Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species.

Abstract

Patch occupancy theory predicts that a trade-off between competition and dispersal should lead to regional coexistence of competing species. Empirical investigations, however, find local coexistence of superior and inferior competitors, an outcome that cannot be explained within the patch occupancy framework because of the decoupling of local and spatial dynamics. We develop two-patch metapopulation models that explicitly consider the interaction between competition and dispersal. We show that a dispersal-competition trade-off can lead to local coexistence provided the inferior competitor is superior at colonizing empty patches as well as immigrating among occupied patches. Immigration from patches that the superior competitor cannot colonize rescues the inferior competitor from extinction in patches that both species colonize. Too much immigration, however, can be detrimental to coexistence. When competitive asymmetry between species is high, local coexistence is possible only if the dispersal rate of the inferior competitor occurs below a critical threshold. If competing species have comparable colonization abilities and the environment is otherwise spatially homogeneous, a superior ability to immigrate among occupied patches cannot prevent exclusion of the inferior competitor. If, however, biotic or abiotic factors create spatial heterogeneity in competitive rankings across the landscape, local coexistence can occur even in the absence of a dispersal-competition trade-off. In fact, coexistence requires that the dispersal rate of the overall inferior competitor not exceed a critical threshold. Explicit consideration of how dispersal modifies local competitive interactions shifts the focus from the patch occupancy approach with its emphasis on extinction-colonization dynamics to the realm of source-sink dynamics. The key to coexistence in this framework is spatial variance in fitness. Unlike in the patch occupancy framework, high rates of dispersal can undermine coexistence, and hence diversity, by reducing spatial variance in fitness.

Full text

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vol. 158, no. 6the american naturalistdecember 2001
Spatial Heterogeneity, Source-Sink Dynamics, and the Local
Coexistence of Competing Species
Priyanga Amarasekare1,*and Roger M. Nisbet1,2,†
1. National Center for Ecological Analysis and Synthesis,
University of California, Santa Barbara, California 93101-5504;
2. Department of Ecology, Evolution, and Marine Biology,
University of California, Santa Barbara, California 93106
Submitted September 18, 2000; Accepted June 19, 2001
abstract: Patch occupancy theory predicts that a trade-offbetween
competition and dispersal should lead to regional coexistence of
competing species. Empirical investigations, however, find local co-
existence of superior and inferior competitors, an outcome that can-
not be explained within the patch occupancy framework because of
the decoupling of local and spatial dynamics. We develop two-patch
metapopulation models that explicitly consider the interaction be-
tween competition and dispersal. We show that a dispersal-compe-
tition trade-off can lead to local coexistence provided the inferior
competitor is superior at colonizing empty patches as well as im-
migrating among occupied patches. Immigration from patches that
the superior competitor cannot colonize rescues the inferior com-
petitor from extinction in patches that both species colonize. Too
much immigration, however, can be detrimental to coexistence.
When competitive asymmetry between species is high, local coex-
istence is possible only if the dispersal rate of the inferior competitor
occurs below a critical threshold. If competing species have com-
parable colonization abilities and the environment is otherwise spa-
tially homogeneous, a superior ability to immigrate among occupied
patches cannot prevent exclusion of the inferior competitor. If, how-
ever, biotic or abiotic factors create spatial heterogeneity in com-
petitive rankings across the landscape, local coexistence can occur
even in the absence of a dispersal-competition trade-off. In fact,
coexistence requires that the dispersal rate of the overall inferior
competitor not exceed a critical threshold. Explicit consideration of
how dispersal modifies local competitive interactions shifts the focus
from the patch occupancy approach with its emphasis on extinction-
colonization dynamics to the realm of source-sink dynamics. The
key to coexistence in this framework is spatial variance in fitness.
Unlike in the patch occupancy framework, high rates of dispersal
* Present address: Department of Ecology and Evolution, University of Chi-
cago, Chicago, Illinois 60637-1573; e-mail: amarasek@uchicago.edu.
†E-mail: nisbet@lifesci.ucsb.edu.
Am. Nat. 2001. Vol. 158, pp. 572–584. ? 2001 by The University of Chicago.
0003-0147/2001/15806-0002$03.00. All rights reserved.
can undermine coexistence, and hence diversity, by reducing spatial
variance in fitness.
Keywords:competition,coexistence,spatialheterogeneity,source-sink
dynamics, immigration, dispersal-competition trade-off.
The issue of how species coexist in patchy environments
is central to both basic and applied ecology. When com-
petition for resources is asymmetric, a life-history trade-
off between competitive and dispersal abilities can lead to
coexistence in a patchy environment (Skellem 1951). This
idea has been formalized in the patch occupancy meta-
population framework (Levins 1969, 1970). The patch oc-
cupancy approach assumes that local competitive inter-
actions occur on a much faster time scale relative to
extinction-colonization dynamics (Cohen 1970; Levins
and Culver 1971; Slatkin 1974; Hastings 1980; Nee and
May 1992; Tilman et al. 1994). For instance, when patches
are colonized by both superior and inferior competitors,
there is rapid exclusion of theinferiorspecies.Thisrestricts
the inferior competitor to patches that the superior com-
petitor cannot colonize. The predicted outcome is regional
coexistence, with the two species occupying mutually ex-
clusive subsets of patches in the metapopulation.
Empirical studies of dispersal-competition trade-offs,
however, reveal apatternthat is atodds withthetheoretical
prediction of regional coexistence. For instance, in Lei and
Hanski’s (1998) study of two parasitoid species that attack
the butterfly Melitaea cinxia, the superior competitor (Co-
tesia melitaearum) was absent fromsomehostpopulations,
but thesuperiordisperser(Hyposoterhorticola)waspresent
in all populationssampled.Anotherhost-parasitoidsystem
consisting of the harlequinbug(Murgantiahistrionica)and
its two egg parasitoids (Trissolcus murgantiae and Ooen-
cyrtus johnsonii) also shows a similar pattern with local
coexistence in some patches and the superior competitor
absent in other patches (Amarasekare 2000a, 2000b).
The mismatch between patch occupancy theory and
data may arise from the separation of time scales inherent
in the patch occupancy framework. The assumption that
local dynamics occur on a faster time scale relative to

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Spatial Heterogeneity and Competition573
spatial dynamics restricts the role of dispersal tocolonizing
empty, or locally extinct, patches. In the absence of any
immigration among occupied patches, dispersal cannot
influence local competitive interactions. This decoupling
of local and spatial dynamics eliminates any possibility of
local coexistence. Empirical observations of local coexis-
tence, however, suggest that dispersal may be sufficiently
rapid to counteract competitive exclusion. In fact, the two
parasitoid species that attack M. cinxia appear to exhibit
a dispersal-competition trade-off most convincingly at the
scale of larval groups within local populations (Lei and
Hanski 1998). Movement between such larval groups is
likely to occur on a time scale comparable to local com-
petitive interactions. The harlequin bug and its two
parasitoidsmoveamong
generation basis (Amarasekare 2000a, 2000b). Harrison et
al.’s (1995) study of the insect herbivores of ragwort (Se-
necio jacobea) also demonstrated rapid dispersal of the
cinnabar moth (Tyria jacobeae) and other herbivores (Bo-
tanophila seneciella and Contarina jacobeae) relative to the
time scale of local competition.
These data present a puzzle for theory. Can a dispersal-
competition trade-off lead to local coexistence when com-
petition and dispersal operate on comparable time scales?
Answering this question necessitates a shift in focus from
extinction-colonization dynamics to source-sink dynam-
ics. For instance, the issue now is not whether a superior
ability to colonize empty patches prevents regional exclu-
sion but whether a superior ability to immigrate among
occupied patches prevents local exclusion.
populationsona per-
The Model
We use atwo-patchmodel withLotka-Volterracompetitive
dynamics within patches and emigration and immigration
between patches. The dynamics are given by the following
system of equations:
dX
dt
X
K
Y
iii
p r X 1 ?
x
? f
?d (X ? X ),
xjix,ii
()
K
x,ix,i
dY
dt
YX
K
iii
p r Y 1 ?
y
? f
?d (Y ? Y),
yj
(1)
iy,ii
()
K
y,iy,i
i, j p 1, 2,i ( j,
where Xiand Yiare the abundances of each species in
patch i; fx, iand fy, iare the competition coefficients, and
Kx, iand Ky, iare the carrying capacities for species 1 and
2 in patch i; rxand ryare the per capita growth rates, and
dxand dyare the per capita emigration rates of species 1
and 2, respectively.
We nondimensionalize equation (1) in order to describe
the system in terms of a minimal set of parameters (Mur-
ray 1993). The following transformations,
X
K
Y
ii
x p
i
,y p
i
,
t p r t,
x
K
x,iy,i
K
K
K
K
y,ix,i
a
p f
,a
p f
,
x,ix,iy,iy,i
x,iy,i
K
K
K
K
r
r
x,jy,jy
k p
x
,k p
y
,
r p
,
x,iy,ix
d
r
d
r
xy
b p
x
,
b p
y
,
xy
yield the nondimensional system
dxi
dt
()
p x 1 ? x ? a
i
y ?b (k x ? x ),
ixxix,iji
dyi
dt
()
p ry 1 ? y ? a
i
x ?rb (k y ? y),
iy
(2)
iy,iy ji
i, j p 1, 2,i ( j.
The quantities xiand yirepresent the densities of species
1 and 2 in the ith patch scaled by their respective carrying
capacities, and ax, iand ay, irepresent the per capita effect
of species 2 on species 1 (and vice versa) scaled by the
ratio of respective carrying capacities. Quantities kxand ky
represent the ratio of carrying capacities in the twopatches
for species 1 and 2, respectively, bxand byare the species-
specific emigration rates scaled by their respective growth
rates, r is the ratio of the per capita growth rates of the
two species, and t is a time metric that is a composite of
t and rx, the growth rate of species 1. The dispersal scheme
is such that individuals leaving one patch end up in the
other patch, with no dispersal mortality in transit. This is
equivalent to the island model of dispersal.
We are interested in a life-history trade-off between
competitive and dispersal abilities. We describe such a
trade-off in terms of competition coefficients (the per cap-
ita effect that a given species has on the other) and per
capita dispersal rates. We assume the species and patches
to be otherwise similar (i.e.,
), which means thatK
p K
x,iy,i
This leads to the following simplified two-patch system:
,,
.
r p 1
p f
k p k p 1
x
anda
y,i
y
a
p f
x,ix,iy,i
dxi
dt
()
p x 1 ? x ? f
i
y ?b (x ? x ),
ixix,iji
dyi
dt
()
p y 1 ? y ? f
i
x ?b (y ? y),
iy
(3)
iy,iji
i, j p 1, 2, i ( j.

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574The American Naturalist
In the absence of dispersal (
interactions within each patch lead to threebasicoutcomes
(Volterra 1926; Lotka 1932): coexistence via niche parti-
tioning (,), exclusion via priority effects
f
! 1 f
! 1
x,iy,i
(,), and exclusion via competitive asym-
f
1 1 f
1 1
x,iy,i
metry (, , or vice versa). Wefocusexclusively
f
! 1 f
1 1
x,iy,i
on competitive asymmetry because this is the situation for
which a life-history trade-off is most relevant.
When patches are linked by dispersal, spatial hetero-
geneity becomes an important consideration (Murdoch et
al. 1992; Nisbet et al. 1993). Wedefinespatialheterogeneity
in termsof factorsthataffectthepatch-specificcompetitive
abilities of the two species fx, iand fy, i. For instance, when
and
f
p f
! 1
f
p f
x,ix,jy,iy,j
petitive environment is spatially homogeneous and one
species is consistently superior within all patches of the
landscape. When
f
( f
x,ix,j
,, , or vice versa), competitive rank-
f
1 1 f
1 1 f
! 1
x,jy,iy,j
ings vary over space such that the species that is the su-
perior competitor in some parts of the landscape is the
inferior competitor in the other parts of the landscape.
This type of spatial heterogeneity can arise due to intrinsic
factors such as genetic variability or phenotypic plasticity
in competitive ability (Huel and Huel 1996; Morrison and
Molofsky 1999). It can also arise via extrinsic factors that
affect the species differently. Examples include spatial var-
iation in microclimatic factors, availability of a second,
critical resource (Tilman and Pacala 1993), disturbances
(Connell 1978), and keystone predation (Paine 1966).
We investigate the conditions under which a trade-off
between competition and dispersal can lead to local co-
existence of superior and inferior competitors. We focus
on three specific situations, each motivated by empirical
studies of dispersal-competition trade-offs.
), competitive
b p b p 0
xy
(or vice versa), the com-
1 1
and(e.g.,,
f
( ff
! 1
y,iy,jx,i
Local Coexistence When the Inferior
Competitor Has a Refuge
The first situation we analyze is motivated by Lei and
Hanski’s (1998) study of the parasitoids of Melitaea cinxia.
Their data showaspatial patternoflocalcoexistenceversus
patches occupied only by the superior disperser (inferior
competitor). Can local coexistence result from a dispersal-
competition trade-off?
We assume that the superior competitor does not move
among occupied patches and is restricted to patch 1. Patch
2 is colonized solely by the inferior competitor and serves
as a refuge from competition for that species. The inferior
competitor also moves between the two patches at a rate
by. This scenario leads to the following version of the
model:
dx1
dt
()
p x 1 ? x ? f
1
y ,
11x,1
dy1
dt
()
p y 1 ? y ? f
1
x ?b (y ? y ),
1y
(4)
1y,121
dy2
dt
()
p y 1 ? y ?b (y ? y ),
22y12
with x denoting the abundance of the superior, immobile
competitor and y denoting that of the inferior, mobile
competitor. Note that
f
! 1
x,1
In the absence of dispersal (i.e.,
sink for the inferior competitor. This is because the su-
perior competitor increases in abundance at the expense
of the inferiorcompetitor,preventingthelatterfrommain-
taining a positive growth rate.
Equation (4) yields four feasible equilibria: first, both
species extinct (,,
xyy ) p (
112
competitor at carrying capacity (1, 0, 0); third, inferior
competitor at carrying capacity (0, 1, 1); and finally, the
coexistence equilibrium.
Local coexistence of inferior and superior competitors
requires that the inferior competitor be able to invade a
patch when the superior competitor is at carrying capacity
and that the coexistence equilibrium be stable to small
perturbations in the abundance of both species.
We first investigate whether the inferior competitor can
invade when rare in both patches. In appendix A, we show
that invasion will succeed if
and .
), patch 1 is a
b p 0
y
f
1 1
y,1
0, 0, 0); second, superior
???
(1 ? f ) ? b (2 ? f ) ! 0.
y,1y
(5)
y,1
The inferior competitor can invade when rare under
two situations: if, invasion can occur as long as
f
! 2
y,1
, and if, then invasion is possible only as
b 1 0
f
1 2
yy,1
long as
b ! (1 ? f )/(2 ? f )
yy,1
(app. A) show that the coexistence equilibrium is stable
when it exists.
The key result is that stable local coexistence of inferior
and superior competitors can occur, but is not guaranteed,
as long as there are patches in the landscape that are col-
onized only by the inferior competitor. Immigration from
such refuge populations rescues the inferior competitor
from exclusion in patches that are colonized by both spe-
cies. Coexistence in the face of competitive asymmetry
depends on both dispersal rates and degree of asymmetry.
When competitive asymmetry in patch 1 is low (e.g.,
and ), coexistence occurs provided the
f
! 11 ! f
! 2
x,1y,1
inferior competitor has a nonzero dispersal rate. When
competitive asymmetry is high (e.g.,
), coexistence occurs only as long as the dispersal rate is2
below a critical threshold. When the dispersal rate exceeds
(fig. 1). Stability analyses
y,1
and
f
! 1
f
1
x,1y,1

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Spatial Heterogeneity and Competition575
Figure 1: Conditions for invasion by the inferior competitor when it has a refuge (eq. [4]). The eigenvalue
defined in app. A) is plotted as a function of the competition coefficient (fy, 1) and dispersal rate (by) of the inferior competitor. The surface depicts
the portion of the parameter space where e1is positive (i.e., the inferior competitor can invade when rare). When
as long as. When, invasion is possible only as long as
b 1 0
f
1 2
b ! [(1? f )/(2? f )]
yy,1y
to local growth (i.e., means that rate of emigration from the patch exceeds the local growth rate).
b 1 1
y
(b and c as
2 1/2
e p [b ? (b ? 4c) ]/2
1
, the invasion can occur
f
! 2
y,1
. Note that byis the ratio of per capita emigration
y,1y,1
this threshold, net emigration from source to sink pop-
ulations causes the source population growth rate to be
negative, and the inferior competitor is excluded from the
entire metapopulation.
Local Coexistence When the Inferior
Competitor Has No Refuge
In Harrison et al.’s (1995) study of the herbivores of rag-
wort, no patches were found that were empty of the su-
perior competitor. This suggests that the superior com-
petitor has a colonization ability comparable to that of the
inferior competitors. Our full two-patch model (eq. [3])
describes the situation where no refuges exist for the in-
ferior competitor and both the superior and inferior com-
petitors are able to move among occupied patches. Now
the issue becomes more challenging: Can a superior ability
to immigrate among occupied patches allow an inferior
competitor to coexist locally with a superior competitor?
We first investigate whether the inferior competitor can
invadewhenthesuperiorcompetitorisatcarryingcapacity
in both patches (i.e.,
show that successful invasion requires
). In appendix B, we
, whereI ! 0
?
1
?
2
x p x p 1
I p (1 ? f )(1 ? f ) ? b [(1 ? f ) ? (1 ? f )].
y,1y,2yy,1y,2
(6)
Note that the quantities
initial growth rates of the inferior competitor in patches
1 and 2 in the absence of dispersal (Pacala and Rough-
garden 1982). Thus, the first term of I represents the prod-
uct of the initial growth rates in the two patches and the
second term their sum. The signs of these two quantities
determine whetheror notinvasioncanoccur.Forexample,
if the sum of the initial growth rates is positive and the
product negative,as long asI ! 0
product are negative, then whether or not
on the actual magnitude of by.
We first consider the situation where the competitive
environment is spatially homogeneous. When f
and
f
p f ! 1
f
p f
p f 1 1
x,jxy,iy,j
perior competitor across the metapopulation. Then the
andare the1 ? f
1 ? f
y,1y,2
. If both sum and
I ! 0
b 1 0
y
depends
p
x,i
, species 1 is the su-
y

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576The American Naturalist
sum of the initial growth rates of the inferior competitor
in the two patches is negative (
uct positive (), which means that[1 ? f ] 1 0
y
. The equilibrium with the superior
f ) ? b (2 ? 2f ) 1 0
yyy
competitor at carrying capacity cannot be invaded by the
inferior competitor. Invasion fails because the superior
competitor increases at the expense of the inferior com-
petitor in both patches, causing the initial growth rate of
the latter to be negative across the metapopulation.
The key result is that when the competitiveenvironment
is spatially homogeneous (i.e., one species is consistently
the superior competitor), and when both species have
comparable colonization abilities such that local refuges
for the inferior competitor do not exist, a superior ability
to migrate among occupied patches is not sufficient to
prevent exclusion of the inferior competitor.
) and the prod-
I p (1 ?
2 ? 2f ! 0
y
2
2
The Role of Spatial Heterogeneity
The above analysis shows that invasion fails in a com-
petitively homogeneous environment because the inferior
competitor has a negative initial growth rate in both
patches. This suggests that invasion may succeed if the
inferior competitor can maintain a positive initial growth
rate in at least one patch. Mathematically, this means that
the product of the initial growth rates in the two patches
should be negative (i.e., [1 ? f ][1 ? f ] ! 0
competition is assumed to be asymmetric (i.e.,
or vice versa;
f
1 1 i p 1
y,i
happen is if there is spatial heterogeneity in competitive
rankings such that the superior competitor suffers a dis-
advantage in at least some parts of the landscape (e.g.,
,;;
f
! 1 f
1 1 i, j p 1, 2 i ( j
x,ix,j
That inferior competitors can flourish in areas disad-
vantageous to superior competitors (e.g., keystone pre-
dation) is well known (Harper 1961; Paine 1966; Connell
1978; Lubchenco 1978). The novel issue we explore is
whether dispersal from such areas allows the inferior com-
petitor to persist in areas where the superior competitor
itself flourishes. Because competitive rankings vary across
space, the average competitive ability of each species be-
comes an important determinant of invasion and
coexistence.
When the competitive environment is spatially heter-
ogeneous, invasion can occur under three biologically dis-
tinct, and significant, circumstances. The first situation
arises when competition is asymmetric at the scale of a
local population but spatial averages of competition co-
efficients are such that niche partitioning occurs at the
scale of the metapopulation. For instance, let
in patch 1 and
f
1 1
f
y,1x,2
the average competitive coefficients be f p [(f
and
f )/2] ! 1
f p [(f
? f )/2] ! 1
x,2y,1
y
). Since
! 1
y,1y,2
and
f
x,i
, 2), the only way this can
).
,
f
! 1
x,1
,in patch 2. Let
1 1 f
! 1
y,2
?
x,1
x
. Then, species 1
y,2
is the superior competitor in patch 1 and species 2 is the
superior competitor in patch 2, but neither species is su-
perior in the sense that interspecific competition is weaker
than intraspecific competition when averaged across the
metapopulation. At the metapopulation scale, the twospe-
cies meet the criteria for classical niche partitioning (Vol-
terra 1926; Lotka 1932).
Under global niche partitioning, the sum of the initial
growth rates is positive and the product negative, which
means that as long asI ! 0
b 1 0
the locally superior competitor at carrying capacity (i.e.,
,,,
p 1, 1, 0, 0 or 0, 0, 1, 1) can be invaded
xxyy
1212
by the locally inferior competitor (species 2 or 1, respec-
tively) as long as it has a nonzero dispersal rate (
, respectively).
b 1 0
x
The important point is that as long as competition is
asymmetric locally and niche partitioning occurs globally,
coexistence can occur even in the absence of a dispersal-
competition trade-off. The patch in which the species has
local competitive superiority acts as asourceofimmigrants
for the patch in which it is locally inferior. Thus, source-
sink dynamics allow each species to maintain small sink
populations in areas of the landscape where it suffers a
competitive disadvantage.
The second situation arises when competition is asym-
metric both locally and globally. For example, species 1 is
the superior competitor in patch 1 (
species 2 is the superior competitor in patch 2 (
), but now species 1 is the superior competitor
f
! 1
y,2
when averaged across the metapopulation (
).
f 1 1
y
The species that is the overall superior competitor can
invade when rare as long as it has a nonzero dispersal rate
(i.e.,). The important issue is whether the overall
b 1 0
x
inferior competitor can invade when rare. Global asym-
metric competition means both the sum and the product
of initial growth rates are negative. Invasibility now de-
pends on the actual magnitude of by. Solving equation (6)
for byshows that the inferior competitor can invade only
if its dispersal rate is below a critical threshold:
. The equilibrium with
y
????
or
b 1 0
y
,) and
1 1
x,2
f
! 1 f
1 1
f
x,1y,1
,
and
f ! 1
x
(1 ? f )(1 ? f )
y,1
(1 ? f ) ? (1 ? f )
y,1
y,2
b ! b
y
p
.
critical
y,2
When competition is asymmetric both locally and glob-
ally, local coexistence does not involve a dispersal-com-
petition trade-off. In fact, local coexistence requires that
the dispersal rate of the overall inferior competitor not
exceed a critical threshold. Once the dispersal rate exceeds
this threshold, the overall inferior competitor cannot in-
crease when rare even when it is competitively superior
in some parts of the landscape.
The critical dispersal threshold depends on spatial het-