Download Allelopathic adaptation can cause competitive coexistence

Theor Ecol (2013) 6:165–171
DOI 10.1007/s12080-012-0168-y
ORIGINAL PAPER
Allelopathic adaptation can cause competitive coexistence
Akihiko Mougi
Published online: 1 August 2012
# Springer Science+Business Media B.V. 2012
Abstract The maintenance of plant diversity is often
explained by the ecological and evolutionary consequences
of resource competition. Recently, the importance of allelopathy for competitive interactions has been recognized. In spite
of such interest in allelopathy, we have few theories for
understanding how the allelopathy influences the ecological
and evolutionary dynamics of competing species. Here, I
study the coevolutionary dynamics of two competing species
with allelopathy in an interspecific competition system, and
show that adaptive trait dynamics can cause cyclic coexistence. In addition, very fast adaptation such as phenotypic
plasticity is likely to stabilize the population cycles. The
results suggest that adaptive changes in allelopathy can lead
to cyclic coexistence of plant species even when their ecological characters are very similar and interspecific competition is
stronger than intraspecific competition, which should destroy
competitive coexistence in the absence of adaptation.
Keywords Competition . Allelopathy . Population cycles .
Adaptive dynamics . Speed of adaptation
Introduction
Competition is a major selective force creating and maintaining plant species diversity (Tilman 1982; Taper and Case
1992; Schluter 2000). Direct resource competition has been
long regarded as one of the most important competitive
Electronic supplementary material The online version of this article
(doi:10.1007/s12080-012-0168-y) contains supplementary material,
which is available to authorized users.
A. Mougi (*)
Ryukoku University,
1-5 Yokoya, Seta Oe-cho,
Otsu 520-2194, Japan
e-mail: [email protected]
mechanisms. Recent research, however, has emphasized
that the competitive mechanisms not related to resource
competition, such as allelopathy (a biological phenomenon
by which an organism produces one or more biochemicals
that influence the growth, survival, and reproduction of
other organisms), also can affect the consequences of competitive interactions (Callaway and Aschehoug 2000; Bais et
al. 2003; Hierro and Callaway 2003; Vivanco et al. 2004;
Callaway and Ridenour 2004). In contrast to the large literature on the evolutionary dynamics of resource competition,
however, there are few theories on the evolutionary dynamics of allelopathy, thus we do not know how the evolution of
allelopathy influences the consequences of competitive
interaction.
The impressive invasion success of exotic plant species
in spite of a very small initial abundance is an ecological
mystery. A major hypothesis is “enemy escape”: enemies in
the original community are absent in the new community
and the exotic plants can allocate their full ability to resource competition (Darwin 1859; Elton 1958). Recently,
allelopathy has been suggested as a mechanism for the
impressive success of invasive plants, and the rapid adaptation of interaction traits, including allelopathy, in response
to a new environment, has also been recognized as playing
an important role of invasion success (Callaway and Ridenour 2004; Lankau and Strauss 2007). However, native species also can evolve in response to invasive species
(Callaway et al. 2005; Strauss et al. 2006). In addition, very
fast allelopathic adaptation such as phenotypic plasticity can
occur. In this context, ecologists need a theoretical foundation describing the relationship between the evolutionary
dynamics of allelopathy and ecological competitive
interactions.
Recent theoretical studies have emphasized the importance of speed of evolutionary adaptation in producing
ecological consequences particularly in antagonistic
166
interaction systems such as predator–prey (Abrams and
Matsuda 1997; Abrams 2003; Dercole et al. 2010; Calcagno
et al. 2010; Mougi and Iwasa 2010, 2011a, b; Mougi et al.
2011; Cortez and Ellner 2010). In competitive interactions,
rapid adaptation also can occur (Hairston et al. 2005; Strauss
et al. 2006), however, there are few studies that examine
how the speed of evolutionary adaptation influences the
outcomes of competitive interactions. In addition to the
speed of evolutionary adaptation, the coevolutionary consequences strongly depend on the forms of coevolutionary
selection, such as interactions mediated by the matching of a
trait in one species to a trait in another (phenotype matching), and the differentiation in the phenotypes of interacting
species, as in classic arms-race dynamics (phenotype differences; Abrams 2000; Yoder and Nuismer 2010). In phenotype matching, the matching of traits maximizes one of the
two interaction coefficients and minimizes another one,
while in phenotype differences, mismatching of traits maximizes one of the two interaction coefficients and minimizes
another one. The theory on evolution of resource competition, such as character displacement, has assumed
that the amount of competition increases with phenotype
matching. In contrast, competitive interaction due to
allelopathy may strengthen with phenotypic differentiation (Yoder and Nuismer 2010). In real competitive
interaction systems, the evolutionary dynamics of both
resource competition and allelopathic competition would
encourage coexistence of competing species, however, I
focus on the evolutionary dynamics of allelopathic competition because until now few studies have examined
the evolutionary dynamics of such competitive interactions (but see Abrams and Matsuda 1994; Law et al.
1997; Kisdi and Geritz 2001).
Here, I study the coevolutionary dynamics of an interspecific competition system with an allelopathic interaction.
It is well known that stable coexistence between competing
species requires that competition be stronger on average
within than between species (Gause 1934; Chesson 2000).
Traits involved in resource competition, such as rates of
nutrient uptake, may be likely to affect intra- and interspecific competitive ability similarly. However, some traits
such as allelopathy may have opposing effects on intraand interspecifc competitive ability (Lankau 2008). For
instance, genotypes that invest heavily in a toxic allelochemical would be strong interspecific competitors, but
may be weak competitors against their own species (which
is unaffected by the chemical) if expressing the trait entails
some production cost (Lankau 2008). As shown in this
example, I assume that (1) allelopathy negatively affects
only heterospecific competing species; (2) increasing of
the level of allelopathy is costly; (3) the allelopathic interaction between species increases with phenotype differences
(Bais et al. 2003; Yoder and Nuismer 2010). Because the
Theor Ecol (2013) 6:165–171
assumptions (1) and (3) are different from those assumed in
classical theory on character displacement (Taper and Case
1992), the evolutionary dynamics of the allelopathic interactions might change the competitive coexistence condition
predicted by classical theory (Gause 1934; Chesson 2000).
The third assumption might be appropriate when one considers two competing plants of physical proximity, each
capable of allelopathy. The plant with the higher production
rate of allelopathic chemical would, on average, kill its
neighbor quicker than its neighbor kills it.
For the ecological dynamics, I adopt the Lotka–Volterra
competition system as the simplest model, and for the evolutionary dynamics, I adopt a quantitative genetic model
(Abrams et al. 1993). The quantitative genetic model has
been used to describe the adaptive trait dynamics such as
genetic evolution, phenotypic plasticity, and behavioral
changes (Abrams et al. 1993; Kondoh 2003). In this system,
I examine the conditions that enable coevolution to produce
a stable equilibrium and large-amplitude oscillations. I particularly focus on how the speed of evolutionary adaptation
affects the evolutionary and population dynamics.
I demonstrate that adaptation of allelopathy can lead to
cyclic coexistence of plant species even when their ecological characters are very similar and interspecific competition
is stronger than intraspecific competition.
Model
Dynamics of population size
I consider a biological scenario of interspecific competition
of plants. The two species negatively affect each other
through allelopathy due to chemicals (Bais et al. 2003;
Callaway and Ridenour 2004; Lankau and Strauss 2007).
In addition, the chemicals do not affect conspecific individuals (Lankau 2008).
I consider the simplest model which reflects the above
scenario by the following Lotka–Volterra two species competition system (May 2001; Law et al. 1997; Kisdi 1999),
dNi ¼ ri ðui Þ Ni aij ui uj Nj Ni ;
dt
ð1Þ
where Ni is population density of species i; ri (i 2 1; 2) is the
per capita growth rate; αij is the competition coefficient,
with αii 01.
Parameter αij is function of certain traits of the two
species. I assume that αij is a function of the difference in
the trait ui and uj in two species. In this scenario, the trait is a
chemical which harms heterospecific plants (Lankau 2008).
Greater differences in trait values imply larger negative
effect on another species due to allelopathy. This can be
realized if the scenarios proposed in introduction is held
Theor Ecol (2013) 6:165–171
(particularly when is θ is large (see below)). For example, it
is appropriate if allelopathy results in mortality at a young
stage in one of the two plants, and the relative production
rates of allelopathic chemicals determine which of two
neighboring plants is likely to survive.
To represent the effects, I use the sigmoidal function,
aij ¼ a0 1 þ exp θ ui uj , where α0 is the basal competition coefficient and θ (>0) is the shape parameter of the
function (Yoder and Nuismer 2010). αij is inversely proportional to (ui −uj). As θ increases, the function approaches a
step function. If the value of species is trait ui is much
greater than that of species j, uj, αij is very small but αji is
very large. In this scenario, interspecific competition, measured by αij is assumed to be larger than intraspecific for the
case of ui 0uj, that is α0 ≥2.
The cost of developing the trait in each species is modeled by assuming that ri is decreasing functions of ui. The
0
rates of decrease ( ri ) indicate the strength of the cost
ρ
constraint. I use a power function, ri 0r0(1 ui i ), where r0 is
the basal per capita growth rate and ρi represents the
strength of the trade-off (where a larger value represents a
smaller cost or weaker constraint).
Dynamics of evolutionary adaptation
I model the coevolutionary dynamics of the population
mean trait values, ui, based on a quantitative trait evolution
model (Abrams et al. 1993) as follows:
dui
@Wi e
¼ Gi
;
ð2Þ
dt
@^ui bui ¼ui
e i is the additive genetic variance which represents
where G
the speed of evolutionary adaptation. Here, the dynamics
expressed by Eq. (2) can reflect an adaptive change in the
traits due to genetic evolution or to an individual’s changing
its phenotype (Abrams et al. 1993; Kondoh 2003). For
ei < 1,
simplicity, I assume the speed is constant. When G
the dynamics of traits are slower than the fitness dynamics
e i > 1 , the dynamics of traits are
(evolution), and when G
faster than the fitness dynamics (plasticity). b
ui is the mutant
trait value. Wi is the mutant fitness, defined as the per capita
ui Þ Ni aij b
ui uj N j
rate of population growth: Wi ¼ ri ðb
: Eq. (2) indicates that the rate of adaptive change in the
traits should be proportional to the selection gradient. If the
selection gradient is positive (negative), selection pushes the
population toward higher (lower) trait values. At evolutionary equilibrium, Eq. (2) becomes zero.
A formula similar to Eq. (2) has also been adopted to
describe phenotypic plasticity, by which individuals can
shift the value of their trait in a direction that will improve
the expected fitness (Abrams et al. 1993). Here, the
167
dynamics expressed by Eq. (2) indicates adaptive change
in traits by either genetic evolution or an individual’s shifting of its phenotype. In the following discussion, I consider
both to be “adaptation” in the broad sense.
The four differential equations, (1) and (2), describe the
coupled coevolutionary and ecological dynamics of two
competing species, which I analyze further below.
Results
In Fig. 1, I show the coevolutionary dynamics of population
sizes Ni and traits ui for several different values of the speed
e . In these examples, cyclic
of evolutionary adaptation G
oscillation occurs, but, depending on conditions, the dynamics may converge to equilibrium. I first discuss the condition
for a stable or unstable equilibrium. Then, I discuss how the
speed of adaptation affects the dynamic behavior of the
population sizes and traits of the two species.
Equilibria or cycles
To ascertain the conditions leading to coevolutionary cycles,
I analyze the local stability of a nontrivial equilibrium in the
coevolutionary system described by Eqs. (1) and (2) (the
results are explained in SI Appendix A).
Figure 2 illustrates the parameter regions in which the
equilibrium is steady state (white) or not (dark gray). In the
unstable regions, the system shows a stable limit cycle.
Figure 2 illustrates the case where u*1 ¼ u*2 (the strength of
the constraints on the two species is same (ρ10ρ20ρ, and they
have the same r0)). The asterisks mean the trait values at
equilibrium. The result shows that the equilibrium tends to
be unstable when α0 is large. In addition, the equilibrium
tends to be unstable when the speed of evolutionary adaptation of two species is similar and slow. However, I find
that the oscillations can occur in light gray regions where the
equilibrium is locally stable but not globally stable. This
type of instability is likely to occur when the speed of
adaptation of one species is very fast and that of another
species is not fast.
Figure 3 illustrates the effects of the strength of the
constraints. When the strength of the constraint on the two
species is weak (ρ>1), the equilibrium tends to be stable
(Figs. 2b and 3a, b). If the strength of the constraint on the
two species is different, that is (ρ1≠ρ2), the equilibrium tends
to be unstable (Fig. 3a, c). Particularly when weaker constraint in one species tends to cause oscillations (Fig. 4). In
this system, strong constraint in one or both species (ρ<1)
tends to show divergent oscillations which cause extinction.
The parameter θ also affects the stability. When θ is large,
the equilibrium tends to be unstable (Figs. 2b and 5a, b).
1
Slow adaptation
a
N2
b
Fast adaptation
c
N1
1
u2
0
u1
200
time
This implies that the population oscillations can occur when
the effect of allelopathic chemicals on the competing species
sensitively depends on the difference in the chemical levels
between species.
In cases of extreme parameter values, I can further analyze local stability (SI Appendix B, C). When evolutionary
adaptation is very fast, I can show that the equilibrium is
locally stable if the following inequality holds
a12 a21 < 1:
ð3Þ
The derivation is explained in SI Appendix B. This result
implies that very fast adaptation such as phenotypic plasticity leads to stability if the interspecific competition is lower
than intraspecific competition. Note that this stability condition corresponds to that in the absence of adaptation
(Gause 1934; Chesson 2000; May 2001). I also can lead to
the stability condition when adaptation is very slow (see SI
Appendix C).
Trait and population oscillations
In Fig. 1, I showed examples of the dynamic behavior of the
system. I observe the dependence of the dynamics on the
speed of adaptation. When the speed of evolution is very
slow for both species, the amplitude of the population oscillation is large and that of the trait cycles is small (Fig. 1a).
Fig. 2 The parameter regions
in which the equilibrium is
stable or unstable. The two axes
e 1 and G
e 2 . The white, dark
are G
gray, and light gray regions
indicate the parameter ranges in
which the equilibrium is
globally stable, locally
unstable, and not globally
stable, respectively. I assume u*1
¼ u*2 . a α0 02, b α0 02.5,
c α0 03. Other parameter values
are r0 01, ρi 02, and θ010
Slow & fast adaptation
0
Trait value
Fig. 1 An example of
nonequilibrium dynamics in
different sets of speed of
e 2 ¼ 0:005
e1 ¼ G
adaptation. a G
e 2 ¼ 0:005.
e 1 ¼ 0:5; G
.bG
e 2 ¼ 0:1. The
e 1 ¼ 0:5; G
cG
solid and dotted lines indicate
the dynamics of the species 1
and 2, respectively. The
parameter values are r0 01,
α0 02.5, ρi 02, and θ015
Theor Ecol (2013) 6:165–171
Population size
168
0
0
50
t im e
When the speed of evolution is very slow for one species
and very fast in another species, the amplitude of the population oscillation is very large and the trait cycles are
relatively large (Fig. 1b). When the speed is very fast for
both species, the amplitude of population fluctuation is
small but that of the trait cycles is very large (Fig. 1c). I
observe that the population cycles tend to show antiphase
oscillations without depending on the speed of adaptation
and other parameters.
Discussion
I analyzed the coevolutionary dynamics of a two-species
competition system with an allelopathic interaction and
examined the conditions under which the coevolution
resulted in oscillations or in a stable equilibrium, and how
the speed of evolutionary adaptation of the two species
controls the population dynamics.
I demonstrated that the coevolutionary consequences
strongly depend on the speed of adaptation. Population
sizes and trait values tend to cycle if the speeds of adaptation of two species are similar and slow or dissimilar
and fast. If the speeds of adaptation of two species are
similar and very fast, the dynamics tends to approach
stable equilibrium.
α =2
a
200
t i me
α = 2.5
b
α =3
c
0
-1
log10 G2
-2
-3
-2
-1
log10 G1
0
-3
-2
-1
log10 G1
0
-3
-2
-1
log10 G1
0
Theor Ecol (2013) 6:165–171
Fig. 3 The parameter regions
in which the equilibrium is
stable or unstable. The two axes
e 1 and G
e 2 a ρ1 0ρ2 01.5,
are G
b ρ1 0ρ2 03, c ρ1 02, ρ2 01.5.
Other parameter values are
same as in Fig. 2b
169
a
ρ1 = ρ 2 = 1.5
ρ1 = ρ 2 = 3
b
c
ρ1 = 2, ρ 2 = 1.5
0
-1
log10 G2
-2
-3
-2
-1
0
-2
-3
log10 G1
I also found that the nature of population and trait cycles
depend critically on the speed of evolutionary adaptation.
The magnitude of fluctuation in the population size is very
large when adaptation is very slow or very fast in one
species, whereas a small amplitude of fluctuation is associated with very fast adaptation. In addition, I also found that
the cycles tend to exhibit antiphase oscillations.
Unlike the case of resource partitioning, cycles in population sizes and trait values can occur in allelopathic coevolution. The mechanism of cycles is the following. Starting
with nearly equal allelopathic trait values in two species,
evolution favors higher values of the allelopathic trait in one
species, which decreases the population size of the other
species. The low population of the second species then
favors lower trait values in the first species because of the
high cost of trait. Thus, the trait value in the first species
evolves to lower values. This in turn favors higher trait
values in the second species, causing the cyclic dynamics.
In a plant community, species coexistence may be maintained by cyclic dynamics. Lankau and Strauss showed that
genetic diversity in the concentration of an allelopathic
Fig. 4 Bifurcation diagrams of
population and trait dynamics
in relation to the ratio of the
strength of the trade-off in two
species. ρ2 changes with respect
to ρ1 01.2. Other parameter
values are same as in Fig. 2b
0.6
-1
0.6
Max
-2
-1
0
log10 G1
b
0.4
Cycle
N1
-3
secondary compound in Brassica nigra is necessary for
the coexistence with its competing species (Lankau and
Strauss 2007). They suggested that a diverse heterospecific
community could be invaded by high-sinigrin genotypes of
B. nigra, and B. nigra increases in abundance, displacing
heterospecific species. Then selection will begin to favor
lower sinigrin concentrations. If sinigrin concentrations decrease low enough, heterospecific species may be able to
reinvade the community, starting the cycle over.
Several theoretical studies have shown that stable evolutionary cycles can occur in a system with asymmetric competition (Brown and Maynard Smith 1986; Abrams and
Matsuda 1994; Law et al. 1997; Kisdi 1999). The intraand interspecific interactions occur through body size in
these models, and in some of these models, the main analysis deals primarily with intraspecific competition. Law et
al. (1997) contain the most extensive analysis of two-species
competition. These authors used the adaptive dynamics
framework (Dieckmann and Law 1996) to demonstrate that
the coevolutionary cycles occur, although the condition of
cycles is relatively narrow and requires the asymmetry of
a
0.4
0
log10 G1
N2
Equilibrium
0.2
0.2
Min
0
1.0
0
1.0
c
d
0.8
0.8
0.6
0.6
u2
u1
0.4
0.4
0.2
0.2
0
0
0.5
0.75
1.0
ρY / ρ X
1.5
2.0
0.5
0.75
1.0
ρY / ρ X
1.5
2.0
170
Theor Ecol (2013) 6:165–171
θ = 15
a
θ = 20
b
0
-1
log10 G2
-2
-3
-1
-2
log10 G1
0 -3
-2
-1
0
log10 G1
Fig. 5 The parameter regions in which the equilibrium is stable or
e 1 and G
e 2 . a θ015, b θ020, Other
unstable. The two axes are G
parameter values are same as in Fig. 2b
parameters. In the model, the interaction trait does not affect
intraspecific competition, and there is no time scales separation between population dynamics and trait dynamics.
Differently from their model, I showed that the cycles are
likely to occur when the parameter values in the two species
are very similar or the same.
The exotic plant species introduced by human often successfully invade native communities and some species become much more dominant, displacing native plants
(Kennedy et al. 2002; Callaway 2002). This has been
explained by classical theory: the escape of invaders from
their traditional natural enemies allows them to utilize their
full potential for resource competition. Recently, allelopathy
has been proposed as an alternative theory for the success of
some invasive plants. In my theory, the evolution of level of
allelopathy can, at first, rapidly displace native species because a potential high level of allelopathy reduces abundance
of the other species with weaker allelopathy. In such a case,
however, my theory also suggests that the abundance of native
species may recover by evolving a higher level of allelopathy.
I assumed that the allelopathic interaction is mediated by the
difference in the phenotypes of interacting species (Yoder and
Nuismer 2010), as in classic arms-race dynamics (Dawkins
and Krebs 1979). This is a simple theory that implicitly
assumed complex biological details related to plants interactions. Hence, the next step is to study the models that explicitly considered the complex allelopathic interactions.
Acknowledgments I am very grateful to P. Abrams for his valuable
comments on this study. This study was supported by a Grant-in-Aid
for a Research Fellow from the Japan Society for the Promotion of
Science and a Research Fellowship for Young Scientists (no.
20*01655) to AM.
References
Abrams PA (2000) The evolution of predator–prey interactions: theory
and evidence. Annu Rev Ecol Syst 31:79–105
Abrams PA (2003) Can adaptive evolution or behavior lead to diversification of traits determining a trade-off between foraging gain
and predation risk? Evol Ecol Res 5:653–670
Abrams PA, Matsuda H (1994) The evolution of traits that determine
ability in competitive contest. Evol Ecol 8:667–686
Abrams PA, Matsuda H (1997) Fitness minimization and dynamic
instability as a consequence of predator–prey coevolution. Evol
Ecol 11:1–20
Abrams PA, Harada Y, Matsuda H (1993) Unstable fitness maxima and
stable fitness minima in the evolution of continuous traits. Evol
Ecol 7:465–487
Bais HP, Vepachedu R, Gilroy S, Callaway RM, Vivanco JM (2003)
Allelopathy and exotic plant invasion: from molecules and genes
to species interactions. Science 301:1377–1380
Calcagno V, Dubosclard M, de Mazancourt C (2010) Rapid exploiter–
victim coevolutin: the race is not always to the swift. Am Nat
176:198–211
Callaway RM (2002) The detection of neighbors by plants. Trends
Ecol Evol 17:104–105
Callaway RM, Aschehoug ET (2000) Invasive plants versus their new
and old neighbors: a mechanism for exotic invasion. Science
290:521–523
Callaway RM, Ridenour WM (2004) Novel weapons: invasive success
and the evolution of increased competitive ability. Front Ecol
Environ 2:436–443
Callaway RM, Ridenour WM, Laboski T, Weir T, Vivanco JM (2005)
Natural selection for resistance to the allelopathic effects of invasive plants. J Ecol 93:576–583
Chesson P (2000) Mechanisms of maintenance of species diversity.
Annu Rev Ecol Syst 31:343–366
Cortez MH, Ellner SP (2010) Understanding rapid evolution in predator–prey interactions using the theory of fast–slow dynamical
systems. Am Nat 176:E109–E127
Darwin C (1859) On the origin of species by means of natural selection, or the preservation of favored races in the struggle for life.
John Murray, London
Dawkins R, Krebs JR (1979) Arms races between and within species.
Proc R Soc B 205:489–511
Dercole F, Ferrière R, Rinaldi S (2010) Chaotic Red Queen coevolution in three-species food chains. Proc R Soc B 277:2321–2330
Dieckmann U, Law R (1996) The dynamical theory of coevolution: a
derivation from stochastic ecological processes. J Math Biol
34:579–612
Elton CS (1958) The ecology of invasions by animals and plants.
Metheun, London
Gause GF (1934) The struggle for existence. Hafner, New York
Hairston NG Jr, Ellner SP, Geber MA, Yoshida T, Fox JA (2005) Rapid
evolution and the convergence of ecological and evolutionary
time. Ecol Lett 8:1114–1127
Hierro JL, Callaway RM (2003) Allelopathy and exotic plant invasion.
Plant Soil 256:29–39
Kennedy TA, Naeem S, Howe KM, Knops JMH, Tilman D, Reich P
(2002) Biodiversity as a barrier to ecological invasion. Nature
417:636–638
Kisdi É (1999) Evolutionary branching under asymmetric competition.
J Theor Biol 197:149–162
Kisdi É, Geritz S (2001) Evolutionary disarmament in interspecific
competition. Proc R Soc B 268:2589–2594
Kondoh M (2003) Foraging adaptation and the relationship between
food-web complexity and stability. Science 299:1388–1391
Lankau RA (2008) A chemical trait creates a genetic trade-off between
intra- and interspecific competitive ability. Ecology 89:1181–1187
Lankau RA, Strauss SY (2007) Mutual feedbacks maintain both genetic
and species diversity in a plant community. Science 317:1561–1563
Law R, Marrow P, Dieckmann U (1997) On evolution under asymmetric competition. Evol Ecol 11:485–501
Theor Ecol (2013) 6:165–171
May RM (2001) Stability and complexity in model ecosystems. Princeton University Press, Princeton
Mougi A, Iwasa Y (2010) Evolution towards oscillation or stability in a
predator–prey system. Proc R Soc B 277:3163–3171
Mougi A, Iwasa Y (2011a) Unique coevolutionary dynamics in a
predator–prey system. J Theor Biol 277:83–89
Mougi A, Iwasa Y (2011b) Green world maintained by adaptation.
Theor Ecol 4:201–210
Mougi A, Kishida O, Iwasa Y (2011) Coevolution of phenotypic
plasticity in predator and prey: why are inducible offenses rarer
than inducible defenses? Evolution 65:1079–1087
Schluter D (2000) Ecological character displacement in adaptive displacement. Am Nat 156:S14–S16
171
Strauss SY, Lau JA, Carroll SP (2006) Evolutionary responses of
natives to introduced species: what do introductions tell us about
natural communities? Ecol Lett 9:357–374
Taper ML, Case TJ (1992) Coevolution among competitors. Oxf Surv
Evol Biol 8:63–109
Tilman D (1982) Resource competition and community structure.
Princeton University Press, Princeton
Vivanco JM, Bais HP, Stermitz FR, Thelen GC, Callaway RM (2004)
Biogeographical variation in community response to root allelochemistry: novel weapons and exotic invasion. Ecol Lett 7:285–
292
Yoder JB, Nuismer SL (2010) When does coevolution promote diversification? Am Nat 176:802–817