance, if parents of type 1 have 10 offspring, and the total number of offspring produced by all parents is 100, the frequency of offspring of type 1 is 10/ 100 ¼ 0.1. Defined in this way, the difference between parental and offspring frequency measures parental fitness – the fittest parental phenotypes are those whose offspring comprise the greatest proportion of the offspring population, compared with the parent’s own frequency. Indexing 2005 Blackwell Publishing Ltd/CNRS 848 J. W. Fox offspring by their own phenotype is inappropriate because doing so ignores which parent gave rise to which offspring, thereby treating the parent and offspring populations as if they were unrelated (no 1 : 1 correspondence). Indexing offspring by parental phenotype allows phenotypic evolution (difference in weighted mean phenotype between parental and offspring populations) to be partitioned into components attributable to natural selection (any nonrandom association between parental phenotype and parental fitness) and fidelity of transmission (any differences between phenotypes of offspring and their parents). However, for our purposes it will be sufficient to examine the special case where each offspring individual has exactly the same phenotype as its parent (perfect transmission). More formally, let the frequency of parents (objects) of type i in the parental population be denoted qi, and let each parent of type i have some character (e.g. mass, length) zi. The offspring of parents of type i have frequency qi0 ; throughout the derivation, primes denote attributes of the offspring population. The average character value of offspring of parents of type i is zi0 ¼ zi , since we are assuming perfect transmission of parental phenotype to offspring (Frank 1997). We can now write the weighted mean character values z and z 0 as X z ¼ qi zi ; ð1Þ i and z 0 ¼ E(x)E(y) + Cov(x, y), where Cov is the covariance operator (Price 1972; Frank 1997). Using these two facts, and the fact that Eðww 1Þ ¼ 0, we can rewrite (4) as w Dz ¼ Cov ; z : ð5Þ w Equation 5 is the Price Equation partition of evolutionary change in mean phenotype, for the special case of perfect transmission (Price 1970, 1972, 1995; Frank 1995, 1997). In this special case, evolutionary change is entirely attributable to natural selection acting on parents, as quantified by the association (covariance) between parental relative fitness and parental phenotype. It will be useful in what follows to recognize a crucial point: the covariance between relative fitness and parental phenotype, calculated using weighted means (i.e. x ¼ P EðxÞ ¼ qi xi ), equals the unweighted covariance between i the difference in parental and offspring frequency and parental phenotype, multiplied by the total number of types. That is, w X qi0 qi zi Cov ; z ¼ w i ¼ N Covuw ðDq; z Þ ¼ N X i¼1 N 1 X Dqi Dqi N i¼1 ! ! N 1 X zi ; zi N i¼1 ð6Þ X qi0 zi0 ¼ i X qi wi i w zi : ð2Þ In eqn 2, the absolute fitness P of parents of type i is wi, mean absolute fitness is w ¼ qi wi , and relative fitness of pari w . Relative fitness is essentially a conents of type i is wi = version factor relating the frequency of parents to the frequency of their offspring. Define the difference in weighted mean character value between the two populations as Dz ¼ z 0 z : ð3Þ Equation 3 defines the amount of evolutionary change between the parental and offspring populations. We wish to express this change in a more meaningful form. To do this, we substitute eqns 1 and 2 into eqn 3 and rearrange: X wi Dz ¼ 1 zi : qi ð4Þ w i We now make use of two facts. First, given the definition of q, the weighted mean (expected) value of any random P variable x is x ¼ EðxÞ ¼ q x , where E denotes the i i i expectation operator. Second, the expected value of the product of two random variables x and y equals the product of their expectations plus their covariance: E(xy) ¼ 2005 Blackwell Publishing Ltd/CNRS qi0 where Dqi ¼ qi , N is the total number of types (i ¼ 1,…,N), and the subscript uw indicates a covariance calculated with respect to unweighted expectations. In going from 5 to eqn 6, I have used the fact that PN eqn 0 q q i zi ¼ NEuw ðDq ÞEuw ðz Þ þ N Covuw ðDq; z Þ, i i¼1 which simplifies to NCovuw(Dq, z) because Euw(Dq) ¼ 0. The point of writing the covariance term as NCovuw(Dq, z) is to emphasize that evolution by natural selection is a zero-sum game. If the offspring of parents of some phenotype i are more frequent than their parents (i.e. Dqi > 0, or equivalently, wwi > 1), then offspring of parents of some other phenotype j „ i must be less frequent than their parents (Dqj < 0). In any process analogous to evolution by natural selection (and natural selection has many analogues outside evolutionary biology; see Price 1995), differences in the frequency of ‘parents’ and their ‘offspring’ will sum to zero, so that Euw(Dq) ¼ 0. Below I demonstrate that the ‘selection effect’ of Loreau & Hector (2001) does not satisfy this condition. RELATIONSHIP BETWEEN THE ‘SELECTION EFFECT’ AND NATURAL SELECTION Loreau & Hector (2001) examine how the total yield of a diverse mixture of plants can deviate from that expected Interpreting the ‘selection’ effect 849 based on species’ yields when grown in monoculture. Loreau & Hector (2001) define the difference in yield, DY, as the difference between the observed total yield of a mixture of plants and the expected total yield under the null hypothesis that all intra- and interspecific interactions are identical. When this null hypothesis holds, the yield of an individual plant does not depend on the species identity of neighbouring plants. Expected total yield under the null hypothesis equals the weighted sum of the monoculture yields of the component species, where species’ monoculture yields are weighted by their expected relative yields. Expected relative yield is the expected yield of a plant species in mixture, divided by its yield in monoculture. Observed total yield is equal to the weighted sum of the monoculture yields of the component species, where species’ monoculture yields are weighted by their observed relative yields in mixture. More formally, define X X DY ¼ YO YE ¼ YOi YEi ¼ X i i RYOi Mi X i RYEi Mi ; ð7Þ i where YO and YE respectively denote total observed and expected yield, YOi and YEi respectively denote observed and expected yield of species i, Mi is the monoculture yield of species i, RYOi (YOi/Mi) is observed relative yield of species i in mixture, and RYEi is expected relative yield of species i in mixture. In a substitutive experimental design, RYEi ¼ ai/atot, where ai is the initial (planted) abundance of species i in mixture, and atot is the total planted abundance of all individuals in mixture or any monoculture. Loreau & Hector (2001) assume a substitutive experimental design, and I make the same assumption. Equation 7 defines the expected mixture as the ‘parental’ population, and the observed mixture as the ‘offspring’ population (compare eqn 3). It would be useful to partition the right-hand side of eqn 7 into additive, ecologically meaningful components. Loreau & Hector (2001) suggest that the deviation from expected yield, DY, has two components. First, species with, e.g., high monoculture yields might produce disproportionately high relative yields in mixture. Loreau & Hector (2001) term such ‘dominance’ by species with high monoculture yields a positive ‘selection effect’. Dominance by species with low monoculture yields produces a negative ‘selection effect’. Loreau & Hector (2001) use the term ‘selection effect’ because of a putative analogy with Price’s (1970, 1972, 1995) general theory of selection: ‘[S]election occurs when changes in the relative yields of species in a mixture are nonrandomly related to their traits (yields) in monoculture’ (Loreau & Hector 2001, p. 73). Second, species’ yields in mixture might be higher (or lower) on average than expected under the null hypothesis, termed a positive (respectively, negative) ‘complementarity effect’. Loreau & Hector (2001) suggest that a positive complementarity effect indicates that plants occupy different niches and/or facilitate one another. Loreau & Hector (2001) quantify ‘selection’ and ‘complementarity’ by rewriting eqn 7 as DY ¼ N Covuw ðDRY ; M Þ þ N Euw ðDRY ÞEuw ð M Þ ð8Þ where DRYi ¼ RYOi ) RYEi is the difference between the observed and expected relative yields of species i, and N is species richness (i ¼ 1,2…,N). The first term on the righthand side of eqn 8 quantifies the ‘selection effect’, while the second term quantifies the ‘complementarity effect’. Equation 8 is the ‘additive partition’ of Loreau & Hector (2001). Although the covariance (selection) term in eqn 8 resembles that in eqn 6, the relationship between the two covariance terms is different than casual inspection suggests. The covariance term in the additive partition is related to the covariance term in the Price Equation, but not in the way suggested by Loreau & Hector (2001). Next I demonstrate that the ‘selection effect’ is only partially analogous to natural selection in the Price Equation, and discuss the implications for the ecological interpretation of the additive partition. Loreau & Hector (2001) define species as ‘types’ of object, and suggest that the relative ‘frequency’ of species can differ between expected and observed ecosystems due in part to a process analogous to natural selection that acts on species’ monoculture yields. Further, the form of the right-hand side of eqn 7 and the passage quoted above indicate that Loreau & Hector (2001) consider relative yields RYE and RYO to be the measures of ‘frequency’ of each species (to see this, substitute eqns 1 and 2 into eqn 3 and compare with eqn 7). The reason that the ‘selection effect’ in eqn 8 is only partially analogous to natural selection relates to the definition of ‘frequency’ in eqn 8. Loreau & Hector (2001) treat relative yield as a measure of ‘frequency’. This definition of ‘frequency’ is problematic because frequencies cannot be < 0 or > 1, and the frequencies of all the types of object in a population must sum to 1. While expected relative P yields are frequencies (i.e. 0 < RYEi < 1 for all i and i RYEi ¼ 1), observed relative yields can be > 1, and need not sum to 1. The fact that, in general, observed relative yields are not frequencies is crucial to the interpretation of the ‘selection effect’. The ‘selection effect’ equals species richness multiplied by the unweighted covariance between monoculture yield (parental trait value) and deviations from expected relative yield DRY (differences in parental and offspring frequency). As shown by eqn 6, the effect of selection sensu Price (1970, 1972, 2005 Blackwell Publishing Ltd/CNRS 850 J. W. Fox Mixture YOA YOB Y DY ‘Selection effect’ 1 2 3 4 5 6 300 330 360 390 420 450 100 110 120 130 140 150 400 440 480 520 560 600 0 40 80 120 160 200 0 2.5 5 7.5 10 12.5 ‘Complementarity effect’ RYOA/ RYEA RYOB/ RYEB 0 37.5 75 112.5 150 187.5 1 1.1 1.2 1.3 1.4 1.5 1 1.1 1.2 1.3 1.4 1.5 Table 1 Observed yields YOA and YOB of two plant species, A and B, planted in a 60 : 40 mixture Total observed yield ¼ Y, and DY is the difference between observed and expected total yield. Monoculture yields are MA ¼ 500 and MB ¼ 250. Expected relative yields are RYEA ¼ 0.6 and RYEB ¼ 0.4. All yields are in g m)2. Also shown are the ‘selection effect’ and ‘complementarity effect’ for each mixture, calculated from eqn 8, and the ‘relative fitnesses’ of A and B (RYOA/RYEA and RYOB/RYEB). 1995) can be expressed as the number of types, multiplied by the covariance between parental trait values and the differences in parental and offspring frequencies – but only if the differences in frequency sum to zero. The ‘selection effect’ in eqn 8 is equivalent to selection sensu Price (1970, 1972, 1995) only if the deviations DRY sum to zero, which in general they do not. The fact that the deviations DRY need not sum to zero implies that ‘selection’ in eqn 8 is not a zero sum game, unlike natural selection in evolutionary biology. High observed relative yield of one species need not come at the expense of other species. A simple numerical example illustrates the consequences of the fact that observed relative yields are not frequencies, and that high observed relative yield of one species need not come at the expense of other species. Consider the yields of various mixtures of two plant species, as described in Table 1. Mixtures 1–6 in Table 1 illustrate increasing observed yield of each species (and thus increasing observed relative yield), but not at the expense of the other species. That neither species increases at the expense of the other is indicated by the fact that, in each mixture, individuals of both species perform equally well (are equally ‘frequent’), relative to their expected performance (frequency) (i.e. RYOA/RYEA ¼ RYOB/RYEB). This implies that the ‘relative fitnesses’ of the two species are equal, since in the Price Equation, relative fitness equals the ratio of q0 offspring to parental frequency: wwi ¼ qii (see eqn 2). Equal ‘relative fitnesses’ imply zero selection sensu Price (1970, 1972, 1995) (note also that the mean ‘relative fitness’ of each species in Table 1 can be > 1, in contrast with evolutionary biology where mean relative fitness necessarily equals 1). However, despite the equal ‘relative fitnesses’, the additive partition eqn 8 finds an increasingly strong ‘selection effect’ as observed yields increase. This illustrates that the ‘selection effect’ of Loreau & Hector (2001) is not the same as selection sensu Price (1970, 1972, 1995). Table 1 also demonstrates that the ‘complementarity effect’ does not 2005 Blackwell Publishing Ltd/CNRS isolate that part of the difference in total yield attributable to processes other than selection sensu Price (1970, 1972, 1995). The entire difference in total yield is attributable to processes other than selection sensu Price (1970, 1972, 1995) for all mixtures in Table 1, but except in the trivial case of mixture 1 the ‘complementarity effect’ is less than the difference in total yield. None of this implies that the additive partition eqn 8 is mathematically invalid – it is not – or that it should not be used. The fact that the ‘selection effect’ in eqn 8 is not analogous to selection in the Price Equation (eqn 5) merely emphasizes the need for careful interpretation. Interpretability of the ‘selection effect’ might be enhanced if it could be partitioned into more easily interpretable subcomponents. To see how this might be done, consider that, in the special case where observed relative yields sum to 1, an increased observed relative yield of species i would come entirely at the expense of other species. In this special case, the ‘complementarity effect’ in eqn 8 is zero, and the ‘selection effect’ equals selection sensu Price (1970, 1972, 1995). This suggests that it might be possible to partition the ‘selection effect’ in eqn 8 into subcomponents, one of which is attributable to ‘natural selection-like processes’ operating on species’ monoculture yields, and the other of which is attributable to other ecological processes that have no evolutionary analogue. Next I develop such a partition. A NEW TRIPARTITE PARTITION P Define RYTO ¼ i RYOi as the observed relative yield RYOi total, and define the observed frequency of species i as RYT . O The observed frequency of species i is simply the proportion of the observed relative yield total comprised RYOi of species i. Note that 0 RYT 1 for all i and O P RYOi i RYTO ¼ 1, as required for a measure of frequency. RYOi Increased RYT for species i necessarily comes at the O expense of other species. Analogously, we could define Interpreting the ‘selection’ effect 851 P expected relative yield total as RYTE ¼ i RYEi and write RYEi the expected frequency of species i as RYTE . However, for the substitutive experimental design assumed here, RYTE ¼ 1, and so we can continue to write the expected frequency of species i as simply RYEi. It may seem unusual to express species’ frequencies as their proportions of the expected and observed relative yield totals, rather than, e.g. their proportions of expected and observed total (absolute) yields YE and YO. However, the above definitions of expected and observed frequencies are necessary in order to follow Loreau & Hector (2001) in writing species’ absolute yields as (in part) the products of species’ frequencies and species’ monoculture biomasses, thereby treating monoculture biomass as a ‘trait’ on which selection can act. Using our new definition of observed frequency, eqn 7 can be written as X X DY ¼ RYOi Mi RYEi Mi i i X X RYOi RYOi ¼ RYOi þ RYEi Mi : Mi RYTO RYTO i i ð9Þ Our goal is to partition eqn 9 into additive components attributable to the effects of processes analogous to natural selection, and effects of other processes with no evolutionary analogue. Selection sensu Price (1970, 1972, 1995) will be quantified by the unweighted covariance between differences in RYOi species’ frequencies, RYT RYEi , and species’ monoculO ture yields Mi, as in eqn 6. Recollecting terms in eqn 9 gives X X RYOi RYOi DY ¼ Mi RYOi Mi RYEi : þ RYTO RYTO i i ð10Þ The second sum on the right-hand side of eqn 10 is that portion of DY attributable to differences between species’ expected and observed frequencies (i.e. to zero-sum dynamics analogous to evolution). The first sum on the right-hand side of eqn 10 is that portion of DY attributable to deviation of observed mixture dynamics from zero sum; such dynamics have no evolutionary analogue. Expressing each sum in eqn 10 as the product of species richness and an unweighted expectation perspecies gives RYO DY ¼ NEuw M RYO RYTO ð11Þ RYO þ NEuw M RYE : RYTO Equation 11 can be rewritten as DY ¼ NEuw ð M ÞEuw ðDRY Þ RYO RYE þ N Covuw M ; RYTO RYO þ N Covuw M ; RYO : RYTO ð12Þ Equation 12 is an additive, tripartite partition of DY. The first (expectation) term on the right-hand side of eqn 12 is the ‘complementarity effect’ of Loreau & Hector (2001) and the sum of the two covariance terms is the ‘selection effect’ of Loreau & Hector (2001) (see eqn 8). I will refer to the expectation term as ‘trait-independent complementarity’. This term quantifies the extent to which species’ observed yields in mixture deviate from a zero sum game, but in a way that is independent of species’ traits (monoculture yields). This term is positive if, e.g. all species produce higher observed yields than expected under the null hypothesis, but all species perform equally well relative to their expected performance. Ecologically, we would expect this term to be large and positive if species occupy different niches and/or facilitate one another, so that individuals of all species perform better in mixture than in monoculture (Loreau & Hector 2001). Negative trait-independent complementarity indicates interspecific interference competition or some other process(es) with the same effect. The first covariance term is the covariance between species’ monoculture yields, and the difference between species’ observed and expected frequencies. This term quantifies the contribution to DY of processes analogous to natural selection sensu Price (1970, 1972, 1995). I refer to this covariance term as the ‘dominance effect’. This term quantifies the extent to which observed species’ relative yields in mixture resemble a zero sum game, with a positive covariance indicating that species with high monoculture yields dominate at the expense of species with low monoculture yields. Negative covariance indicates dominance by species with low monoculture yields at the expense of others. Ecologically, we would expect this term to be large and positive when species occupy similar niches, so that those species better-adapted to the niche perform better when grown in monoculture, and competitively exclude others when grown in mixture. The second covariance term is the covariance between monoculture yield and the deviation of observed relative yield from observed frequency. This term quantifies the extent to which species’ observed yields in mixture deviate from a zero sum game in a way that depends on species’ traits (monoculture yields). This term is positive when species with high monoculture yield attain high observed relative yields, but not at the expense of other species, and negative when species with low monoculture yields attain 2005 Blackwell Publishing Ltd/CNRS 852 J. W. Fox 20 (a) 0 –20 20 Effect size high observed relative yields, but not at the expense of other species. I refer to this term as quantifying ‘trait-dependent complementarity’. The interpretation of trait-dependent complementarity is best illustrated by contrast with the more familiar traitindependent complementarity. Positive trait-independent complementarity is ‘two-way’ or ‘mutual’ complementarity that occurs when all species perform better when grown with different species (Loreau & Hector 2001). Positive trait-independent complementarity suggests lack of niche overlap. In contrast, trait-dependent complementarity is ‘one-way’ complementarity that only benefits species with certain traits (monoculture yields). Positive trait-dependent complementarity suggests that species have ‘nested’ niches, so that species with ‘large’ niches attain high monoculture biomass and high relative yield in mixture, but not at the expense of species with ‘small’ (included) niches (see Chase 1996). Trait-dependent complementarity generates the non-zero ‘selection effect’ in Table 1. Going from mixtures 1–6, species A (the species with the largest monoculture biomass) attains an increasingly large observed relative yield compared with the observed relative yield of species B. Meanwhile the observed frequencies of each species remain constant at their expected values. (b) 0 –10 30 (c) 0 APPLICATION OF THE TRIPARTITE PARTITION The BIODEPTH experiment was a multisite field experiment testing the effect of grassland plant diversity (species richness) on total aboveground biomass (yield) (Hector et al. 1999, Loreau & Hector 2001; Spehn et al. 2005). Here I apply the tripartite partition to data from three sites (Silwood, Sheffield, Sweden) from the second year of the experiment. These three sites exhibit a range of effects of plant diversity on yield (Hector et al. 1999, Loreau & Hector 2001; Spehn et al. 2005), and so provide a useful illustration of the application of the tripartite partition. A complete reanalysis of the BIODEPTH experiment is beyond the scope of this article. Each multispecies plot at each site comprised two to 12 species (Sweden, Sheffield) or two to 11 species (Silwood) chosen randomly from a site-specific species pool, with the most species-rich plots comprising all species in the pool. Species were planted in a substitutive design. All species in each site’s species pool were also grown in monoculture (for details see Hector et al. 1999). The single most important result is that trait-dependent complementarity is often a large effect (Fig. 1). This shows that the tripartite partition identifies a new effect that is not only important in principle, but important in practice. In fact, all three components of the tripartite partition are broadly similar in terms of their average absolute 2005 Blackwell Publishing Ltd/CNRS –20 0 2 4 8 Species richness (log2 scale) 16 Figure 1 Application of the tripartite partition to data from three sites of the BIODEPTH experiment: Sweden (a); Sheffield (b); and Silwood (c). For each multi-species plot at each site there are three data points, giving the values of the dominance effect (squares), trait-independent complementarity effect (diamonds), and the traitdependent complementarity effect (triangles), plotted against planted species richness. All effect sizes are square-root transformed with original signs preserved, as in Loreau & Hector (2001). Lines are statistically-significant linear regressions reported in the text. Solid lines, dominance effect; dashed line, traitindependent complementarity; dotted line, trait-dependent complementarity. Some points are slightly offset horizontally for clarity. magnitudes, and in their ranges of variation, indicating that all three are often important determinants of the difference between observed and expected total yield at the three BIODEPTH sites examined here (Fig. 1). Total yields often increase with increasing plant diversity (species richness) in BIODEPTH and other experiments, although this variation typically occurs against a background of substantial variation among species compositions within Interpreting the ‘selection’ effect 853 Table 2). The two components of the ‘selection effect’, traitdependent complementarity and dominance, are significantly negatively correlated across all plots at Sweden, but significantly positively correlated at the other two sites (Table 2). Because trait-dependent complementarity can be a large component of the ‘selection effect’ in terms of absolute magnitude (Fig. 1), the ‘selection effect’ provides an inaccurate estimate of the importance of dominance by species with high (or low) monoculture biomass at the expense of others 20 (a) 0 1: 1 –15 –15 20 Selection effect diversity levels (Hector et al. 1999, Hooper et al. 2005). Examining how the components of the tripartite partition vary with diversity and composition can aid interpretation of variation in total yield. Inspection of the BIODEPTH data suggests that variation in effect sizes among plots within sites is dominated by variation among species compositions, rather than by variation among species richness levels (Fig. 1). I tested for directional effects of species richness on effect sizes at each site by regressing effect sizes (square-root transformed with original signs preserved, as in Loreau & Hector 2001) against log2-transformed planted species richness. While not the most statistically sophisticated test for a directional effect of species richness on effect size, this regression test is adequate for revealing strong directional trends. At Sweden, dominance tends to increase with increasing species richness, but this trend is marginally non-significant (P ¼ 0.056, R2 ¼ 0.11; Fig. 1a). At Sheffield, all three effects increases significantly with increasing richness (traitindependent complementarity, P < 0.001, R2 ¼ 0.45; dominance, P ¼ 0.002, R2 ¼ 0.29; trait-dependent complementarity, P < 0.001, R2 ¼ 0.57; Fig. 1b). However, inspection of residuals for Sheffield suggests that both dominance and trait-dependent complementarity are actually unimodal functions of species richness (Fig. 1b). Effect sizes do not vary significantly with species richness at Silwood (Fig. 1c). At all three sites, results for traitindependent complementarity (which equals the ‘complementarity effect’ in the additive partition) match those reported by Loreau & Hector (2001) using a more sophisticated statistical test. The three effects do not vary independently among plots within sites, and the correlations among them vary among sites (Table 2). Trait-dependent and trait-independent complementarity are significantly positively correlated at Sweden and Sheffield (Table 2), indicating that the ‘complementarity effect’ in the additive partition generally underestimates the absolute magnitude of ‘total complementarity’ (traitdependent + trait-independent) at these sites. Dominance and trait-independent complementarity are significantly positively correlated at Sheffield, and exhibit a marginally non-significant negative correlation at Sweden (P ¼ 0.099; 0 20 (b) 0 1: –10 –10 20 1 0 15 (c) 0 Table 2 Pearson’s correlations between the dominance and trait-independent complementarity effects (rd-ti), dominance and trait-dependent complementarity effects (rd-td), and between the trait-independent and trait-dependent complementarity effects (rti-td), at three BIODEPTH sites 1: –20 –20 1 0 Dominance effect 20 Site rd-ti rd-td rti-td Figure 2 The dominance effect vs. the ‘selection effect’ at three Sweden Sheffield Silwood )0.30 0.59** )0.05 )0.51* 0.79*** 0.62** 0.46* 0.67** 0.21 BIODEPTH sites: Sweden (a); Sheffield (b); and Silwood (c). Data transformation as in Fig. 1. Solid lines indicates 1 : 1 relationships. The two effects are positively correlated at each site, but statistical tests would be inappropriate because the two effects are not independent (dominance is a component of the ‘selection effect’). *P < 0.05, **P < 0.01, ***P < 0.001. 2005 Blackwell Publishing Ltd/CNRS 854 J. W. Fox (Fig. 2). Differences between the ‘selection effect’ and the dominance effect vary in magnitude from negligible to > 50% of the magnitude of the dominance effect (Fig. 2). The largest inaccuracies occur at Silwood in plots strongly dominated by high-yielding species (Fig. 2a). DISCUSSION Theoretical insights The analogy between the ‘selection effect’ in the additive partition and natural selection in Price Equation is only partial. This does not invalidate the additive partition, which is mathematically valid. Indeed, my tripartite partition confirms the two central insight embodied in the additive partition. First, dominance of a mixture by plants with particular traits, at the expense of other species, is analogous to natural selection favouring individuals with particular traits. Second, differences between observed and expected total yield also are partially attributable to niche complementarity, a factor with no evolutionary analogue and no counterpart in the Price Equation. My tripartite partition refines these insights by showing how to separate the effects of processes analogous to natural selection from other effects with no evolutionary analogue. Petchey (2003) first pointed out that the ‘selection effect’ of Loreau & Hector (2001) does not distinguish the effects of processes analogous to natural selection from the effect of any association between species’ monoculture yields and their degree of niche differentiation from other species. My tripartite partition formalizes Petchey’s (2003) verbal argument, and shows how to distinguish these two classes of effect (dominance and trait-dependent complementarity). Following Petchey (2003), Hooper & Dukes (2004) noted that the ‘complementarity effect’ in the additive partition only puts a minimum bound on the degree to which mixture dynamics deviate from zero sum. However, Hooper & Dukes (2004) lacked a way to quantify traitdependent complementarity, which my tripartite partition provides. Quantifying the three effects in the tripartite partition is important because these effects likely arise from different underlying mechanisms. A strong dominance effect suggests that species lack niche differentiation, so that the ‘fittest’ species tends to exclude the others, with the sign of the dominance effect depending on whether the fittest species have high or low monoculture biomass. A strong positive trait-independent complementarity effect suggests that species occupy different, non-overlapping niches and/or facilitate one another, so that all species perform better than expected when grown in mixture. A strong negative traitindependent complementarity effect suggests mechanisms 2005 Blackwell Publishing Ltd/CNRS that cause species to compete more strongly interspecifically than intraspecifically. A strong trait-dependent complementarity effect suggests that species occupy ‘nested’ niches; a possible example might be shallow-rooted plants vs. plants with both shallow and deep roots. Growth in mixture rather than monoculture might benefit the species with the ‘larger’ niche (e.g. by reducing competition for nutrients in deep soil) but not at the expense of the species with the ‘smaller’ niche (e.g. since shallow-soil competition is equally intense in both mixture and monoculture). The trait-dependent complementarity effect will be positive when species with ‘larger’ niches perform best in monoculture, and negative otherwise. Application of the tripartite partition does not itself test for any particular mechanism, but can aid the interpretation of data already collected and suggest mechanistic hypotheses to be tested with further experiments (see below). The tripartite partition suggests a refined interpretation of several proposed measures of ‘niche complementarity’. Many studies use the relative yield total (RYTO) or the mean proportional deviation from expected yield (D, which equals RYTO ) 1) to quantify niche complementarity or its effects on total yield (Hector 1998; Loreau are proportional 1998; Petchey 2003). Both RYTO and D to the complementarity effect in the additive partition (Loreau & Hector 2001). It follows that both RYTO and D measure trait-independent complementarity, like the complementarity effect in the additive partition. Trait-independent complementarity is probably the form of complementarity that most ecologists think of when they think of ‘complementarity’, because this form of complementarity describes species with mutually non-overlapping niches (Petchey 2003). However, if species are considered to be ‘complementary’ whenever they have non-identical (e.g. nested) niches, then ‘total complementarity’ equals traitindependent plus trait-dependent complementarity. RYTO, and related measures do not estimate ‘total compleD, mentarity’. The tripartite partition is best able to aid interpretation of data when its own ecological interpretation is clear. The heuristic interpretations suggested above (e.g. traitdependent complementarity as arising from ‘nested niches’) are useful, but could be made more precise. Theoretical studies examining how the tripartite partition behaves when applied to simulated data generated by known underlying models would enhance the interpretability of the tripartite partition. Theoretical studies would be particularly helpful in refining the heuristic interpretations of trait-dependent complementarity and trait-independent complementarity as respectively quantifying ‘oneway’ and ‘two-way’ niche differentiation. Classical notions of ‘niche differentiation’ and ‘niche overlap’ are not always applicable in the context of contemporary niche theory (Leibold 1995). Interpreting the ‘selection’ effect 855 Empirical insights ACKNOWLEDGEMENTS The BIODEPTH data illustrate that trait-dependent complementarity can be a substantial component of the ‘selection effect’, and that all three components of the tripartite partition can make important contributions to the difference between observed and expected total yield. These results show that the tripartite partition is an important practical as well as conceptual advance, suggesting new interpretations of empirical data. It would be interesting to apply the tripartite partition to other experiments on plant diversity and total yield and compare the relative magnitudes of the three effects across studies. It would also be interesting to interpret the effect sizes estimated from the tripartite partition in light of what is known about the ecologies of the species in different plots. Does the tripartite partition correctly identify those plots that would have been expected (based on knowledge of species’ ecologies) to exhibit strong trait-independent complementarity, strong trait-dependent complementarity, or strong dominance? Much previous work on plant diversity and total plant biomass focuses on whether the presence/absence of legumes explains variation in trait-independent complementarity (e.g. Loreau & Hector 2001). It would be interesting to know if presence/absence of legumes also can drive variation in trait-dependent complementarity. For instance, if species with low monoculture yields are poorly adapted to low-N soil and so benefit more than other species from being grown in mixture with legumes, we might expect to observe negative trait-dependent complementarity in plots with legumes. Other trait-based hypotheses proposed in recent work might be usefully refined via application of the tripartite partition. Dimitrakopoulos & Schmid (2004) found that total biomass of plant mixtures increased with species richness more strongly when mixtures were planted in deeper soil, a result they attributed primarily to increased trait-independent complementarity in deeper soil. However, Dimitrakopoulos & Schmid (2004) also found an increasing ‘selection effect’ with increasing soil depth, which might be attributable to increasingly strong trait-dependent complementarity in deeper soil. Hille Ris Lambers et al. (2004) found that superior N competitors performed better than expected when grown in mixture in a Minnesota grassland. It would be interesting to know the extent to which this result was attributable to dominance vs. trait-dependent complementarity. Hooper & Dukes (2004) found that negative ‘selection effects’ counterbalanced positive trait-independent complementarity in a California grassland. It would be interesting to know whether negative ‘selection effects’ arose because of negative dominance effects, negative trait-dependent complementarity, or both. The tripartite partition provides a meaningful, interpretable bridge between species’ ecologies and wholeecosystem functioning. Thanks to Andy Hector and the BIODEPTH team for permission to use the BIODEPTH data, and to Austin Burt for sharing his insight into the Price Equation. The comments of three referees and the editor greatly improved the manuscript. REFERENCES Chase, J.M. (1996). 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