Download How variation between individuals affects species coexistence

Ecology Letters, (2016)
doi: 10.1111/ele.12618
IDEA AND
PERSPECTIVE
Simon P. Hart,1* Sebastian J.
Schreiber2 and Jonathan M.
Levine1
1
Institute of Integrative Biology,
€ rich (Swiss Federal Institute
ETH Zu
€trasse 16,
of Technology), Universita
€ rich, Switzerland
8092 Zu
2
Department of Evolution & Ecol-
ogy and the Center for Population
Biology, One Shields Avenue,
University of California, Davis, CA
95616, USA
*Correspondence: E-mail: simon.
[email protected]
How variation between individuals affects species coexistence
Abstract
Although the effects of variation between individuals within species are traditionally ignored in
studies of species coexistence, the magnitude of intraspecific variation in nature is forcing ecologists to reconsider. Compelling intuitive arguments suggest that individual variation may provide
a previously unrecognised route to diversity maintenance by blurring species-level competitive differences or substituting for species-level niche differences. These arguments, which are motivating
a large body of empirical work, have rarely been evaluated with quantitative theory. Here we
incorporate intraspecific variation into a common model of competition and identify three pathways by which this variation affects coexistence: (1) changes in competitive dynamics because of
nonlinear averaging, (2) changes in species’ mean interaction strengths because of variation in
underlying traits (also via nonlinear averaging) and (3) effects on stochastic demography. As a
consequence of the first two mechanisms, we find that intraspecific variation in competitive ability
increases the dominance of superior competitors, and intraspecific niche variation reduces specieslevel niche differentiation, both of which make coexistence more difficult. In addition, individual
variation can exacerbate the effects of demographic stochasticity, and this further destabilises
coexistence. Our work provides a theoretical foundation for emerging empirical interests in the
effects of intraspecific variation on species diversity.
Keywords
community assembly, competition, demographic heterogeneity, demographic stochasticity, functional traits, genetic diversity, individual variation, intraspecific variation, nonlinear averaging,
species coexistence.
Ecology Letters (2016)
INTRODUCTION
Population and community dynamics are emergent properties
of the demographic rates and interactions of individuals. Consequently, solving important problems in ecology requires
understanding how processes operating at the level of individuals translate into population and community dynamics.
Nonetheless, attempts to understand one of the central problems in ecology – species coexistence – have traditionally
excluded consideration of intraspecific variation, instead relying on average differences between species to explain diversity
maintenance. Ecologists are now being forced to reconsider
this approach because of growing empirical evidence that a
large fraction of the trait variation in nature occurs within,
not just between species (Messier et al. 2010; Violle et al.
2012). Indeed, approximately 25% of the total trait variation
within plant communities worldwide is found within species
(Siefert et al. 2015). This is a striking pattern with unexplored
but likely important consequences for species coexistence that
deserve greater theoretical attention.
Although a large theoretical literature on character displacement focuses on the evolutionary processes that influence species diversity via selection on individual variation (Slatkin
1980; Taper & Case 1985; Vellend 2006; Pfennig & Pfennig
2012), the implications of this work for the purely ecological
consequences of individual variation are rarely considered.
More generally, the theoretical foundation for understanding
the ecological mechanisms by which individual variation influences species coexistence is limited (Begon & Wall 1987; Lichstein et al. 2007). Nevertheless, ecologists have recently
argued that individual variation may provide a previously
unrecognised route to species coexistence (Bolnick et al. 2003,
2011; Hubbell 2005; Fridley & Grime 2009; Clark 2010; Clark
et al. 2010; Jung et al. 2010; Messier et al. 2010; Pfennig &
Pfennig 2012; Violle et al. 2012). This conjecture is motivating
a rapid change in focus in field assessments of diversity maintenance away from average differences between species to concentrate on differences between individuals within species
(Jung et al. 2010; Messier et al. 2010; Violle et al. 2012; Siefert et al. 2015). However, in the absence of general quantitative theory, common expectations that individual variation
promotes species coexistence may be premature.
One compelling argument for a positive effect of individual
variation on coexistence is that differences in competitive ability between individuals should ‘blur’ differences in competitive
ability between species. Under this scenario, individual variation is expected to break down competitive hierarchies, which
should promote coexistence by making competitive exclusion
less likely or less rapid (Hubbell 2005; Fridley et al. 2007). It
has also been suggested that niche variation evident at the
individual level is critical to resolving species-level differences,
thereby allowing coexistence of seemingly similar species
(Clark 2010). These arguments have generated significant
recent interest in the potential positive effects of individual
© 2016 John Wiley & Sons Ltd/CNRS
2 S. P. Hart et al.
Idea and Perspective
variation on coexistence. In contrast, an opposing prediction
emerges from classical niche theory and suggests that niche
variation between individuals should increase species-level
niche overlap and therefore decrease the likelihood of coexistence in the absence of evolutionary change (Roughgarden
1972; Slatkin 1980; Taper & Case 1985; Doebeli 1996). When
taken together, these arguments suggest contrasting expectations for how intraspecific variation should influence diversity
maintenance.
Experimental studies on the effects of intraspecific variation
for competitive dynamics remain rare, and tend to investigate
the consequences of genetic rather than trait diversity (Hughes
et al. 2008). Results of these studies are equivocal with respect
to species coexistence (Booth & Grime 2003; Crutsinger et al.
2008; Fridley & Grime 2009), and reconciling these results is
difficult because the ecological and/or evolutionary mechanisms
underlying the effects are unknown. Thus, theory should be
particularly useful to guide expectations for how high levels of
intraspecific variation in nature influence species coexistence.
Here, we develop a quantitative framework and general predictions for understanding the ecological effects of individual
variation on coexistence. To do this, we incorporate individual variation into a simple model of competition. We focus
explicitly on variation in the traits and/or demographic and
competitive rates that determine species’ competitive ability or
niche differentiation, which together determine the outcome
of competition. In doing so, three pathways by which individual variation affects coexistence become clear. First, because
the dynamics of competing populations are often nonlinearly
dependent on the demographic and competitive rates of individuals, variation changes dynamics via nonlinear averaging.
Second, because demographic and competitive rates are themselves nonlinearly dependent on underlying traits, intraspecific
variation changes the mean strength of competition (also via
nonlinear averaging). And third, because individuals are discrete, intraspecific variation combines with demographic
stochasticity to cause population fluctuations that change
expected population trajectories of competing species. We find
that counter to common expectations of beneficial effects of
individual variation on coexistence (Clark 2010; Violle et al.
2012), these mechanisms cause intraspecific variation to
reduce the likelihood of species coexistence.
MODEL AND APPROACH
We base our analysis on an annual plant competition model
(Beverton & Holt 1957; Leslie & Gower 1958) and provide
model-independent generalisations in Appendix S1. The model
is well studied analytically (Cushing et al. 2004), and describes
competitive population dynamics in plant communities in the
field (Godoy & Levine 2014). In the absence of individual
variation, the dynamics of species 1 (and with subscripts
reversed, species 2) can be expressed as:
n1;tþ1 ¼ n1;t
g1 k1
1 þ a11 g1 n1;t þ a12 g2 n2;t
ð1Þ
where n1,t is the density of seeds of species 1 at time t, g1 is
the fraction of seeds that germinate and k1 is the per germinant fecundity at low density. The interaction coefficients a11
© 2016 John Wiley & Sons Ltd/CNRS
and a12 describe per capita effects of conspecifics and heterospecifics on seed production respectively. The model assumes
that seeds that do not germinate die, although the addition of
a seed bank does not qualitatively change the results
(Appendix S1).
An advantage of using this model is that we have previously
identified the quantities that describe niche differences and differences in competitive ability (average fitness differences)
between species (Godoy & Levine 2014; Godoy et al. 2014),
both of which are central to the leading hypotheses about the
effects of individual variation on coexistence. The interaction
between species is stabilised when intraspecific effects (e.g.
a11) are greater than interspecific effects (e.g. a21) and the
ratio of these effects for both species determines the niche difference. Such niche differences emerge from underlying trait
differences that allow species to exploit resources, for example, in different ways.
To isolate the determinants of competitive ability we first
remove the possibility for niche differentiation. We do so by
assuming that the per capita effect of species 2 on species 1,
for example, can be expressed as the product of species 2’s
generic competitive effect e2 on all species and species 1’s generic response to competition r1, such that a12 = r1e2. Competitive ability then becomes a trait of the species independent of
the identity of its competitors, as occurs, for example, with
competition for a single limiting resource (i.e. R* in Tilman
1982). Previous work shows that competitive ability of species
1 (and with subscripts reversed, species 2) is then defined as
(Appendix S4 in Godoy et al. 2014):
g1 k1 % 1
r1
ð2Þ
The demographic and competitive parameters in expression (2) determine the outcome of competition in the absence
of niche differentiation; the species with the largest value of
expression (2) excludes all other species.
To bound the problem and to focus on the effects of individual variation per se, we concentrate on cases where species
have the same amount of variation between individuals, and
where there is no trade-off between the amount of intraspecific variation in a trait and the mean value of the same or a
different trait. This approach is consistent with the focus of
the recent literature, which rarely invokes different levels
of variation or trade-offs to make predictions for the effects
of individual variation on species diversity. However, in specific cases we relax these assumptions to explore a wider range
of dynamics. Furthermore, the approaches we develop are
general so that one can explore any of the many possible
trade-offs or levels of variation that might occur in different
species (Appendix S2).
We divide the paper into three sections. In the first section,
we assess the effects of individual variation in competitive
ability via nonlinear averaging. In the second section, we
assess the effects of individual niche variation via changes in
both the mean and the variance of the interaction coefficients.
In the third section, we assess the effects of individual variation when individuals are discrete and populations’ finite. We
provide analytic support for our results in Appendices S1 and
S6, and compare our analytic and numerical results in
Intraspecific variation & species coexistence 3
Idea and Perspective
Appendix S3. We provide R code for implementing our methods in Appendix S2.
itself an individual-level trait. Rather, it determines only the
total number of germinants competing in a population, which
does not vary between individuals.
In contrast to g and k, per capita seed production is nonlinearly dependent on competitive sensitivity (Fig. 1b). Therefore, to quantify the effects of individual variation in
competitive sensitivity we extend the no-variation model
(eqn (1)) to include individual variation in r. We include variation between individuals in competitive sensitivity by allowing r to be described by a distribution of values. When
individuals of species 1 vary in competitive sensitivity according to a distribution p1(r1), dynamics become:
ð
g1 k1
ð3Þ
p1 ðr1 Þdr1
n1;tþ1 ¼ n1;t
1 þ r1 ðe1 g1 n1;t þ e2 g2 n2;t Þ
EFFECTS OF INDIVIDUAL VARIATION IN COMPETITIVE
ABILITY ON COMPETITIVE EXCLUSION
Superior competitive ability is conferred by high demographic
potential (high g and high k) and low sensitivity to competition (low [non-negative] r; eqn (2)). Therefore, if the intuition
that individual variation in competitive ability promotes coexistence is correct, then it is variation in these determinants of
competitive ability that must mediate the effect. Furthermore,
because differences in competitive ability promote exclusion, if
individual variation is to promote coexistence the effects of
variation must benefit weak competitors more than strong
competitors so that any species-level competitive disadvantage
can be overcome.
In general, individual variation will influence dynamics
when that variation changes the mean per capita offspring
production and therefore the realised population growth rate
(Bjørnstad & Hansen 1994). According to Jensen’s inequality
(Jensen 1906), these changes will occur when per capita offspring production depends nonlinearly on the demographic
rate that is varying (Bjørnstad & Hansen 1994). Following
these principles we can rule out any effects of individual variation in germination (g) or low-density fecundity (k) on
dynamics because both of these demographic rates are linearly
related to per capita seed production (Fig. 1a). Note that the
germination term, g, in the denominator of eqn (1) is not in
This expression is simply eqn (1) with interaction coefficients (a11, a12) replaced by the products of the generic competitive sensitivity of species 1 (r1) and the species’ generic
competitive effects (e1, e2), and with competitive sensitivity
following a distribution of values describing individual variation. Thus, annual population growth is the integrated seed
production of all individuals (differing in competitive sensitivity) in the population. The form of eqn (3) follows approaches
previously used in studies of variation on population dynamics (e.g. Hassell et al. 1991; Bjørnstad & Hansen 1994; Schreiber et al. 2011).
We explore the effects of individual variation on competitive dynamics with numerical simulations of eqn (3). To determine if variation can promote coexistence by assisting weaker
3
gj j
1+ rj (e j g j n j + ek gk nk )
gj j
gjnj +
gn
jk k k
Seed production
(b)
jj
1+
Seed production
(a)
2
2
1
1
0
Realized 0
seed
production
2
4
6
8
10
0
λ
0
.01
.02
.03
.04
.05
r
Intraspecific variation in
demographic parameters
Figure 1 Illustration of the influence of linear and nonlinear relationships on seed production when individuals within species vary in their demographic
rates. The normal distributions below the x-axes of each panel represent variation within a ‘blue’ species and a ‘grey’ species in (a) low-density seed
production, k, and (b) competitive sensitivity, r. Distributions to the left of the y-axes of each panel were generated numerically using the annual plant
model and show realised seed production after accounting for the effects of a fixed density of competitors. Linear relationships (a) mean that the higher
seed production of superior individuals is offset by the lower seed production of inferior individuals such that individual variation causes no net change in
species-level seed production. In contrast, nonlinear, concave-up relationships (b) cause the gains in seed production from low-sensitivity individuals to
more than outweigh the losses in seed production from high-sensitivity individuals – notice positive skew of both distributions left of the y-axis in (b).
Therefore, at a fixed density of competitors per capita seed production is greater with individual variation, and the effect is stronger for species with lower
mean r (the on-average competitively superior blue species has greater positive skew than the on-average competitively inferior grey species). The heights of
the distributions on the y-axis in (b) have been rescaled for clarity. Parameters in (a) where b = blue species and g = grey species: germination = 1,
kb ¼ 4:5, kg ¼ 2:5, r2kb ¼ r2kg ¼ 0:09, Nb = Ng = 50, abb = agg = 0.012, abg = agb = 0.006 (b): kb ¼ kg ¼ 3, Nb = Ng = 165, eb = eg = 1, rb ¼ 0:008, rg ¼ 0:015,
r2rb ¼ r2rg ¼ 1:7e % 6.
© 2016 John Wiley & Sons Ltd/CNRS
4 S. P. Hart et al.
competitors to overcome their competitive disadvantage, we
compete an on-average inferior competitor (higher mean r)
with an on-average superior competitor (lower mean r), keeping all other demographic rates equal between species. Initially
we assume that inferior and superior species have the same
amount of intraspecific variation, and we relax this assumption in a subsequent analysis. We describe individual variation
in competitive sensitivity with symmetric, four-parameter beta
distributions, which allows us to define reasonable positive
minimum and maximum values for r (Appendix S3). Our
qualitative results are robust to the type of distribution used
to describe individual variation (Appendices S1, S3).
We quantify the effects of intraspecific variation on competitive outcomes by estimating the growth rate of a low-density
invader in the presence of a resident species at its single-species equilibrium density. The winner in competition will have
a positive invasion growth rate and the loser in competition
will have a negative invasion growth rate. We quantify the
invasion growth rate of the species that loses in competition,
where a less negative growth rate indicates a smaller difference
between species in competitive ability. This metric quantifies
competitive dominance in a way that follows from the ‘average fitness difference’ of Chesson (2000).
To provide general, analytic support for our results, we
develop small-variance approximations to describe the effects
of variation for a range of different competition models
(Appendix S1). We compare our numerical simulations and
analytic approximations in Appendix S3. Importantly, the
value of the second-order term of the Taylor approximations,
which describes the concavity of the per capita growth rates
with respect to the varying parameter (r, in this case), can be
used to determine the effect of individual variation on population growth (positive, zero or negative depending on the sign
of the second derivative). When this value is compared
between species it can be used to quantify whether the effect
of variation differentially impacts inferior or superior competitors. We use this feature of the second term of the Taylor
approximation to interpret our simulation results.
Effects of individual variation in competitive ability: Results
Two of the three determinants of competitive ability – germination probability and low-density fecundity – are linearly
related to per capita seed production, such that individual
variation in these demographic rates has no effect on dynamics via nonlinear averaging (Fig. 1a). Individual variation in
the third determinant of competitive ability – sensitivity to
competition – strengthens the competitive superiority of the
dominant competitor, making coexistence more difficult
(Fig. 2a, 2b). The magnitude of this effect increases at an
increasing rate as variation between individuals increases
(Fig. 2c).
The key to understanding the influence of individual variation in competitive sensitivity is to understand how nonlinear
relationships between individual-level and population-level
demographic rates influence dynamics via Jensen’s inequality
(Fig. 1). The relationship between competitive sensitivity (r)
and per capita seed production is concave up (positive second
derivative; Fig. 1b). By Jensen’s inequality, this nonlinear
© 2016 John Wiley & Sons Ltd/CNRS
Idea and Perspective
relationship implies that individual variation in competitive
sensitivity increases population-level per capita seed production. To see why, consider a single species – the blue (or grey)
species in Fig. 1b. Individuals that are less sensitive to competition (r lower than the mean) contribute more to total seed
production and individuals that are more sensitive to competition (r higher than the mean) contribute less to total seed production. The key point, however, is that due to the
nonlinearity the higher seed production of competitively superior (less sensitive) individuals within a species exceeds the
losses of competitively inferior (more sensitive) individuals.
Most importantly, the effect of nonlinear averaging is not
equal for species with different mean competitive sensitivities
(Fig. 1b). This is because the magnitude of this effect depends
on the strength of the nonlinearity (the degree of curvature)
around the mean demographic rate. In our model, the degree
of curvature decreases with increasing values of competitive
sensitivity (i.e. the third derivative with respect to r is negative
for all competitor densities; Fig. 1b; Appendix S1). Consequently, individual variation in competitive sensitivity causes
larger increases in per capita seed production in the species
with lower mean sensitivity to competition – in other words,
the better competitor (Fig. 1b and 2b). As the better competitor receives the greater fitness benefit from individual variation, it competitively excludes the inferior species more
rapidly (Fig. 2a). More generally, the invasion growth rate of
the inferior species becomes more negative due to the negative
effect of higher heterospecific densities exceeding the positive
effect of nonlinear averaging on its own per capita seed production. Higher levels of individual variation thereby increase
the dominance of the superior competitor, as indicated by the
increasingly negative invasion growth rates of the inferior
competitor as variation increases (Fig. 2c). Only when inferior
competitors have substantially more individual variation than
superior competitors can individual variation weaken or
reverse a competitive hierarchy (Fig. 2d).
Our analytical approximations in Appendix S1 support
these results by demonstrating that these effects are general
across parameter values and different models. Indeed, similar
results emerge in other common models of competition (e.g.
Ricker, Hassell-May; Appendix S1) because species’ performance naturally tends to decline monotonically and approach
an asymptote with increasing competitive sensitivity (as in
Fig. 1b).
EFFECTS OF INDIVIDUAL NICHE VARIATION ON
COEXISTENCE
In two-species systems niche differences arise when individuals
within species limit the performance of conspecifics more than
they limit heterospecifics (i.e. a11 > a21). This could occur, for
example, if species differ in the soil depth at which they access
resources. Implicit in most formulations of species-level niche
differentiation is that individuals of the same species share the
same niche and so compete equally, but this need not be the
case. For example, a relatively deep-rooted individual of a
shallow-rooted species will be less sensitive to intraspecific
competition than will the other conspecifics on average.
Meanwhile, the same individual will, on average, be more
Intraspecific variation & species coexistence 5
Idea and Perspective
(a)
(b)
350
σ =0
300
2
σ = 4e−05
Density of blue species
250
200
150
100
50
0
σ2 = 0
σ2 = 4e−05
200
150
100
50
0
0
20
40
60
80
0
100
(c)
100
150
200
250
(d)
0.00
0.00
−0.05
−0.10
−0.15
−0.20
−0.25
−0.30
0e+00
2e−05
4e−05
6e−05
8e−05
Variance in r (σ2r,inferior = σ2r,superior)
−0.05
−0.10
species 1 wins
ln(inv. gr. rate) of losing sp.
ln(invasion growth rate)
50
Density of grey species
Time
−0.15
−0.20
−0.25
species 2 wins
Population density
250
2
σ2r,sp1
−0.30
0e+00
2e−05
4e−05
6e−05
8e−05
Variation in species 2 (σ2r,sp2),
the on−average inferior competitor
(i.e. sp. 2 has higher mean r)
Figure 2 The effect of individual variation in sensitivity to competition on coexistence. (a) Representative dynamics of two competing species (blue and
grey) with (solid lines) and without (circles) individual variation in competitive sensitivity. Because variation in r increases the single-species equilibrium
densities of both species, initial population sizes are set at half the equilibrium single-species density of the species with the lowest mean competitive
sensitivity (blue species); initial conditions do not change the outcome of competition. (b) Zero-net-growth isoclines for a two-species system with and
without individual variation. Isoclines were generated by numerically solving eqn (3) for zero growth of a focal species across a range of densities of its
competitor. After the inclusion of individual variation, the isoclines remain perfectly parallel but the distance between them increases, which indicates that
individual variation causes a larger difference in average fitness between the two species. (c) Effect of increasing variance in competitive sensitivity on the
invasion growth rate of the species with the lower mean competitive sensitivity. The increasingly negative invasion growth rates of the on-average inferior
species indicate increasing differences in competitive ability between species as individual variation increases; (d) Effect of unequal variance on invasion
growth rates. As intraspecific variation in the species that is the on-average poorer competitor (species 2, which has higher mean r) increases relative to its
competitor, it reduces and eventually reverses the difference in competitive ability between species. Individual variation was described using four-parameter
beta distributions. Parameter values for all plots, where b = blue species and g = grey species: germination = 1, kb = kg = 3, rbðsuperiorÞ ¼ 0:011,
rgðinferiorÞ ¼ 0:012, max/min values for beta distributions: rb & 0:0109, rg & 0:0109.
sensitive to interspecific competition from a deep-rooted species. This example implies that niche variation between individuals causes individuals within species to vary in their intraand interspecific interaction coefficients, and causes these
interaction coefficients to be correlated as emerges in models
of character displacement (Slatkin 1980; Vasseur et al. 2011).
We assess the effects of variation in the interaction coefficients
in the first part of our analysis of individual niche variation
on coexistence.
Niche variation is also expected to affect the mean value of
the interaction coefficients. This is most obvious when considering intraspecific interaction coefficients, which are maximised when all individuals interact with the environment (e.g.
use resources) in exactly the same way. Intraspecific niche
variation should therefore reduce the mean species-level
intraspecific interaction coefficient, and for similar reasons,
also affect the mean interspecific interaction coefficient. We
address the simultaneous effects of individual niche variation
on both the mean and variance of the interaction coefficients
in a biologically justified manner using an underlying niche
model. This forms the second part of our analysis.
Part 1: Effects of variation in the interaction coefficients
We first ask: how does variation between individuals in their
response to intra- and interspecific competition affect the
dynamics of species that are niche differentiated on average?
Even though this question only addresses part of the effects
of niche variation on coexistence, answering it is important
for interpreting the effects of empirically observed individual
© 2016 John Wiley & Sons Ltd/CNRS
6 S. P. Hart et al.
Idea and Perspective
variation in interactions. To explore these effects, we add variation in a11 and a22 and in a21 and a12, to the no-variation
model by allowing each of these parameters to follow a distribution. Dynamics of species 1 (and, with subscripts reversed,
species 2) can then be expressed as:
ðð
g1 k1
pða11 ; a12 Þda11 da12
ð4Þ
n1;tþ1 ¼ n1;t
1 þ a11 g1 n1;t þ a12 g2 n2;t
where p(a11, a12) describes the joint distribution of individual
variation in the intra- and interspecific interaction coefficients.
The joint distribution is necessary to account for the correlated nature of the interaction coefficients that emerge in Part
2. In Part 1, we assume the interaction coefficients are uncorrelated to concentrate on the effects of variance per se. The
population growth of each species is the integrated seed production of all individuals, accounting for their different
responses to intra- and interspecific competition.
We use eqn (4) to simulate competitive dynamics between
species with variation in their interaction coefficients. We are
interested in the effects of individual variation on species-level
niche differentiation and so we assess the case where species
are symmetric (i.e. parameters of the two species are equal)
and niche differentiated on average (i.e. a11 ¼ a22
[ a12 ¼ a21 ), and have equal levels of individual variation. We
investigate departures from these conditions in Appendix S4.
To assess the effects of variation on coexistence we quantify the
ability of a rare species, which experiences only interspecific
competition, to increase from low density in the presence of a
resident, which experiences only intraspecific competition. In
these analyses we assume that the invader is at low density, but
with large enough population size to ensure deviations from
deterministic dynamics are inconsequential (an assumption we
relax in our assessment of individual variation in the context of
stochastic dynamics, below). In our simulations we describe
individual variation in interaction coefficients using symmetric,
four-parameter beta distributions, although the choice of
distribution does not change our results (Appendices S1, S3).
Analytic, small-variance approximations of the invasion
dynamics support our results for a larger class of models
(Appendix S1).
Part 1: Results
When species are niche differentiated on average and the variance in the intra- and interspecific interaction coefficients is
equal, variation between individuals in their response to intraand interspecific competition strengthens stabilised coexistence
by increasing the ability of species to recover from low density
(Fig. 3).
The effect is again driven by Jensen’s inequality, and in particular, the different effects of nonlinear averaging on the
growth rates of common vs. rare species. Population growth
is a nonlinear function of the interaction coefficients with positive second derivative (concave-up relationship; Fig. 3a).
Therefore, variation between individuals in their response to
intra- or interspecific competition as expressed in the interaction coefficients increases the mean per capita seed production
of both species at any given density of competitors. The net
effect on coexistence again depends on the strength of the
nonlinearity (degree of curvature) of the growth function
around the mean interaction coefficient experienced by each
competitor. In our model, curvature decreases as the magnitude of the interaction coefficients increase (i.e. the third
derivative is negative; Fig. 3a, Appendix S1). When the variance in both interaction coefficients is equal, the positive
effect of variation on seed production is larger for the species
experiencing the smaller of the two interaction coefficients.
When species are niche differentiated on average
(i:e: a11 [ a21 ; a22 [ a12 ), the invader experiences the smaller
of the two interaction coefficients because it experiences only
(a)
(b)
400
2.5
Population density
Per capita seed production
3.0
2.0
1.5
1.0
0.5
0.0
300
σ2 = 1e−05
σ2 = 2.5e−06
200
σ2 = 0
100
0
0.000
0.005
0.010
0.015
0.020
Interaction coefficient (α11, α21)
0
20
40
60
80
Time
Figure 3 Effect of variation between individuals in their intra- and interspecific competition coefficients. (a) The relationship between the magnitude of a
competition coefficient and per capita seed production is nonlinear, with positive second derivative and negative third derivative such that variation results
in a larger increase in per capita seed production in the species experiencing the smaller average competition coefficient. (b) Dynamics when individuals
vary in their intra- and interspecific interaction coefficients. Individual variation increases per capita seed production of both species, which results in higher
equilibrium densities of the resident (blue species) and a higher invasion growth rate of the invader that more than compensates for the higher level of
competition it experiences from the resident, thus strengthening coexistence. Individual variation was described using four-parameter beta distributions.
Parameter values: (a) k = 3, germination = 1, total density = 350; (b), where b = blue species and g = grey species: germination =1, kb = kg = 3,
abb ¼ agg ¼ 0:006, agb ¼ abg ¼ 0:004, max/min values for beta distributions: mean interaction coefficient &0:0039.
© 2016 John Wiley & Sons Ltd/CNRS
Intraspecific variation & species coexistence 7
Idea and Perspective
interspecific competition. Consequently, an invader’s higher
per capita growth more than offsets the higher competition
exerted by a more abundant resident (Fig. 3b). In this case,
coexistence is strengthened by variation between individuals in
their response to competition. Importantly, this effect on
coexistence does not arise because individual variation favours
one competitor over another, but instead emerges because the
magnitude of the effect of individual variation is different for
species at high vs. low density.
species 1: Z1 ' N ðlZ1 ; r2Z1 Þ), assuming that species are niche
differentiated on average (i.e. lZ1 6¼ lZ2 ), have the same
amount of niche variation between individuals (i.e. r2Z1 ¼ r2Z2 )
and are equal in all other parameters. Our simulations incorporate the influence of niche variation on the mean, variance
and covariance of species’ interaction coefficients. In
Appendix S1 we provide analytic support for our numerical
results.
Part 2: Results
Part 2: Effects of niche variation between individuals on
coexistence
Niche variation involves more than variation in interaction
coefficients; it also involves changes in the mean values of
those interaction coefficients. To determine how niche variation between individuals affects coexistence, we need to understand how this variation translates into intra- and interspecific
interaction coefficients in a biologically justified manner. To
do this we rely on classic models of individual-level niche variation based on resource-utilisation functions traditionally used
to explore evolutionary character displacement (Roughgarden
1972; Slatkin 1980; Taper & Case 1985; Doebeli 1996). These
models begin with the premise that the strength of the interaction between two individuals is negatively related to their distance along a niche axis, such as the soil depth at which
plants access resources. Following classic evolutionary models
we formalise this thinking with the expression:
aðz; z0 Þ ¼ a0 e
0
2
%ðz%z0Þ
2r2
a
ð5Þ
where aðz; z Þ describes the strength of the interaction between
individuals at locations z and z0 along the niche axis. The
interaction takes its maximum value, a0, when two individuals
have zero niche difference (i.e. when z % z0 ¼ 0), and the
strength of the interaction declines monotonically with
increasing distance between individuals along the niche axis
following a Gaussian function with variance r2a .
The individual-to-individual interactions described in
eqn (5) can be used to calculate an intraspecific and interspecific interaction coefficient for each individual in the population (Appendix S5). The total intraspecific competition
experienced by a focal individual is the sum of each of its
interactions with all conspecific individuals. Because individual-to-individual interactions are additive, an individual’s
intraspecific interaction coefficient is equal to the mean of its
conspecific interactions. The interspecific interaction coefficient for each individual is determined the same way, but this
time averaging interactions with all heterospecific individuals.
Repeating for all individuals generates a distribution of
intraspecific and interspecific interaction coefficients that
reflect underlying niche differentiation between individuals.
To evaluate the effects of niche variation on coexistence, we
simulated an invasion scenario using eqn (4), but with numerically generated (and correlated) distributions of the interaction coefficients following from the niche model (eqn (5),
Appendix S2, S5). In these simulations, we describe niche
variation between individuals within species along a onedimensional niche axis using Gaussian distributions (i.e. for
Niche variation between individuals in our model weakens
stabilised coexistence by reducing the ability of species to
increase from low density in the presence of a competitor
(Fig. 4a). The negative effect on coexistence is largely driven
by reductions in the mean intraspecific competition coefficient
with increasing niche variation (Fig. 4b). This reduction
occurs because intraspecific interactions are strongest when all
individuals have the same niche, and thus niche variation necessarily reduces the mean intraspecific interaction coefficient
(Fig. 4b). Lower intraspecific competition increases the equilibrium density of the resident species (Fig. 4a), which reduces
growth of the low-density invader. In our model, mean interspecific interaction coefficients can increase or decrease
because of niche variation (explained in more detail in Appendices S1, S5), but in either case, the mean interspecific interaction coefficient is not reduced to the same extent as the mean
intraspecific interaction coefficient (Fig. 4b). Thus, the ratio
of mean intraspecific to mean interspecific effects declines with
individual niche variation, causing coexistence to be less stabilised.
The negative effects of changes in the mean values of the
interaction coefficients on invasion growth rates overwhelm
any positive effects that arise because of variation in the interaction coefficients (the effects demonstrated in Part 1;
Fig. 4c). Nevertheless, variance in the interaction coefficients
as a consequence of niche variation still influences dynamics.
For example, the negative consequences of changes in the
mean values of the interaction coefficients on invasion growth
rates are partially offset by the positive effects of variance in
the interaction coefficients that occur via nonlinear averaging
(Fig. 4c). This result emerges in part because individual niche
variation causes more variation in interspecific than
intraspecific interaction coefficients in our model (Appendices
S1, S5).
EFFECT ON COEXISTENCE OF VARIATION BETWEEN
DISCRETE INDIVIDUALS
As with most deterministic models of species interactions, our
analyses thus far assume that competition occurs on wellmixed landscapes of infinite size. Therefore, while species densities can fluctuate, population sizes are infinite. However,
important dynamical effects of individual variation emerge
explicitly because individuals are discrete and populations are
finite. Because individuals are discrete, births and deaths are
probabilistic events such that individuals that are otherwise
identical to each other can differ by chance in their realised
fecundity or survival. Such demographic stochasticity causes
© 2016 John Wiley & Sons Ltd/CNRS
8 S. P. Hart et al.
Idea and Perspective
(a)
Figure 4 The effect of intraspecific niche variation on coexistence. (a)
Dynamics of two competing species (blue and grey) with (solid lines) and
without (circles) individual niche variation. Niche variation reduces the
recovery rate of an invader in the presence of a resident at its singlespecies equilibrium density. (b) Effect of niche variation on the mean
strength of the intra- and interspecific interaction coefficients. (c)
Partitioning the effects of intraspecific niche variation on coexistence via
changes in the means and variances of the interaction coefficients on the
invasion growth rate. The dashed line shows effects that arise as a
consequence of changes in the mean values of the interaction coefficients,
while the solid line shows the combined effects of the changes in the
means and the variances. The difference between the lines represents the
positive effects of variance in the interaction coefficients on invasion
growth rates. The invasion growth rates for the dashed line were
calculated with eqn (1) using the mean values of the distributions of the
interaction coefficients after accounting for intraspecific niche variation,
but ignoring the variance of those distributions. Invasion growth rates for
the solid line were calculated using eqn (4), which accounts for changes in
both means and variances. Individual variation was described using
Gaussian distributions. Parameter values: a) where b = blue$ species$ and
g = grey species, germination $= 1, kg$ = kb = 3, a0 = 0.008, $zb % zg $ ¼ 1,
r2a ¼ 0:5; b) and c) a0 = 0.008,$z1 % z2 $ ¼ 1, r2a ¼ 0:5.
Population density
700
600
500
400
σ=1
300
σ = 0.5
200
σ=0
100
0
0
20
40
60
80
Time
Mean competition coefficient
0.008
(b)
α11
α21
0.006
0.004
0.002
0.000
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Individual−level niche variation (σ)
ln(invasion growth rate)
0.6
(c)
Effect of change in:
0.5
Mean & var.
0.4
Mean
0.3
single model is a two-step process. The first step determines
how many seeds germinate at the beginning of the year, and
therefore how many discrete plants compete and produce
seeds. The second step determines the discrete number of
seeds produced by each plant at the end of the year. While
most implementations of stochastic competitive dynamics
assume that all individuals within species have the same
expected demographic rates, we allow individuals to vary continuously in their expected germination probability and lowdensity fecundity. We concentrate on variation in these traits
to determine how the discrete nature of individuals mediates
the effects of individual variation, independent of the effects
of nonlinear averaging demonstrated in the sections above.
The stochastic model describing dynamics on finite landscapes of discrete individuals that vary in their expected
demographic rates can be expressed as:
0.2
G1 ¼
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Individual−level niche variation (σ)
annual growth rates of finite populations to vary between
years, which reduces expected population growth rates for single species (Lande 1998). Importantly, individual variation in
expected demographic rates – or demographic heterogeneity –
may modify this effect (Kendall & Fox 2003). However, the
effects of demographic heterogeneity on the stochastic dynamics of competing populations have received little attention. To
study these effects, we analyse a stochastic version of the
annual plant model that admits continuous variation in
expected demographic rates between discrete individuals.
There are two sources of demographic stochasticity for an
annual plant whose life cycle is described in eqn (1): germination and seed production. Incorporating both sources into a
© 2016 John Wiley & Sons Ltd/CNRS
N1;t
X
i¼1
N1;tþ1 ¼
" #
Bernoulli X1;i
G1
X
i¼1
Poisson
Y1;i
1 þ a11SG1 þ a12SG2
!
ð6aÞ
ð6bÞ
where i represents an individual, G1 describes the number of
successfully germinating seeds of species 1 and N1,t is the
number of discrete individuals of species 1 at time t. In
eqn (6a), X1,i is a germination probability independently
drawn from a fixed distribution (X1) that describes continuous
variation between individuals in expected germination probabilities. The Bernoulli process then ensures each seed either
germinates or does not. In eqn (6b), total seed production
(N1,t+1) is the sum of the Poisson distributed seed production
of each individual. Y1,i is a single value of expected low-density fecundity drawn from a distribution Y1 of expected lowdensity fecundities. S describes landscape size and consequently GS1 is the density of germinants in the landscape. In
the limit of landscapes of infinite size there are essentially an
infinite number of invading individuals (though at density
approaching zero) and the dynamics of the stochastic model
Intraspecific variation & species coexistence 9
Idea and Perspective
(6) converge on the dynamics of the mean-field model
(eqn (3); Kurtz 1981; Faure & Schreiber 2014). A second set
of equations with species subscripts reversed describes dynamics of species 2.
Because germination probabilities are chosen randomly and
independently for each individual, the probability of a seed
germinating is the expected value X1 of X1, and therefore the
number of germinating seeds is a binomial distribution, i.e.
G1 ' BinomialðN1;t ; X1 Þ. Because the number of germinating
seeds in this expression depends only on the mean germination rate X1 across all individuals, individual variation in
expected germination probabilities has no effect on dynamics
(see Kendall & Fox 2003 for detailed discussion). In contrast,
with stochasticity in individual-level seed production described
by a Poisson distribution, individual variation in expected
fecundity increases the total variance in realised fecundity
(Kendall & Fox 2003; Melbourne & Hastings 2008).
Equation (7) can be used to evaluate the effects of individual
variation in demographic rates between discrete individuals on
the expected growth rate of the invader. Crucially, notice that
variation between individuals (r21 ) reduces the growth rate of the
invading species because the sign of the second term is negative.
This effect occurs because growth is a multiplicative process and
any factor that causes variation in growth between years – individual variation in this case – will tend to reduce the expected
growth rate (Lande et al. 2003). The approximation also shows
that the negative effect of individual variation is small and
declines as population size increases (Fig. 5a). Overall, in systems
of competing finite populations composed of discrete individuals,
populations will suffer from being reduced to low numbers
because individual variation causes fluctuations in annual growth
rates that depress expected growth.
Analytical approximation of invasion growth rates
Our analytical approximation (7) assumes that variation
between individuals is small and that the resident assumes a
fixed density in a large landscape and so cannot respond to
changes in invader population size, nor fluctuate itself. We are
able to relax these assumptions in stochastic simulations using
eqn (6). Furthermore, the analytical approximation does not
distinguish between the sources of the variation between individuals. Importantly, the total amount of individual variation
comes from both expected and stochastic differences between
individuals (Kendall & Fox 2003). Because we are explicitly
interested in the consequences of individual variation in
expected demographic rates and how these effects are mediated by the discrete nature of individuals, we explore this
dynamic using simulations.
In our simulations we assess the effect of variation in
expected low-density fecundity (k) between discrete individuals. We simulate an invasion scenario, which requires first
simulating resident dynamics to generate a range of potential
(stochastic) resident population sizes. We then initialise each
invasion scenario by drawing one value from this resident
abundance distribution as the initial resident population size,
while the invader begins with two individuals. We then use
eqn (6) to project population sizes of both species forward in
time. We use the median trajectory of 10 000 runs to compare
stochastic invasion dynamics with and without individual variation. We also record the proportion of 100 000 realisations
in which the invader reaches half of its equilibrium population
size in the two-species system, which should provide a good
indication of the likelihood of longer term coexistence (Turelli
1980; Faure & Schreiber 2014). In our analyses the resident
and invader have the same mean demographic rates, and the
same variance in low-density fecundity, and we set intraspecific competition coefficients greater than interspecific competition coefficients to ensure that deterministic low-density
growth rates of the invader are positive. Individual variation
in expected low-density fecundity (k) is described using
gamma distributions. From the first section of the paper it is
clear that in the absence of demographic stochasticity individual variation in k has no effect on coexistence because it is
linearly related to population growth. Therefore, any effects
of variation in expected low-density fecundity in our
To derive an analytical approximation of the stochastic growth
rate of a rare invader we assume the landscape is sufficiently
large (S ≫ 1) that resident dynamics tend to remain near the
deterministic single-species equilibrium density of the meanfield model (Faure & Schreiber 2014). Residents are thus
affected by variation between individuals in their expected
demographic rates (as in the deterministic models in the sections
above), but are negligibly influenced by dynamics that emerge
as a consequence of the discrete nature of individuals in populations of small size. With the density of the resident species (species 2, for example) fixed at its equilibrium, we can describe the
stochastic dynamics of an invading species (species 1), assuming
the system is not so large that the invader itself is immune to
the effects of demographic stochasticity.
While realisations of stochastic population trajectories can
vary substantially, we are interested in the most likely population trajectory of an invader. When considering population
trajectories driven by variable fates of discrete individuals, a
square-root transformation of population size tends to produce a more symmetrical distribution of potential population
sizes (Lande 1998). This means that the expected population
size on the square-root scale describes the change in the median/mode population size. For an invader (species 1) that
remains rare, the dynamics of its expected population size on
the square-root scale can be expressed (Appendix S6):
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi r21 l%3=2
1
ffiffiffiffiffiffiffiffi
E½ N1;tþ1 ) * l1 N1;t % p
8 N1;t
ð7Þ
where l1 is the arithmetic mean individual-level seed production after accounting for competition from the resident and r21
describes the total variance in individual-level seed production, including both expected (i.e. innate) and stochastic differences between individuals in their demographic rates. When
individuals vary in their low-density fecundity, l1 equals
g1 k1
^
^2 a12 , where N2 is the equilibrium density of the resident
1þg2 N
&
'2
and r21 equals r2k1 1þg gN1^ a
, where r2k1 is the variance
2
2 12
between individuals in low-density fecundity.
Stochastic simulations of invasion trajectories
© 2016 John Wiley & Sons Ltd/CNRS
10 S. P. Hart et al.
(a)
0.025
0.00
(b)
Stochastic
0.020
−0.01
Density
Inv. growth w/ind. var − w/out ind. var
Idea and Perspective
−0.02
−0.03
Stoch. + ind. var.
Deterministic
0.015
0.010
0.005
−0.04
0
10
20
30
40
50
0.000
0
100
Population size
(c)
0.7
Deterministic
Prop. successful invasions
Median invader pop. size
50
Stochastic
40
200
300
400
Resident carrying capacity
Stoch. + ind. var.
30
20
10
(d)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
0
2
4
6
8
10
Time
0
2
4
6
8
10
Individual variation
in low−density fecundity (σ2)
Figure 5 The effect of individual variation between discrete individuals on coexistence. (a) The effect on the invasion growth rate of variation between discrete
individuals from eqn (7). The negative effect of individual variation declines as population size increases. (b) Distributions of stochastic resident carrying
capacity with and without individual variation. When the deterministic resident carrying capacity is large, carrying capacity is relatively unaffected by variation
between discrete individuals (n = 100 000 simulations). (c) The effect of variation in expected fecundity between discrete individuals on invasion trajectories.
The deterministic trajectory (red line) was evaluated using eqn (1). Because k is linearly related to population growth (Fig. 1a) the deterministic trajectory is the
same with or without individual variation. Demographic stochasticity has a negative effect on the median trajectory (dashed line), and this effect is exacerbated
with variation between individuals in their expected fecundity (solid black line) (n = 10 000 simulations). (d) The effect of individual variation on the
proportion of successful invasions of a species at low abundance in the presence of a resident competitor. Higher levels of individual variation reduce the
likelihood of the invader reaching half its equilibrium population size (n = 100 000 simulations). Individual variation for the simulations was described using
gamma distributions. Parameter values:k1 ¼ k2 ¼ 3, g1 = g2 = 1, a11 = a22 = 0.008, a12 ¼ a21 ¼ 0004; a) r2k ¼ 0:9; b & c) r2k1 ¼ r2k2 ¼ 5.
stochastic simulations are mediated by demographic stochasticity arising from the discrete nature of individuals.
Results of stochastic simulations
Our simulations support the central result from the analytical
approximation – variation between discrete individuals in
expected low-density fecundity reduces the median population
trajectory of a low-density invader (Fig. 5c). Higher levels of
individual variation increase the negative effect on invasion
trajectories, which decreases the likelihood of successful invasion (Fig. 5d).
We know from the deterministic model that individual variation in low-density fecundity has no effect on competitive
dynamics via nonlinear averaging (Fig. 1a). However, following Lande (1998), when individuals are discrete and populations finite, variation between individuals causes fluctuations
between years in annual population growth rates and this
reduces likely growth trajectories. Because individual variation
© 2016 John Wiley & Sons Ltd/CNRS
in expected fecundity increases the total amount of variation
between discrete individuals in realised fecundity the negative
effects on population growth are exacerbated. For an invading
species with low numbers of individuals, between-year growth
fluctuations reduce population growth and so reduce the most
likely invasion trajectory (eqn (7), Fig. 5a and c). In contrast,
the resident population is much larger and so suffers little
from this effect (Fig. 5b). Individual variation thereby favours
resident species with large population size over invading species with small population size. Because these effects induce
positive frequency dependence (i.e. favouring species at high
abundance over species at low abundance), individual variation in low-density fecundity (k) between discrete individuals
destabilises coexistence.
DISCUSSION
Our work suggests that the purely ecological effects of individual variation do not promote species coexistence. If
Idea and Perspective
anything, intraspecific variation impedes the maintenance of
species diversity. Working with a common model of competition we find three important results: (1) variation between
individuals in the drivers of competitive ability either has no
effect on coexistence or tends to reinforce competitive dominance via the effects of nonlinear averaging (Fig. 2); (2)
while individual variation in the response to intra- and interspecific interactions can stabilise coexistence (Fig. 3), niche
variation between individuals tends to weaken coexistence by
strongly reducing species-level niche differentiation (Fig. 4);
and (3) variation between individuals combines with demographic stochasticity to reduce the likelihood of long-term
coexistence by favouring abundant competitors over species
recovering from small population sizes (Fig. 5). Only when
additional conditions are invoked, such as mean–variance
trade-offs (e.g. Lichstein et al. 2007; Bolnick et al. 2011),
including higher levels of individual variation in otherwise
inferior competitors (Fig. 2d), do we find that individual
variation increases the potential for coexistence. In sum, our
results suggest that the ecological effects of individual variation create additional hurdles for species to overcome if they
are to coexist.
Individual variation in competitive ability promotes competitive
exclusion
When individuals within species vary in competitive ability,
differences in competitive ability between species are less
obvious (Hubbell 2005; Clark 2010; Messier et al. 2010).
Intuition might therefore suggest that individual variation in
traits that determine competitive ability should reduce differences in competitive ability between species. Our work suggests the opposite is true (Fig. 2). Why doesn’t the intuition
work?
Addressing this question requires understanding if gains
from individuals that are relatively strong competitors offset
losses from individuals that are weak competitors, and
whether the balance differs between species with different
mean competitive abilities. When the relationship between
individual-level traits and population growth is linear (as for
low-density fecundity and germination), there is no net change
in competitive ability at the species level and therefore no
effect of variation in these traits on coexistence. But for nonlinear relationships individual variation changes the potential
growth rate of both superior and inferior species at any fixed
level of competition. When individuals vary in their sensitivity
to competition, species gain more (in terms of seed production) from individuals that are stronger competitors than they
lose from individuals that are weaker competitors (Fig. 1). To
promote coexistence, the effects of individual variation must
benefit inferior species more than superior species, but in our
model and other common models of competition the opposite
occurs. This is because the curvature of the relationship (i.e.
the positive effect of the nonlinearity) around the mean trait
value is greater for superior than inferior species regardless of
the densities of competitors, giving the net competitive advantage to the superior species (Fig. 2). Only when there is more
variation between individuals of an on-average inferior species
can a competitive disadvantage be overcome (Fig. 2d).
Intraspecific variation & species coexistence 11
Therefore, in the absence of a priori reasons to expect more
variation in inferior than superior competitors, individual
variation in sensitivity to competition reinforces competitive
hierarchies, making coexistence more difficult.
Individual niche variation tends to weaken coexistence
Individual-level niche variation causes changes in both the
mean and the variance of the interaction coefficients that ultimately determine species-level niche differentiation. In our
model, while increasing the variance in the interaction coefficients assists in the recovery of species from low density via
the effects of nonlinear averaging (Fig. 3), individual variation in niches ultimately weakens coexistence by reducing the
mean strength of intraspecific relative to interspecific competition (Fig. 4). This occurs because niche variation can only
weaken average per capita intraspecific competition, but will
increase the strength of interspecific competition in at least
some individuals. Consequently, decreases in the strength of
intraspecific effects cannot be offset by decreases in interspecific effects. Our result is consistent with those of character displacement models showing that selection favours reductions
in trait variation leading to less niche overlap (Pfennig &
Pfennig 2012).
Although understanding the effects of niche variation on
coexistence requires recognising that the negative effect of
changing mean interaction coefficients overwhelms any effects
of variation in these coefficients, the latter effect remains
empirically relevant. Quantifying niche variation in a theoretically justified manner is difficult (Kraft et al. 2015), whereas
measuring interaction coefficients and their variation between
individuals is feasible. Further empirical approaches are
needed to evaluate the effects of individual niche variation per
se.
Individual variation between discrete individuals further destabilises
coexistence
Variation in expected fecundity between discrete individuals
carries a small but unavoidable demographic cost that
reduces the likelihood of long-term coexistence (Fig. 5; Turelli 1980). This is because variation between discrete individuals causes fluctuations in population-level demographic
rates between years, which reduces average population
growth (Lande 1998). Because this effect is disproportionately experienced by rare species, individual variation results
in positive frequency-dependent dynamics (abundant species
are favoured), impeding coexistence (Fig. 5). The more individual variation, the greater the demographic cost of being
rare and the more vulnerable species are to competitive
exclusion.
Although variation between discrete individuals incurs a
cost, not all demographic rates are equal in this respect. While
individual variation in low-density fecundity increases the
costs of demographic stochasticity, individual variation in germination success has no effect over and above the negative
effects of simple demographic stochasticity (but see Kendall &
Fox 2002). Finally, we note that although the effects of individual variation combined with demographic stochasticity are
© 2016 John Wiley & Sons Ltd/CNRS
12 S. P. Hart et al.
consistently negative for a population depressed to low numbers, effects tend to be weak and diminish rapidly as population size increases (Fig. 5a). Nevertheless, these effects will
increase the demand on the species-level niche differences
required for coexistence to occur.
Idea and Perspective
species population dynamics (Kendall & Fox 2002), with
unknown consequences for coexistence. Finally, we note that
our results in no way undermine the central importance of
individual variation as the material on which selection can act
to promote coexistence (Taper & Case 1985; Doebeli 1996;
Vellend 2006).
Limitations and extensions
Much of our work has focused on individual variation in
demographic rates and competitive parameters, while most
field studies quantify intraspecific variation in species’ functional traits, such as wood density or specific leaf area (Messier et al. 2010; Jung et al. 2014; Siefert et al. 2015).
Nonlinear relationships between specific functional traits and
the demographic rates that we consider will add yet another
layer of nonlinear averaging, which can affect the distributions (means, variances and higher moments) of demographic
rates in complex ways. Indeed, our niche model provides one
such example, where effects on dynamics are influenced by the
nonlinear relationship between niche position and the interaction coefficient (eqn (5)), such that the distribution of the
underlying niche traits affects the mean species-level interaction strengths (Fig 4b, Appendix S2). Empirically accounting
for these additional nonlinearities between traits and demographic rates, which may not simply be concave or convex, is
important because they will modify, and can sometimes
reverse, the predicted effects of variation (e.g. Drake 2005).
Until these relationships are understood, our approach provides a default set of expectations for the effects of individual
variation on coexistence.
To resolve the effects of individual variation when individuals are discrete we applied the logic of the invasion condition,
developed for infinite systems, to finite systems. Although this
condition is not proven in competitive systems with finite
numbers of individuals (but see Faure & Schreiber 2014), we
believe the underlying logic should still apply. Moreover,
empirical studies commonly make this assumption when relying on inference from the fate of small numbers of discrete
individuals to understand coexistence (Seabloom et al. 2003;
Levine & Hille Ris Lambers 2009; Hart & Marshall 2013).
One lesson from our results for such studies is that ecologists
may need to account for the effects of demographic stochasticity and individual variation on predicted dynamics. Indeed,
recent theoretical work formally includes extinction risk as a
consequence of stochasticity into assessments of species coexistence, which is more consistent with empirical realities (Turelli 1980; Tilman 2004; Adler & Drake 2008; Gravel et al.
2011; Kramer & Drake 2014).
Two important extensions of our work will require understanding how spatial processes and structured variation
between individuals mediate the influence of individual variation on multispecies interactions. We have assumed a wellmixed system and while our own preliminary analyses suggest
that spatially explicit neighbourhood competition does not
change our qualitative results, additional analyses are
required. In addition, we assume that individuals within and
between generations are independent. However, variation in
demographic rates that is correlated between discrete individuals can modify the effects of individual variation for single© 2016 John Wiley & Sons Ltd/CNRS
Implications for empirical studies
There is currently an unprecedented push to quantify
intraspecific trait variation in nature and the majority of this
work appears motivated by the potential positive effects of
individual variation on species diversity (Violle et al. 2012).
Arguably, the most prominent recent empirical work on the
positive effects of individual variation on coexistence is by
Clark and colleagues (e.g. Clark 2010; Clark et al. 2011). This
work describes analytical techniques that demonstrate that
empirical differences between species only become clear when
variation between individuals is appropriately accounted for
(Clark 2003). However, there is a large difference in aim
between using individual-level data to infer species-level differences, and explicitly testing the causal effects of individual
variation on species dynamics. Our contribution addresses the
latter – the effects of individual variation on dynamics – and
suggests that individual variation is more likely to prevent
coexistence than promote it, and may often have little or no
effect.
Our results have clear implications for future empirical
work. First, empirical evidence that intraspecific variation is
significant and obscures species-level differences is insufficient for inferring how such variation affects species diversity. Second, our results suggest that moving forward
requires not only quantifying the magnitude of intraspecific
trait variation, but also defining the potentially nonlinear
relationship between the trait of interest and the population
growth rates of the competing species. This latter step
requires integrating field data into mathematical models of
competition, which also allows one to quantify the effects
of individual variation on stochastic dynamics. Only when
the relationship between traits and demographic parameters
is explicitly described can robust hypotheses about the
effects of individual variation on competitive outcomes be
accurately formulated.
ACKNOWLEDGEMENTS
We thank David Viola, Mark Rees, Peter Chesson, Jacob Usinowicz, Sabine G€
usewell, Jake Alexander and Christopher
Johnson for conversations on the topic and assistance, and
Bruce Kendall, Gy€
orgy Barab"
as, and Ian Carroll for comments
on an earlier version of the manuscript. SJS was supported by
US National Science Foundation Grant DMS-1313418.
AUTHORSHIP
SPH and JML conceived the problem; SPH did the research,
with JML and SJS providing input; SJS derived the approximations; SPH wrote the manuscript and all authors contributed to revisions.
Idea and Perspective
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SUPPORTING INFORMATION
Additional Supporting Information may be found online in
the supporting information tab for this article.
© 2016 John Wiley & Sons Ltd/CNRS
Idea and Perspective
Editor, Tim Coulson
Manuscript received 10 March 2016
First decision made 6 April 2016
Manuscript accepted 20 April 2016
Appendix S1
In this Appendix, we derive the main approximations for the deterministic models with individual
variation. To highlight the main assumptions under which our results hold, we consider a more general
class of models than presented in the main text. For these models, let i denote the maximal number
of seeds produced by individuals of species i, gi the fraction of successfully geminating seeds, si survivorship of non-germinating seeds of species i to the next year, and ↵ij the competition coefficients.
If ni,t denotes the seed density of species i in year t, then
(S1.1)
n1,t+1 =
1 g1 n1,t f (↵11 g1 n1,t
+ ↵12 g2 n2,t ) + s1 (1
g1 )n1,t
n2,t+1 =
2 g2 n2,t f (↵11 g1 n1,t
+ ↵12 g2 n2,t ) + s2 (1
g2 )n2,t .
where f (x) is a decreasing, convex, function (i.e. f 0 (x) < 0 and f 00 (x) > 0 for all x), and f 000 (x) < 0 for
1
all x. Functions f (x) satisfying these assumptions include the Leslie-Gower type model f (x) = 1+x
as
used in the main text with si = 0, Ricker type models f (x) = exp( x), and Hassell-May type models
1
f (x) = (1+x)
1.
b with b
As in the main manuscript, our analysis is divided into three parts. First, we consider the case in
which there are no niche di↵erences and examine the e↵ect of individual variation in the determinants of
competitive ability on the competitive dynamics. Second, we examine the e↵ect of individual variation
on intra- and inter-specific competition coefficients on the competitive dynamics. Finally, we examine
the e↵ect of niche variation on the mean and variance of the competition coefficients. As discussed in
the main text, individual variation in i , gi , and si have no e↵ect on the dynamics of equation (S1.1).
Hence, in both cases, we focus on the e↵ect of individual variation in ↵ij on the competitive dynamics.
Furthermore, while our analysis focuses on the case when there is equal amounts of individual variation
2
for both species, we do indicate how the conclusions change if the variances di↵er between the
species.
It is important to note that the assumptions of the approximations may not hold when additional
nonlinearities (e.g. nonlinear mappings of measured traits to competition coefficients or niche values)
need to be accounted for. For these additional nonlinearities, however, one can use a similar approach
as developed here to identify how these nonlinearities influence competitive outcomes.
E↵ects of individual variation in competitive ability on competitive exclusion. As in the
main text, we assume that ↵ij = ri ej , there is variation in ri , and 1 = 2 = , g1 = g2 = g, s1 = s2 = g,
and e1 = e2 = e. To allow for individual variation in the ri parameters, let Z be a random variable
with mean 0 and variance 1. Let 2 = 12 = 22 be the variance in the ri parameters and define
h(r, C) = E[ gf ((r + Z)C) + s(1
g)]
where E denotes the expectation. Then the model with individual variation becomes
(S1.2)
n1,t+1 = n1,t h(r1 , Ct ) where Ct = e gn1,t + e gn2,t
n2,t+1 = n2,t h(r2 , Ct ).
The case in the main text corresponds to choosing Z to be a continuous random variable in which case
the expectation in the definition of h yields an integral with respect to the probability density function
of Z.
A second order Taylor expansion of h with respect to yields
(S1.3)
h(r, C) ⇡ gf (rC) +
1
g 00
f (rC)C 2
2
2
+s
2
for all C. As f 00 (rC) > 0, variation 2 in ri increases the per-capita growth rate for both species for
all competitive feedbacks i.e. all C values. Furthermore, our assumption that f 000 < 0 implies that
if r1 < r2 , then f 00 (r1 C) > f 00 (r2 C) and, thus, the per-capita growth rate of competitively superior
species (species 1) increases more with than the per-capita growth rate of inferior species (species
2). As this added advantage for the superior species occurs at all population densities, the superior
competitor will exclude the inferior competitor more quickly.
If the variances i2 di↵er for the two species, we can still use equation (S1.3) with replaced by i2 .
In this case, if the competitive advantage of the superior species (species 1) isn’t too great and the
inferior species has a sufficiently larger variance ( 2
1 ), then (S1.3) implies that the inferior species
may get the larger fitness boost and thereby exclude the superior species.
E↵ects of individual variation in intra- and inter-specific competition coefficients on coexistence. Next, we consider the case that 1 = 2 , s1 = s2 = s, g1 = g2 = g, ↵11 = ↵22 =: ↵ and
↵12 = ↵21 =: . To allow for individual variation in ↵ and , let Y and Z be random variables with
mean 0 and variance 1. Let 2 = 12 = 22 be the variance in the competition coefficients and define
h(x, y, ) = E[ gf ((↵ + Y )gx + ( + Z)gy) + s(1
g)].
The model with individual variation becomes
n1,t+1 = n1,t h(n1,t , n2,t , )
(S1.4)
n2,t+1 = n2,t h(n2,t , n1,t , ).
The special case in the main text corresponds to choosing Y and Z to be continuous random variables
in which case the expectation in the definition of h yields a double integral with respect to the joint
probability density function of Y and Z. As Y and Z have mean 0 and variance 1, we have
@h
(x, y, 0) = 0
@
@ 2h
(x, y, 0) = gf 00 (↵gx + gy)g 2 (x2 + ⇢xy + y 2 )
2
@
where ⇢ = E[Y Z] is the correlation between Y and Z.
To examine the e↵ects of individual variation on coexistence, we determine its e↵ects on per-capita
growth rate of each species when it is rare and the other species is at equilibrium. Without loss of
generality, let species 1 be at equilibrium and let species 2 be rare. To understand the e↵ect of individual
variation on the equilibrium abundance of species 1, let n̂1 ( ) denote the non-zero equilibrium as a
function of . This equilibrium satisfies h(n̂1 ( ), 0, ) = 1. Implicit di↵erentiation with respect to
yields
@h
@h
0=
(n̂1 (0), 0, 0)n̂01 (0) +
(n̂1 (0), 0, 0) .
@x
|@
{z
}
=0
Hence,
n̂01 (0)
= 0. Applying implicit di↵erentiation a second time yields
✓ 2
◆
@ h
@ 2h
@h
0
0 =
(n̂1 (0), 0, 0)n̂1 (0) +
(n̂1 (0), 0, 0) n̂01 (0) + (n̂1 (0), 0, 0)n̂001 (0)
2
| {z } @x
@x
@x@
=0
@ 2h
@ 2h
+ 2 (n̂1 (0), 0, 0) +
(n̂1 (0), 0, 0)n̂01 (0) .
@
|@ @x
{z
}
=0
3
Hence,
n̂001 (0)
=
@2h
(n̂1 (0), 0, 0)
@ 2
@h
(n̂1 (0), 0, 0)
@x
=
f 00 (↵gn̂1 (0))n̂1 (0)2 g 2
>0
f 0 (↵gn̂1 (0))↵g
as f is decreasing and convex. By Taylor’s theorem
f 00 (↵gn̂1 (0))n̂1 (0)2 g
.
2
f 0 (↵gn̂1 (0))↵
2
n̂1 ( ) ⇡ n̂1 (0)
Hence, individual variation in the intraspecific competition coefficient increases the equilibrium abundance of species 1.
The per-capita growth rate of species 2 at the equilibrium determined by species 1 equals h(0, n̂1 ( ), )
which we call R( ). To get a small variance approximation of this term, we first compute the first
derivative at = 0
@h
@h
R0 (0) =
(0, n̂1 (0), 0)n̂01 (0) +
(0, n̂1 (0), 0) = 0.
@y
@
Taking the second derivative yields
✓ 2
◆
@ h
@ 2h
@h
00
0
R (0) =
(0, n̂1 (0), 0)n̂1 (0) +
(0, n̂1 (0), 0) n̂01 (0) + (0, n̂1 (0), 0)n̂001 (0)
2
| {z } @y
@y
@y@
=0
2
+
2
@ h
@ h
(0, n̂1 (0), 0) +
(0, n̂1 (0), 0) n̂01 (0)
2
| {z }
@
@ @y
=0
2
=
@h
@ h
(0, n̂1 (0), 0)n̂001 (0) + 2 (0, n̂1 (0), 0)
@y
@
where the first component corresponds to the change in R due to the increase in the density of the
common species (species 1) and the second component corresponds to the change in R due to individual
variation in the rare species (species 2). Hence, the change in the per-capita growth rate due to
individual variation is approximated by
✓ 00
◆
2
2
f (↵gn̂1 (0))n̂1 (0)2 g
0
(S1.5)
gf ( gn̂1 (0)) g
+
gf 00 ( gn̂1 (0))g 2 n̂1 (0)2
2
f 0 (↵gn̂1 (0))↵
2
✓ 00
◆
2
f ( gn̂1 (0))
f 00 (↵gn̂1 (0))
3
2 0
=
g n̂1 (0) f ( gn̂1 (0))
2
f 0 ( gn̂1 (0))
↵ f 0 (↵gn̂1 (0))
Hence, this change in the rare species’ per-capita growth rate is positive if and only if
f 00 ( n̂1 (0)) f 0 (↵n̂1 (0))
>
f 0 ( n̂1 (0)) f 00 (↵n̂1 (0))
↵
When intraspecific competition is greater than interspecific competition (i.e. ↵ > ), this inequality
1
is satisfied for the Leslie-Gower type model f (x) = 1+x
as used in the main text with si = 0, Ricker
1
1. Hence, mutual
type models f (x) = exp( x), and Hassell-May type models f (x) = (1+x)
b with b
invasibility is enhanced by individual variation in these cases. We conjecture that this conclusion holds
more generally under the assumption f 000 (x) < 0 for all x. We also note that the reverse inequality
holds if > ↵ for the aforementioned functions. Hence, in this case, individual variation strengthens
priority e↵ects.
If individual variation di↵ers among the species (i.e. 1 6= 2 ), then equation S1.5 still holds but
with the first 2 term being replaced by 12 and the second 2 term being replaced by 22 . In this case,
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Figure S1.1. Changes in the means and variances of the competition coefficients
due to variation
in the niche values. Circles correspond to exact values calculated numerically, and lines correspond to the analytical approximations. Parameters:
(|z z 0 |2 ) = exp( |z z 0 |2 /2) and Xi normally distributed with means 0 and variance
1.
if ↵ < but close to
species 2.
and
1/ 2
1, then coexistence may be disrupted with species 1 excluding
E↵ects of niche variation on competition coefficients. To examine the e↵ects of niche variation
on coexistence, we assume that the two species compete along a one-dimensional niche axis. For two
individuals with niche values, z and z 0 , let ↵(z, z 0 ) = (|z z 0 |2 ) represent the strength of competition
between the two individuals. Consistent with the strength of competition decreasing with di↵erences in
niche value, we assume that is a positive, decreasing, concave up function. For example, in the main
text, we use (|z z 0 |2 ) = ↵0 exp( |z z 0 |2 /(2⌧ 2 )) where ↵0 is the strength of competition experienced
by individuals with the same niche value and ⌧ determines how strongly individuals with similar niche
values compete. In order to perform our approximations, we assume that is twice, continuously
di↵erentiable. To represent the distribution of niche values exhibit by species i, let Zi = µi + Xi be
random variables where Xi have mean 0 and variance 1.
The competition coefficient for an individual with niche value z due to interacting with species i is
given by ↵i (z) = E[↵(z, Zi )]. The intra-specific competition coefficients for species 1 is distributed like
e1 is independent from but identically distributed like Z1 . In particular,
↵1 (Ze1 ) where Ze1 = µ1 + X
e1 has mean 0 and variance 1. To understand the mean and variance of this distribution for small
X
variance 2 , we can use the approximation (x2 ) ⇡ (0)+ 0 (0)x2 for x ⇡ 0. Thus, the mean intraspecific
5
competition coefficient is approximately
E[↵1 (Ze1 )] ⇡ E[ (0) +
=
=
0
(0)(Ze1
Z1 ) 2 ]
e 1 )2 ]
(0) + 0 (0)E[( X1
X
(0) + 2 0 (0) 2 .
As 0 (0) < 0, niche variation reduces the mean of the intraspecific competition coefficient by a factor
proportional 2 . In contrast, the variance of the intraspecific competition coefficients is approximately
e i )2 ]
Var[↵1 (Ze1 )] ⇡ 0 (0)2 4 Var[(Xi X
which is of order 4 .
The interspecific competition coefficients for species 1 are distributed like ↵2 (Z1 ). As
(x2 ) ⇡ (a2 ) + 2a 0 (a2 )(x
a) + (2a2
00
(a2 ) +
0
(a2 ))(x
a)2 for x ⇡ a,
the mean interspecific competition coefficient is approximately
E[↵2 (Z1 )] ⇡ E[ (a2 ) + 2a 0 (a2 ) (X1 X2 ) + (2a2 00 (a2 ) +
= (a2 ) + (2a2 00 (a2 ) + 0 (a2 )) 2 E[(X1 X2 )2 ]
= (a2 ) + (4a2 00 (a2 ) + 2 0 (a2 )) 2 .
0
(a2 )) 2 (X1
X2 ) 2 ]
where a = µ2 µ1 . As 00 (a2 ) > 0 and 0 (a2 ) > 0 (0), the change in the mean interspecific coefficient due
to individual variation is greater (up to order 2 ) than the change in the mean intraspecific coefficient.
Moreover, variation in niche values can actually increase the mean interspecific competition coefficient,
but always decrease the mean intraspecific competition coefficient. The variance of the interspecific
competition coefficients is approximately (up to order 2 )
Var[↵2 (Z1 )] ⇡ Var[2a 0 (a2 ) (X1
= 8a2 0 (a2 )2 2 .
X2 )]
Hence, unlike the variance of the intraspecific competition coefficient which is negligible up to order
2
, the variance of the interspecific competition coefficient is of the same order 2 as the means of the
competition coefficients. Numerical simulations (see Figure S1.1), however, suggest that this term is
always smaller than the changes in the mean and, in fact, is negligible for large or sufficiently small
mean niche di↵erences µ1 µ2 .
While our analysis focused on “smooth” competition kernels, stronger conclusions can be drawn
for certain non-smooth competition kernels (e.g. Barabás et al. 2013, Theoretical Ecology 6:1-19).
Specifically, consider ↵(z, z 0 ) = (|z z 0 |) where satisfies the same assumptions as before. In this case,
e1 |].
the mean intraspecific competition coefficient is approximately E[↵1 (Ze1 )] ⇡ (0) + 0 (0)E[|X1 X
Hence, there is an order reduction in the intraspecific competition coefficient. In contrast, the change
in the mean interspecific competition coefficient is of order 2 as the function (|x|) is smooth at x > 0.
Hence, at the level of mean competition coefficients, niche di↵erentiation is eroded more starkly by
individual variation for these non-smooth kernels.
Appendix(S3:(Comparison(of(the(effects(on(dynamics(of(describing(intraspecific(variation(
with(different(probability(distributions(or(with(an(analytical(approximation.(
(
0.000
0.005
0.010
0.015
0.020
0.025
150
200
250
no variation
approximation
beta
uniform
100
population density
800
600
400
0
200
probability density
sp.1
sp.2
mean
0
200
400
600
800
sp.1
sp.2
mean
c. Competitive population dynamics
50
1200
b. Variance = 1e−07; four parameter Beta
0
probability density
a. Variance = 1e−07; Uniform
0.000
0.005
0.010
r
0.015
0.020
0.025
0
20
40
r
80
100
e. Variance = 1e−05; four parameter Beta
f. Competitive population dynamics
0.000
0.005
0.010
0.015
0.020
0.025
250
200
150
100
population density
0
50
100
80
60
40
probability density
0
20
80
60
40
20
0
probability density
120
d. Variance = 1e−05; Uniform
60
time
0.000
0.005
0.010
r
0.015
0.020
0.025
0
20
40
r
80
100
h. Variance = 2e−05; four parameter Beta
k. Competitive population dynamics
0.005
0.010
0.015
0.020
0.025
200
150
100
population density
250
0.000
0
50
60
40
0
20
probability density
50
40
30
20
10
0
probability density
60
80
g. Variance = 2e−05; Uniform
60
time
0.000
0.005
0.010
r
0.015
0.020
0.025
0
20
40
r
80
100
i. Variance = 4e−05; four parameter Beta
l. Competitive population dynamics
0.005
0.010
0.015
r
0.020
0.025
200
150
100
population density
250
0.000
0
50
40
30
20
0
10
probability density
40
30
20
10
0
probability density
50
j. Variance = 4e−05; Uniform
60
time
0.000
0.005
0.010
0.015
r
0.020
0.025
0
20
40
60
80
100
time
(
Figure(S3.1(Effects(of(intraspecific(variation(in(competitive(sensitivity(in(two(competing(
species((blue(and(grey).(Moving(across(a(single(row(of(plots(shows,(for(a(given(level(of(
variance,(Uniformly(distributed(intraspecific(variation((left),(fourFparameter(Beta(
distributed(variation((center)(and(the(dynamics(that(arise(from(describing(individual(
variation(using(these(distributions,(as(well(as(the(analytical(approximation((right;(Appendix(
S1).(Dynamics(with(no(intraspecific(variation(are(also(shown((solid(line(in(right(panels).(All(
implementations(show(the(same(qualitative(result,(and(there(is(negligible(difference(
between(dynamics(that(arise(when(using(the(Uniform(or(Beta(distributions(to(describe(
intraspecific(variation.(As(expected,(the(analytical(approximation(is(most(accurate(when(
variance(is(small.((Parameter(values:(!"#$ = !"#& (=(3;('"#.$ = 0.012;,'"#.& = 0.011;(max/min(
values(for(fourFparameter(Beta(distribution:('"#.- ± ,0.0109.(
(
0.04
0.05
0.01
200
0.02
0.03
0.04
0.05
0
0.04
0.05
0.03
0.04
0.05
0
0.05
0.03
0.04
0
20
40
0.05
80
100
200
l. Competitive population dynamics
population density
20
15
10
0
5
probability density
0.04
60
time
0
0.03
150
0.05
25
25
20
15
10
0.02
competition coefficient
100
200
0.02
k. Variance = 1.325e−04; four−parameter Beta
5
probability density
0.01
competition coefficient
0
0.01
80
0
0.00
j. Variance = 1.325e−04; Uniform
60
100
population density
30
25
20
15
10
5
probability density
0.04
competition coefficient
0.00
40
i. Competitive population dynamics
0
0.03
20
time
30
25
20
15
probability density
10
0.02
150
200
0.02
h. Variance = 1e−04; four−parameter Beta
5
0.01
100
population density
0.01
competition coefficient
0
0.00
100
0
0.00
competition coefficient
g. Variance = 1e−04; Uniform
80
50
120
100
80
60
40
probability density
0.03
60
f. Competitive population dynamics
0
20
80
60
40
probability density
20
0.02
40
time
e. Variance = 1e−05; four−parameter Beta
0
0.01
20
competition coefficient
d. Variance = 1e−05; Uniform
0.00
150
0
0.00
50
0.03
150
0.02
competition coefficient
100
0.01
50
0.00
sp.1
sp.2
no variation
approximation
beta
uniform
100
population density
800
600
400
probability density
c. Competitive population dynamics
0
200
800
600
400
200
αi i
αi j
mean
50
1200
b. Variance = 1e−07; four−parameter Beta
αi i
αi j
mean
0
probability density
a. Variance = 1e−07; Uniform
0.00
0.01
0.02
0.03
competition coefficient
0.04
0.05
0
20
40
60
80
100
time
(
Figure(S3.2(Effects(of(intraspecific(variation(in(the(interaction(coefficients(on(dynamics.(
Moving(across(a(single(row(of(plots(shows,(for(a(given(level(of(variance,(Uniformly(
distributed(intraspecific(variation((left),(fourFparameter(Beta(distributed(variation((center)(
and(the(dynamics(that(arise(from(describing(intraspecific(variation(using(these(
distributions,(as(well(as(the(analytical(approximation((right;(Appendix(S1).(Dynamics(with(
no(intraspecific(variation(are(also(shown((solid(line(in(right(panels).(As(for(competitive(
sensitivity,(all(implementations(show(very(similar(quantitative(results,(particularly(at(low(
variance.(As(expected,(the(approximation(becomes(less(accurate(at(higher(levels(of(variance.(
Parameter(values:(!"#$ = !"#& (=(3;(0$& = 0&$ = 0.020;,0$$ = 0&& = 0.025;(max/min(values(
for(fourFparameter(Beta(distribution:(0$$ ± ,0.019,,0&& ± ,0.019,,0$& ± ,0.019,,0&$ ± ,0.019(
(
(
Appendix(S4:(The(effect(of(unequal(variance(in(intra8(and(interspecific(
competition(coefficients(on(invasion(growth(rates(across(a(range(of(levels(of(
average(niche(differentiation.((
(
2.0
1.8
variance in intra (σ211)
variance in inter (σ221)
1.6
1.4
1.2
1.0
0.8
0.6
0.8
0.9
1.0
1.1
1.2
1.3
mean intra (α11)
mean inter (α21)
(
Figure(S4.1(Contour(plot(showing(log(invasion(growth(rates(of(species(2(
competing(against(species(1,(which(is(at(it’s(single8species(equilibrium(density(
after(accounting(for(the(effects(of(individual(variation.(In(general,(higher(variance(
in(the(intraspecific(relative(to(interspecific(competition(coefficients(reduces(
invasion(growth(rates,(and(can(result(in(exclusion(of(the(invader(for(species(that(
are(weakly(niche(differentiated(on(average.(Alternatively,(higher(variance(in(the(
interspecific(competition(coefficients(causes(low8density(growth(rates(of(an(
invader(to(more(strongly(offset(the(higher(equilibrium(densities(of(the(resident,(
thus(promoting(coexistence.(Individual(variation(is(described(using(four8(
parameter(Beta(distributions.(Parameter(values(for(the(intraspecific(competition(
coefficient((!"" )(are(fixed(whereas(those(of(the(interspecific(competition(
coefficient((!#" )(are(allowed(to(vary(according(to(the(ratios(on(the(x8(and(y8axes.(
Parameter(values:($" = $# = 3;(!"" = 0.008;(range(of(values(for(!#" :(0.006(to(
0.010;*+,#-- =*1.5e805;(range(of(+,#.- :(7.5e806(to(3e805;(max/min(values(for(four8
parameter(Beta(distributions:(!"" ± *0.0059,*!#" ± *0.0059.(
Appendix(S5:((
(
The$annual$plant$model$accounting$for$intraspecific$niche$variation.$
(
We(start(with(the(premise(that(the(strength(of(the(interaction(between(two(
individuals(is(negatively(related(to(their(distance(along(a(niche(axis.(Following(
classic(evolutionary(models(we(can(formalize(this(thinking(with(the(expression(
(Taper(&(Case(1985;(Doebeli(1996):(
(
! ", "′ = !& '
( )()* +
,-.+ (
(
S5.1((≡0eq.(5(in(main(text)(
(
where(! ", "′ (describes(the(strength(of(the(interaction(between(individuals(at(
locations(z(and(z’#along(the(niche(axis.#The(interaction(takes(its(maximum(
value, !& ,(when(two(individuals(have(zero(niche(difference((i.e.(when(" − "′ = 0),(
and(the(strength(of(the(interaction(declines(monotonically(with(increasing(
distance(between(individuals(along(the(niche(axis(following(a(Gaussian(function(
with(variance(34, .(((
(
The(total(amount(of(competition(experienced(by(a(focal(individual(in(competition(
with(species(5(is(equal(to(the(sum(of(each(of(its(individualOlevel(interactions(with(
the(individuals(of(species(j.(Therefore,(the(interaction(coefficient#experienced(by(
a(focal(individual(in(competition(with(species(5(is(simply(equal(to(the(average(of(
all(individualOtoOindividual(interactions(experienced(by(that(focal(individual(from(
individuals(of(species(j.#Assuming(z(for(species(j(is(distributed(according(to(a(
distribution(67 " ,(then,(the(average(interaction(strength(experienced(by(an(
individual(with(niche(value("(due(to(interactions(with(all(individuals(of(species(5(
equals:(
(
!7 " = ! ", " * 67 " * 8" * ( (
(
(
S5.2((
(
Equation(S5.2(simply(describes(the(strength(of(competition(from(species(j(on(an(
individual(with(niche("(as(the(weighted(average(of(the(individualOtoOindividual(
interaction(coefficients,(with(weights(according(to(the(proportional(abundance(of(
competitor(individuals(with(niche("’.(Because(interaction(strengths(between(
individuals(are(additive,(equation(S5.2(accurately(describes(an(individual’s(
intraspecific((when(competing(with(all(conspecific(individuals)(and(interspecific(
(when(competing(with(all(heterospecific(individuals)(competition(coefficient.(By(
implementing(equation(S5.2(for(all(individuals(in(a(focal(population(one(can(
generate(a(distribution(of(intraO(and(interspecific(interaction(coefficients.((
(
In(the(manuscript(we(assume(that(Z#for(species(j(is(normally(distributed(with(
mean(:7 (and(variance(37, .(In(this(case,(equation(S5.2(can(be(expressed(as:(
(
!7 " =
(
;
!
(;
", " * <7 ("′)8"′(
(
(
S5.3((
where(<7 (is(the(density(function(for(the(normal(distribution(with(mean(:7 (and(
variance(37, :(
(
<7 " * =
?
,@-A+
exp −
) E (FA
+
(
,-A+
(
(
(
(
S5.4((
(
(
(
S5.5(0
0
(
in(which(case:(
(
!7 " = 0
0
4G -.
-.+ H-A+
exp −
)(FA
+
,(-.+ H-A+ )
(
0
(
The(dynamics(of(species(j(are(then(given(by:(
I7,JH? = IK,J
LA MA
;
<
(; ?H4N ) LN ON,P H4+ ) L+ O+,P 7
" 8"(
(
(
S5.6(0
(
Equivalently,(if(we(define(6 !? , !, (as(the(joint(density(function(of(the(random(
vector((!? Q7 , !, Q7 )(where(Q7 (is(normally(distributed(with(mean(:7 (and(
variance(37, ,(then:(
(
I7,JH? = I7,J
LA MA
;
6
& ?H4N LN ON,P H4+ L+ O+,P 7
!? , !, 8!? 8!, (
(
S5.70
Notice(that(this(representation(is(a(special(case(of(the(general(form(we(use(to(
account(for(the(effects(of(individual(variation((i.e.(equation(4(in(the(manuscript).(
Importantly,(with(intraspecific(niche(variation(there(are(correlations(between((
!? 0and0!, ,(which(are(accounted(for(in(our(formulation.(
(
(
The$effects$of$intraspecific$niche$variation$on$the$mean$interaction$
coefficients(
Individual(variation(in(niches(reduces(the(mean(strength(of(intraspecific(
competition(within(species((Fig.(4b).(The(reason(is(straightforward.(If(all(
individuals(have(the(same(niche(all(pairwise(interactions(will(be(of(maximum(
intensity((=(!0(in(equation(5).(However,(when(individuals(have(different(niches(
the(pairwise(interaction(strengths(between(at(least(some(individuals(will(be(
reduced(below(the(maximum(value,(which(reduces(the(average(strength(of(
competition(across(all(individuals(within(a(species.(Individual(variation(in(niches(
initially(increases(the(mean(strength(of(interspecific(competition(because(as(
variation(increases(niche(overlap(between(species(increases.(However,(when(
individual(variation(increases(further,(all(individuals(are(essentially(niche(
differentiated(from(all(other(individuals,(and(consequently(the(average(strength(
of(interspecific(competition(declines.(Of(course,(this(assumes(that(intraspecific(
variance in competition coefficient (x10−6)
niche(variation(is(unconstrained.(If(we(consider(a(periodic(niche(axis(such(that(
niche(space(is(constrained,(then(higher(levels(of(intraspecific(variation(will(
continue(to(result(in(higher(levels(of(niche(overlap(and(therefore(stronger(
interspecific(competition.(Such(a(scenario(only(exacerbates(the(negative(effects(
on(coexistence(of(intraspecific(niche(variation((results(not(shown).((
(
$
The$effect$of$intraspecific$niche$variation$on$the$variance$of$the$
interaction$coefficients.$
(
(
3.0
αii
αji
2.5
2.0
1.5
1.0
0.5
0.0
1.2
1.4
0.0
0.5
1.0
1.5
2.0
individual−level niche variation (σ)
2.5
3.0
(
(
Figure(S5.1(The(effect(of(intraspecific(niche(variation(on(the(variance(in(the(intra(
and(interspecific(competition(coefficients.(Parameter(values:(mean(niche(
,
,
difference,(|:V,? − :V,, | = 1;03V,?
= 3V,,
= 1;(!& = 0.008;(34, = 0.5.(
(
(
Variation(between(individuals(in(their(niches(results(in(nonOzero(variance(
in(the(speciesOlevel(interaction(coefficients((Fig.(S5.1).(With(no(intraspecific(
variation,(there(is(zero(variance(in(both(of(the(interaction(coefficients(as(all(
individuals(experience(the(same(level(of(intraO(and(interspecific(competition.(At(
high(levels(of(variation,(all(individuals(tend(to(be(niche(differentiated(from(all(
other(individuals(and(so(experience(more(or(less(uniformly(low(levels(of(intraO(
and(interspecific(competition(–(with(low(variance.(Between(these(two(extremes,(
variance(in(the(interaction(coefficients(peaks(at(intermediate(levels(of(
intraspecific(niche(variation.((Variance(is(higher(in(the(interspecific(interaction(
coefficient(but(this(is(insufficient(to(overturn(the(negative(effects(of(reduced(
average(niche(differentiation(on(coexistence((Fig.(4(in(main(text).(Also(see(
Appendix(S1.((
(
(
1.(
Doebeli,(M.((1996).(An(Explicit(Genetic(Model(for(Ecological(Character(
Displacement.(Ecology,(77,(510O520.(
2.(
Taper,(M.L.(&(Case,(T.J.((1985).(Quantitative(Genetic(Models(for(the(Coevolution(
of(Character(Displacement.(Ecology,(66,(355O371.(
(
1
Appendix S6
Here we derive the discrete-time Lande approximate for the individual-based model when species 1
(the resident) is at equilibrium (i.e. the system size S is very large) and species 2 (the invader) is at
very low densities. In this case, we can approximate the stochastic dynamics of the invader using a
discrete-time branching process given by
N2,t+1 =
N2,t
X
Fi,t+1
i=1
where Fi,t+1 are i.i.d. non-negative, integer-valued random variables which independent of N2,t . For
example, for the main model in the text without individual variation, these Fi,t+1 are Poisson-distributed
2 2
with mean 1+↵g12
where N1⇤ is the equilibrium density of the resident species. Now for the general
g1 N1⇤
Fi,t+1 , let µ = E[Fi,t+1 ] be the expected number of o↵spring, and 2 = Var[Fi,t+1 ] the variance. As the
conditional expectations and variances satisfy
E[N2,t+1 |N2,t ] = N2,t µ and Var[N2,t+1 |N2,t ] = N2,t
2
,
the noise is not isotropic with respect to the standard coordinates i.e. the variance varies with the
population state.
Assume N2,t is not too small in which case the Central Limit Theorem implies
!
N2,t
X
p
Zt+1
Fi,t+1 ⇡ N2,t µ +
N2,t Zt+1 = N2,t µ + p
N2,t
i=1
where Zt+1 is a standard normal random variable independent of N2,t
p. To make the noise isotropic, we
follow Lande (1998) and introduce the change of coordinates Xt = N2,t . As
p
p
1
1 3/2
x+ x⇡ x+ p x
x
( x)2
8
2 x
we get that the conditional expectation E[Xt+1 |Xt ] can be approximated as follows
2v
3
u N2,t
u
6 X
7
E[Xt+1 Xt ] = E 4t
Fi,t+1 Xt 5
i=1
⇡ E
"s
= Xt E
N2,t
"r

✓
Zt+1
µ+
Xt
Zt+1
µ+
Xt
Xt
◆
#
1
Zt+1
p
⇡ Xt E
µ+ p
2 µ Xt
✓
◆
1 3/2 2
p
= Xt
µ
µ
8
Xt2
1 3/2 2
p
= Xt µ
µ
8
Xt
Xt
#
1
µ
8
3/2
2
2
Zt+1
Xt
Xt2
2
and the conditional variance satisfies
1 2
.
4 µ
Hence, the noise is approximately isotropic in this coordinate system, and we can think the of the
dynamics as roughly corresponding to
Var[Xt+1 |Xt ] ⇡
Xt+1 =
µ 3/2
+“isotropic noise.”
8Xt
|
{z
}
“deterministic part”
p
2
µXt