Download Interpreting the `selection effect` of biodiversity on ecosystem function

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.
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Editor, George Hurtt
Manuscript received 14 March 2005
First decision made 21 April 2005
Manuscript accepted 16 May 2005