Theor Ecol (2014) 7:367–379
DOI 10.1007/s12080-014-0224-x
ORIGINAL PAPER
Evolutionary dynamics through multispecies competition
Aysegul Birand · Ernest Barany
Received: 27 November 2013 / Accepted: 27 March 2014 / Published online: 25 May 2014
© Springer Science+Business Media Dordrecht 2014
Abstract Disruptive selection, emerging from frequencydependent intraspecific competition can have very exciting
evolutionary outcomes. One such outcome is the origin of
new species through an evolutionary branching event. Literature on theoretical models investigating the emergence of
disruptive selection is vast, with some investigating the sensitivity of the models on assumptions of the competition and
carrying capacity functions’ shapes. What is seldom modeled is what happens once the population escapes its effect
via increase phenotypic or genotypic variance. The expectation is mixed: disruptive selection could diminish and
ultimately disappear or it could still exist leading to further
speciation events through multiple evolutionary branching
events. Here, we derive the conditions under which disruptive selection drives two subpopulations that originated at
a branching point to other points in trait space where each
subpopulation again experiences disruptive selection. We
show that the general pattern for further branchings require
that the competition function to be even narrower than what
is required for the first evolutionary branching. However, we
also show that the existence of disruptive selection in higher
Both authors contributed equally to this manuscript.
A. Birand ()
Department of Biology, Middle East Technical University,
Dumlupinar Blv., No. 1, Cankaya, Ankara 06800 Turkey
e-mail: [email protected]
E. Barany
Department of Mathematical Sciences, New Mexico State
University, 3MB, P.O. Box 30001 Las Cruces, NM 88003 USA
e-mail: [email protected]
dimensional systems is also sensitive to the shapes of the
functions used.
Keywords Adaptive dynamics · Adaptive radiation ·
Competitive speciation · ESS · Evolutionarily stable
coalitions · Lotka–Volterra
Introduction
Disruptive selection can occur when individuals utilize a
continuously varying resource and exploit the most abundant resource, assuming that phenotypically similar individuals compete more strongly with each other than dissimilar phenotypes. Potential evolutionary consequences of
disruptive selection are numerous, leading to increases in
phenotypic and genotypic variances (reviewed in Rueffler
et al. 2006). One very appealing, albeit potentially uncommon, outcome of disruptive selection is speciation (e.g.,
Dieckmann and Doebeli 1999), a topic that continues to
be of central interest in evolutionary biology (Howard and
Berlocher 1998; Gavrilets 2004; Coyne and Orr 2004;
Doebeli 2011). Similar to what was observed previously
in studies of limiting similarity (e.g., May 1974a, 1974b;
Roughgarden 1630), the potential for disruptive selection
is also sensitive to the shapes of the competition, carrying capacity and utilization functions used (Abrams et al.
2008; Baptestini et al. 2009; Doebeli et al. 2007; Leimar
et al. 2008). Most commonly assumed carrying capacity
and competition functions are Gaussian (Dieckmann and
Doebeli 1999; Prout 1968; Christiansen 1991; Roughgarden
1972; Slatkin 1980) which lead to the very well-known
368
condition for sympatric speciation (or protected polymorphism, evolutionary branching) through competition: that
speciation is possible if the variance of the carrying capacity function is greater than that of the competition function.
In that case, the monomorphic population evolves toward
the optimum phenotype where it can utilize the most abundant resource, but then experiences disruptive selection due
to increased competition.
After the population escapes this fitness minimum by
evolutionary branching, the resulting bimorphic population could still be exposed to disruptive selection due
to continued competition for resources and experience
further branching events. Understanding multiple branching events is important since they could provide insight
into adaptive radiations, where theoretical work remains
scarce (Gavrilets and Losos 2009; Gavrilets and Vose 2005;
Bolnick 2006; Ito and Dieckmann 2007). Another interesting question is whether the phenotypes would split into so
many branches that the phenotypic distribution of the population could ultimately match the continuous resource distribution (Polechova and Barton 2005). Surprisingly, thus far,
only a few models investigate multiple branching events
analytically (Geritz et al. 1998, 1999; Kisdi 1999; Doebeli
2011), or even numerically (Bolnick 2006; Ackermann and
Doebeli 2004).
In his very accessible review, Doebeli (2011) briefly
investigates the branching behavior in higher dimensions.
Here, we build on his analysis and investigate the conditions
for which disruptive selection continues to exist with multiple carrying capacity and competition functions. We used a
family of exponential functions that includes the commonly
employed Gaussian functions, as well as quadratic and
“box-like” hyperbolic tangent functions used by Baptestini
et al. (2009). Unlike Doebeli (2011), we obtain some general trends about branchings in higher dimensions and show
that the conditions for further branchings become more
restrictive than the conditions required for the first branching. Typically, the competition function must be much more
narrower for branchings in higher dimensions than what is
required for the first branching.
We should point out that throughout the text, we use
the terms species and speciation loosely. Even though the
terms are appropriate for clonal organisms (Doebeli 2011),
assortative mating should evolve among sexually reproducing organisms in order for the diverged populations to
remain distinct. In the strict sense, the populations resulting from successful invasions of residents by mutants could
be considered polymorphisms rather than distinct species.
Our analytical solutions should be considered as necessary
conditions for speciation that individual-based simulations
should comply with, even though they may not be sufficient when more realistic genetic mechanisms are included
Theor Ecol (2014) 7:367–379
(Burger and Schneider 2006; Burger et al. 2006; Johansson
and Ripa 2006; Gavrilets 2005).
Model
Following Dieckmann and Doebeli (1999), we model the
invasion dynamics of a mutant through its ecological
interactions with residents using the Lotka–Volterra competition model:
M
dnj
k=1 C(xj , xk )nk
= rj 1 −
nj
dt
K(xj )
(1)
where j = 1, 2, ...M indexes the competing types or
species. The function C(xj , xk ) describes competition
between the two phenotypes xj and xk , and the function
K(xj ) gives the carrying capacity for the phenotype xj ,
which is a measure of the resource availability for that
phenotype. Both of these functions depend on xj , which
specifies a particular location on the resource spectrum that
type j utilizes. This can also be thought of as a morphological or a behavioral trait of the type j; typically, a trait such as
beak size that determines the efficiency of utilization of different sized seeds. The constant rj is the growth rate. Since
the specific values of rj are not relevant to the stability of the
results as long as they are positive, rj > 0 ∀j , we assume
rj = 1 for simplicity. Lastly, nj is the number of types that
have phenotype xj .
We will consider two different sets of functions for competition and carrying capacity. In the first set, we assume
that the competition function is Gaussian, which is typically assumed in much of the literature (e.g., Dieckmann
and Doebeli 1999; Bolnick 2006):
C(xj , xk ) = e−(xj −xk )
2 /2σ 2
c
(2)
According to this model of competition, the strength of
competition between two individuals depends both on the
distance between their resource utilization traits, xj and xk ,
and on the variance σc , which determines how fast the effect
declines with this distance.
With Gaussian competition function, we will consider
two different carrying capacity functions. First, we look at
functions that decrease exponentially with some power 2+
of the phenotypic value (Doebeli et al. 2007):
K(x) = K0 e−(x)
2+ /2σ 2+
k
(3)
where K0 is a constant that scales the carrying capacity
function K(x), and σk determines how fast it declines from
Theor Ecol (2014) 7:367–379
369
the optimum. Note that when = 0, the resulting function is
a Gaussian function, which is a common choice for carrying
capacity functions (e.g., Bolnick 2006; Ito and Dieckmann
2007). When = 2, the resulting function is an exponential
quartic function, which we refer to as quartic function for
simplicity (Fig. 1).
The second carrying capacity function that we are going
to use with the Gaussian competition function is quadratic
over a finite range of phenotype differences and that vanish
for larger values (Doebeli 2011):
2 K0 1 − xa
, |x| ≤ a
K(x) =
(4)
0,
|x| > a
these functions, ξ is a function
√ For
√ of σ (ξ =
2β ∗ σ/coshβ ∗ , where β ∗ = 0.5ln(2+ 3) ≈ 0.658 and σ
is a measure of the width) and β is a parameter that affects
the shape of the function near the maximum. Specifically,
the larger β is, the broader the function is near the critical
point, i.e., the more ‘box-like’ the function is (Fig. 1).
Evolutionary invasion analysis typically investigates
whether a population that is fixed for a trait or an allele can
be invaded by a mutant. A necessary property a trait must
satisfy is that it should be evolutionarily singular, i.e., x ∗
at which the local fitness gradient ∂(x1 , x2 )/∂x2 should
vanish (Otto and Day 2007):
where a is a measure of the region of parameter space where
the capacity is nonzero (Fig. 1).
For the second set, we consider the family of “box-like”
functions based on hyperbolic tangents used by Baptestini
et al. (2009):
(8)
∂(x1 , x2 ) =0
∂x2
x1 =x2 =x ∗
where
Of particular interest are two questions: whether there are
trait values that both resist invasion by all possible mutants
(fitness maximum, evolutionarily stable strategy, or ESS)
and are convergence stable or whether there are traits that
are convergence stable but are invadable (convergent stable fitness minimum, i.e., evolutionary branching points).
The condition for ESS states that (x ∗ , x2) regarded as
a function of the invader phenotype x2 must be a local
fitness maximum to resist invasions (that is, (x ∗ , x ∗ ) ≥
(x ∗ , x2 ) for all x2 = x ∗ and sufficiently close to x ∗ ):
1
x
x
H (x, β, ξ ) =
tanh β
+1
− tanh β
−1
2 tanh β
ξ
ξ
(7)
∂ 2 (x1 , x2 ) ∂x22
x
C(xj , xk ) = H (xj − xk , βc , σc )
(5)
K(x) = K0 H (x − x0 , βk , σk )
(6)
<0
(9)
∗
1 =x2 =x
The reverse of the condition (9) above describes an evolutionarily singular strategy where the fitness is minimized:
1
0.8
0.6
0.4
0.2
0
Fig. 1 Five different carrying capacity functions describing a single
continuous resource that has its maximum value at the location x = 0
(chosen without loss of generality) and declines symmetrically with
distance from that point. They are plotted assuming σk = 0.7 for the
Gaussian (black) and the quartic (blue) carrying capacity functions,
a = 1 for the quadratic (green) carrying capacity function, and σ =
0.9 for the “box-like” carrying capacity functions with β = 0.658 (red)
and β = 5 (magenta)
∂ 2 (x1 , x2 ) ∂x22
x
>0
(10)
∗
1 =x2 =x
which means that the singular strategy x ∗ can be invaded by
all nearby phenotypes. Since the two conditions Eqs. 9 and
10 determine the curvature of the fitness function at the singular strategies, we refer to them as “curvature conditions”
for short.
The above conditions determine the existence of a fitness
maximum or a minimum in the landscape; however, these
characterizations beg the question of whether an evolving
population can ever get close enough to the singular value
for it to be relevant. This is the property of convergence stability, which states that a population with a phenotype close
to x ∗ can be invaded by a mutant whose phenotype is even
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Theor Ecol (2014) 7:367–379
closer to x ∗ than the resident population. If x ∗ is convergence stable and an ESS, then the population does approach
the singular strategy asymptotically and is called continuously stable strategy. However, if x ∗ is a fitness minimum
and is convergent stable, then it is referred to as an evolutionary branching point. The condition for convergence
stability is (Geritz et al. 1998):
∂ 2 (x1 , x2 ) ∂x12
x1 =x2
=x ∗
∂ 2 (x1 , x2 ) >
∂x22
x1 =x2
(11)
=x ∗
which assumes that the second derivatives in Eq. 11 do not
all vanish. In the event that the second derivatives in Eq. 11
are all identically zero, a higher order analysis is needed
(Appendix D).
Most of the analyses using adaptive aynamics (AD)
found in the literature are explicitly restricted to two phenotype interactions (cf. Geritz et al. 1998, 1999 for analytical
solutions). As discussed by Geritz et al. (1998), the conditions above can be immediately generalized to an invader
introduced into a resident population with any number of
phenotypes (also see Doebeli 2011). Here, we consider circumstances in which the case of a dimorphic resident (that
is, a three-dimensional population dynamical system) can be
reduced to a two-dimensional system. We do this by invoking the property of symmetry (see Appendix A), which for
our purposes means that the phenotypes of the bimorphic
population are mirror images of each other relative to the
optimal monomorphism. That is, if the singular monomorphism is at x = 0, then a symmetric bimorphism is a
population with phenotypes x and −x, such that the each
subpopulation with nonzero phenotype are equal. Under
these conditions, the branching and convergence conditions
become:
∂(x, −x, x3) =0
(12)
∂x3
x3 =x=x ∗
∂ 2 (x, −x, x3) ∂x32
conditions. Note that if condition (13) is reversed, the above
are the conditions of evolutionarily stable coalitions.
Results
First branching Even though the invasion dynamics of a
single resident and a mutant are well understood, we present
the results here to illustrate our methods and terminology.
Moreover, it is not meaningful to search for further branching events unless we know that there is a first branching
event. A monomorphic resident reaches its equilibrium population size n∗1 = K(x1 ) in the absence of any mutants. The
requirements for a branching point (Eqs. 8, 10, and 11) are
further reduced to (see Eqs. 29, 30, and 31 in Appendix B):
K (x ∗ ) = 0
(15)
K (x ∗ ) − K(x ∗ )C o > 0
(16)
K (x ∗ ) < 0
(17)
o
2
2
where C = ∂ C(x2 , x)/∂x2 x =x=x ∗ . Eqs. 15, 16, and 17
2
represent the singularity, fitness minimum, and convergence
stability conditions, respectively.
Second branching After the invasion, the two (sister)
species settle to their equilibrium population sizes in the
absence of a new mutant (see Appendix C). The requirements for the second branching (Eqs. 12, 13, and 14) can be
written as (Eqs. 32, 33, 34 in Appendix C):
K (x ∗ )
C∗
=
K(x)
1 + C∗
(18)
Co + C∗
K(x ∗ ) > 0
1 + C∗
2
2C ∗
C∗
K (x ∗ )
−
−
>0
1 + C∗
1 + C∗
K(x ∗ )
K (x ∗ ) −
(19)
(20)
>0
(13)
x3 =x=x ∗
∂ 2 (x, −x, x3) ∂ 2 (x, −x, x3) >
∂x 2
∂x32
x3 =x=x ∗
where C ∗ = C(x, −x)|x=x ∗ , C ∗ = ∂C(x3 , −x)/
∂x3 |x3 =x=x∗ , C ∗ = ∂ 2 C(x3 , −x)/∂x32 x =x=x ∗ and C o =
3
∂ 2 C(x3 , x)/∂ 2 x3 x =x=x ∗ . Eqs. 18, 19, and 20 represent
3
the singularity, fitness minima, and convergence stability
conditions, respectively.
Gaussian competition and Gaussian ( = 0) carrying
capacity functions
x3 =x=x ∗
(14)
Note that the above conditions hold both for x3 = x ∗ and
x3 = −x ∗ . Due to the symmetry of the K and C functions (Appendix A), these turn out to be not independent
First branching If both the competition and the carrying capacity function are Gaussian ( = 0 in Eq. 3),
the monomorphic evolutionarily singular strategy (15) is
x ∗ = 0, where the carrying capacity function is maximized. This strategy is also convergent stable (17). The
Theor Ecol (2014) 7:367–379
371
ε=0
1
0.8
c
0.6
σ
curvature condition (16) leads to the very well-known condition that the coexistence of two competitors on a single
continuous resource is possible when the variance of the
carrying capacity function is greater then that of the competition function, σk2 > σc2 (Roughgarden 1972; Slatkin 1980).
This is the result that was obtained by Dieckmann and
Doebeli (1999) for sympatric speciation, which was previously obtained by others as protected polymorphisms (Prout
1968; Christiansen 1991; Waxman and Gavrilets 2005a).
0.4
0.2
0
0
0.2
0.4
σ
0.6
0.8
1
0.8
1
0.8
1
0.8
1
0.8
1
k
ε=0.5
1
0.8
c
0.6
σ
Second branching When the evolutionarily singular strategy x ∗ = 0 becomes unstable and is replaced by a
dimorphic solution, no uninvadable two species community
emerges. To see all this, note that the evolutionarily singular
strategy condition for a two-species community is given by
Eq. 18. When the carrying capacity function is Gaussian, we
have K (x ∗ )/K(x ∗ ) = −x ∗ /σk2 , which along with Eq. 18,
can be solved to obtain:
σc x ∗ = √ ln(2ρ − 1)
(21)
2
0.4
0.2
0
0
0.2
0.4
σk
0.6
ε=1.0
1
0.8
0.6
0.2
0
0
0.2
0.4
σ
0.6
k
ε=1.5
1
0.8
0.6
c
Figure 2 shows that the condition is automatically satisfied for ρ > 1, which means that the two species solution
can be invaded by all nearby species. Doebeli (2011) also
0.4
σ
where ρ =
From this, it is seen that the critical symmetric two species solution exists only if ρ > 1, which is the
same as the condition for the first branching. These strategies are also convergent stable (Eq. 35 in Appendix E). The
condition for fitness minima at this strategy is (19):
1 − 2ρ
ρ − 1 + ln(2ρ − 1)
>0
(22)
2ρ
σ
c
σk2
.
σc2
0.4
0.2
0
0
0.2
0.4
σ
0.6
k
ε=2.0
0.2
1
0.8
0.1
σ
c
0.6
0
0.4
−0.1
0.2
−0.2
0
0
0.2
0.4
σ
0.6
k
−0.3
−0.4
−0.5
0.5
1
1.5
2
σ2k /σc2
Fig. 2 The fitness minima condition (22) for the two-sister species
after the first branching event as a function of ρ = σk2 /σc2 . Since the
solution is greater than zero for all ρ > 1, the two-sister species are
always invadable. Also, note that ρ > 1 is required for the first branching event. This implies that the condition that allows the first branching
automatically satisfies the condition for the second branching
Fig. 3 Evolutionarily stable coalitions and second branching points
observed with the carrying capacity function given by Eq. 3 for different values of . Filled circles represent parameter combinations
when evolutionarily singular strategies are evolutionary branching
points, and empty circles represent evolutionarily stable coalitions.
The parameter combinations that lack evolutionarily singular strategies
and/or convergence stability do not have any circles. When = 0, the
carrying capacity function is the Gaussian function and the evolutionary singular strategies are always branching points as long as σc < σk .
For 0 < < 2, the conditions for the existence of disruptive selection
becomes more restrictive, and competition function has to be much
narrower than the carrying capacity function (roughly σc < 2σk ), but
unlike when = 0, evolutionarily stable coalitions are possible
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Theor Ecol (2014) 7:367–379
derived the singularity condition above (see his Eq. 3.27)
and mentioned them as being fitness minima. However, he
did not explicitly derive the condition (22).
Gaussian competition and quartic ( = 2) carrying capacity
function
First branching Evaluating Eqs. 15 and 16 with the Gaussian competition and the quartic-carrying capacity ( = 2
in Eq. 3) functions shows that the evolutionarily singular
strategy is x ∗ = 0, and that it is a branching point if (16):
1
>0
σc2
(23)
This strategy is also convergent stable; however, it is not
possible to determine that with Eq. 17 since K (x ∗ ) = 0
(see Appendix D). It is clear that the above condition is
always satisfied, which means that x ∗ = 0 is always a
fitness minimum when the carrying capacity is quartic. In
contrast to the Gaussian case, x ∗ = 0 is always a fitness
minimum for all parameter values, i.e., there is no “ecological” restriction (such as σk2 > σc2 ) for the existence of the
branching point.
Second branching After the first branching event, the singular coalition (18) is implicitly given by:
2 (x ∗ )2 σc2
(x ∗ )2
1−
= tanh
σc2
σk4
Fig. 4 σc vs. the curvature
condition for the fitness maxima
(27) and the singular coalition x ∗
with quadratic-carrying capacity
function. The evolutionary
singular coalition (26) requires
that σc < 0.7071, which
satisfies the fitness maxima
condition. Thus, evolutionarily
stable coalition exist as long as
0.4410 < σc < 0.7071. Also,
note the value when the
curvature condition starts to fail
at ∼ 0.4410 when x ∗ is at its
maximum value; the reason for
this phenomenon requires
further investigation
(24)
The above equation requires x ∗ > 0, which reinforces the
fact that the new strategies (x ∗ = x1 = −x2 ) begin to exist
only if the strategy at x ∗ = 0 is no longer stable. The evolutionary singular strategies given by Eq. 24 are convergent
stable (Eq. 36 in Appendix E) and are branching points if
(19):
−6 (x ∗ )2
4 (x ∗ )6
4 (x ∗ )4
1
+
−
+ 2 >0
4
8
4
2
σc
σk
σc σk
σk
(25)
Contrary to the case above where both the competition and
carrying capacity are Gaussian, the evolutionarily singular
strategies here could be evolutionarily stable coalitions or
branching points depending on the parameters σc and σk
(Fig. 3). However, the conditions for evolutionary branching
points become more restrictive, and the competition function has to be much narrower than the carrying capacity
function (roughly σc < 2σk ).
Branchings with carrying capacity functions when 0 < < 2
Evolutionarily singular strategy at x ∗ = 0 (15) is always a
fitness minimum for all > 0 (16). This is indeed true for
other competition functions as well provided that C < 0.
After the first branching event, the singular coalitions
could be evolutionarily stable coalitions or branching points
depending on the parameters σc and σk (Fig. 3). One thing
that is clear is that for the second branching, the competition function needs to be much narrower compared the
carrying capacity function (note the degeneracy of Gaussian
case with = 0 where all the singular strategies are always
fitness minima).
Theor Ecol (2014) 7:367–379
373
Second branching After the first branching, the singular
coalitions (18) are implicitly given by the solutions to:
∗ 2
−2σc2
(x )
(26)
+
1
=
tanh
∗
2
1 − (x )
σc2
The solution for the above begins to exist at σc = 0.7071.
Contrary to the Gaussian case, the new two (sister) species
can resist invasion by new mutants if (reverse of Eq. 19):
−5(x ∗ )2 − 2σc2 + 1 < 0
(27)
This coalition is also convergent stable (Eq. 37 in Appendix
E). Doebeli (2011) also briefly considered the second
branching with these set of functions (see his Fig. 3.2),
however, but did not obtain full results.
An interesting picture emerges when the singularity and
invadability conditions are investigated at the same time
(Fig. 4). The evolutionarily singular coalition (26) requires
that σc < 0.7071, and for those values, the curvature condition (27) is also negative, which means that an evolutionarily
stable coalition exists for 0.4410 < σc < 0.7071. It is
intriguing that the value of σc for which the curvature condition fails is also the value that maximizes x ∗ ; the reason
for this phenomenon requires further investigation.
Second branching Box-like functions result in singular
coalitions that might be evolutionarily stable coalitions or
branching points depending on the parameters σc , σk , βc ,
and βk (Fig. 5, the conditions are given by Eqs. 39 and 40 in
Appendix F). Different from the scenarios where carrying
capacity functions were quartic or quadratic, which resulted
in singular coalitions with two singular strategies after the
first branching event (i.e., x ∗ and −x ∗ , where x ∗ > 0),
here, the system goes through a pitchfork bifurcation and
results in a coalition with three strategies (i.e., in addition to
x ∗ and −x ∗ , where x ∗ > 0, there is a third singular strategy at x ∗ = 0; however, this strategy at x ∗ = 0 is always
a fitness minimum and is not convergent stable; Eq. 38 in
Appendix E).
β =β =β*=0.658
c
k
1
0.8
0.6
c
First branching If the competition function is Gaussian,
and the carrying capacity function is quadratic (assuming
that a = 1 in Eq. 4), the evolutionarily singular strategy is
x ∗ = 0. This convergent stable strategy is a branching point
if σc < 0.7071 (see Fig. 3.2 and Eq. 3.31 in Doebeli 2011).
function (e.g., β = 5 in Fig. 1); however, contrary to the
quartic function, Eq. 28 pose a restriction on the existence of
disruptive selection and the evolutionarily singular strategy
at x ∗ = 0, which is not a default fitness minimum.
σ
Gaussian competition and quadratic carrying capacity
function
0.4
0.2
0
0
0.2
0.4
σ
0.6
0.8
1
0.8
1
0.8
1
k
*
β =1.146, β =β =0.658
c
k
1
0.8
σ
c
0.6
Box-like competition and carrying capacity functions
0.4
0.2
First branching The condition for a fitness minimum (15)
at the evolutionarily singular strategy x ∗ = 0 have been
derived by Baptestini et al. (2009) when both the competition and the carrying capacity functions are box-like
(Eq. 5, 6):
0
0.2
0.4
σk
0.6
*
β =0.2, β =β =0.658
c
k
1
0.8
0.6
(28)
This strategy is also convergent stable (convergence stability was not mentioned by Baptestini et al. (2009) and also
note that the cosh term should be squared). Note that when
βc = βk , the first branching condition reduces to σc < σk ,
which is the same solution obtained with the Gaussian carrying capacity and competition functions (Baptestini et al.
2009). When βc > βk , the opportunity for disruptive selection is reduced (Baptestini et al. 2009). Higher values of β
lead to more platykurtic distributions similar to the quartic
σc
βc2
>
cosh2 βc σc2
βk2
.
cosh2 βk σk2
0
0.4
0.2
0
0
0.2
0.4
σ
0.6
k
Fig. 5 Evolutionarily stable coalitions and second branching points
observed with “box-like” competition and carrying capacity functions.
Filled circles represent parameter combinations when evolutionarily
singular strategies are evolutionary branching points, and empty circles
represent evolutionarily stable coalitions. The parameter combinations
that lack evolutionarily singular strategies and/or convergence stability
do not have any circles. a βc = 0.658, b βc = 1.146, c βc = 0.2. For
all the figures, βk = 0.658
374
The parameter space that allows for second branching points or stable coalitions is reduced compared to the
parameter space of the first branchings (Fig. 5). When βc =
βk , the first branching condition σc < σk (Baptestini et al.
2009) is no longer sufficient; the opportunity for disruptive
selection is more restricted (Fig. 5a). As in the first branching, when βc > βk , the opportunity for disruptive selection
is further reduced (Fig. 5b). Surprisingly, when βc = 0.2,
and βk = 0.658, the resulting coalitions (roughly when
3σc < σk ) are always fitness minima, and evolutionarily
stable coalitions do not exist (Fig. 5c).
Discussion
Disruptive selection could have exciting evolutionary outcomes such as new species and sexual bimorphisms
(reviewed in Rueffler et al. 2006) or even multiple species
if the resulting bimorphic population continues to compete for resources (e.g., Bolnick 2006; Ackermann and
Doebeli 2004; Geritz et al. 1999; Doebeli 2011). Here,
we derived the well-understood conditions under which
individuals in an initially monomorphic population should
experience disruptive selection and checked whether the
resulting bimorphic populations continue to experience disruptive selection. Even though these conditions may not be
sufficient to result in speciation or adaptive radiations due to
the genetic mechanisms (Gourbiere 2004; Gavrilets 2005;
Waxman and Gavrilets 2005b; Burger and Schneider 2006;
Burger et al. 2006), there is some evidence suggesting that
frequency-dependent selection through competition could
result in adaptive speciation in clonal organisms (Spencer
et al. 2008; Herron and Doebeli 2013; Doebeli 2011).
Multiple branchings are seldom studied analytically
(Geritz et al. 1998, 1999; Doebeli 2011), and most of
the analyses are restricted to numerical simulations (Kisdi
1999; Bolnick 2006; Ackermann and Doebeli 2004; Ito and
Dieckmann 2007). We have used various unimodal competition and carrying capacity functions (Fig. 1) while
deriving the analytical solutions for multiple branchings. It
has already been demonstrated that the occurrence of disruptive selection is sensitive to the assumptions about the
shapes of the functions used (Baptestini et al. 2009; Abrams
et al. 2008; Leimar et al. 2008). The preliminary analysis of Doebeli (2011) suggested the same for branchings
in higher dimensions. Here, we build on the analysis of
Doebeli (2011) and investigate the conditions for which disruptive selection continues to exist with various carrying
capacity and competition functions to obtain some general
trends about branchings in higher dimensions.
We use two sets of functions. In the first set, we
assumed that the competition function is Gaussian, which
is typically assumed in much of the literature (e.g.,
Theor Ecol (2014) 7:367–379
Dieckmann and Doebeli 1999; Bolnick 2006 and considered
two different carrying capacity functions. The first carrying capacity function was a family of exponential functions
with 2 + power Doebeli et al. 2007), and the second was
quadratic. In the second set, we considered the family of
box-like functions based on hyperbolic tangents used by
Baptestini et al. (2009). Even though these functions are not
exhaustive, it allowed us to determine some trends in second
branchings: the ratio of the width of the competition functions (σc ) to the width of the carrying capacity functions
(e.g., σk for Gaussian or a for quadratic) has to be smaller
than the value for the first branching. Unsurprisingly, we
also observed some degenerate behaviors in addition to the
now well-known degeneracy of Gaussian function.
One of the interesting results of our analysis is the
demonstration that the Gaussian competition and carrying
capacity function pair results in bimorphic populations that
are always invadable (Fig. 2). This simultaneous branching behavior was noted by Bolnick (2006) in his numerical
simulations. The degeneracy of this Gaussian pair has been
noted previously with analyses on quantitative characters
with a density distribution (Meszena et al. 2006; Doebeli
et al. 2007; Szabo and Sznaider 2004). An immediate question that arises is whether this behavior could be observed
in higher dimensional systems with more species since that
might suggest that communities might eventually approach
the continuous solution matching that of the resources as
originally proposed by Roughgarden (1972) and recently by
others (Barton and Polechova 2005; Polechova and Barton
2005; Doebeli 2011).
There is some degeneracy with the quartic-carrying
capacity function (Doebeli et al. 2007) as well, which leads
to an interesting result where the evolutionarily singular
strategy of the monomorphic population is always a fitness minimum. Contrary to the conditions derived with the
Gaussian-carrying capacity function, the fact that the singularity is always a fitness minimum prevents one from
making an argument that is often made when both functions
are Gaussian: speciation occurs when the strength of disruptive selection exceeds that of stabilizing selection. With the
quartic-carrying capacity function, as long as the variance of
the competition function is positive, the monomorphic population will experience disruptive selection. The resulting
bimorphic population, however, can resist further invasions,
roughly if σk < 2σc (Fig. 3).
The quadratic-carrying capacity function leads to a fit2
ness minimum only if σc2 < a2 (we assumed without loss
of generality that a = 1, Eq. 4), which also corresponds to
the condition when the coalition starts to exist. The resulting bimorphic population could resist further invasion if
0.4410 < σc < 0.7071.
Box-like functions, used by Baptestini et al. (2009), lead
to the first branching condition that could be reduced to the
Theor Ecol (2014) 7:367–379
same solution obtained with the Gaussian-carrying capacity and competition functions assuming that βc = βk
(Baptestini et al. 2009), where β is a parameter that affects
the shape of the function near the maximum. The conditions of branching becomes more restrictive when βc >
βk (Baptestini et al. 2009). Higher values of β lead to
more platykurtic distributions similar to the quartic function; however, unlike the quartic function, the evolutionarily
singular strategy is not always a fitness minimum. Box-like
functions result in coalitions that might be evolutionarily stable coalitions or branching points depending on the
parameters, but the parameter space that allows for second
branching points or stable coalitions is reduced compared
to the parameter space of the first branchings. For example, when βc = βk , the first branching condition is no
longer sufficient for further branchings. Baptestini et al.
(2009) argued that using a box-like competition function
is more appropriate since it represents an abrupt decline in
competition with increasing phenotypic distance.
There are two issues that might be worthwhile to investigate further. First is the effect of multidimensional phenotypes. Doebeli and Ispolatov (2010) extended the classical
model where a single phenotypic trait determines an individual’s resource preferences to include multidimensional
phenotypic individuals. Their analysis showed that even if
frequency-dependent selection through competition along a
single phenotypic trait is not strong enough to induce evolutionary branching, interactions between multiple phenotypic
traits relax the conditions and generate evolutionary branching much more easily (Doebeli and Ispolatov 2010). The
overall pattern that emerges from our analyses is that further branchings require even stronger frequency-dependent
selection. It would be interesting to check the extent to
which the introduction of multidimensional phenotypes
ease the requirements of further branchings (Doebeli and
Ispolatov 2010).
Second is the effect of explicit resource dynamics. Even
though a large majority of theoretical work use the Lotka–
Volterra model derived from the consumer-resource system
of MacArthur (1970, 1972), the model may not be general when resource dynamics is modeled explicitly (Abrams
et al. 2008). Not only is the generality of using Gaussian
competition function is called into doubt (Ackermann and
Doebeli 2004) but also the analysis of Abrams et al.
(2008) suggests that disruptive selection may cease to exist
under both strong and weak competition depending on the
shape of the consumer’s resource utilization functions and
resource dynamics.
As a final note, we stress the fact the degenerate behavior of some of the functions should not be overlooked.
We observed strikingly different dynamics with functions
that were qualitatively similar. Since it may not be possible to distinguish the shapes of these functions in nature,
375
we urge theorists to check the sensitivity of their models’
results to changes in the details of the competition and carrying capacity functions used. Once fully understood, these
results could bear some insight into the role of ecology in
diversity-dependent speciation rates (Rabosky et al. 2012;
Ezard et al. 2011).
Acknowledgments We thank Erol Akcay, Mary Ballyk, Bill
Boecklen, Sergey Gavrilets, Dan Howard, Samraat Pawar, Lenny Santisteban, and Xavier Thibert-Plante for the valuable discussions and
comments on the various versions of this manuscript. We also thank A.
Hastings, M. Doebeli, and the anonymous reviewers for their detailed
comments that improved the manuscript greatly. A.B. was supported
by the Joaquin O. Loustaunau Memorial Graduate Fellowship at
NMSU when this study was initiated.
Appendix A
Here, we briefly investigate and justify the idea of symmetry to reduce the high dimensional systems with multiple
species or branching events to more manageable systems.
We exploit conditions that lead to special branching patterns
that allow a dimorphism to be described in terms of a single number. We do this by assuming symmetry by which
we assume one of the coalition phenotypes is parameterized by the other; thereby, the three-dimensional problem
is reduced to two. The mathematical theory of symmetry is
vast, and the details are not needed for our purposes (Weyl
1952; Golubitsky et al. 1988; Hamermesh 1989).
For our purposes, it is sufficient to interpret symmetry
as referring to a branching pattern in which the dimorphic mutant invaders are equally distant from the resident
in phenotype space (that is, x1∗ = −x2∗ = x ∗ , for some
x ∗ yet to be determined, and x3 = 0). The mathematical
point here is that if the K and C functions are symmetric,
then finding symmetric solutions is a much easier problem
numerically than finding nonsymmetric branching patterns.
For the Lotka–Volterra system (1), this can be described as
follows: if the conditions K(x) = −K(−x), C(x1 , x2 ) =
C(x2 , x1 ) and C(0, x) = C(0, −x) are satisfied (this will be
so if, for example, K(x) and C(x1 , x2 ) depend on x through
the quantities x 2 and (x1 − x2 )2 , respectively), then finding
symmetric branching patterns require the solution of only
a scalar nonlinear equation rather than solution of a set of
coupled nonlinear equations.
Lastly, we would also like to point out that the foregoing
methods are easily generalizable to nonsymmetric solutions
if the symmetry conditions are not exactly satisfied by the C
and K functions. The argument is based on continuity and
serves as the basis of an approach for the computation of the
branching patterns, either using numerical methods or analytically as a power series in a small parameter. Then, it can
be seen that the symmetric solution x2 = −x1 is replaced by
376
Theor Ecol (2014) 7:367–379
a branching pattern (x1 , x2 ) = (x1∗ , x2∗ ) where xj∗ are nonsymmetric functions that reduce to the symmetric solution
under appropriate conditions. All this implies that the conditions obtained are expected to hold for all systems in an
open neighborhood of the symmetric system in parameter
space. Therefore, the results obtained are not restrictive.
species x1 and x2 have nonzero populations at the resident equilibrium (n3 = 0). From N = A−1 K, where
N and K are the vectors of population sizes and carrying capacity functions, respectively, and A−1 is the inverse
of the community matrix, the solution for the two resi1 )−C(x1 ,x2 )K(x2 )
dents can be found to be: n1 = K(x
1−C(x1 ,x2 )C(x2 ,x1 ) and
2 )−C(x2 ,x1 )K(x1 )
n2 = K(x
1−C(x1 ,x2 )C(x2 ,x1 ) . The mutant’s fitness measures its
ability to invade the existing two species community:
Appendix B
Here, we further simplify the conditions for the first branching given by Eqs. 8, 10, and 11. Successful invasion by
the mutant requires that the corresponding eigenvalue for
the invader dynamics (or invasion fitness) be greater than
zero. Hence, the requirement for successful invasion by the
mutant is:
(x1 , x2 ) = K(x2 ) − C(x2 , x1 )K(x1 ) > 0
We assume that C(xj , xj ) = 1 for all j and
is the
∂C(x1 ,x2 ) maximum from which it follows that
=
∂x1
x1 =x2
∂C(x1 ,x2 ) = 0. A necessary condition for an evo
∂x2
x2 =x1
lutionarily singular strategy is (from Eq. 8 which further
reduces to Eq. 15):
∂(x1 , x2 ) ∂C(x2 , x1 ) ∗
=
K
(x
)
−
K(x ∗ ) = 0
∂x2
∂x2
x1 =x2 =x ∗
x1 =x2 =x ∗
(29)
This evolutionarily singular strategy is a fitness minimum if
(from Eq. 10, which further reduces to Eq. 16):
∂ 2 (x1 , x2 ) ∂x22
x1 =x2 =x ∗
2
= K (x ∗ ) − ∂ C(x22,x1 ) ∂x2
x1 =x2 =x ∗
K(x ∗ ) > 0
C(x3 , x1 ) − C(x3 , x2 )C(x2 , x1 )
K(x1 )
1 − C(x1 , x2 )C(x2 , x1 )
C(x3 , x2 ) − C(x3 , x1 )C(x1 , x2 )
K(x2 ) > 0
−
1 − C(x1 , x2 )C(x2 , x1 )
(x1 , x2 , x3 ) = K(x3 ) −
due to symmetry properties outlined in Appendix A, this
further reduces to the following at x1 = x and x2 = −x:
C(x3 , x) + C(x3 , −x)
K(x)
1 + C(x, −x)
(x, −x, x3) = K(x3 ) −
Evolutionarily singular strategies (12) are:
∂C(x3 ,−x ∗ )
|x3 =x ∗
∂(x, −x, x3 ) ∂x3
∗
=
K
(x
)
−
K(x ∗ ) = 0
∗
∂x3
1 + C(x , −x ∗ )
x=x3 =x ∗
(32)
which reduces to Eq. 18. These strategies are fitness minima
(13) if:
∂ 2 (x, −x, x3 ) ∂x32
x=x3 =x ∗
= K (x ∗ ) −
∂ 2 C(x3 ,x ∗ ) ∂x32
∂ 2 C(x3 ,−x ∗ ) ∂x32
x3 =x ∗
x3 =x ∗
∗
∗
1 + C(x , −x )
+
K(x ∗ ) > 0
(33)
(30)
Convergence stability (from Eq. 11, which further reduces
to Eq. 17) is:
∂ 2 C(x2 , x1 ) −
∂x12
K(x ∗ ) − 2
x1 =x2 =x ∗
∂C(x2 , x1 ) K (x ∗ )
∂x1
x1 =x2 =x ∗
−C(x2 , x1 )|x1 =x2 =x ∗ K (x ∗ )
∂ 2 C(x2 , x1 ) > K (x ) −
∂x22
x
∗
K(x ∗ )
1 =x2 =x
(31)
∗
Appendix C
Here, we further simplify the conditions for the second
branching given by Eqs. 12, 13, and 14. Only the resident
which reduces to Eq. 19. For convergence stability (lefthand side of the inequality in Eq. 14):
∂C(x3 ,−x)
3 ,x)
− ∂C(x
∂(x, −x, x3) ∂x
∂x
=
K(x)
∂x
1 + C(x, −x)
x3 =x=x ∗
∂C(x, −x) C(x3 , x) + C(x3 , −x)
∂x
(1 + C(x, −x))2
C(x3 , x) + C(x3 , −x)
− K (x)
1 + C(x, −x)
2
∂ (x, −x, x3) 3C ∗ − C o
=
K(x ∗ )
1 + C∗
∂x 2
∗
+2K(x)
x3 =x=x
− K (x ∗ ) − 4K(x ∗ )
(C ∗ )2
C∗
+ 2K (x ∗ )
∗
2
(1 + C )
1 + C∗
Theor Ecol (2014) 7:367–379
377
where C ∗ = C(x, −x)|x=x ∗ , C ∗ = ∂C(x∂x3 ,−x)
,
3
x3 =x ∗
2
2
3 ,−x)
3 ,x)
C ∗ = ∂ C(x
, and C o = ∂ C(x
. Note
∂x3
∂x3
∗
∗
x3 =x
x3 =x
that the right-hand side of the inequality in Eq. 14 is given
by Eq. 33. Putting the two together:
−K (x ∗ ) +
+ C∗
1 + C∗
D (0) =
∂ 3 (x1 , x2 )
∂ 3 (x1 , x2 )
∂ 3 (x1 , x2 )
+2
+
∂x12 ∂x2
∂x1 ∂x22
∂x23
=0
x1 =x2 =x ∗ +θ
at θ = 0. And:
K(x ∗ ) > 0
D (0) =
3C ∗ − C o
(C ∗ )2
C∗
K(x ∗ ) − K (x ∗ ) − 4K(x ∗ )
+ 2K (x ∗ )
∗
∗
2
1+C
(1 + C )
1 + C∗
Co
occur, the leading order term must be of odd order and the
coefficient must be negative. In that case:
∂ 4 (x1 , x2 )
∂x13 ∂x2
=
∂ 4 (x1 , x2 )
∂x12 ∂x22
∂ 4 (x1 , x2 ) +
∂x24
∗
4C ∗
(C ∗ )2
C∗
K(x ∗ ) − 2K (x ∗ ) − 4K(x ∗ )
+ 2K (x ∗ )
>0
∗
∗
2
1+C
(1 + C )
1 + C∗
(34)
which is further reduced to Eq. 20 since K (x)
K(x)C ∗ /(1 + C ∗ ) from Eq. 32.
+3
+3
∂ 4 (x1 , x2 )
∂x1 ∂x23
<0
x1 =x2 =x +θ
These last two conditions are satisfied when the carrying
capacity function is quartic.
Appendix D
Appendix E
Here, we derive conditions of convergence stability when
the second derivative of the carrying capacity function
equals zero. Convergence stability to a local minimum or
a maximum is determined by the fitness gradient near the
evolutionarily singular strategy x ∗ (Eshel 1983):
Here, we derive the convergent stability conditions (20). For
the Gaussian competition and carrying capacity functions:
2exp
∂(x1 , x2 ) D(θ) =
∂x2
x2 =x1 =x ∗ +θ
1 + exp
1
1
D(θ) = D(0) + D (0)θ + D (0)θ 2 + D (0)θ 3 + . . .
2
3!
Since x ∗ is an evolutionarily singular strategy, D(0) = 0,
and convergence stability condition is satisfied if:
D (0) =
∂ 2 (x1 , x2 ) ∂ 2 (x1 , x2 )
+
∂x1 ∂x2
∂x22
−
which should be negative for small values of θ (or positive
for small negative values of θ ). Taylor expansion of D(θ ) is:
−2(x ∗ )
σc2
<0
x1 =x2 =x ∗ +θ
−2(x ∗ )
σc2
(x ∗ )2
2
1
+
σk4
4 (x ∗ )2
1
+
2
σc
σc4
σk2
−
2
(x ∗ )2
x∗
tanh
−
1
σc2
σc2
>0
(35)
Gaussian competition and quartic-carrying capacity functions:
2exp
1 + exp
−2(x ∗ )
σc2
2
−2(x ∗ )
σc2
−
2
4 (x ∗ )6
σk8
4 (x ∗ )2
1
+
2
σc
σc4
+
−
x∗
tanh
σc2
(x ∗ )2
σc2
2
−1
6 (x ∗ )2
>0
σk4
(36)
Gaussian competition and quadratic-carrying capacity functions:
Note that the above condition results in Eq. 11 when the
cross-derivatives are eliminated (Geritz et al. 1998). If the
second derivatives are equal to zero, this derivation fails, and
convergence stability depends on the leading order nonvanishing terms of D(θ). In order for convergence stability to
2
2exp
1 + exp
−2(x ∗ )
σc2
2
−2(x ∗ )2
σc2
−
2
1 − (x ∗ )2
4 (x ∗ )2
1
+
2
σc
σc4
>0
−
x∗
tanh
σc2
(x ∗ )2
σc2
2
−1
(37)
378
Theor Ecol (2014) 7:367–379
Box-like competition and carrying capacity functions:
∗
βc
2 2x ∗ + 1 + tanh 2x ∗ − 1 sech2 2x ∗ − 1
tanh 2x
+
1
sech
σc
σc
σc
σc
2σc tanhβc
∗
∗
1
1 + 2σc tanhβ
tanh βc 2x
− tanh βc 2x
σc + 1
σc − 11
c
⎛
⎞2
∗
∗
βc
2x
2x
tanh
+
1
−
tanh
−
1
σc
σc
⎜
⎟
2σc tanhβc
∗
∗
⎟
+⎜
⎝
⎠
1
1 + 2σc tanhβ
tanh βc 2x
− tanh βc 2x
σc + 1
σc − 11
c
∗
∗
∗
∗
x
x
x
x
− βk2 tanh βk
+1
sech2 βk
+1
− tanh βk
−1
sech2 βk
−1
>0
σk
σk
σk
σk
(38)
Appendix F
Here, we give the conditions for evolutionarily stable coalitions (or branching points) with box-like functions (Eqs. 18
and 19):
∗
∗
βk sech2 βk xσk + 1 − sech2 βk xσk − 1
∗
∗
σk tanh βk xσk + 1 − tanh βk xσk − 1
∗
∗
βc
2x
2 β
sech2 βc 2x
+
1
−
sech
−
11
c
σ
σ
c
c
2σc tanhβc
=
∗
∗
1
2x
1+
tanh βc 2x
+
1
−
tanh
β
−
11
c
σc
σc
2σc tanhβc
∗
∗
∗
∗
−βk2 tanh βk xσk + 1 sech2 βk xσk + 1 − tanh βk xσk − 1 sech2 βk xσk − 1
∗
∗
σk2 tanh βk xσk + 1 − tanh βk xσk − 1
∗
−βc tanhβ ∗ sech2 β ∗
2 2x ∗ + 1 + tanh 2x ∗ − 1 sech2 2x ∗ − 1
− tanh 2x
+
1
sech
σc
σc
σc
σc
σc tanhβc
>
1
∗
∗
1+
tanh βc 2x
− tanh βc 2x
σc + 1
σc − 11
2σc tanhβc
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