vol. 160, no. 3the american naturalistseptember 2002
Evolutionary Consequences of Asymmetric Dispersal Rates
Tadeusz J. Kawecki1,*and Robert D. Holt2,†
1. Institute of Zoology, University of Basel, Rheinsprung 9, CH-
4051 Basel, Switzerland;
2. Department of Ecology and Evolutionary Biology, University of
Kansas, Lawrence, Kansas 66045
Submitted June 25, 2001; Accepted February 15, 2002
abstract: We study the consequences of asymmetric dispersalrates
(e.g., due to wind or current) for adaptive evolution in a system of
two habitat patches. Asymmetric dispersal rates can lead to over-
crowding of the “downstream” habitat, resulting in a source-sink
population structure in the absence of intrinsic quality differences
between habitats or can even cause an intrinsically better habitat to
function as a sink. Source-sink population structure due to asym-
metric dispersal rates has similar consequencesforadaptiveevolution
as a source-sink structure due to habitat quality differences: natural
selection tends to be biased toward the source habitat. We demon-
strate this for two models of adaptive evolution: invasion of a rare
allele that improves fitness in one habitat but reduces it in the other
and antagonistic selection on a quantitative trait determined by five
additive loci. If a habitat can sustain a population without immi-
gration, the conditions for adaptation to that habitat are most fa-
vorable if there is little or no immigration from the other habitat;
the influence of emigration depends on the magnitude of the allelic
effects involved and other parameters. If, however, the population is
initially unable to persist in a given habitat without immigration,
our model predicts that the population will be most likely to adapt
to that habitat if the dispersal rates in both directions are high. Our
results highlight the general message that the effect of gene flowupon
local adaptation should depend profoundly on thedemographiccon-
text of selection.
Keywords: adaptation, dispersal, gene flow, marginal habitats, het-
erogeneous environments, source-sink structure.
It is a trivial fact of ecology that a change in the population
density at a given locality reflects the balance between local
* Corresponding author. Present address: Department of Biology, Unit of
Ecology and Evolutionary Biology, University of Fribourg 10, Chemin du
Muse ´e, CH-1700 Fribourg, Switzerland; e-mail: email@example.com.
†Present address: Department of Zoology, University of Florida, Gainesville,
Am. Nat. 2002. Vol. 160, pp. 333–347. ? 2002 by The University of Chicago.
0003-0147/2002/16003-0005$15.00. All rights reserved.
births, local deaths, immigration, and emigration. A local
population can thus remain stable even though its local
birth rate does not equal its local death rate. Populations
living in heterogeneous environments can thus exhibit a
source-sink structure, with a net flow of dispersers from
some (source) habitats to other (sink) habitats (Lidicker
1975; Holt 1985; Pulliam 1988; Kawecki 1995; Dias 1996).
According to the broad definition we adopt here, a habitat
is a sink if the number of immigrants it receives from
other habitats exceeds the number of individuals that suc-
cessfully emigrate from it to other habitats. Our definition
of sink habitats thus includes habitats unable to sustain a
population without immigration (absolute sinks) as well
as habitats good enough to sustain a population that be-
come sinks as a result of high immigration and density
dependence (relative sinks; Watkinson and Sutherland
1995 call them “pseudosinks”). Conversely, sourcehabitats
are net producers of migrants (for a caveat concerning the
above definition when there are more than two habitats,
see Rousset 1999).
As a consequence of a source-sink structure, a stable
population may be maintained in an absolute sink (i.e., a
habitat unable to sustain a population in the absence of
immigration; Lidicker 1975; Ja ¨rvinen and Va ¨sa ¨inen 1984;
Pulliam 1988; Robinson et al. 1995). In a habitat that is
a relative sink, the population is maintained above the
local carrying capacity (the density at whichdeathsbalance
births), while the reverse holds for a source habitat (e.g.,
Holt 1985; Watkinson and Sutherland 1995; Dias 1996).
Some models predict that source-sink dynamics can have
a stabilizing effect on population dynamics (Holt and Has-
sell 1993; Doebeli 1995; Gomulkiewicz et al. 1999).
Source-sink population structure also has important
consequences for adaptive evolution. Source habitats tend
to contribute more to the future gene pool of the entire
population than do sink habitats. Most individuals in a
sink habitat trace their ancestry to immigrants from a
source habitat. The resulting asymmetric gene flow makes
natural selection on performance in a sink habitat rela-
tively ineffective (Holt and Gaines 1992; Kawecki and
Stearns 1993; Kawecki 1995; Holt 1996a, 1996b). A pop-
ulation in a sink habitat may thus persist in a state of
permanent maladaptation, a pattern confirmed by an in-
334The American Naturalist
creasing number of empirical studies (e.g., Stearns and
Sage 1980; Dhondt et al. 1990; Blondel et al. 1992; Stanton
and Galen 1997).
Predictions concerning the evolutionary consequences
of source-sink population structure have to date largely
been based on models in which the net flow of dispersers
from source to sink habitats results from a difference in
habitat quality. Under that scenario, more offspring are
produced in the better habitat at equilibrium. Therefore,
if the propensity to disperse is habitat independent, more
individuals move from the better to the poorer habitat
than in the opposite direction, simply because the pool of
potential dispersers is greater in the former than in the
However, the number of dispersing individuals is the
product of the number of potential dispersers(propagules)
and their propensity to disperse (dispersal rate). An asym-
metry in the number of dispersing individuals will thus
also arise, even in the absence of differences in habitat
quality if the dispersal rate from habitat i to habitat j (i.e.,
the probability that a propagule produced in habitat i dis-
perses to habitat j) is greater than the dispersal rate in the
opposite direction. As a result of this asymmetry, the pop-
ulation density in habitat i will become reduced below the
local carrying capacity, while habitat j will become over-
crowded. The overcrowding will depress the reproductive
success in habitat j until, at equilibrium, the difference
between the reproductive outputs of the two habitatscom-
pensates for the asymmetric dispersal. Habitat i will thus
become a source and habitat j a sink (Doebeli 1995). This
can be true even if habitat i has the lower carryingcapacity.
In general, whether there is a net flow of dispersers (i.e.,
whether there is a source-sink structure), and in which
direction, will depend on differences in both habitat qual-
ity and dispersal rates. If there are no costs of dispersal
and the individuals have perfect control over their dis-
persal, natural selection is expected to adjustdispersalrates
in such a way that there is no sink structure; that is, for
each habitat, the number of immigrants from other hab-
itats equals the number of emigrants that move to other
habitats (McPeek and Holt 1992; Doebeli 1995; Lebreton
et al. 2000; Holt and Barfield 2001). When habitats differ
in quality, the dispersal rates required for this “balanced
dispersal” scenario are asymmetric: a propagule produced
in a poorer habitat must be more likely to move to another
habitat than a propagule produced in a better habitat. This
asymmetry of dispersal rates must be so adjusted that dif-
ferences in population density exactly compensate for dif-
ferences in habitat quality and the expected lifetime re-
productive success is the same in all habitats (Lebreton et
al. 2000; Holt and Barfield 2001). The outcome is an ideal
free distribution (Fretwell and Lucas 1970).
Since organisms are usually neither ideal nor free, the
balanced dispersal scenario is likely to be uncommon in
nature (though examples seem to exist for vertebrates, e.g.,
Diffendorfer 1998; Lin and Batzli 2001). Asymmetries in
dispersal rates can also arise from social interactions (Pul-
liam 1988; Holt 1996a) or the directional influence of
environmental agents of dispersal, for example, gravity,
river or ocean current, or prevailing wind direction. For
instance, asymmetric dispersal due to ocean currents has
long been recognized as an important factor that affects
the distribution, abundance, and genetic variation of ma-
rine benthic invertebrates with planktonic larvae (e.g.,
Scheltema 1986). A recent study of mtDNA in two bar-
nacle and one sea urchin species off California found a
signature of an excess of southward over northward dis-
persal, consistent with the pattern of currents (Wares et
al. 2001). Similar patterns should be expected in river
invertebrates. On land, asymmetric dispersal due to wind
should play an important role on dunes and other exposed
coastal habitats. Among the few cases of well-studied
source-sink dynamics in natural populations, asymmetric
dispersal due to wind has a crucial impact on distribution
and abundance in the sand dune plant Cakile edentula
(Keddy 1981, 1982; Watkinson 1985). In that system, the
windward (seaward) side of the dunes is the sourcehabitat,
where most of the seeds are produced, but because most
of them are transported by wind to the sink habitataround
the dune crests, plant density in the latter habitat is con-
siderably higher than in the windward habitat. At the same
time, the seed emigration from the windward habitat re-
duces competition and boosts the reproductive output
from that habitat.
An asymmetry of dispersal rates can thus create a
source-sink structure in the absence of differences in hab-
itat quality. Where habitats do differ in quality, asymmetry
of dispersal rates can amplify the source-sink structure if
its direction is opposite to that required for balanced dis-
persal, as in the C. edentula example mentioned above.
Dispersal rates asymmetric in the direction predicted by
the balanced dispersal scenario may reduce the source-
sink structure, but if the asymmetry is greater than re-
quired under the balanced dispersal model, the source-
sink structure will be reversed: the poorer habitat will
become the source, and the better one, the sink. Unbal-
anced dispersal is also likely in recently changed land-
scapes, where dispersal syndromes may be maladapted to
In this article we study the effect of asymmetries in
dispersal rates on adaptation in the face of a trade-off in
fitness between habitats. Only a few models have explicitly
addressed this issue. Holt (1996b) studied the conse-
quences of asymmetric dispersal rates in a two-patch
source-sink model and concluded that the conditions for
adaptation to the sink are most favorable when the dis-
Asymmetric Dispersal and Adaptation 335
persal rates in both directions are high. That model as-
sumes, however, that one of the two habitats is unable to
sustain a population without immigration and thus must
be a sink if the population persists at all, irrespective of
dispersal rates. Furthermore, the fitness sensitivity ap-
proach taken by Holt (1996a, 1996b) is equivalent to con-
sidering the fate of rare alleles with infinitesimal effects
on fitness. Yet, dispersal rate may affect the fate of rare
alleles with small and large effects in qualitatively different
ways (Kawecki 2000a). A special, extreme case of dispersal
rate asymmetry is one-way dispersal, resulting in a “black
hole” sink, which receives dispersers from a source but
sends no successful dispersers back. Theoretical aspects of
the population dynamics and adaptive evolution in black
hole sinks have been extensively studied by Gomulkiewicz
and coworkers (Holt and Gomulkiewicz 1997; Gomulk-
iewicz et al. 1999; LoFaro and Gomulkiewicz 1999). They
showed that this form of source-sink structure has qual-
itatively similar consequences for adaptive evolution in the
sink as the source-sink structure because of differences in
habitat quality. Yet they have not studied whether allowing
for some dispersal back into the source (i.e., relaxing the
“black hole” assumption) makes the conditions for adap-
tive evolution in the sink more or less stringent.
Our aim in this article is to generalize the qualitative
conclusions reached in those papers regarding the evo-
lutionary consequences of the source-sink structure to
source-sink structure caused or modified by dispersal rate
asymmetry. We combine a two-patch source-sink model
that has an arbitrary combination of dispersal rates with
explicit one-locus and multilocus population genetic ap-
proaches. In the next section we describe the model of
population dynamics that we assume. In the following
section genetic variance is introduced in the form of a rare
allele that improves fitness in one habitat but reduces it
in the other. We study how the conditions under which
this allele is favored depend on the combination of the
dispersal rates. The last section considers the evolution of
a quantitative trait with antagonistic effects on fitness in
the two habitats. These two approaches—invasion of a
mutant allele and the evolution of a quantitative
trait—provide two complementary ways to study whether
and how asymmetric dispersal rates and the resulting
source-sink structure promote or hinder habitat-specific
The Model of Population Dynamics
Our model of population dynamics follows that of Doebeli
(1995). We consider a population with discrete genera-
tions, living in two habitat patches connected by passive
dispersal. Each individual completes its development
within a single habitat patch, during which time it is sub-
ject to density dependence and selection (the latter intro-
duced inthe following section, “InvasionofaRareAllele”).
The equations for population dynamics are compatible
with three modes of dispersal. First, adults mate and re-
produce in the habitat in whichtheydeveloped;thezygotes
(seeds, eggs, etc.) then disperse. Second, adultsmatewithin
the habitat of origin; mated females disperse and oviposit
after dispersal. And third, adults of both sexes disperse;
mating and reproduction take place after dispersal. In the
two latter cases, the performance of the dispersing adults
is assumed to be only affected by the conditions and den-
sity in the habitat of origin and not by the habitat of
destination. The census takes place at the zygote stage
(after dispersal if zygotes disperse). Conditions are con-
stant in time, and the population is assumed to be large
enough for stochastic effects to be ignored. The dynamics
of the two subpopulations can be described by the matrix
equation (Doebeli 1995)
f (N )(1 ? m )
f (N )m
f (N )m
f (N )(1 ? m )
? ( ) (
where Niis the size of the subpopulation in habitat i at
census time, the prime denotes population size in the next
generation, fi(Ni) is the density-dependent expected life-
time reproductive success of individuals developing in
habitat i, and mijis the dispersal rate from habitat i to
habitat j (i.e., the probability than an individual present
in habitat i before dispersal migrates to habitat j). Density
dependence here may act on survival from zygote to adult.
Alternatively, it can be interpreted as affecting adult fe-
cundity; in the latter interpretation, however, the density-
dependent component is a function of the number of ju-
veniles present in the habitat at the census time, not the
number of breeding adults. To obtain the numerical re-
sults, we implement a form of density dependence pro-
posed by Maynard Smith and Slatkin (1973):
1 ? cNi
where c and b are constants, for simplicity assumed to be
equal for both habitats.
Throughout the article we restrict our analyses to pa-
(1) and (2) has a stable nonzero equilibrium. The con-
dition for the existence of a nonzero equilibrium (i.e.,
persistence of the population) is derived in the appendix.
If the equilibrium exists, it is always stable if
conditions for stability of the equilibrium were studied by
In the absence of dispersal, the form of density depen-
; theb ≤ 2
346 The American Naturalist
The minimum R1needed to maintain nonzeroequilibrium
densities increases with increasing m12and with decreasing
m21. A sufficient condition for persistence is R (1 ?
; in this case, the species can persist in habitat 1,m ) 1 1
regardless of what happens in habitat 2. Finally, for given
R1and R2, the maximum m12that permits persistence is
a linearly increasing function of m21.
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