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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: tadeusz.kawecki@unifr.ch.

†Present address: Department of Zoology, University of Florida, Gainesville,

Florida 32611-8525.

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-

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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

latter.

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

current conditions.

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-

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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

adaptation.

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 )

11

f (N )m

11

f (N )m

2

f (N )(1 ? m )

22

N

N

N

N

122 21

1

1

p

,(1)

? ( ) (

2

)( )

12 21

2

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):

Ri

f(N) p

ii

,(2)

b

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-

rameter valuesforwhichthesystemdescribedbyequations

(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

Doebeli (1995).

In the absence of dispersal, the form of density depen-

; theb ≤ 2

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336The American Naturalist

dence described by equation (2) leads to equilibrium den-

sities of

1/b

K p [(R ? 1)/c] .

ii

(3)

We refer to Kias the carrying capacity of habitat i. When

there is dispersal, the equilibrium densities (

equal the carrying capacities if the dispersal rates satisfy

, which, given equation (3), simplifies to K m

p K m

1 122 21

, ) will

ˆˆ

N N

12

1/b 1/b

(R ? 1) m

1

p (R ? 1) m .

2

(4)

12 21

This is the special case of balanced dispersal: the number

of migrants in both direction is the same, and there is

thus no source-sink structure (McPeek and Holt 1992;

Doebeli 1995; Diffendorfer 1998; Lebreton et al. 2000). If

equation (4) is not satisfied and its left-hand side is greater

than its right-hand side, habitat 1 will be a source and

habitat 2, a sink; otherwise, the opposite holds. Except for

special cases (e.g., one-waydispersal),theequilibriumden-

sities (,) can only be found numerically.

ˆˆ

N N

12

Invasion of a Rare Allele

The Model

In this section we consider a mutant allele beneficial in

habitat 2 but deleterious in habitat 1 and study how the

conditions for invasion of such an allele are affected by

dispersal rates in the two-patch system described above.

As a caveat on the results described here, it should be

noted that a rare allele that satisfies the condition for in-

vasion derived in this deterministic model is still prone to

be lost because of drift. Therefore, the condition for in-

vasion might be better interpreted as the condition for the

allele not to be deterministically eliminated by selection.

We consider a nonrecessive mutant allele, present at a

low frequency in an otherwise monomorphic equilibrium

population whose dynamics follow the assumptions of the

previous section (“The Model of Population Dynamics”).

The absolute fitness (expected reproductive success) of the

heterozygous carriers of the mutant allele in habitat 1 is

. The multiplicative relation between the den-f (N )(1 ? s )

111

sity-dependent and the genotype-dependent components

of fitness means that the selection coefficient against the

mutant allele in habitat 1 is s1, irrespective of density. The

allele improves performance in habitat 2; there the fitness

of the heterozygotes isf (N )(1 ? s )

22

invasion of this allele can thus be formulated in terms of

a critical value of s1, denoted

is favored when rare if and only if

is the maximum selection coefficient against the rare

allele habitat 1 that is still outweighed by the beneficial

. The conditions for

2

, such that the mutant allele

. In other words, s ! s

11

∗s1

∗

∗s1

effect on fitness in habitat 2, s2, such that the rare allele

can invade. If, the rare allele will invade even if it

1

is lethal in habitat 1. The question addressed in thissection

can thus be formulated like this: How is

dispersal rates m12, m21, as well as by s2and the population

growth parameters defined above?

As usual when considering the fate of a rare allele, the

frequency of mating between two heterozygotes and the

frequency of homozygotes for the mutant allele are as-

sumed negligible. With this assumption it does not matter

whether mating takes place before or after dispersal; that

is, the results hold for all three modes of dispersal con-

sidered in the previous section (“The Model of Population

Dynamics”). Also, because the allele is rare, the effect of

the heterozygotes on the population densities can be ne-

glected. That is, as long as the mutant allele is rare, the

population densities remain at

densities of a population monomorphic for the common

homozygote. With these assumptions, the dynamics of the

density of heterozygous individuals in the two habitats,

M1and M2, are described bythe setof recurrenceequations

∗s p 1

affected by the

∗s1

and, the equilibrium

ˆˆ

NN

12

?

ˆ

f (N )(1 ? s )(1 ? m )

11

ˆ

f (N )(1 ? s )m

11

M

( ) (

M

2

112

1p

?

1 12

ˆ

f (N )(1 ? s )m

22

ˆ

f (N )(1 ? s )(1 ? m ) M

222

M

221

1,

2

(5)

)( )

21

where the prime again denotes the number in the next

generation. (This equation can easily be reformulated in

terms of allele frequencies, leading to identical results.)

Equation (4) is linear in (M1, M2). This means that, after

converging to a stable distribution between the habitats,

M1and M2will grow (or decline) exponentially with the

rate given by the dominant eigenvalue

in equation (5). Because the densities of the resident ho-

mozygotes ( ,) remains constant at the equilibrium,

ˆˆ

N N

12

gives the rate of change of both the number of het-

lM

erozygotes and the frequency of the mutant allele, as long

as and. The allele is thus favored when

ˆˆ

M K N M K N

1122

rare if . Note that

l 1 1

l p 1

MM

, where A is the matrix in equation (5) and I is an identity0

matrix. This equation can be solved for s1to yield

of the matrix

lM

implies det(A ? I) p

ˆ

1?f (N )(1?m )(1?s )

22

ˆ

f (N )[1?m ?f (N )(1?m ?m )(1?s )]

111222

21 2

∗s p 1?

1

;

ˆ

1221 2

(6)

the rare allele can invade if the selective cost it experiences

in habitat 1 satisfies .

s ! s

11

∗

Page 5

Asymmetric Dispersal and Adaptation337

Analytical Considerations

Although expression (6) is general, it is not sufficient for

analyzing the net effect of dispersal rates on evolution,

because the magnitudes of dispersal enter expression (6)

not only directly but also indirectly through effects on

,. Before using expression (6) to obtain numerical

ˆˆ

NN

12

results, we describe some analytical insights concerning

the relative weight of the two selection coefficients and

some special cases.

The stable (asymptotic) fraction of carriers of the rare

allele present in each habitat

genvector corresponding to

lM

sum of its elements equal 1 (Caswell 1989). Note that, by

definition,

is the right ei-

q p (q , q )

, normalized to make the

12

Aq p l q. (7)

M

By adding up the elements of the resulting vectors on both

sides of the equation and using the fact that q ? q p

, one can show that1

12

ˆˆ

l p q f (N )(1 ? s ) ? q f (N )(1 ? s )

M1 111

(8)

2 222

(see Liberman 1988 for a formal derivation). In deter-

mining the ultimate growth rate of the frequency of the

allele, the two selection coefficients are thus weighted by

the products of the density-dependent components of fit-

ness (which convert the relative selection coefficients into

absolute differences in reproductive success) and the pro-

portions of carriers exposed to each habitat. Note that the

carriers of the rare allele not only have local fitnesses dif-

ferent from the common genotype but also are differently

distributed between the habitats: q is a function of the

selection coefficients s1, s2. The dispersal rates affect the

fate of the rare allele through

, but alsothroughtheireffectonhowthedistribution

ˆ

f (N )

22

of the carriers of the rare allele between the habitats de-

viates from that of the common genotype. If m ?

, which should usually be the case in nature, q1ism

! 1

21

smaller than the analogous proportion for the common

resident genotype (assuming, as we do throughout, that

), and the difference increases as the sums , s 1 0

12

decreases. The reason is that if dispersal is limited,m21

the distribution of the allele will tend to be biased toward

the habitat where it is relatively more fit. The reverse holds

when . At the borderline (m ? m

1 1

12 21

which includes the complete mixing case (m

), it can be shown that0.5(q , q ) p (m , m )

1

that

ˆˆ

f (N )m ? f (N )m

p 1

11 2122 12

is the weight given to the two selection coefficients in-

dependent of the coefficients themselves, and equation (6)

simplifies to

,, and thus ,

ˆˆˆ

NNf (N )

1121

12

m ?

12

),m ? m

12

p 1

21

p m

and, thus,

p

1221

2

. Only in this special case

2112

ˆ

ˆ

f (N )m

2

f (N )m

1

2 12

∗s p

1

s .

2

(9)

121

In other cases (

still allows the rare allele to invade is not proportional to

s2. In the limit of weak selection, after expanding

Taylor series around (,s p 0 s p 0

1

), the critical value of s1thatm ? m

12

( 1

21

in a

lM

), one can show that

2

∗

1

ˆ

ˆ

s1 ? f (N )(1 ? m )

11

1 ? f (N )(1 ? m )

22

12

lim

s r 0 s

2

p

;(10)

( )

2 21

numerical results show this to be a good approximation

for alleles with s2of the order of 0.01 or less.

Another special case is for the common genotype at

equilibrium to exhibit balanced dispersal (i.e., to satisfy

eq. [4]). This implies that f (N ) p f (N )

1

selection coefficients of the rare allele are now weighted

by proportion of carriers exposed to each habitat (q1, q2).

Note however, that q1, q2will not, in general, be propor-

tional to the carrying capacities (equilibrium densitiespre-

dicted under no dispersal) of the common genotype nor

to those for the rare genotype. Although the common

genotype is at balanced dispersalequilibrium,dispersalwill

not ultimately be balanced for the carriers of the rareallele:

when the rare allele settles into its stable habitat distri-

bution, it will be relatively more common in the habitat

where it is fitter, leading, in turn, to a net flow of the

carriers from habitat 2 to habitat 1. In the limit of weak

selection with balanced dispersal, we have

, which states that selection is biased inm /m

p K /K

1221 21

favor of the habitat with the higher carrying capacity.

Finally, we consider the special case of unidirectional

dispersal. At the limit m

p 0

12

1 has no influence on habitat 2, so we can consider habitat

2 alone. Thus,

ˆ

f (N )(1 ? m ) p 1

2221

the growth rate of a rare heterozygote is

, which is independent of s1and whichexceedsm )(1 ? s )

212

unity for all. Hence, the scope for adaptation to s 1 0

2

habitat 2 is unaffected by the coupling between habitat 2

and habitat 1. Habitat 1 is a black hole sink, so any del-

eterious effects of local adaptation in habitat 1 are irrel-

evant to evolution in habitat 2 because of the absence of

feedback. Note, however, that for the population topersist,

the intrinsic growth rate in habitat 2, R2, must exceedunity

by a margin large enough to compensate for the loss of

individuals to emigration. If the population persists only

in habitat 1, the allele is eliminated for any

analogous limit , habitat 2 is a black hole sinkm

p 0

21

(again assuming that the local population in habitat 1

persists). As shown by Holt and Gomulkiewicz (1997), the

rare allele will only increase if

, and the two

ˆˆ

122

∗s /s p

12

, what happens in habitat

at the equilibrium, and

ˆ

f (N )(1 ?

22

. At thes 1 0

1

. Since

ˆ

f (N )(1 ? s ) 1 1

222