A dynamical theory of speciation on holey adaptive landscapes

Abstract
The metaphor of holey adaptive landscapes provides a pictorial representation of the process of speciation as a consequence of genetic divergence. In this metaphor, biological populations diverge along connected clusters of well-fit genotypes in a multidimensional adaptive landscape and become reproductively isolated species when they come to be on opposite sides of a ``hole'' in the adaptive landscape. No crossing of any adaptive valleys is required. I formulate and study a series of simple models describing the dynamics of speciation on holey adaptive landscapes driven by mutation and random genetic drift. Unlike most previous models that concentrate only on some stages of speciation, the models studied here describe the complete process of speciation from initiation until completion. The evolutionary factors included are selection (reproductive isolation), random genetic drift, mutation, recombination, and migration. In these models, pre- and post-mating reproductive isolation is a consequence of cumulative genetic change. I study possibilities for speciation according to allopatric, parapatric, peripatric and vicariance scenarios. The analytic theory satisfactorily matches results of individual-based simulations reported by Gavrilets et al. (1998). It is demonstrated that rapid speciation including simultaneous emergence of several new species is a plausible outcome of the evolutionary dynamics of subdivided populations. I consider effects of population size, population subdivision, and local adaptation on the dynamics of speciation. I briefly discuss some implications of the dynamics on holey adaptive landscapes for molecular evolution.

arXiv:adap-org/9807005v1 27 Jul 1998
A dynamical theory of speciation
on holey adaptive landscapes
Sergey Gavrilets
Departments of Ecology and Evolutionary Biology
and Mathematics,
University of Tennessee,
Knoxville, TN 37996-1610, USA
phone: (423) 974-8136
fax: (423) 974-3067
e-mail: gavrila@tiem.utk.edu
key words: evolution, speciation, holey adaptive land-
scapes, mathematical models
running head: Dynamical Theory of Speciation
ABSTRACT: The metaphor of holey adaptive land-
scapes provides a pictorial representation of the process of
speciation as a consequence of genetic divergence. In this
metaphor, biological populations diverge along connected
clusters of well-ﬁt genotypes in a multidimensional adap-
tive landscape and become reproductively isolated species
when they come to be on opposite sides of a “hole” in the
adaptive landscape. No crossing of any adaptive valleys is
required. I formulate and study a series of simple models
describing the dynamics of speciation on holey adaptive
landscapes driven by mutation and random genetic drift.
Unlike most previous models that concentrate only on
some stages of speciation, the models studied here de-
scribe the complete process of speciation from initiation
until completion. The evolutionary factors included are
selection (reproductive isolation), random genetic drift,
mutation, recombination, and migration. In these mod-
els, pre- and post-mating reproductive isolation is a conse-
quence of cumulative genetic change. I study possibilities
for speciation according to allopatric, parapatric, peri-
patric and vicariance scenarios. The analytic theory sat-
isfactorily matches results of individual-based simulations
reported by Gavrilets et al. (1998). It is demonstrated
that rapid speciation including simultaneous emergence
of several new species is a plausible outcome of the evo-
lutionary dynamics of subdivided populations. I consider
eﬀects of population size, population subdivision, and lo-
cal adaptation on the dynamics of speciation. I brieﬂy
discuss some implications of the dynamics on holey adap-
tive landscapes for molecular evolution.
Speciation has traditionally been considered to be one
of the most important and intriguing processes of evolu-
tion. In spite of this consensus and signiﬁcant advances
in both experimental and theoretical studies of evolution,
understanding speciation still remains a major challenge
(Mayr 1982a; Coyne 1992). The main reason for such
a discouraging situation is that direct experimental ap-
proaches, which are widely used for solving other prob-
lems of evolutionary biology, are not eﬀective for studying
speciation because of the time scale involved. Experi-
mental work necessarily concentrates on distinct parts of
the process of speciation intensifying and simplifying the
factors under study (Rice and Hostert 1993; Templeton
1996). In situations where direct experimental studies are
diﬃcult or impossible, mathematical modeling has proved
to be indispensable for providing a unifying framework.
Although numerous attempts to model parts of the pro-
cess of speciation have been made, a quantitative theory
of the dynamics of speciation is still missing. Currently,
verbal theories of speciation are far more advanced than
mathematical foundations. As often is the case with ver-
bal theories (both scientiﬁc and otherwise), diﬀerent de-
duced (or induced) aspects of speciation are emphasized
by diﬀerent workers resulting in confusion and contro-
versy. The situation is not helped by the absence of gen-
eral agreement on a species deﬁnition (e.g., Claridge et
al. 1997).
Here, I attempt to develop some foundations for a gen-
eral dynamical theory of speciation. One possible ap-
proach to this goal would be to begin with a species deﬁni-
tion, then to deﬁne speciation accordingly and to develop
an appropriate dynamical model. I do not think such ap-
proach would be very useful, due to a lack of generality.
My models are not based on a speciﬁc “species concept“.
I reason that species are “diﬀerent” with respect to some
characteristics, and that whatever these diﬀerences, they
have a genetic basis. Thus, modeling the dynamics of spe-
ciation is equivalent to modeling the dynamics of genetic
divergence. I use a bottom-up approach: begin with a
model incorporating a range of factors thought to lead to
speciation (e.g. selection, mutation, population subdivi-
sion etc.) and then try to interpret its dynamic behavior
in terms of diﬀerent “species concepts”. As expected,
many aspects of speciation that are emphasized by dif-
ferent species concepts (such as reproductive isolation,
separate genotypic clusters or common evolutionary tra-
jectories) emerge from the same processes. This clearly
indicates that diﬀerent species concepts are not mutually
exclusive.
The choice of a modeling approach depends upon the
purpose of the model. A common view in (evolution-
ary) biology is that mathematical models are mainly use-
ful for making predictions that can be used in experi-
mental work. Although such a pragmatic approach is
probably what should be expected in contemporary so-
ciety, a model’s testable predictions are not necessarily
its main contribution to science. Insights provided by
models, their ability to train our intuition about complex
phenomena, to provide a framework for studying such
phenomena and to identify key components in complex

systems are at least as important as speciﬁc predictions.
For these purposes the most useful tools are simple mod-
els and metaphors.
Sewall Wright’s (1932) metaphor of “rugged adaptive
landscapes” is a well-known and widely used metaphor
in evolutionary biology. In the standard interpretation,
a rugged adaptive landscape is a surface in a multidi-
mensional space that represents the mean ﬁtness of the
population as a function of gamete (or allele) frequencies
which characterize the population state (see Fig.1a). It is
envisioned that this surface has many peaks and valleys
corresponding to diﬀerent adaptive and maladaptive pop-
ulation states, respectively. The population is imagined
as a point on the surface which is driven by selection up
hill but can get stuck on a local peak. Two general points
about scientiﬁc metaphors should be kept in mind. The
ﬁrst is that speciﬁc metaphors (as well as mathematical
models) are good for speciﬁc purposes only. The second is
that accepting a speciﬁc metaphor necessarily inﬂuences
and deﬁnes the questions that are considered to be im-
portant. The metaphor of “rugged adaptive landscapes”
is very useful for thinking about adaptation. However,
its utility for understanding speciation is questionable.
From a pragmatic point of view, the process of splitting a
population into two diﬀerent species is impossible to de-
scribe in a framework where a population is the smallest
unit. Finer resolution is necessary for describing the split-
ting of populations. Accepting the metaphor of rugged
adaptive landscapes immediately leads to a problem to
be solved: how can a population evolve from one adap-
tive peak to another across an adaptive valley when selec-
tion opposes any changes away from the current adaptive
peak? Wright’s solution to this problem, his shifting-
balance theory (Wright, 1931, 1982), does not seem to
be satisfactory (Gavrilets 1996; Coyne et al. 1997).
Provine (1986), Barton and Rouhani (1987), Whitlock
et al. (1995), Gavrilets (1997a) and Coyne et al. (1997)
discuss other weaknesses of Wright’s metaphor. I have
argued elsewhere (Gavrilets 1997a) that his metaphor of
rugged adaptive landscapes with its emphasis on adap-
tive peaks and valleys is, to a large degree, a reﬂection of
the three-dimensional world we live in. Both genotypes
and phenotypes of biological organisms diﬀer in numerous
characteristics, and, thus, the dimension of “real” adap-
tive landscapes is much larger than three. Properties of
multidimensional adaptive landscapes are very diﬀerent
from those of low dimension. Consequently, it may be
misleading to use three-dimensional analogies implicit in
the metaphor of rugged adaptive landscapes in a multi-
dimensional context. I believe that understanding speci-
ation requires a diﬀerent metaphor.
The metaphor of “holey” adaptive landscapes . An in-
dividual organism can be considered as a combination of
genes. All possible combinations of genes form a geno-
fitness
genotype space
fitness
a
b
population state
Figure 1: Adaptive landscapes. a. A rugged
adaptive landscape. The main emphasis is on
“peaks” and “valleys” corresponding to diﬀerent
well-adapted and maladaptive genotypes. b. A
holey adaptive landscape. The main emphasis
is on clusters of well-ﬁt genotypes that extend
throughout the genotype space. All other geno-
types are treated as “holes”.
type space (which, mathematically, can be represented
by a hypercube). In discussing the evolution of popula-
tions, it is useful to visualize each individual as a point in
this genotype space. Accordingly, a population will be a
cloud of points, and diﬀerent populations (or species) will
be represented by diﬀerent clouds. Selection, mutation,
recombination, random drift and other factors change the
size, location and structure of these clouds. To construct
an adaptive landscape one assigns “ﬁtness” to each geno-
type (or each pair of genotypes) in genotype space. Dif-
ferent forms of selection and reproductive isolation can
be treated within this conceptual framework. For exam-
ple, ﬁtness can be a genotype’s viability (in the case of
viability selection); it can be fertility or the probability
of successful mating between a pair of genotypes (in the
case of fertility selection or premating isolation, respec-
tively). A ﬁnite population subject to mutation is likely
to be represented by genotypes with ﬁtnesses within a
ﬁtness band determined by the balance of mutation, se-
lection and random drift. A general property of adap-
tive landscapes with a very large number of dimensions is
that genotypes with ﬁtnesses within a speciﬁed band form
connected “clusters” that extend throughout the geno-

type space (Gavrilets 1997; Gavrilets and Gravner 1997).
Thus, populations can evolve and diverge along bands of
highly-ﬁt genotypes without going across the states with
a large number of low-ﬁt genotypes (that is without cross-
ing any adaptive valleys).
The metaphor of “holey” adaptive landscapes puts spe-
cial emphasis on these clusters of highly-ﬁt genotypes, dis-
regarding ﬁtness diﬀerences between them and treating
all other genotypes as “holes” (Gavrilets 1997; Gavrilets
and Gravner 1997). The justiﬁcation for the latter is a
belief that selection will be eﬀective in moving the popu-
lation away from these areas of genotype space on a time
scale that is much faster than the time scale for speciation.
Accordingly, microevolution and local adaptation can be
viewed as the climbing of the population from a “hole” to-
wards the holey adaptive landscape, whereas macroevolu-
tion can be viewed as a movement of the population along
the holey landscape with speciation taking place when the
diverging populations come to be on opposite sides of a
“hole” in the adaptive landscape. In this scenario, there is
no need to cross any “adaptive valleys”; reproductive iso-
lation between populations evolves as an inevitable side
eﬀect of accumulating diﬀerent mutations. The metaphor
of holey adaptive landscapes can be traced to a verbal
two-locus two-allele model of reproductive isolation pro-
posed by Dobzhansky (1937) and similar ideas discussed
by Bateson (cited in Orr, 1997), Muller (1942), Maynard
Smith (1970, 1983), Nei (1976), Barton and Charlesworth
(1984), Kondrashov and Mina (1986). For more discus-
sion of this metaphor see Gavrilets (1997ab) and Gavrilets
and Gravner (1997). Orr (1995) and Gavrilets (1997a)
gives a summary of relevant experimental evidence. The
metaphor of “holey” adaptive landscapes is illustrated
graphically in Figure 1b.
Mathematical models for holey adaptive landscapes.
Here I brieﬂy review previously published work on the
evolutionary dynamics on “holey” adaptive landscapes.
Nei (1976) and Wills (1977) were the ﬁrst to present for-
mal analyses of the Dobzhansky model. Nei et al. (1983)
studied one- and two-locus multi-allele models with step-
wise mutations and considered both postmating and pre-
mating reproductive isolation. In their models geno-
types were reproductively isolated if they were diﬀerent
by more than 1 or 2 mutational steps. In these situations,
speciation was very slow. They conjectured, however,
that increasing the number of loci may signiﬁcantly in-
crease the rate of speciation. Bengtsson and Christiansen
(1983) presented a deterministic analysis of mutation-
selection balance in the Dobzhansky model. Bengts-
son (1985), Barton and Bengtsson (1986) and Gavrilets
(1997b) analyzed the properties of hybrid zones arising
under Dobzhansky-type epistatic selection. Wagner et al.
(1994) considered a two-locus, two-allele model of stabi-
lizing selection acting on an epistatic character. For a
speciﬁc set of parameters, the interaction of epistasis in
the trait and stabilizing selection on the trait resulted
in a ﬁtness “ridge”. The existence of this ridge sim-
pliﬁed stochastic transitions between alternative equilib-
ria. Gavrilets and Hastings (1996) formulated a series
of two- and three-locus Dobzhansky-type viability selec-
tion models as well as models for selection on polygenic
characters. They studied these models in the context of
founder eﬀect speciation and noticed that the existence
of ridges in the adaptive landscape made stochastic di-
vergence much more plausible. Similar conclusions were
reached by Gavrilets and Boake (1998) who studied the
eﬀects of premating reproductive isolation on the plau-
sibility of founder eﬀect speciation. Higgs and Derrida
(1991, 1992) proposed a model where the probability of
mating between two haploid individuals is a decreasing
function of the proportion of loci at which they are diﬀer-
ent. Here, any two suﬃciently diﬀerent genotypes can be
considered as sitting on opposite sides of a hole in a ho-
ley adaptive landscape. These authors as well as Manzo
and Peliti (1994) studied this model numerically assum-
ing that the number of the loci is inﬁnite, the loci are
unlinked and highly mutable, and mating is preferential.
Orr (1995) and Orr and Orr (1996) studied speciation
in a series of models in which viability of a diploid or-
ganism depends on the number of heterozygous loci. All
these papers postulated the existence of ridges of highly-
ﬁt genotypes. Gavrilets and Gravner (1997) studied a
general class of multilocus selection models and showed
the existence of ridges to be inevitable under fairly gen-
eral conditions. Independently, a similar conclusion was
reached in Reidys et al. (1997). Most previous studies of
the dynamics of speciation on holey adaptive landscapes
were numerical. To develop a dynamical theory of speci-
ation it is desirable to have a simple model that can be
treated analytically.
The Model
I consider ﬁnite populations of haploid individuals with
discrete, non-overlapping generations. I assume that re-
production involves gene exchange (amphimixis) between
individuals. The restriction to haploids is for algebraic
simplicity. Models for diploids will be discussed later. In-
dividuals are diﬀerent with respect to L possibly linked
diallelic loci. Without any loss of generality each individ-
ual’s genotype can be represented as a sequence of 0’s and
1’s. Letlα = (lα
1,...,l α
L) wherelα
i is equal to 0 or 1, be such
a sequence for an individual α . In standard population
genetics models, the population state is usually described
in terms of gamete frequencies. In systems with many
loci such an approach is not practical. For instance, with
10 diallelic loci there are 2 10 diﬀerent gametes. Thus, one
would need to analyze more than 1000 coupled equations.

Another complication follows from the fact that even in
very large populations with hundreds of thousands indi-
viduals each speciﬁc genotype is represented only by a
small number of copies or is not represented at all. Thus,
the notion of a gamete frequency in multilocus evolution
might be very diﬃcult to justify. Here, I will be inter-
ested in the levels of genetic variation within subpopu-
lations and genetic divergence between subpopulations.
Both can be characterized in terms of genetic distance
d deﬁned as the number of loci at which two individu-
als are diﬀerent. More formally, the genetic distance dαβ
between individuals α and β is
dαβ =
L∑
i=1
(lα
i − lβ
i )2. (1)
Genetic distance d is the standard Hamming distance.
It is analogous to the number of segregating sites in a
sample of two gametes, which is widely used in molecu-
lar evolutionary genetics (Li, 1997), and to the number
of heterozygous loci in a diploid organism. Genetic dis-
tanced is also closely related to the notion of the overlap
q between two sequences, d = L
2 (1 − q), which is com-
monly used in statistical physics (e.g., Derrida and Peliti
1991). I model the expected dynamics of average genetic
distances within and between populations, using Dw for
the former and Db for the latter.
I assume that reproductive isolation is caused by cumu-
lative genetic change. I will use a very simple symmetric
model which is closely related to the models discussed
above and which allows one to treat both pre- and post-
mating isolation within the same framework. I posit that
an encounter of two individuals can result in viable and
fecund oﬀspring only if the individuals are diﬀerent at
no more than K loci. Otherwise the individuals do not
mate (premating reproductive isolation) or these oﬀspring
are inviable or sterile (postmating reproductive isolation).
More formally, I assign “ﬁtness” w to each pair of individ-
uals depending on the genetic distance d between them
w(d) =
{
1 for d ≤ K,
0 for d>K. (2)
(see Appendix for an outline of more complicated ap-
proaches). In this formulation, any two genotypes diﬀer-
ent at more than K loci can be conceptualized as sitting
on opposite sides of a hole in a holey adaptive landscape
(cf. Higgs and Derrida 1991, 1992). At the same time, a
population can evolve to any reproductively isolated state
by a chain of single locus substitutions. The adaptive
landscape corresponding to this model is both “holey”
and “correlated”. The latter means that the probability
that two genotypes are reproductively isolated correlates
with the genetic distance between them. In Nei et al.
(1983) and Gavrilets and Boake (1998) models individ-
uals separated by more than one mutational step were
reproductively isolated which corresponds to K = 1. The
neutral case (no reproductive isolation) corresponds to K
equal to the number of loci.
The mathematical model presented above was inter-
preted as describing sexual haploid populations with ﬁt-
nesses assigned to pairs of individuals depending on the
genetic distance between them. However, there is an al-
ternative interpretation in that the model describes ran-
domly mating diploid populations. In the diploid case,
the genetic distance (1) between the two gametes forming
an individual is equivalent to individual’s heterozygosity,
and ﬁtness function (2) speciﬁes ﬁtness as a function of
individual heterozygosity. Therefore, most conclusions of
this paper will also be applicable to situations when post-
mating reproductive isolation is in the form of reduced (or
zero) viability of hybrids due to incompatibility of the
genes they receive from their parents (Wu an Palopoli
1994).
Dynamics in the neutral case
Before developing a theory for the dynamics of specia-
tion in the above model, it is illuminating to start with
the neutral case. Here I summarize some relevant results
that are presented in (or follow directly from) classical
papers (Watterson 1975; Li 1976; Slatkin 1987; Strobeck
1987). Let µ be the probability of mutation per locus per
generation. The approximations below assume that mat-
ing is random, the number of loci L is large, but µ is very
small so that the probability of mutation per individual
per generation v ≡ Lµ<< 1. The migration rate m and
the inverse of the population size 1 /N are small as well.
Genetic variation within an isolated population . Let
us consider an isolated population of size NT . The ex-
pected change in the average genetic distance within the
population per generation is
∆ Dw = 2v − Dw
NT
, (3)
where the ﬁrst term in the right-hand side is the contribu-
tion of mutation whereas the second term is the random
drift reduction of Dw. Asymptotically, a mutation-drift
equilibrium is reached with
D∗
w =θ ≡ 2vNT. (4)
Genetic divergence between isolated populations . Let
us consider several isolated populations of arbitrary size.
The probability that a speciﬁc mutation gets ﬁxed in a
population is one over the population size. Diﬀerent mu-
tations will get ﬁxed in diﬀerent populations resulting in
their genetic divergence. The average genetic distance

between any two of them increases with the rate equal
twice the mutation rate per gamete
∆ Db = 2v. (5)
The rate of neutral divergence does not depend on pop-
ulation sizes. In particular, it is the same independent of
whether there are many small populations or a few large
populations. Because the number of loci L is ﬁnite, an
indeﬁnite increase of Dw, which is implied by equation
(5) is impossible. This equation as well as equation (3)
above and equations (6) below approximate the dynamics
when genetic distances Dw and Db are small relative to
the number of loci L. To treat the general case, one has to
substitute v for v(1 − 2Dw/L ) in equations (3) and (6a)
and for v(1 − 2Db/L ) in equations (5) and (6b). With
a ﬁnite number of loci, the genetic distance Db between
isolated populations approaches L/ 2 asymptotically.
Subdivided populations. The eﬀect of migration on the
average genetic distances depends on the spatial structure
of populations. Assume that a population of size NT is
subdivided into n subpopulations of size N =NT/n and
that a proportion m> 0 of individuals migrate to any of
the other n − 1 subpopulations. The expected changes
in the average genetic distances within and between sub-
populations are
∆ Dw = 2v + 2m(Db − Dw) − Dw
N , (6a)
∆ Db = 2v + 2m
n − 1 (Dw − Db). (6b)
Equations (6) assume that v,m and 1 /N are small.
Asymptotically, a mutation-migration-drift equilibrium is
reached with
D∗
w =θ, (7a)
D∗
b =θ + (n − 1) 2v
m, (7b)
where θ is given by equation (4). The average genetic
distance within a subpopulation (of size N ) does not de-
pend on the number of subpopulations n and migration
ratem and is the same as is expected in a single popula-
tion with size NT . The average genetic distance between
sub-populations increases with the population subdivi-
sion and decreasing migration. Figure 2 illustrates the
dynamics of neutral divergence in a system of two sub-
populations.
Peripheral population . Assume a “peripheral” popu-
lation of size N is receiving migrants from a very large
“main” population. Genetic variation in the main popu-
lation is assumed to be constant (and not inﬂuenced by
migration from the peripheral population). The expected
changes in the average genetic distances within the pe-
0 500 1000 1500 2000
generation
40genetic distances Dw
Dw
Db
Db
Figure 2: Dynamics of Dw and Db in the neutral case.
Population size N = 100 (solid lines) and N = 200 (dashed
lines). The rate of migration is m = 0 . 01, the mutation rate
per individual is v = . 0384. Initially, Dw = Db = 0.
ripheral population, Dw, and between the peripheral and
main populations, Db are
∆ Dw = 2v + 2m(Db − Dw) − Dw
N , (8a)
∆ Db =v +m(D0 − Db) (8b)
whereD0 is the average genetic distance within the main
population and m is the proportion of individuals in the
peripheral population replaced by migrants from the main
population. Asymptotically, a mutation-migration-drift
equilibrium is reached with
D∗
w = 2Nm
1 + 2Nm D0, (9a)
D∗
b =D0 + v
m, (9b)
where the former equation assumes that genetic variation
in the main population is suﬃciently large ( D0 >> vN)
and the number of migrants, Nm , is not too small. The
average genetic distance within the peripheral population
is always larger than that for an isolated population of
its size ( D∗
w > 2vN ). If the number of migrants is large
(Nm>> 1), the average genetic distance within the pe-
ripheral population is about the same as in the “main”
population.
Dynamics with reproductive isolation
The main feature of the model for reproductive isola-
tion introduced above and other models of holey adaptive
landscapes is the existence of chains of equally-ﬁt combi-
nations of genes separated by single substitutions that
extend throughout the genotype space. These chains can
be though of as ”neutral paths” in the adaptive land-
scape. It is important to realize, however, that the exis-

tence of “holes” in a holey adaptive landscape makes the
actual dynamics of genetic divergence not neutral. In this
section I summarize some analytical results on the evo-
lutionary dynamics in the case of reproductive isolation
described by equation (2). Gavrilets et al. (1998) have
studied the possibilities for speciation in this model nu-
merically. Details of the analytical methods used are out-
lined in the Appendix. To derive the dynamic equations
below, I have used the same assumptions as described
above at the beginning of the section on the neural case
substituting the assumption of random mating for the
assumption of random encounters. In addition, I have
assumed that the distributions of genetic distances both
within and between populations are Poisson. There are
several sets of approximations resulting in Poisson distri-
bution of genetic distances. In the present contest, the
weakest set seems to be the assumption that genetic vari-
ation at each locus is small most of the time (rare-alleles
approximation) and that the population is approximately
at linkage equilibrium. These assumptions are standard
in analyzing the dynamics of multilocus systems under
the joint action of selection, mutation, and random drift
(e.g., Barton 1986; Barton and Turelli 1987; B¨ urger et al.
1989; Gavrilets and de Jong 1993). The ﬁt of individual-
based simulations with analytic predictions is satisfactory
both at qualitative and quantitative levels (see below and
Gavrilets et al. 1998).
Genetic variation within an isolated population . After
the population becomes polymorphic at K loci, new mu-
tations are selected against when rare because individuals
carrying them have a reduced probability of producing vi-
able and fecund oﬀspring. Selection experienced by indi-
vidual loci underlying reproductive isolation is frequency-
dependent (and is similar to that arising in the case of
underdominant selection on a diploid locus). The change
in Dw per generation in an isolated population of size N
is approximately
∆ Dw = −sDw + 2v − Dw
N , (10)
where
s = e− DwDK
w
Γ(K + 1,D w) (11)
and Γ( x,y ) is an incomplete gamma function (e.g.,
Gradshteyn and Ryzhik, 1994). The value of D∗
w at
the mutation-drift-selection equilibrium can be found by
equating the right-hand side of (10) with zero and solving
for Dw. Figure 3a illustrates the dependence of D∗
w on
the parameters of the model. This ﬁgure indicates that
the equilibrium values of Dw are close to the correspond-
ing neutral predictions (4) if K is larger than 2 or 3 times
θ (whereθ = 2Nv is the average genetic distance within a
ﬁnite population in the neutral case). Figure 3b gives the
values of the eﬀective selection coeﬃcient s. With moder-
ately large K (that is with K ≥ 10),s is very small. The
eﬀective selection coeﬃcient s can also be thought of as
the strength of induced selection on each locus underlying
reproductive isolation. Figure 3b shows that very strong
selection on the whole genotype (implied by the existence
of complete reproductive isolation at ﬁnite values of K)
results in very weak selection at the level of individual
loci.
0 10 20 30 40 50
K
0.00
0.01
0.02
0.03
0.04
0.05
Genetic load
-4.0
-3.0
-2.0
-1.0
0.0
S
40Dw
v=0.0384
v=0.00384
v=0.00384
v=0.0384
N=100
N=200
N=400
N=800a
b
c
Figure 3: (a) Average genetic distance Dw maintained by
mutation-selection-drift balance in an isolated populati on of
size N as a function of K for v = 0. 0384. The circles, squares,
diamonds and triangles give estimates from individual-bas ed
simulations for N = 100, 200, 400 and 800, respectively (thirty
runs for each parameter conﬁguration). (b) Eﬀective select ion
coeﬃcient s in the case of inﬁnite population size for two values
of v. (c) The genetic load 1 − w for parameters values as in
Figure b.
The mean ﬁtness of the population, ww, can be deﬁned
as the proportion of pairs of individuals that can mate
and produce fertile and viable oﬀspring (cf., Nei et al.
1983). For a population with an average genetic distance
Dw,
ww = Γ(K + 1,D w)
Γ(K + 1) , (12)
where Γ( x + 1) is a gamma function (e.g., Gradshteyn
and Ryzhik, 1994. For integer x, Γ(x + 1) = x!). Figure
3c shows that in spite of relatively high levels of genetic
variation maintained in the population, the genetic load
(that is the proportion of reproductively isolated pairs
of individuals, 1 −
ww) is very low. This seems to be a
general property of holey adaptive landscapes (cf. Wills

1977; Bengtsson and Christiansen 1983).
Genetic divergence between isolated populations . Even
after the genetic distance within an isolated population
has reached an equilibrium level, the population keeps
evolving as diﬀerent mutations get ﬁxed. As a conse-
quence, isolated populations will continuously diverge ge-
netically. The asymptotic rate of divergence of two iso-
lated populations of size N each is
∆ Db = 2vR, (13a)
where
R = 2e− S√
S√ π erf(
√
S)
(13b)
is the rate of divergence relative to the neutral case. Here,
S = Ns/ 2, s is deﬁned by equation (11) with Dw corre-
sponding to the mutation-selection-drift equilibrium, and
erf (x) is the error function (= 2 / √
π
∫x
0 exp(−y2)dy). In
the neutral case, s = 0, U = 1/N and equation (13a) re-
duces to equation (5). Figure 4 illustrates the dependence
of the relative rate of divergence R on model parame-
ters. In the neutral case, the rate of genetic divergence
∆ Db does not depend on the population size (equation
5). In contrast, with reproductive isolation the rate of
divergence decreases with increasing population size. Af-
ter the population becomes polymorphic at K loci, new
mutations are selected against when rare. Genetic drift
operating in ﬁnite populations overcomes the eﬀect of se-
lection and allows genetic divergence. For example with
K = 20 and v =. 0384, a population of size N = 800 will
accumulate about 5 substitutions per 1000 generations.
A few thousand generations will be suﬃcient for Db to
exceed K signiﬁcantly. In contrast, very large randomly
mating populations will diverge very slowly.
Figure 4 indicates that the rate of substitutions is close
to the corresponding neutral predictions if K is larger
than 2-3 times θ (θ = 2Nv ). Note that as in the neutral
case considered above, an implicit assumption in equation
(13) is that genetic distance Db is small relative to the
number of loci L. In the general case, ∆ Db = 2v(1 −
2Db/L )R and Db approachesL/ 2 asymptotically.
At what moment can the two diverging population be
considered as two diﬀerent species? The answer obviously
depends on what one means by a species. Let us say that
the two populations are diﬀerent species if the proportion,
wb, of encounters between individuals from diﬀerent pop-
ulations that can result in mating and viable and fertile
oﬀspring is less than a small number γ. (This deﬁnition
uses the biological species concept.) During initial stages
of divergence, this proportion can be approximated by
the right-hand side of equation (12) with Db taking the
place of Dw. Figure 5 shows the minimum genetic dis-
tance between populations required for speciation as a
function of K for several values of γ. One can see that a
0 10 20 30 40 50
K
0.0
0.2
0.4
0.6
0.8
1.0
R
N=100
N=200
N=400
N=800
Figure 4: The rate of difergence R relative to the neu-
tral case in an isolated population of size N as a function of
K for v = . 0384. The circles, squares, diamonds and tri-
angles give estimates from individual-based simulations f or
N = 100 , 200, 400 and 800, respectively (thirty runs for each
parameter conﬁguration).
genetic distance between the populations on the order of
2 or 3 times K will be suﬃcient for the status of separate
“biological” species. Note that there is very little eﬀect
of the magnitude of γ.
Speciation in a subdivided population . In the determin-
istic limit (that is with NT → ∞ ), the genetic variation
of a subdivided population can be maintained by migra-
tion. This can happen if initially alternative alleles are
close to ﬁxation in diﬀerent subpopulations and selection
is suﬃciently strong relative to migration (e.g., Karlin
and McGregor 1972). Let k be the number of loci at
which alternative alleles are close to ﬁxation in diﬀerent
subpopulations. Respectively, L − k will be the number
of loci at which the same allele is close to ﬁxation in dif-
ferent subpopulations. In the deterministic limit, k does
not change. In the n-island model the dynamics of Dw
and Db are described by equations
∆ Dw = −sDw + 2v + 2me(Db − Dw), (14a)
∆ Db = −s(Db − k) + 2v + 2me
n − 1 (Dw − Db), (14b)
where s is deﬁned by equation (11), and the “eﬀective”
migration rate
me =m wb
ww
(15)
if k ≤ K and me = 0 otherwise. Here, ww is given by
equation (12) above whereas wb = Γ( K + 1 − k,D b −

0 10 20 30 40 50
K
80Minimum distance for speciation
Figure 5: Minimum genetic distance between populations
for speciation for γ = . 0001, . 001 and . 01 (solid lines from top
to bottom). Also shown is the diagonal D = K (dashed line)
k)/ Γ(K + 1− k) is the probability that two randomly cho-
sen individuals from diﬀerent populations are not repro-
ductively isolated. The eﬀective migration rate me can
be thought of as half the probability of mating between
individuals from diﬀerent subpopulations. With no re-
productive isolation (with very large K) or no genetic di-
vergence between subpopulations (with Db ≈ Dw,k = 0),
the eﬀective migration rate is equal to the actual migra-
tion rate (me =m). Comparing equations (14) with their
neutral analogs (8) shows that reproductive isolation re-
sults in two eﬀects. First, it directly reduces genetic vari-
ation within subpopulations and genetic divergence be-
tween subpopulations. These eﬀects are described by the
ﬁrst terms in the right-hand side of equations (14). Also,
reproductive isolation reduces the gene ﬂow between pop-
ulations. Given that Db ≥ Dw, then me ≤ m reﬂecting
the fact that genes brought by migrants have a smaller
probability of being incorporated in the resident popula-
tion. In the deterministic limit, both Dw and Db always
evolve to ﬁnite equilibrium values.
Random genetic drift results in two eﬀects. First, it re-
duces the genetic variation within sub-populations by the
amountDw/N . The dynamic equation for Dw becomes
∆ Dw = −sDw + 2v + 2me(Db − Dw) − Dw
N . (16)
Second, genetic drift might change k. The expected
change in k can be approximated as
∆ k = 2vR2− Nm e − 2kmeR(e/ 2)Nm e, (17)
where R is given by equation (13b). The ﬁrst term in
the right-hand side of equation (17) can be thought of as
the rate at which an allele that is initially rare in both
sub-populations becomes close to ﬁxation in one of the
subpopulations. This rate was found by Lande (1979)
using a diﬀusion approximation and assuming that mi-
gration is weak. The second term is the rate at which
the loci with diﬀerent alleles initially close to ﬁxation in
diﬀerent subpopulations become ﬁxed for the same allele
in both of them. To ﬁnd this term I used Barton and
Rouhani’s (1987) method.
Depending on parameter values and initial conditions
there are two diﬀerent dynamic regimes. In the ﬁrst
regime, both Dw and Db evolve to ﬁnite values, which
are smaller than those in the neutral case (and which are
much smaller than the number of loci L). Here, selec-
tion (reproductive isolation) reduces genetic divergence
both within and between subpopulations. In the sec-
ond regime, Dw stays small (relative to L) whereas Db
increases eﬀectively indeﬁnitely (to values order L/ 2).
Here, selection (reproductive isolation) reduces the ef-
fective migration rate to zero resulting in speciation.
These dynamics can be understood in the following way.
Changes in Dw andDb induced by selection are expected
to happen on a faster time scale than changes in k in-
duced by random genetic drift. Thus, Dw and Db values
should be close to the equilibrium values predicted by
equations (14b,16) when k is treated as a constant. The
dynamic behavior depends on whether k reaches a ﬁnite
equilibrium value or keeps increasing. In the latter case,
the eﬀective migration rate me reduces to zero, the rate
of change of k approaches 2vR withDb increasing at the
same rate (cf. equation 13b).
Figure 6 illustrates the dynamics observed by numer-
ically iterating the model equations. The iterations
started with all N individuals identical. During the ﬁrst
1000 generations there were no restrictions on migra-
tion between subpopulations and the whole population
evolved as a single randomly mating unit (cf. Gavrilets
et al. 1998). The average genetic distance within the
populationDw evolves according to equation (10). Start-
ing with generation 1000, restrictions on migration were
introduced and the dynamics are described by equations
(14b-17) afterwards. After generation 1000, each of these
ﬁgures has two sets of three curves corresponding to two
diﬀerent values of the parameter(s) under consideration.
The curves within each set represent Db,D w andk. With
migration rate m = 0. 01, all these variables evolve to-
wards ﬁnite equilibrium values (see Fig.6a) whereas with
a smaller migration rate ( m = 0. 001), Db and k increase
eﬀectively indeﬁnitely signifying that speciation has taken
place. Thus, reducing migration makes speciation more
plausible. Figure 6b shows that increasing mutation rate
(from v = 0. 0384 to 5 times this value) has a similar ef-

fect. These two ﬁgures describe the dynamics expected
in a system of two subpopulations. Figure 6c compares
the dynamics observed in a population subdivided into 2
and 4 subpopulations. This ﬁgure shows that increasing
population subdivision makes speciation more plausible.
Note that the process of genetic divergence described in
Fig.6c results in a simultaneous emergence of 4 species.
In the cases where speciation takes place (as signiﬁed by
continuous increase in the genetic distance between sub-
populations), the curves representing Db and k are par-
allel meaning that asymptotically the genetic divergence
is due to ﬁxation of diﬀerent mutations in diﬀerent sub-
populations. In the cases where speciation does not take
place, k is close to zero.
Altogether, at the qualitative level the results presented
in Fig.6 correspond to both biological intuition and the
results of individual-based simulations in Gavrilets et al.
(1998). At the quantitative level, there is a very good
ﬁt between simulations and analytical predictions for lev-
els of genetic variation maintained in subpopulations and
the asymptotic rate of divergence between subpopula-
tions. However, the conditions for speciation as predicted
by iterating equations (14b-17) appear to be more strict
than those observed in the individual-based simulations
performed by Gavrilets et al. (1998). For example, for
parameter values used in Figure 6c, no speciation in a
system of four subpopulations occurs if m >0. 0035. In
contrast, in individual-based simulations speciation was
observed form = 0. 01 (Figure 3b in Gavrilets et al. 1998).
The main reason for this disperancy seems to be an inad-
equacy of equation (17) at moderate levels of migration
(e.g. Lande 1979; Barton and Rouhani 1987).
Speciation in a peripheral population . Here I consider
the case of a peripheral population of size N receiving
migrants from a very large main population. The dynam-
ics of the average genetic distance within the peripheral
population,Dw, the average genetic distance between the
peripheral and main populations, Db, and the number of
diverged loci k are approximated by equations
∆ Dw = − sDw + 2v + 2me(Db − Dw) − Dw
N , (18a)
∆ Db = − s
2 (Db − k) +v +me(D0 − Db), (18b)
∆ k =vR2− Nm e − kmeR(e/ 2)Nm e. (18c)
Here D0 is the average genetic distance within the main
population. Figure 7 illustrates the dynamics observed
by numerically iterating equations (18). For D0 I used
mutation-selection balance values for a very large isolated
population predicted by equations (10, 11). The initial
values of Dw and Db were equal to D0. The parameter
values in Fig.7a and Fig.7b are the same as those that
resulted in speciation in Fig.6a and 6b, respectively. The
outcome of the dynamics is the same - speciation - but the
rate of divergence is smaller than when all subpopulations
are uniformly small. This is apparent from the level of
genetic distance between subpopulations achieved after
1000 generations of divergence which are about twice as
small in Figures 7a and 7b relative to those in Figures 6a
and 6b.
Discussion
The theory developed above together with earlier numer-
ical simulations (see references above) show that rapid
speciation is a plausible outcome of the evolutionary dy-
namics in subdivided populations. Here speciation is a
consequence of two fundamental factors. The ﬁrst fac-
tor is the existence of various and possibly signiﬁcantly
diﬀerent highly-ﬁt combinations of genes underlying di-
verse solutions (genetical, ecological, behavioral, devel-
opmental etc.) to the problem of survival and reproduc-
tion. In multidimensional genotype space these combina-
tions of genes tend to form connected clusters that extend
throughout genotype space. At the same time these geno-
types are not mutually compatible - they are separated
by “holes”. The second factor is mutation pressure. Be-
cause the population size is ﬁnite and the number of loci
is very large whereas the probability of a speciﬁc mu-
tation is very small, diﬀerent mutations tend to appear
(and increase in frequency) in diﬀerent subpopulations
(cf., Barton 1989; Mani and Clarke 1990). Metaphori-
cally speaking, mutation tends to tear apart the cloud
of points representing the population in genotype space.
Combining genes from two diﬀerent organisms in one oﬀ-
spring can counteract the disruptive eﬀect of mutation,
keeping a single randomly mating population together in
genotype space. But restricting gene exchange as a conse-
quence of limited migration between subpopulations gives
mutation a signiﬁcant advantage. Eventually the popula-
tion cloud will be broken and smaller clouds representing
the subpopulations will drift apart in genotype space -
an event representing speciation. Given suﬃcient genetic
divergence, restoring migration to high levels will not re-
turn the system back to the state of free gene exchange
between subpopulations which now can be considered as
diﬀerent species. It is not necessary to invoke strong selec-
tion for local adaptation to explain speciation in a subdi-
vided population, as studied here, or after a founder event
(Gavrilets and Hastings 1996; Gavrilets and Boake 1998).
Mutation is ubiquitous. Population size is never inﬁnite
and, thus, genetic drift is always present. Speciation as
caused by mutation and random drift should represent
a null model against which speciation as caused by local
adaptation can be tested (cf., Nei 1976; Lande 1976).

Unlike most previous models that concentrate only on
some stages of speciation, the model studied here de-
scribes the complete process of speciation from initiation
until completion. I assumed that reproductive isolation
is caused by cumulative genetic change. The model is
described in terms of dynamic equations for the vari-
ables analogous to those used in molecular evolution-
ary biology - the average genetic distances between and
within subpopulations. Average genetic distances within
(sub)populations always evolve towards ﬁnite equilibrium
values. Depending on parameter values and initial con-
ditions average genetic distances between subpopulations
either converge to a ﬁnite equilibrium or increase eﬀec-
tively indeﬁnitely. The former regime is interpreted as no
speciation. In the latter regime, three eﬀects take place
simultaneously: (1) genetic distances between subpopula-
tions signiﬁcantly exceed genetic distances within them,
(2) encounters between individuals from diﬀerent sub-
populations do not result in viable and fertile oﬀspring,
(3) evolutionary changes in a subpopulation do not aﬀect
other subpopulations. Thus, subpopulations form sepa-
rate genotypic clusters in genotype space, become repro-
ductively isolated and undertake changes as evolutionary
independent units. This regime is interpreted as specia-
tion according to any of the species concept common in
the literature (e.g., Mallet 1995; Claridge et al. 1997).
The dynamic equations derived above describe the ex-
pected changes in the average genetic distances neglect-
ing stochastic ﬂuctuations around the expected values.
The predicted dynamics have two clearly distinct regimes:
convergence towards a ﬁnite equilibrium (no speciation)
or eﬀectivly indeﬁnite divergence (speciation). Stochas-
tic ﬂuctuations around the expected values, which are
present in natural populations (and individual-based sim-
ulations), make the boundary between these two regimes
less strict and may result in the population escaping the
ﬁrst regime and entering the second regime after some
time (see Gavrilets et al. 1998). My analysis has been
based on approximations which are standard in study-
ing multilocus systems. I assumed that alleles are rare,
that linkage disequilibrium, mutation and migration rates
are small and used a theory developed by Lande (1979),
Walsh (1982) and Barton and Rouhani (1987) for de-
scribing stochastic transitions driven by random genetic
drift. The analytic theory presented here ﬁts satisfacto-
rily with the results of individual-based simulations. The
model can be used to evaluate qualitative eﬀects of dif-
ferent factors on the dynamics of speciation, the order of
magnitude of parameters resulting in or preventing spe-
ciation, and the time scale involved. According to both
biological intuition and previous numerical simulations,
increasing mutation rate and decreasing migration pro-
mote speciation. Increasing the number of loci has sig-
niﬁcantly increased the plausibility of speciation relative
to that in earlier models (Nei et al. 1983; Wagner et al.
1995; Gavrilets and Hastings 1996). Note that the actual
number of loci inﬂuences the dynamics only through the
mutation rate per gamete, v, and parameter K. For re-
alistic parameter values the time scale for speciation can
be as short as a few thousands or even hundreds of gener-
ations. This is compatible with rates observed in several
cases of rapid speciation in natural populations described
recently (Schluter and McPhail 1992; Yampolsky et al.
1994; Johnson et al. 1996; McCune 1996, 1997) includ-
ing the most spectacular case - the origin of hundreds of
species of Lake Victoria cichlids in 12,000 years (Johnson
et al. 1996). The model has demonstrated the plausibility
of speciation with relatively low levels of both initial ge-
netic variation and new genetic variation introduced into
the population each generation (both supplied by muta-
tion). With higher levels of the former (as in laboratory
experiments on speciation, reviewed by Rice and Hostert
1993, and Templeton, 1996) or of the latter (for instance
as a result of natural hybridization, reviewed by Bullini
1994, Rieseberg 1995, Arnold 1997), the rate of speciation
is expected to be even higher.
Local adaptation and speciation
The model analyzed above shows that rapid speciation in
a subdivided population can occur even without any dif-
ferences between selection regimes operating in diﬀerent
subpopulations (that is without selection for local adap-
tation). An important question is how genetic changes
brought about by selection for local adaptation would af-
fect the dynamics of speciation (e.g., del Solar 1966; Ayala
et al. 1974; Kilias et al. 1980; Dodd 1989; Schluter 1996;
Givnish and Sytsma 1997). These eﬀects will depend on
whether the genes responsible for local adaptation are
diﬀerent from or are the same as the genes underlying
reproductive isolation.
Assume ﬁrst that the two sets of genes are completely
diﬀerent. Let the strength of selection per locus induced
by reproductive isolation be very small so that these loci
can be considered as eﬀectively neutral. (For the model
studied here, this seems to be the case if K is larger that
2-3 times θ, where θ is the average genetic distance main-
tained by mutation in a ﬁnite population in the neutral
case.) Then, Birky and Walsh’s (1988) results tell us
that the rate of substitution in these loci will not be af-
fected by selection on other loci independently of linkage.
However, given that reproductive isolation is a result of
genetic incompatibilities, the loci underlying reproduc-
tive isolation will be under frequency-dependent selection
against rare alleles which is analogous to underdominant
selection in diploid populations. Birky and Walsh (1988)
have shown that linkage to advantageous alleles slightly
increases the rate of ﬁxation of detrimental mutations.

This suggests that selection on linked loci will increase
the rate of substitutions in the loci underlying reproduc-
tive isolation and, thus, will promote speciation to some
degree. No results seem to be known on how linkage to
advantageous alleles increases the rate of ﬁxation of un-
derdominant mutations or alleles experiencing frequency-
dependent selection. No quantitative predictions can be
made here, but most likely if the two sets of loci are not
extremely tightly linked, eﬀects of selection for local adap-
tation on the rate of speciation will not be signiﬁcant.
Assume now that the loci under consideration
pleiotropically aﬀect both survival in a given environment
and reproductive isolation. For instance, this may be
the case if disruptive selection acts on habitat preferences
which also deﬁne mating patterns (e.g., Rice 1984; Rice
and Salt 1988) or if the probability of mating between
individuals depends on the diﬀerence in their morpho-
logical traits which are under direct selection. Let sLA
be the average strength of selection per locus induced by
selection for local adaptation. Using Walsh’s (1982) re-
sults, the relative rate of ﬁxation of new mutations in an
isolated population of size N can be approximated (see
Appendix) as
R = 4e− S(1− α )2√
S
√ π
[
erf (
√
S(1 +α )) +erf (
√
S(1 − α ))
] , (19)
where S = Ns/ 2, s is the strength of selection per lo-
cus induced by reproductive isolation, and α = sLA/s .
With α = 0, equation (19) reduces to (13b). Figure 8
illustrates the dependence of R on S and α . Increasing
α always increases R. Thus, selection for local adapta-
tion always increases the rate of substitutions and pro-
motes speciation. With suﬃciently strong selection for
local adaptation ( sLA > s), the net eﬀect of new alleles
will be advantageous and their frequencies will tend to
increase even when rare. In the limit of large popula-
tion size, the probability of ﬁxation is 2( sLA − s). This
is analogous to the classical results on the probability of
survival of an advantageous mutant in a very large pop-
ulation (Haldane 1927; Walsh 1982). The rate of accu-
mulation of genetic diﬀerences will be 2( sLA − s)Nv and
can be signiﬁcant. Very strong artiﬁcial selection for local
adaptation has been shown to result in rapid evolution of
reproductive isolation (e.g., del Solar 1966; Kilias et al.
1980; Dodd 1989). However, the changes brought about
by moderately strong artiﬁcial selection may not exceed
those resulting from random genetic drift only (e.g. Ringo
et al. 1985).
Population subdivision and speciation
In the models considered here, speciation is a by-product
of ﬁxation of diﬀerent alleles in diﬀerent subpopulations.
It is well known that the rate of ﬁxation of neutral alleles
does not depend on population size, that of advantageous
alleles increases with population size, and that of delete-
rious or underdominant alleles decreases with population
size (e.g., Gillespie 1991; Ohta 1992). At the level of
individual loci, selection induced by reproductive isola-
tion in the form considered here is similar to underdom-
inant selection (or frequency-dependent selection against
rare alleles). Thus, in the absence of selection for local
adaptation (or with independent loci controlling traits
for local adaptation) decreasing population size will in-
crease the rate of substitutions and promote speciation
(see equation 13b). Eﬀects of the population size on the
plausibility of speciation will be similar even if the same
loci control both reproductive isolation and locally ben-
eﬁcial traits given that selection for local adaptation is
not too strong ( sLA < s, see equation 19 and Fig.9). In
all these cases, speciation will be driven by mutation and
random genetic drift and will be fastest if the popula-
tion is subdivided into small subpopulations. This con-
clusion about the eﬀect of population subdivision on the
probability of speciation in Dobzhansky-type models dif-
fers from that of Orr and Orr (1996). They argued that
the degree of population subdivision has no eﬀect on the
rate of speciation if speciation is caused by mutation and
random drift. Orr and Orr did not consider the actual
process of ﬁxation of new mutations assuming that it
will be a simple neutral process. However, the existence
of holes in the adaptive landscape makes the process of
substitution non-neutral and new mutations are selected
against when rare. Such mutations are ﬁxed more easily
in smaller subpopulations. For the discussion of the ex-
isting experimental evidence regarding eﬀects of random
genetic drift on the plausibility of speciation see Rice and
Hostert (1993) and Templeton (1996). The time scale
for speciation is short meaning that restrictions on mi-
gration between subpopulations do not need to be long
lasting; several hundreds of generations may be suﬃcient
for a signiﬁcant divergence and evolution of reproductive
isolation. It is quite possible that several new species
will emerge from a highly subdivided population within
a short period of time (see above; Gavrilets et al. 1998).
These theoretical conclusions are consistent with a verbal
“micro-allopatric” model of speciation suggested for cich-
lid ﬁshes in the East African Great Lakes (e.g., Reinthal
and Meyer, 1997). Hoelzer and Melnick (1994) have em-
phasized that the possibility of simultaneous emergence
of several new species should be incorporated more ex-
plicitly in the contemporary methods for reconstructing
phylogenies.
If the same loci control both reproductive isolation and
locally beneﬁcial traits and selection for local adaptation
is suﬃciently strong ( sLA >s ), increasing the population
size will result in increasing the rate of substitutions (see

Fig.9). In this case, speciation will be driven by selection
and will be fastest if the population is subdivided into
a small number (say, two) of large subpopulations (Orr
and Orr 1996) as implied by the vicariance scenario (e.g.
Wiley 1988).
Many species are thought to be represented by a few
large populations and many smaller “peripheral” popula-
tions. Mayr (1963, 1982b) proposed the theory of peri-
patric speciation arguing that speciation is typically ini-
tiated in small peripheral populations and he attributed
a special role to genetic drift in this process. Gavrilets
(1996) has shown that an invasion of a new adaptive com-
bination of genes is most successful if it is initiated in a
peripheral population. The results presented here bear
out Mayr’s argument(see Fig. 8). Small peripheral pop-
ulations will rapidly diverge genetically from the “main”
large population and speciate. Although diﬀerences in se-
lection regimes between peripheral and main populations
can accelerate divergence, random genetic drift will be the
most important factor. On the other hand, if a periph-
eral population is large enough and is under a selection
regime that is suﬃciently diﬀerent from the one operating
in “main” populations, then disruption of gene ﬂow can
cause evolutionary divergence, perhaps leading to rapid
speciation, in the absence of contributions from random
genetic drift (Garcia-Ramos and Kirkpatrick 1997).
Summarizing, large randomly mating populations will
diverge genetically and speciate only if there is strong se-
lection for local adaptation (for instance after a change
in the environment). In contrast, small populations will
diverge and speciate even without diﬀerences in selec-
tion regimes between them. Possibilities for speciation
strongly depend on the geographic structure of the pop-
ulation. Here, analysis was restricted to the island model
and the continent-island model. Manzo and Peliti (1994)
and Gavrilets et al. (1998) present numerical results for
stepping-stone models.
Relationship to other speciation models
Using genetic distance (1) implies the equivalence of loci.
A general case of non-equivalent loci can be described by
introducing a ( L × L) matrix G = {Gij} of weights and
deﬁning a generalized distance between individuals α and
β as
dαβ = (lα − lβ )TG(lα − lβ ), (20)
where lα and lβ are vectors deﬁning the corresponding
genotypes and superscript T means transpose. Consid-
ering haploid populations and premating isolation only,
the model assumes that individuals can mate only if they
are not too diﬀerent genetically. Here, the degree of
reproductive isolation was controlled by cumulative ge-
netic diﬀerence. However, using the generalized distance
(20) allows one to treat models for reproductive isola-
tion controlled by quantitative traits as well as models
for sexual selection within the same framework (see Ap-
pendix). The close relationship between the models of
speciation as a consequence of “quasi-neutral” divergence
along ridges in the adaptive landscapes and as a conse-
quence of sexual selection was already recognized by Bar-
ton and Charlesworth (1984).
A fundamental reason for speciation on a holey adap-
tive landscape is mutation which tends to break the pop-
ulation into reproductively isolated pieces. Population
subdivision and the resulting reduction in gene exchange
facilitates this process. Here, migration rates compati-
ble with rapid speciation were small (that is speciation
was allopatric or parapatric). An interesting question is
whether speciation is possible with much higher migration
rates. In other words, is sympatric speciation by muta-
tion and random genetic drift on a holey adaptive land-
scape possible? Numerical simulations of similar models
of sympatric speciation where mutation rates were higher
(Higgs and Derrida 1991, 1992) or the time span studied
was longer (Wu 1985) or the population size was smaller
(Gavrilets and Boake 1998) than here, provide an aﬃr-
mative answer. Adding disruptive selection due to either
abiotic factors (say, diﬀerent resources) or biotic factors
(competition) should create additional pressure on the
population cloud which might result in rapid sympatric
speciation.
Beyond holey landscapes
Gavrilets and Gravner’s (1997) results have suggested
that clusters of well-ﬁt genotypes that extend through-
out genotype space are plausible. If this is so, biological
populations are expected to evolve mainly within these
clusters and consist most of the time of well-ﬁt geno-
types with ﬁtnesses within some band. The metaphor
of “holey” adaptive landscapes neglects the ﬁtness diﬀer-
ences between genotypes in the cluster but these diﬀer-
ences are supposed to exist and should be apparent on a
ﬁner scale. If one applies a ﬁner resolution, the movement
along the cluster will be accompanied by slight increases
or decreases in ﬁtness. Evolution will proceed by ﬁxa-
tion of weakly selected alleles which can be advantageous,
deleterious, over- and underdominant, or apparently neu-
tral depending on the speciﬁc area of genotype space the
population passes through. Smaller populations will pass
faster through the areas of genotype space correspond-
ing to ﬁxation of slightly deleterious mutations whereas
larger populations will pass faster through the areas cor-
responding to ﬁxation of (compensatory) slightly advan-
tageous mutations. This pattern of molecular evolution,
as predicted from the general properties of multidimen-
sional adaptive landscapes, is similar to the patterns re-
vealed by the methods of experimental molecular biology,

which form the empirical basis for the nearly neutral the-
ory of molecular evolution (Ohta 1992). From general
considerations, one should not expect complete symme-
try of “real” adaptive landscapes which are supposed to
have areas varying with respect to the “width” and con-
centration of ridges of well-ﬁt genotypes. Sexual popula-
tions are expected to spend more time in areas of high
concentration of well-ﬁt genotypes (Peliti and Bastolla
1994). One of the biological manifestations of this eﬀect
will be apparent reduction in the probability of harmful
mutations, that is, evolution of genetic canalization (cf.,
Wagner 1996). The metaphor of holey adaptive land-
scapes may be useful for thinking about these and other
evolutionary problems.
Acknowledgments.
I am grateful to Chris Boake, Marc Camara, Mitch
Cruzan, Nick Barton and G¨ unter Wagner for very help-
ful comments and suggestions. Nick Barton and G¨ unter
Wagner forced me to extend the generality of the math-
ematical approximations developed here. Collaboration
with Hai Li and Michael Vose was crucial in developing
a computer program used to obtain results of individual-
based simulations presented in Figures 3 and 4. This
work was partially supported by grants from Universit´ e
P. et M. Curie and ´Ecole Normale Sup´ erieur, Paris, and
by National Institutes of Health grant GM56693.
Appendix
Eﬀects of mutation, migration and drift on the dynam-
ics of the average genetic distances within and between
subpopulations have been previously studied thoroughly
(e.g., Watterson 1975; Li 1976; Slatkin 1987; Strobeck
1987). What is left is to add reproductive isolation (that
is selection) to the model. I will use the deterministic
framework assuming that the population size N → ∞ .
The distribution of Dw under rare-alleles and linkage
equilibrium approximation. I will use the standard no-
tations Ai and ai for alternative alleles at the i-locus
(i = 1,...,L ). Let pi be the frequency of allele Ai at the
i-th locus, qi = 1 − pi, and ψw,i = 2piqi. Variable ψw,i can
be thought of as the probability that two randomly cho-
sen individuals (sequences) from the same subpopulation
are diﬀerent at the i-th locus. Let dw,i = (lα
i − lβ
i )2 be the
genetic distance at the i-th locus between two randomly
chosen individuals α and β . Note that dw,i = 1 with
probability ψw,i and dw,i = 0 with probability 1 − ψw,i .
Because dw,i is a Binomial random variable, its gener-
ating function is γdw,i(s) = ψw,is + 1 − ψw,i , which can
be approximated as eψ w,i(s− 1) if ψw,i << 1 (rare-alleles
approximation). Under approximate linkage equilibrium,
the generating function of dw = ∑ dw,i is
γdw (s) = Π ieψ w,i(s− 1) =e
∑
iψ w,i(s− 1) =eDw (s− 1) (A1)
where Dw = ∑
iψw,i . This shows that random variable
dw has approximately Poisson distribution with parame-
ter Dw and, thus,
P (dw =i) = exp(−Dw)Di
w
i! . (A2)
Selection within an isolated population . Let w(d) be
the expected number of fertile and viable oﬀspring that
can be produced by a pair of individuals diﬀerent in d
loci. The average ﬁtness of the population is
w =
∑
j
w(j)P (d =j).
The dynamics of the general model of fertility selection
and premating isolation in a haploid population consid-
ered here are identical to that of a symmetric viability
selection model for a diploid population with viabilities
w(d) depending on the number of heterozygous loci d.
Under approximate linkage equilibrium, changes in allele
frequencies are described by Wright’s equation
∆ spi = piqi
∂ lnw
∂pi
. (A3)
(Wright 1969). Using the equalities ∂ lnw/∂p i = 2(qi −
pi)∂ lnw/∂ψ i and Dw = ∑
iψi, equation (A3) can be
rewritten as
∆ spi =spiqi(pi − qi), (A4a)
with
s = d lnw
dDw
. (A4b)
To describe the dynamics of allele frequencies one needs
to know the mean ﬁtness of the population.
Truncation selection. This is a selection scheme ana-
lyzed in detail in the main body of the paper. Here
w(d) =
{
1 for d ≤ K,
0 for d>K. (A5)
Using the Poisson approximation (A1), the mean ﬁtness
is
wthreshold =
K∑
i=0
exp(−Dw)Di
w
i! = Γ(K + 1,D w)
Γ(K + 1) ,
where the last equality follows from equation (8.352) in
Gradshteyn and Ryzhik (1994), resulting in s given by
equation (11). To ﬁnd equation (10), one starts with
(A4a) and proceeds using the fact that ∆ ψi ≈ 2(qi −

pi)∆ pi and that Dw = ∑ ψi.
Other selection schemes can be considered in a simi-
lar way, and some of them result in relatively compact
expressions for
w and s.
Linear selection. Here
w(d) =
{ 1 − ad for d ≤ K,
0 for d>K. (A6)
The mean ﬁtness is
wlinear =wthreshold − aDwΓ(K,D w)
Γ(K) .
Quadratic selection. Here
w(d) =
{
1 − ad − bd2 for d ≤ K
0 for d>K (A7)
The mean ﬁtness is
w =wlinear − b [Dw(Dw − K)
+ Dw(K + 1)Γ(K,D w)
Γ(K)
−exp(−Dw)DK+2
w H(2,K + 2,D w)
Γ(K + 2)
]
,
whereH is the hypergeometric function (Gradshteyn and
Ryzhik 1994).
Exponential selection. Here
w(d) =
{ exp(−ad) for d ≤ K,
0 for d>K. (A8)
The mean ﬁtness is
w =exp(−Dw(1 − e− a)) Γ(K + 1,D we− a)
Γ(K + 1) .
I have not explored how assuming these selection schemes
would aﬀect the outcome of the dynamics.
Stochastic transitions in an isolated population .
Adding mutation results in equation
∆ pi =spiqi(pi − qi) +µ (qi − pi), (A9)
whereµ is the rate of mutation (assumed to be equal for
forward and backward mutations). Equation (A9) is sim-
ilar to the classical equation describing underdominant
selection on a single locus in a diploid population. This
allows one to use Lande’s results (1979; see also Hedrick
1981; Walsh 1982; Barton and Rouhani 1987) to ﬁnd the
rate of stochastic divergence. This rate is twice the ex-
pected number, vN , of new mutations in a population
times the probability that a given one will be ﬁxed, U .
Using the diﬀusion approximation, U is deﬁned by equa-
tions (1a) and (2) in (Lande 1979). Lande used some
approximations to evaluate U . However, the integrals in
his equation (1a) can be found exactly resulting in
U = 1

1 −
erf
[ √
S(1 − 2
N )
]
erf
[ √
S
]

, (A10)
where S = Ns/ 2 (Walsh 1983). Expanding in a Taylor
series under the assumption that 1 /N << 1 results in
(13b) which is equivalent to Lande’s (1979) formula. The
diﬀerence between Lande’s approximate formula and the
exact equation (A10) is negligible.
The distribution of Db under rare-alleles and linkage
equilibrium approximation. Let us consider two subpop-
ulations. Let pi and Pi be the frequencies of allele Ai
in the ﬁrst and second subpopulations, respectively. The
genetic distance db,i at the i-th locus between two ran-
domly chosen sequences from two diﬀerent subpopula-
tions is a Binomial variable taking values 1 and 0 with
probabilities ψb,i =piQi +qiPi and 1 − ψb,i , respectively
(qi = 1 − pi,Q i = 1 − Pi). I will assume that genetic varia-
tion within each subpopulation is low so that ψb,i is close
to either 0 or 1. Let δi =db,i ifψb,i ≈ 0 and δi = 1 − db,i if
ψb,i ≈ 1. The genetic distance between individuals α and
β can be represented as db =k − ∑
1δi + ∑
2δi where the
ﬁrst sum is over k loci at which ψb,i ≈ 1 and the second
sum is over L − k loci at which ψb,i ≈ 0. Using the as-
sumption of linkage equilibrium, the generating function
of db becomes
γdb(s) =E{sk−
∑
1δi+
∑
2δi }
=skΠk
i=1e− φ i(s− 1)ΠL
i=k+1eφ i(s− 1)
= sk
ek(s− 1)eDb(s− 1),
whereφi is the expectation of δi andDb is the expectation
of db. Using the properties of generating functions, the
distribution of db is
P (db =i) =
{
0 if i<k,
(Db− k)i− k
(i− k)! e− (Db− k) if i ≥ k. (A11)
With ﬁtness function (A5), the probability that two ran-
domly chosen individuals from diﬀerent subpopulations
are not reproductively isolated is
wb =
K∑
i=0
P (db =i) = Γ(K − k + 1,D b − k)
Γ(K − k + 1) , (A12)
if k ≤ G and wb = 0 if k>K .
Deterministic dynamics in a subdivided population .
With no reproductive isolation and with equal forward

and backward migration rates and equal population sizes,
the change in pi due to migration is ∆ mpi =m(Pi − pi).
The corresponding change in Dw is ∆ mDw = 2m(Db −
Dw). With reproductive isolation and given that Db ≥
Dw, individuals migrating from other subpopulations will
have reduced probability of mating. Let
ww and wb be
the expected numbers of fertile and viable oﬀspring that
can be produced as a result of within and between sub-
populations encounters. For simplicity I will omit the in-
dex specifying the locus under consideration. With equal
population sizes and migration rates, the change in the
allele frequency due to migration becomes
∆ mp =me(P − p), (A13a)
where the “eﬀective” migration rate is
me =m
wb
ww
. (A13b)
(compare with the models with migration between popu-
lations of unequal size where the eﬀective migration rate
ism times the ratio of the population sizes, e.g. Gavrilets
1996). The corresponding change in Dw is
∆ mDw = 2me(Db − Dw). (A13c)
Changes ∆ mpi can be thought of as changes in allele fre-
quencies brought about by selection between groups of in-
dividuals (migrants and residents) whereas the ﬁrst term
in the right-hand side of equation (A9) can be thought of
as the change in p brought about by individual selection.
The dynamics of allele frequencies at a speciﬁc locus
under the joint action of selection, mutation and migra-
tion are described by
∆ p =spq(p − q) +me(P − p) +µ (q − p), (A14a)
∆ P = ˜sPQ (P − Q) +me(p − P ) +µ (Q − P ).(A14b)
With s = ˜s = const and me = const and me < s/ 6,
dynamic system (A14) has two types of stable equilibria:
mutation-selection balance equilibria with p ≈ P, pq ≈
µ/s, ψ w ≈ 2µ/s,ψ b ≈ 2µ/s and migration-selection equi-
libria with p ≈ Q, pq ≈ µ/s + m/s, ψ w ≈ 2µ/s +
2m/s,ψ b ≈ 1 − 2µ/s − 2m/s . I assume that in the deter-
ministic limit, k out of L loci evolve towards migration-
selection balance equilibria whereas the remaining L − k
loci evolve towards mutation-selection balance equilibria.
In the latter L − k loci, the dynamics of ψw and ψb are
approximated by equations
∆ ψw = −sψw + 2µ + 2me(ψb − ψw), (A15a)
∆ ψb = −sψb + 2µ + 2me(ψw − ψb). (A15b)
In the former k loci, the dynamics of ψw are described as
before by (A15a) whereas the dynamics of ψb are approx-
imated by equation
∆ ψb =s(1 − ψb) − 2µ + 2me(ψw − ψb). (A16)
Selection always reduces ψw whereas mutation always
increases it (see equation A15a). Selection and muta-
tion have the same eﬀects on ψb for the loci evolving
toward mutation-selection balance equilibria (see equa-
tion A15b). However, for the loci evolving toward
migration-selection balance equilibria, selection increases
ψb whereas mutation decreases it (see equation A16).
Summing up over all loci, one ﬁnds equations (14) of the
main text. Equations (18) are derived in a similar way as-
suming that the allele frequencies in the main population
do not change.
Stochastic transitions in a subdivided population . In
a subdivided population, migration tends to reduce ge-
netic diﬀerentiation. Given that migration is suﬃciently
strong relative to selection, the same allele will be close
to ﬁxation in both subpopulations. If, by a chance, an
alternative allele approaches ﬁxation in one of the sub-
populations creating signiﬁcant diﬀerentiation at a given
locus, such diﬀerentiation will be quickly lost. The num-
ber of loci k at which alternative alleles are close to ﬁx-
ation in diﬀerent subpopulations will be close to zero on
average. However, if migration is relatively weak, then
the diﬀerentiation created by random genetic drift will
not be lost quickly and actually can even accumulate.
Let us consider a locus at which initially the same allele
is close to ﬁxation in both subpopulations (that is both
p ≈ 0 and P ≈ 0). Neglecting the changes in P , the de-
terministic change in p due to selection and migration is
approximately
∆ p =spq(p − q) − mep. (A17)
Lande (1979; see also Barton and Rouhani 1987) has
shown that the rate at which allele A becomes close to
ﬁxation in the ﬁrst subpopulation while its frequency is
about zero in the second population is approximately
2− Nm e times the rate of ﬁxation in the absence of im-
migration. Assuming that alleles A are brought about
by mutation at rate µ and summing up over L − k loci,
one ﬁnds the ﬁrst term in the right-hand side of equation
(17). Once alternative alleles are close to ﬁxation in dif-
ferent subpopulations, random drift can remove genetic
diﬀerentiation. Let us consider a locus at which initially
p ≈ 0 but P ≈ 1. Neglecting the changes in P , the deter-
ministic change in p due to selection and immigration is
approximately
∆ p =spq(p − q) − me(1 − p). (A18)

Using Barton and Rouhani’s (1987) method one ﬁnds that
the rate at which allele A becomes close to ﬁxation in
both subpopulations is approximately (e/ 2)Nm e times the
rate of ﬁxation in the absence of immigration. Assuming
that alleles A are brought about by migration at rate me
and summing up over k loci, one ﬁnds the second term in
the right-hand side of equation (17).
Stochastic divergence with local adaptation . Let us as-
sume that the allele under consideration is favorable in
a given environment with selective advantage sLA. The
change in this allele frequency as deﬁned by the joint ac-
tion of selection induced by reproductive isolation and
selection for local adaptation is
∆ sp =spq(p − q) +sLApq. (A19)
This equation is identical to the one describing meiotic
drive in the Appendix of Walsh (1982). Following Walsh,
the ﬁxation probability is
U =
erf
[ √
S(1 − α )
]
− erf
[ √
S(1 − α − 2
N )
]
erf
[ √
S(1 − α )
]
+erf
[ √
S(1 +α )
] , (A20)
where S = Ns/ 2 and α = sLA/s . Expanding the nu-
merator in a Taylor series under the assumption that
1/N << 1 and multiplying the results by the expected
number of mutants, vN , results in the relative rate of
ﬁxation given by equation (19).
Genetic distance (20) and some other models . Genetic
distance (1) is recovered by assuming that G is an iden-
tity matrix. Assuming that G is a diagonal matrix with
non-equal diagonal elements is a simple way to introduce
non-equivalence of loci. The case when the probability of
mating depends on the diﬀerence in a quantitative trait
can be treated within the same framework. Let ci be the
contribution of the i-th locus to a quantitative trait z.
Neglecting microenvironmental eﬀects, the trait values for
individualsα andβ arexα = ∑ cilα
i andzβ = ∑ cilβ
i , re-
spectively. The square of the diﬀerence of zα andzβ is re-
covered from equation (20) by assuming that Gij =cicj.
A common way to model sexual selection is to assume
that the probability of mating between a male and a fe-
male depends on the diﬀerence in a female phenotypic
trait, zf , and a male phenotypic trait, zm, which are
controlled by two diﬀerent sets of loci (e.g., Lande 1981;
Kirkpatrick 1982; Nei at al. 1983; Wu 1985; Turner and
Burrows 1995). Let zm = ∑ cm
i li and zf = ∑ cf
ili where
the sums are taken over the corresponding sets of loci.
The value (zm − zf )2 is recovered from (20) by assuming
that matrix G has a block form
G =
(
0 Gs
Gs 0
)
.
The diagonal Lm × Lm and Lf × Lf zero matrices cor-
respond to the interactions within the set of Lm genes
controlling the male trait and within the set of Lf genes
controlling the female trait (L =Lf +Lm), and matrix Gs
describing the interactions between the two sets of genes
has elements Gs
ij =cf
icm
j .

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0 500 1000 1500 2000
generation
50genetic distances
50genetic distances
50genetic distances
a
b
c
Db
k
Dw
Db k
Dw
Db
k
Dw
Figure 6: Dynamics of speciation in a subdivided popula-
tion. Unless speciﬁed otherwise, K = 20 , v = 0 . 0384, n = 2.
a. Eﬀects of migration rate. Stronger migration, m =
0. 01 (dashed lines; no speciation), and weaker migration,
m = 0 . 001 (solid lines; speciation). Total population size
NT = 200. b. Eﬀects of mutation rate. Weaker mutation,
v = 0 . 0384 (dashed lines; no speciation), and stronger muta-
tion, v = 5 × 0. 0768 (solid lines; speciation). Other param-
eters: NT = 400, m = 0 . 005. c. Eﬀect of population sub-
division. n = 2 subpopulations (dashed lines; no speciation)
and n = 4 subpopulations (solid lines; speciation). Other pa-
rameters: NT = 800, m = 0 . 0033. Dashed lines represent Db
(top line), Dw (middle line) and k (bottom line), respectively.
During the ﬁrst 1000 generations there are no restrictions o n
migration.

0.0 200.0 400.0 600.0 800.0 1000.0
generation
50genetic distances
50genetic distances
a
b
Db
k
Dw
Db
k
Dw
Figure 7: Dynamics of speciation in a peripheral population.
a. Speciation with m = 0 . 001; N = 100; K = 20; v = 0 . 0384
(cf. Fig.7a). b. Speciation with m = 0 . 005; N = 200; K =
20; v = 5 × 0. 0768 (cf. Fig.7b).
0.0 2.0 4.0 6.0 8.0 10.0
S
0.0
2.0
4.0
6.0
8.0
10.0
R
α=2
α=1
α=0.5
α=0
Figure 8: Relative rate of ﬁxation in the case with local
adaptation.