the role of density-dependent dispersal in source–sink dynamics
TRANSCRIPT
Journal of Theoretical Biology 226 (2004) 159–168
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The role of density-dependent dispersal in source–sink dynamics
Priyanga Amarasekare*
Department of Ecology and Evolution, The University of Chicago, 1101 East 57th Street, Chicago IL 60637, USA
Received 10 June 2003; received in revised form 13 August 2003; accepted 19 August 2003
Abstract
I investigate two aspects of source–sink theory that have hitherto received little attention: density-dependent dispersal and the cost
of dispersal to sources. The cost arises because emigration reduces the per capita growth rate of sources, thus predisposing them to
extinction. I show that source–sink persistence depends critically on the interplay between these two factors. When the emigration
rate increases with abundance at an accelerating rate, dispersal costs to sources is the lowest and risk of source–sink extinction the
least. When the emigration rate increases with abundance at a decelerating rate, dispersal costs to sources is the highest and the risk
of source–sink extinction the greatest. Density-independent emigration has an intermediate effect. Thus, density-dependent dispersal
per se does not increase or decrease source–sink persistence relative to density-independent dispersal. The exact mode of dispersal is
crucial. A key point to appreciate is that these effects of dispersal on source–sink extinction arise from the temporal density-
dependence that dispersal induces in the per capita growth rates of source and sink populations. Temporal density-dependence due
to dispersal is beneficial at low abundances because it rescues sinks from extinction, and detrimental at high abundances because it
drives otherwise viable sources to extinction. These results are robust to the nature of population dynamics in the sink, whether
exponential or logistic. They provide a means of assessing the relative costs and benefits of preserving sink habitats given three
biological parameters.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Source–sink dynamics; Density-dependent dispersal; Population persistence; Cost of dispersal; Rescue effect
1. Introduction
The theory of source–sink dynamics predicts thatemigration from populations with positive per capitagrowth rates (sources) can rescue populations withnegative per capita growth rates (sinks) from extinction(Pulliam, 1988; Pulliam and Danielson, 1991; Watkin-son and Sutherland, 1995; Dias, 1996). This rescue effectprovides a mechanism for species persistence in spatiallyheterogeneous environments.Empirical studies highlight two directions in which
source–sink theory could be advanced. The first is therole of density-dependence in dispersal. Data show thatdispersal is often density-dependent in a wide variety oftaxa. For example, an increase in the dispersal rate withincreasing abundance has been observed in insects (e.g.Denno and Peterson, 1995; Fonseca and Hart, 1996),birds (e.g. Veit and Lewis, 1996) and territorial
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i.2003.08.007
mammals (e.g. Wolff, 1997; Aars and Ims, 2000). Adecrease in the dispersal rate with increasing abundancehas been observed in insects (e.g. Herzig, 1995), birds(e.g. Doncaster et al., 1997) and mammals (e.g. Wolff,1997; Diffendorfer, 1998). Despite the ubiquity ofdensity-dependent dispersal in nature, most theory onmetapopulation and source–sink dynamics considersdispersal to be density-independent (e.g. Levin, 1974;Pacala and Roughgarden, 1982; Shmida and Ellner,1984; Holt, 1985; Hanski and Gilpin, 1997; Amarase-kare and Nisbet, 2001). Only a handful of studies haveinvestigated density-dependent dispersal (Pulliam, 1988;Howe et al., 1991; Saether et al., 1999).The nature of density-dependence in dispersal can
have significant consequences for source–sink dynamics.For example, an increase in the emigration rate withincreasing density will lead to negative density-depen-dent dispersal, which will strengthen self-limitation andincrease a species’ ability to increase when rare. Incontrast, a decrease in the emigration rate withincreasing density will lead to positive density-depen-dent dispersal, which will weaken self-limitation and
ARTICLE IN PRESSP. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168160
reduce a species’ ability to increase when rare. Thisinterplay between density-dependence in dispersal anddensity-dependence in local demographic processes islikely a crucial determinant of population persistence viasource–sink dynamics.The second theoretical direction suggested by data
involves the cost of dispersal to source populations. Arecent empirical study has shown that sustainedemigration can cause per capita growth rates of sourcepopulations to become negative, despite high localreproduction and recruitment (Gundersen et al., 2001).While the benefits of the rescue effect to sink popula-tions has received a great deal of attention (see Gilpinand Hanski, 1991; Hanski and Gilpin, 1997; Hanski,1999), its costs in terms of increased extinction risk ofsources has been under-appreciated (but see Holt, 1993).This becomes a crucial consideration when applyingsource–sink theory to conservation problems. Preserva-tion of sink habitats has been advocated as a con-servation priority because they contain rare speciesmaintained by immigration from source habitats (Howeet al., 1991). However, preserving sinks is a viableoption only if the sinks can be rescued withoutincreasing the extinction risk of the sources themselves.Although previous theoretical studies have suggested
that source abundance can be reduced below carryingcapacity by emigration (e.g. Holt, 1985; Wootton andBell, 1992; Doak, 1995), there is no formal theoreticalanalysis of the conditions under which the rescue effectincreases the extinction risk of source populations.Moreover, most theoretical studies that suggest areduction in source abundance due to emigrationconsider dispersal to be density-independent (e.g. Holt,1985; Wootton and Bell, 1992; Doak, 1995). The studiesthat do consider density-dependent dispersal (e.g. Pull-iam, 1988; Howe et al., 1991; Saether et al., 1999) do notaddress the issue of dispersal-induced extinction risk tosources. This leaves the open question of how density-dependent dispersal influences the cost of dispersal tosource populations.Here I present a single species model that investigates
both density-dependent dispersal and the cost ofdispersal to sources. I show that the nature of density-dependence in dispersal is key to the viability of sourcesand hence, the long-term persistence of source–sinksystems. I discuss the implications of these results forconservation.
2. The model
Consider a patchy environment inhabited by multiplepopulations of a given species. Some populations aresinks, i.e. they have a negative per capita growth rateand would go extinct in the absence of dispersal,while other populations are sources, i.e. they have a
positive per capita growth rate and provide a sourceof immigrants that rescue sink populations fromextinction.The dynamics of a two-patch source–sink system are
given by:
dN1
dt¼ r1N1 1�
N1
K1
� �� a1
N1
K1
� �s
N1 þ a2N2;
dN2
dt¼ r2N2 � a2N2 þ a1
N1
K1
� �s
N1; ð1Þ
where N1 and N2 are the abundances of the source andsink populations, respectively, and ri are their intrinsicgrowth rates ði ¼ 1; 2Þ: The parameter ai is the per capitaemigration rate of population i; and s denotes thestrength of density-dependence in dispersal. The sourcepopulation undergoes logistic-type population growth inthe absence of dispersal, i.e. given a positive intrinsicgrowth rate it will increase to carrying capacity K1: Thesink, in contrast, exhibits exponential decay in theabsence of dispersal (i.e. r2o0). Because sinks aretypically suboptimal habitats within which speciescannot maintain themselves, both local dynamics anddispersal are considered to be density-independent in thesink. Dispersal from the source can be density-indepen-dent ðs ¼ 0Þ or density-dependent ðsa0Þ:Eq. (1) can be expressed in non-dimensionalized
quantities. Non-dimensional analysis reduces the num-ber of parameters that describe the system. It also shedslight on the scaling relations among processes thatdetermine population dynamics (Murray, 1993).I use the following substitutions:
xi ¼Ni
Ki
; bi ¼ai
r1; r ¼
r2
r1; t ¼ r1t ði ¼ 1; 2Þ
to transform Eq. (1) to its non-dimensionalized form.The non-dimensional time metric t expresses time in
terms of the source population’s intrinsic growth rate.This time-scaling allows for easy comparison amongsystems that vary in their natural time-scales. Theabundances of source and sink populations are ex-pressed as a fraction of the source’s carrying capacity.While a particular value of the source’s carryingcapacity may not be all that informative, knowinghow close source and sink populations are to sourcecarrying capacity is. For example, x1; x251 indicatesthat the source and sink abundances are well below thesource’s carrying capacity, while x1; x2-1 indicates theopposite.The metric r is the ratio of the intrinsic growth rates
in sink and source. The sink, by definition, has anegative intrinsic growth rate and hence ro0 always.The magnitude of r however is informative: r ¼ �1indicates that growth rates in source and sink are ofequal magnitude; r > �1 means that the source has apositive growth rate of greater magnitude than thenegative growth rate of the sink, and ro� 1 implies the
ARTICLE IN PRESSP. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168 161
opposite. The magnitude of r is thus a measure of thequality of the source. The metric bi is the per capitaemigration rate from the source or sink populationscaled by the intrinsic growth rate of the sourcepopulation. It measures the cost of dispersal, i.e. it isthe magnitude of emigration relative to local reproduc-tion and mortality in the source.Substituting the non-dimensional quantities into
Eq. (1) yields the following system of equations. Unlessotherwise noted, all parameters from this point on areexpressed as scaled quantities.
dx1
dt¼ x1ð1� x1Þ � b1x
sþ11 þ b2x2;
dx2
dt¼ rx2 � b2x2 þ b1x
sþ11 : ð2Þ
The nature of density-dependence in dispersal ismediated by the parameter s (Fig. 1). When s ¼ 0;emigration is density-independent and occurs at the percapita rate bi:When �1oso0 emigration increases withdensity at a decelerating rate and approaches the percapita rate bi; akin to a Type II functional response inconsumer–resource dynamics (Holling, 1959). Thismode of dispersal is likely to occur when individualstend to emigrate from low density populations, due todifficulties in mate finding or reduced predator vigilance.Examples include insects (Herzig, 1995; Kuussaari et al.,1998), birds (Birkhead, 1977) and mammals (Wolff,1997; Diffendorfer, 1998). When s > 0 emigrationincreases with density at an accelerating rate towardsbi; akin to a Type III functional response at low resourceabundances (Murdoch and Oaten, 1975). This mode ofdispersal is likely to occur when individuals tend to leavefrom high density populations due to strong intra-specific competition. Examples include insects (Dennoand Peterson, 1995; Fonseca and Hart, 1996; Rhainds
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Population
Em
igra
tion
rate
s=0
s>0
-1<s<0
(a)
B
Fig. 1. Modes of density-dependent dispersal. The x-axis is population abund
of density-independent dispersal ðs ¼ 0Þ; emigration rate increases in proportioccurs when the emigration rate increases with abundance at a deceleratin
emigration rate increases with abundance at an accelerating rate. When so�rate decreasing with increasing abundance.
et al., 1998), and territorial birds and mammals (Veitand Lewis, 1996; Wolff, 1997; Aars and Ims, 2000).Exploiting the dynamical analogy with functional
responses, I will refer to the three dispersal modes asType I, Type II and Type III. Note that when so� 1emigration is inversely density-dependent, i.e. it declineswith increasing density to the density-independent ratebi (Fig. 1). Inverse density-dependent dispersal does notseem biologically realistic, given the weight of empiricalevidence for emigration rates increasing with increasingdensity (Wolff, 1997; Aars and Ims, 2000; Sutherlandet al., 2002). I therefore do not consider this case indetail.In the analyses that follow, I assume that the per
capita emigration rate is species-specific and not habitat-specific (i.e. dimensional quantities a1 ¼ a2 ¼ a ) non-dimensional quantities b1 ¼ b2 ¼ b). Relaxing thisassumption makes the analyses much less tractable,but does not alter any of the conclusions.Table 1 characterizes the attractors of the system
described by Eq. (2) for various combinations of r; s andb: The key point to emphasize is that for small b thesystem attains a locally stable equilibrium with the sinkpersisting at a small positive abundance and the sourceat an abundance below its carrying capacity ½ðx�1 ; x
�2 Þ ¼
ðo1; > 0Þ�: For large b and r-0; both source and sinkcontinue to persist at this interior equilibrium. For largeb and r50; extinction of both source and sink results.
2.1. Density-independent dispersal
Previous source–sink models have investigated den-sity-independent dispersal in depth (e.g. Holt, 1985;Wootton and Bell, 1992; Doak 1995). The key point Iwant to emphasize is that source–sink dynamics resultsfrom the negative density-dependence that dispersal
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
abundance
s=-1.5
s=-2.0
(b)
B
ance ðxÞ; and the y-axis, the emigration rate ðbxsþ1Þ: In the baseline caseon to density (panel a). Type II density-dependent dispersal ð�1oso0Þg rate. Type III density-dependent dispersal ðs > 0Þ occurs when the
1 dispersal is inversely density-dependent (panel b), with the emigration
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Table 1
Effects of local demographic parameters ðk; rÞ and dispersal ðbÞ on equilibrium abundances of source and sink populationsa
Exponential population dynamics in the sink [Eq. (2)]
q 0obo1 b > 1
r-0 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ðo1; > 0Þ
r50 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ð0; 0Þ
Logistic population dynamics in the sink [Eq. (3)]
k q 0obo1 b > 1
ko1 r-0 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ðX1;X1Þb
r50 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ð0; 0Þ
k ¼ 1 r-0 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ð1; 1Þ
r50 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ð0; 0Þ
k > 1 r-0 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ðo1; > 0Þ
r50 ðx�1 ;x�2 Þ ¼ ðo1; > 0Þ ðx�1 ;x
�2 Þ ¼ ð0; 0Þ
aThis characterization of equilibria holds for Type I ðs ¼ 0Þ; Type II ðs > 0Þ and Type III ð�1oso0Þ dispersal, except in the case s > 0 and k > 1
where ðx�1o1;x�2 > 0Þ is stable even for very large b:bWhen ko1 and r-0 no interior equilibrium exists for intermediate values of b; implying extinction of both source and sink; at sufficiently high b;
both source and sink persist at their respective carrying capacities.
P. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168162
induces in the per capita growth rates of source and sinkpopulations (see also Holt, 1985, 1993). Paradoxically,this density-dependence can occur even when dispersal isdensity-independent. In the absence of dispersal, thesource population’s per capita growth rate declineslinearly with increasing abundance, becoming negativeonce the source’s abundance exceeds carrying capacity.The sink experiences exponential decay in the absence ofdispersal. The negative density-dependence induced bydispersal causes the per capita growth rates of bothsource and sink to decline with abundance in a non-linear fashion (Fig. 2). Thus, dispersal induces self-limitation in the sink, and increases the strength ofself-limitation already experienced by the source. Thishas two crucial consequences. First, it causes the percapita growth rates of both source and sink to be higherat lower abundances compared to isolated populations(Fig. 2a, b). This is the mechanism underlying the rescueeffect: dispersal increases the ability of species toincrease from low abundances, thus reducing thetendency for small populations to go extinct when rare.Second, the negative density-dependence induced bydispersal causes the per capita growth rate of the sourceto be lower at higher abundances compared to anisolated population (Fig. 2a). This is the cost of dispersalto the source. When x1 > x2; the source suffers a net lossdue to emigration and its per capita growth rate fallsbelow that in isolation, becoming negative well before itreaches carrying capacity (Fig. 2a). The higher the percapita emigration rate relative to the source’s intrinsicgrowth rate (i.e. higher the b), the lower the abundanceat which the source’s per capita growth rate becomesnegative. At sufficiently high b emigration exceeds localreproduction even when the source’s abundance is wellbelow carrying capacity, causing extinction of the sourcepopulation (Fig. 2c).
Thus, the negative density-dependence induced bydispersal is beneficial at low abundances because it canrescue sink populations from extinction, but detrimentalat high abundances because it can drive otherwise viablesource populations to extinction. The key point to noteis that population persistence via source–sink dynamicsrequires the cost of rescuing sinks to be sufficiently lowthat source populations can maintain positive per capitagrowth rates. The above results show that density-independent dispersal can pose significant costs to asource population. A key question is what effect density-dependent dispersal has on source population growth.I investigate this issue next.
2.2. Density-dependent dispersal
The demographic parameter r and the two dispersalparameters b and s provide a complete description of thedynamics of the source–sink system (Eq. (2)). For anymode of dispersal (Type I, II, III), source quality r (ratioof intrinsic growth rates in sink and source) and cost ofdispersal b (per capita emigration rate relative to thesource’s intrinsic growth rate) determine the conditionsfor source population persistence, and hence source–sink persistence.I first present results that are general to all three
modes of dispersal. When the source quality is relativelylow ðro� 1Þ; source–sink persistence requires that thecost of dispersal ðbÞ be sufficiently low that the source’sper capita growth rate does not become negative. Once bexceeds a critical threshold, extinction of the sourcepopulation, and hence the source–sink system, results(Fig. 3). When source quality is relatively highð�1oro0Þ; source–sink persistence is possible evenwhen the cost of dispersal to the source is relatively large(bX1; Fig. 3). This is because the source’s intrinsic
ARTICLE IN PRESS
0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0.5
1
1.5
2
0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0.5
1
1.5
2SinkSource
Per
cap
ita g
row
th r
ate
Source abundance (x1) Sink abundance (x2)
(a) (b)
(c) (d)
0 < Beta < 1
B = 0
Beta >> 1
Beta = 0
0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0.5
1
1.5
2
0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0.5
1
1.5
20 < Beta < 1
Beta = 0
Beta = 0
Beta >> 1
Fig. 2. Per capita growth rate vs. abundance for source and sink populations. In the absence of dispersal the source’s per capita growth rate declines
with abundance linearly. The sink’s per capita growth rate is constant. With dispersal, per capita growth rates of both source and sink decline with
abundance nonlinearly, causing them to be higher at lower abundances, and lower at higher abundances, compared to isolated populations (panels a
and b). The former constitutes the benefit of the rescue effect to the sink and the latter, the cost of dispersal to the source. When b is sufficiently high
emigration exceeds local production even when source abundance is well below carrying capacity (panels c and d), causing the source population to
go extinct. Parameter values are as follows: s ¼ 0; r ¼ �1:5; b ¼ 0:5 for panels a and b, and b ¼ 2:8 for panels c and d.
-2.5 -2 -1.5 -1 -0.5 0
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0
1
2
3
4
5
E
E
P
P
Type I(s = 0)
Type II(-1 < s < 0)
Type III(s > 0)
Cos
t of d
ispe
rsal
(B
eta)
Source quality (Rho)
-2.5 -2 -1.5 -1 -0.5 0
1
2
3
4
5
P
(a) (b) (c)
Fig. 3. Effects of source quality and cost of dispersal on source–sink persistence when the sink exhibits exponential decay in the absence of dispersal.
In all panels, the region marked E depicts the portion of the r� b parameter space that leads to extinction of the source population, and the region
marked P, source–sink persistence with the sink at a low positive abundance and the source below its carrying capacity. In general, source–sink
persistence is greatest for Type III density-dependent dispersal and least for Type II density-dependent dispersal. Parameter values are s ¼ 0;�0:1and 0.05 for panels a, b and c, respectively.
P. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168 163
growth rate is sufficiently high that emigration does notexceed local reproduction. When the source has a highreproductive and/or low mortality rate, benefits of therescue effect outweigh costs to the source.Density-dependent dispersal has an important effect
on the source’s extinction threshold. With Type IIdensity-dependent dispersal, the cost to sources is
much greater compared to density-independent disper-sal (Fig. 3b). This is because the emigration rate isgreater than linear at low abundances (Fig. 1). Becauselosses due to emigration are disproportionately higher atlower abundances, the source’s per capita growth ratealso becomes negative at a lower abundance comparedto density-independent dispersal. The high cost of
ARTICLE IN PRESSP. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168164
dispersal to sources makes the probability of source–sink extinction greatest under Type II density-dependentdispersal. With Type III density-dependent dispersal,cost to sources is much lower compared to density-independent dispersal (Fig. 3c). This is because theemigration rate is less than linear at low abundances, somost emigration occurs at higher abundances whenintra-specific competition is likely to be the strongest.Because it incurs very little cost to sources, theprobability of source–sink extinction is the least underType III density-dependent dispersal (Fig. 3).
2.3. Density-dependent population growth in the sink
The above results were derived for the situation wherethe sink experiences an exponential decline in abun-dance in the absence of dispersal. This is a reasonableassumption when the sink constitutes an inhospitablehabitat of low resource availability within which aspecies cannot maintain itself. However, a populationmay experience a negative per capita growth rate due tofactors other than resource availability. For example,predators or parasites may cause the per capita deathrate to exceed the per capita birth rate regardless ofresource availability. In such a situation it is reasonableto assume density-dependent (e.g. logistic) populationdynamics in the sink, albeit with a negative intrinsicgrowth rate ðr2o0Þ: One may also consider emigrationas being density-independent or density-dependent.Considering logistic growth in the sink leads to the
following system of equations for source–sink dynamics:
dx1
dt¼ x1ð1� x1Þ � b1x
sþ11 þ kb2x
sþ12 ;
dx2
dt¼ rx2ð1� x2Þ � b2x
sþ12 þ
b2k
xsþ11 ð3Þ
with the metric k denoting the ratio of carryingcapacities in the sink and source ðk ¼ K2
K1Þ:
When both source and sink exhibit logistic growth,spatial variation in carrying capacity (i.e. ka1) hasinteresting and somewhat counterintuitive effects onsource–sink dynamics. If the sink has a negative intrinsicgrowth rate due to factors other than resourceavailability (e.g. predators or parasites) then its carryingcapacity may be greater ðk > 1Þ or smaller ðko1Þ thanthat of the source. When k > 1; intra-specific competi-tion is stronger in the source than the sink. In thissituation both source and sink persist at abundancesbelow their respective carrying capacities even whensource quality is high (Figs. 4a–c). This is because thestronger intra-specific competition in the source pre-vents it from pulling the sink into the domain ofattraction of the equilibrium with both populations atcarrying capacity. It is important to note that when thesink has logistic dynamics and a greater carryingcapacity than the source, effects of source quality and
dispersal costs on source extinction are qualitativelyindistinguishable from the case where the sink experi-ences exponential dynamics (compare Figs. 3a–c withFigs. 4a–c).When ko1; the source experiences weaker intra-
specific competition than the sink due to greaterresource availability. Then neither the source nor thesink can persist at intermediate values of r except undervery low levels of dispersal (b-0; Figs. 4g–i). This effectis strongest for Type II dispersal and least for Type IIIdispersal. Dynamically, persistence is impossible be-cause no interior equilibrium exists at intermediatevalues of r unless b is vanishingly small (Fig. 5). Whenthe source quality is high and b not too low, not onlydoes the source rescue the sink from extinction, but italso provides a sufficiently high number of emigrantsthat the sink population can approach its carryingcapacity. For all three modes of dispersal when r-0and kp1; the system moves from the domain ofattraction of the equilibrium ðx�1 ;x
�2 Þ ¼ ðo1; > 0Þ to
that of ðx�1 ; x�2 ÞXð1; 1Þ (Figs. 4g–i).
2.4. Evaluating source population persistence in practice
The conditions that guarantee source populationpersistence are a crucial consideration when applyingsource–sink theory to conservation. The model pre-sented here (Eq. (2)) provides a complete descriptionof source–sink dynamics for a single species with onlythree parameters (four when the sink exhibits logisticdynamics; Eq. (3)). Even so, measuring some of theparameters may prove difficult in practice. The aboveanalyses show that source quality ðrÞ and dispersal costðbÞ are the most important parameters. If the growthrates of sources and sinks can be measured and somemeasure of dispersal exists, one can use the model toevaluate the costs and benefits of preserving sinks. Inpractice, the nature of density-dependence in dispersal isunlikely to be known, but density-independent dispersalprovides a fairly conservative estimate of the extinctionrisk to sources (Figs. 3a, 4a, d, g). If minimizing risks isof the highest priority then the most conservativeapproach would be to assume Type II density-depen-dent dispersal, since this mode of dispersal imposes thehighest extinction risk to sources. It is also biologicallyreasonable for endangered species that persist as smallsink populations from which individuals may preferen-tially leave because of Allee effects.Demographic and environmental stochasticity are
important considerations when estimating extinctionrisk in conservation situations. Although the currentmodel is deterministic, it is relatively straightforward tomodify it to accommodate demographic and environ-mental stochasticity. It should be noted that while thesestochastic forces may have a quantitative effect onsource–sink dynamics by increasing the extinction
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0.5 1 1.5 2 2.5
0.5
1
1.5
2
2.5
0.5 1 1.5 2 2.5
0.5
1
1.5
2
2.5
source abundance (x1)
sin
k a
bu
nd
an
ce
(x
2)
dx2 dx1 dx2 dx1
(a) (b)
Fig. 5. Phase plots for the source–sink system with logistic population dynamics in both source and sink. The zero growth isocline for the source
population is denoted by dx1 and the zero growth isocline for the sink population, dx2: Panel a depicts the situation when there is no interior
equilibrium for intermediate values of r for ko1 and b not vanishingly small (the region marked NQ in Fig. 4, bottom row). Panel b depicts the
situation when the system bifurcates to an interior equilibrium with both source and sink at their respective carrying capacities (the region marked
HP in Fig. 4, bottom row). This requires that the source quality be relatively high, i.e. r-0: Parameter values are: b ¼ 0:5; k ¼ 0:5; s ¼ 0:2; withr ¼ �0:5 for panel a and r ¼ �0:05 for panel b.
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
Source quality (Rho)
Cos
t of d
ispe
rsal
(B
eta)
Type I(s = 0)
Type II(-1 < s < 0)
Type III(s > 0)
k=1
k>1
k<1
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
E E E
E E
E
LP LP LP
HP HPHP
-2.5 -2 -1.5 -1 -0.5 0-3
LPLPLP
1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
E
NQ
LP1
2
3
4
5
-2.5 -2 -1.5 -1 -0.5 0-3
HP
E
NQ
LP
1
2
3
4
5HP
E
NQ
LP
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 4. Effects of source quality and cost of dispersal on source–sink persistence when the sink exhibits logistic dynamics in the absence of dispersal.
In all panels, the region marked E depicts the portion of the parameter space that leads to source extinction, and the region marked LP, source–sink
persistence with the sink at a low positive abundance and the source below its carrying capacity. For kp1 (middle and bottom rows), high source
quality allows source and sink to persist at their respective carrying capacities (region marked HP). For ko1 (bottom row) the area marked NQ
represents the region of the parameter space where no interior equilibrium exists. As with exponential dynamics (Fig. 3), source–sink persistence is
greatest for Type III density-dependent dispersal and least for Type II density-dependent dispersal. Parameter values are: k ¼ 0:5 for panels a, b, andc, k ¼ 2 for panels g, h and i, and s ¼ 0;�0:1 and 0.1, respectively for Type I, II and III dispersal.
P. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168 165
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threshold, they do not alter the qualitative dynamicsthat arise from the interaction between dispersal andlocal dynamics.
3. Discussion
Source–sink dynamics provide a framework forpopulation persistence in spatially heterogeneous envir-onments. Despite much theoretical progress, unresolvedissues exist that hamper the application of source–sinktheory, particularly to conservation problems. Thispaper investigates two such issues: cost of dispersal tosources and the role of density-dependent dispersal.The key new finding is that source–sink persistence
depends on the interplay between source quality and thecost of dispersal, which in turn is mediated by the modeof dispersal. Type III density-dependent dispersal, wherethe emigration rate increases with abundance at anaccelerating rate, imposes the lowest cost to sourcepopulations and hence the least likelihood of source–sink extinction. In contrast, Type II density-dependentdispersal, where the emigration rate increases withabundance at a decelerating rate, imposes the highestcost to sources and the greatest likelihood of source–sink extinction. Thus, density-dependent dispersal per sedoes not enhance source–sink persistence relative todensity-independent dispersal. The exact form of den-sity-dependence is crucial. These results are unaffectedby the nature of population dynamics in the sink,whether exponential or logistic.Previous studies of density-dependent dispersal have
focused on a single dispersal mode where only thoseindividuals forced out of the source emigrate to the sink(e.g. Pulliam, 1988; Howe et al., 1991). While this modeof dispersal is realistic for territorial species that engagein intra-specific interference competition, it does notapply to species who have a greater tendency to emigratefrom lower-density populations due to other types ofsocial phenomena such as reduced predator vigilance ordifficulties in finding mates. A comparative analysis ofdifferent dispersal modes is thus essential for developinga framework for source–sink dynamics that applies to avariety of different taxa.A second important result of the study is that the
negative density-dependence induced by dispersal canhave both positive and negative consequences forsource–sink dynamics. Although dispersal-induced den-sity-dependence has been demonstrated in source–sinkmodels with density-independent dispersal (Holt, 1985,1993), its consequences for source–sink dynamics whendispersal itself is density-dependent have not previouslybeen explored. Dispersal has a positive effect by causingthe per capita growth rate to be higher at lowerabundances compared to an isolated population. Thisis in fact the mechanistic basis for the rescue effect. The
negative density-dependence induced by dispersalincreases the ability of species to increase when rare,thus reducing their extinction risk (Holt, 1993). At thesame time, negative density-dependence induced bydispersal causes the per capita growth rate to be lowerat higher abundances compared to an isolated popula-tion. This is the cost of dispersal to source populations.Because dispersal causes the per capita growth rate todecline with density at a greater than linear rate, thesource’s per capita growth rate can become negative atabundances well below its carrying capacity. Thus,dispersal can cause the sudden extinction of otherwiseviable source populations. The crucial point to note isthat dispersal can induce negative temporal density-dependence even when it is density-independent. Den-sity-dependent dispersal can either enhance or weakenthis effect, depending on the exact nature of the density-dependence.The cost of dispersal to sources is an issue that has
received much less attention than the benefit of therescue effect to sinks. Yet, it is likely to be the mostcrucial consideration when applying source–sink theoryto conservation problems. Preservation of sink habitatshas been advocated as a conservation priority becausethey contain many rare species that are maintained byimmigration (Howe et al., 1991). The analyses presentedhere show that preservation of sinks is a viable optiononly as long as dispersal costs are not so high that thesource itself faces risk of extinction. This prediction isborne out by recent empirical studies showing thatsustained emigration can cause per capita growth ratesof source populations to become negative, despite highrates of local reproduction and recruitment (Gundersenet al., 2001). The model presented here provides a way toassess the costs and benefits of preserving sink habitatswith the use of a few critical parameters. For instance, ifthe empirical estimate of source quality is high (i.e.r-0), connecting sources and sinks would be beneficialunless the cost of dispersal is prohibitively high, i.e. theestimated emigration rate greatly exceeds the source’sintrinsic growth rate ðbb1Þ: If source quality is lowðr50Þ; the most conservative approach would be topreserve the source habitats by themselves instead oflinking them with sinks unless it is known that costs ofdispersal are minimal ðb51Þ or it is possible totranslocate individuals at a rate that would notnegatively affect the source’s per capita growthrate.The model also predicts the biological attributes of
species that influence the extinction risk of sourcepopulations. Type III density-dependent dispersal, whichposes the lowest extinction risk to sources, is most likelyto occur in species that exploit defendable and spatiallywell-defined resources such as nest sites or breedingterritories (e.g. territorial invertebrates, birds andmammals; Birkhead, 1977; Wolff, 1997; Diffendorfer,
ARTICLE IN PRESSP. Amarasekare / Journal of Theoretical Biology 226 (2004) 159–168 167
1998; Sutherland et al., 2002). For instance, in Pulliam’s(1988) source–sink model, individuals emigrate from thesource to the sink only after all breeding sites in thesource are occupied. Because emigration involves only‘‘surplus’’ individuals, it does not affect the source’s percapita growth rate and hence poses no extinction risk tothe source. The Type III density-dependent dispersalmode considered here is a generalization of the dispersalmode assumed in Pulliam’s model to other types ofresources (e.g. food) that are not as well defined ordefendable. Type II density-dependent dispersal, whichposes the greatest extinction risk to sources, may beprevalent in species that experience difficulties in findingmates or avoiding predators at low densities. Forexample, female goldenrod beetles (Trirhabda virgata)emigrate at a higher rate at lower mate densities,regardless of host plant quality (Herzig, 1995). Type IIdensity-dependent dispersal may also occur in animalpollinators that preferentially leave small or low-qualityplant patches (Lamont et al., 1993; Groom, 1998).When evaluating the long-term viability of a source
population it is crucial to measure the per capita growthrate as well as the local reproductive rate. If rates oflocal reproduction and recruitment continue to be highbut the per capita growth rate shows a decline, it couldbe an indication of high emigration losses. High ratesof local reproduction may give a misleading impressionof source quality, if emigration rates are also high(Watkinson and Sutherland, 1995).The model presented here focuses on the role of
source–sink dynamics in population persistence.Source–sink dynamics can also promote species coex-istence (Levin, 1974; Holt, 1985, 1993; Pacala andRoughgarden, 1982; Shmida and Ellner, 1984; Hanskiand Gilpin, 1997; Amarasekare and Nisbet, 2001).However, most theory on species coexistence viasource–sink dynamics considers dispersal to be den-sity-independent. Investigating the role of density-dependent dispersal in species coexistence is an im-portant next step.
Acknowledgements
This research was supported by NSF grant DEB-0129270 and the Louise R. Block Fund from theUniversity of Chicago.
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