Vol. 108, No.

960 The American Naturalist March-April 1974

DISPERSION AND POPULATION INTERACTIONS*

SIMON A. LEVIN
Section of Ecology and Systematics,
Department of Theoretical and Applied Mechanics, and Center for Applied Mathematics,
Cornell University,Ithaca, New York 14850

The distributionof a species over its range of habitats is a fundamental

and inseparableaspect of its interactionwith its environment, and no com-
plete study of population dynamicscan affordto ignoreit. This point was
emphasizedover 20 years ago by Skellam (1951) and Hutchinson (1951);
and yet, until recently,the mathematicaltheoryof population dynamics
has largely ignoredspatial considerations.The rise of the theoryof island
biogeographyhas given rise, however,to a renewedinterestin such ques-
tions,particularlywithreferenceto the coexistenceof speciesin a patchwork
environmentof similar habitats (Cohen 1970; Levins and Culver 1971;
Horn and MacArthur1972; Slatkin,in preparation). The generalapproach
of theseinvestigatorsis to focus attentionsimplyon the numberof patches
in which each species is found,intentionallyignoringboth the densitiesof
the species withinindividual patches and the identitiesof occupied patches.
Horn and MacArthur (1972) extend the approach of Levins and Culver
(1971) to considercompetitionbetweentwo species over a mosaic of patches
of two habitat types, labeled 1 and 2, developing for that purpose the
equations
dq1.
= ql[(ll - ell) - rnllql - clP] + rn21q2(- ql)

dq2 (1)
dt (1= q2[ (M22 - e2) - M222- 2P2] + M12q (1- q2)
fortherespectivefractions,q1 and q2, of available patchesactually occupied
by species 2. Analogous equations apply for species 1 and its occupation
fractions,Pi and P2. Here, mijis a coefficient
governingthe rate of coloniza-
tion of habitatj by individualscomingfrompatchesof habitati, so that,for
example,M12q1 (1 - q2) is the rate of colonizationof patchesof habitat2 by
individualsfromhabitat1. Further,et is the rate of local extinctionin type
i patchesin the absence of interspecific
competition,and capj the increasein
that rate due to competition.Horn and MacArthur (1972) introducecs
throughthe statement"species 1 outcompetesspecies 2 in a fractioncl of
the habitats of type 1 that both species co-occupy," and later identifycl
- 1 with the situationwhen species 2 always loses in habitat 1. This some-

* This manuscript is dedicated to the memory of the late Robert H. MacArthur,

whose ingenious effortsin stimulating the development of a mathematical ecology were
unsurpassed. He will be sorely missed.

207
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208 THE AMERICAN NATURALIST

whatmisstatesel's true role,as a decay rate whichmustbear referenceto a

timescale, but the confusionis not centralto theirmain points.
Horn and MacArthur (1972) furtherrestricttheir attentionto the case

-2211 -= M12M21, (2)

since theirmodel is based on random colonizationof patches. Little would
be lost, however,in extendingtheir resultsto the more general case when
thereis no restrictionon the mnjor when,say, colonizationof typei patches
mightconceivablybe most likelyto occur by colonistsfromtype i patches.
In the latter case, the weakenedrestrictionbecomessimply

11m22> M12m21. (3)

Such a relaxationof the model would be particularlyrelevantif the patches

were not randomlyintermingledbut tended to be clumped according to
type. Similarly,habitat selection in a coarse-grainedenvironmentwould
necessitatea conditionsuch as (3) in place of (2).
The model discussed above ignores direct referenceto the sizes of the
individual coloniesor the numbersof habitable patches,these variables by
implicationexertingtheir influencethroughthe various colonizationand
extinctionparameters.
The mainresultof theworkof Horn and MacArthur(1972) is thecriterion
for species 2 to be able to invade a patchworkin whichspecies 1 is already
presentin equilibrialoccupancyfractionsj and 12. That conditionis given
as the inequality
n11 M22
--+ >1 (4)
el + c1pe c*!-f
+ C.,3
If eitherterm on the left exceeds unity by itself,species 2 could invade a
habitat made up entirelyof patches of the one correspondingtype; but
otherwise,it is the existence of both sets of patches which permits the
invasion.
Note that the potentialcoexistencedoes not depend on the fact that there
are two types of patches. What condition (4) requires is simply that the
colonizationcoefficients in11 and 22 be sufficiently high with respect to
e1 + c13i and e2 + c232. Indeed, in an environmentin which there is no
ecological distinctionbetweenthe two types of patches and the labeling is
an arbitraryclassificationwhich labels half of the patches type 1 and half
type 2, it will be the case that ini, = in22, el = e2, and cl = C2. Assuming
P1 ==P2, (4) becomessimplyin1l > -1 (e1 + ciiX), whereasthe appropriate
conditionwhen only the type 1 patches are habitable is the more stringent
MI > el + elp1. Since type 1 patches are ecologicallyidentical to type 2
patches, the only altered ingredientwhen type 2 patches are present is
that there are twice as many patches. What this means is that coexistence
may become possible simply due to a doubling of the numberof patches,
leading to a doubling of the colonizationrates througha doubling of the

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 DISPERSION AND POPULATION INTERACTIONS 209

numberof potentialcolonists.There is no real connectionto the existence

of two types of patches. The result is more a kind of Allee effect(Allee
1939) in that invasion of a regionis only possible if sufficientnumbersof
supportive colonies of the species are present to reinforceand replace
colonies lost to extinction.When the species is low in numbers,growthis
slow because recolonizationis slow. As it expands,so does the recolonization
rate and hence the overall growthrate.
Condition (4) appears at firstglance to be independentof the inter-type
colonizationcoefficientsM12 and Mn21;but this is an artifactcaused by as-
sumption (2). Without (2), (4) is replaced by the conditionthat either
ml,/(el+ cl1ap)> 1 or M22/(e2 + C2i32)> 1 (species 2 can persistin one
patch typealone) or else
Bull n122 VlhIII22 - M2M21
+ .1 + A Cc2--. (5)
l + ClP1i + c.p2 (e1 + c 1 1) (2 , 2)
Thus, if inter-typecolonization for the invader (species 2) is reduced
without a correspondingincrease in intra-typecolonization,invasion be-
comes more difficult, again due to a reductionin the overall recolonization
rates. Condition (5) is howevernot dependenton condition(3).
Note that a generalloweringof the barriersto migrationfor both species
will not necessarilymake the task easier for an invader, since it is to be
expectedthat such a loweringof the barrierswould lead to an increase in
the equilibrial values P1 and P2. Success in invasion by species 2 is thus
indirectlyrelated to differentialcolonization,that is, to the colonization
rate of species 2 relativeto that of species 1.
Slatkin (in preparation.), in reconsideringthe problem analyzed by
Levins and Culver (1971), has made points which apply equally well to
model (1). He points out that the model assumes the probabilityof extinc-
tion,say, of species 2 in patches of a particulartype to be related to inter-
specificcompetitiondirectlythroughthe fractionp of patches of that type
occupied by species 1. Slatkin argues correctlythat a more appropriate
schemeis an extensionof Cohen's approach (Cohen 1970), which considers
four possible states for each patch: with or withoutspecies 1 and with or
withoutspecies 2.
It is worthwhileto rephraseand examinein more detail the point raised
by Slatkin. Denote by (i, j) the four possible states of a patch, where
i, j
- 0 or 1. [Here, (1, 0) indicatesthe presenceof species 1 alone; (0, 1),
only species 2; (1, 1), both; (0, 0), neither.] Slatkin's objectionwould not
be important if the covariance D =E(ij) - E(i)E(j) were zero. In
general,however, this is not the case, and such an assumption,in fact, is
not consistentwith the premisesof the model. Hence, the objectioncannot
be easily dismissed.Indeed, D is undergoingdirected change and may be
expected to stabilize in the negative region,facilitatingcoexistence.Model
(1) is valid for the considerationof invasion,the case of most interestto
Horn and MacArthur (1972), since the random colonizationhypothesis
implies that initially D -0. For the question of coexistence,however,a

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210 THE AMERICAN NATURALIST

modifieddescriptionis necessary which includes considerationof the D

values for the two habitats.The problemsuggestsvery stronglyan analogy
with the population genetics problem of the considerationof two-locus
gameticfrequencies,with,for example,the "gamete" Ab correspondingto
patchesin state (1, 0). In the contextof this generalapproach,the fraction
of patches with both species present will be pq + D, species 1 alone
p(1 - q) - D, species 2 alone q(1 - p) - D, neither species (1 - q)
(1 - p) + D. The existenceof two types of patches as in the Horn and
MacArthur (1972) model,of course,would add a furtherminor complica-
tion.
When the modificationssuggested are made, the general approach de-
scribed above presentssome strikingpotentialadvantages. For example, it
allows one to confrontthe question of extinction,a stochasticevent at the
level of the individual patch, and to treat it deterministically
by considera-
tion of a large aggregateof patches. Thus, whereas individual patches may
be constantly entertainingnew tenants, including " fugitives" whose
survival depends on rapid dispersal and effectiverecolonization,the over-
all species densities for the mosaic should be more predictable and may
indeed reach steady state. These ideas are not restrictedto the fugitive
fromcompetition,such as the insect Cori=a dentipes cited by Hutchinson
(1959). They apply equally well in other contexts,for example, to the
"fugitive" prey mites in Huffaker's laboratory predator-preysystem
(Huffaker 1958) or to the prickly pear Opuntia in its flightfrom the
herbaceousmothCactoblastiscactorum(Dodd 1940, 1959; Nicholson1947).
Huffaker'sexperimentsprovide a classic example of the role of differential
dispersal abilitiesin allowingtwo species to coexist.
Althoughthe coexistencedescribedabove is based on a balance between
local extinctionsand recolonizations,complete obliterationof local demes
is not a requirement.The essential elementis the temporalfluctuationof
local population densities,which translatesinto fluctuationin the environ-
mentfacing the various species. Moreover,it is not a concept restrictedto
discreteenvironments, being equally relevantin continuousones. The only
analytical modificationnecessarywhen local fluctuations,rather than ex-
tinctions,providethe means for coexistenceis the replacementof the gross
"occupancy fraction" by some more general spatial average. The fugitive
survivessimplydue to its abilityto take advantage of local fluctuationsin
interspeciespressures, fluctuationswhich the fugitive's own movements
may help maintain. A spatially uniform"equilibrium" would mean ex-
tinctionof the fugitive.
In the patch occupancy models, population densityvariations within a
patch are ignored.Indeed, what is being assumedis that over the shortrun,
the densities within each patch will reach an equilibrium which can be
entirelycharacterizedby the presence or absence of the various species.
On the slower time scale, equilibrium is viewed as a balance between
colonizationand extinction.The technique,*hen valid, simplifieswhat may
be a very complicated mathematicalproblem when population densities

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 DISPERSION AND POPULATION INTERACTIONS 211

are considered. For example, when one is dealing with fugitive species,
considerationof the equations which are directedto the "fast" time scale,
and thus to the transientdynamicsof the intrapatchdensitiesof the in-
dividual species, must involve the difficultsearch for stable nonconstant
solutions (e.g., limitcycles). On the slowertimescale, however,equilibrium
can be attained in termsof the fractionof patches occupied, althoughthe
specificdispersionpatternswill necessarilyvary with time. For such cases,
therefore,this approach has real advantages. It has apparent advantages
as well for the considerationof invasion by introducedspecies, especially
in situations where the sizes of individual colonies are relatively unim-
portant.When, however,the specificdispersal patternis crucial, as would
be the case if migrationrates between patches depended on geographical
proximity,a different approach is mandated. This mightnot be a difficulty
for the type of epidemic problemsconsideredby Cohen (1970) where in-
dividuals (or perhaps families or communes) are highly mobile patches
withoutfixedgeographicalposition. However, for infestationproblemsas
posed for example by the gypsy moth,the geographical spread is of es-
sential importance.
The migration-extinction approach is similarly not adequate when the
possible equilibriumcolonies may have more than the limited number of
characterizationsindicated, nor obviously when interestmust be focused
on the fast time scale. Moreover, even when the approach is valid, it
representsa simplificationof the full equations,that is, those which allow
for considerationof intrapatchdensities.For all of these reasons, I now
discuss the full equations.

GENERAL FORMULATION

The approach describedin this sectionis the classical one used in dealing
with "diffusion-reaction"systemsin many applied contexts; and similar
"diffusionequations" may be found in Skellam's work (Skellam 1951).
Further,the understandingof the integrationof spatial and temporalor-
ganization is one of the central problemsof theoreticalbiology and the
subject of much currentresearch (e.g., Goodwin and Cohen 1969; Keller
and Segel 1970; Othmer and Scriven 1971, 1973; Karlin and McGregor
1972; Gierer and Meinhardt1972). Much currentwork (Keller and Segel
1970; Othmerand Scriven 1971, 1973; Gierer and Meinhardt 1972) has
taken inspirationfromthe seminal paper of Turing (1952), which was an
attemptto explain the breakdownof symmetryin morphogenesis.
In the formdevelopedby Othmerand Scriven (1971), the equations are
very general and serve as a startingpoint for the developmentgiven here.
Considern species distributedover an interconnected networkof m patches.
The densityof speciesi (i = 1, . n. , n) in patch [t ([t = 1, ... , m) is denoted
xA. Withinpatch ,t,ignoringmigrationsbetweenpatches,the overallgrowth
rate of species i is labeled fA(x), where x1 is the vector (X11a,.. . .,xn) of
densitiesof all speciesin patch st.The functionsfig are arbitrary,exceptthat

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 212 THE AMERICAN NATURALIST

theyare assumedto be definedand continuouslydifferentiable on an open set

containingthe biologicallyrealisticregionR (xi > 0 for all i), and further
thaton R, fg'(X/h) > 0 if xji - 0. The last conditionis commonsense,stating
simplythat a nonexistentspecies is in no danger of declining.Migrations
into the patch from outside the network,however,might establish the
species there,so that fiAf
(XA') need not be zero. The net migrationof species i
fromotherpatchesto patch , is denotedJi.A.
In the simplestcase, the net exchangefrompatch v to patch [t is propor-
tional to xi,"- xA, a passive migrationwith nonnegativeconstantof pro-
portionalityDiat (Dew'ulis arbitrarilydefinedas zero for all [X). With this
simplification,Jjatakestheform

Ju E
DillD (xi, -d xie). (6)
v'=>
More generally,if for example predators do not diffuserandomly,but
ratherin responseto a prey gradient,or if fugitivespecies do not colonize
randomly,but are able to seek out unoccupied areas, then a differentform
of (6) would result (see [8] and [9]). This would similarlybe true in the
importantcase when emigrationis density-dependent.However, if (6)
wereto be employed,thegoverningequationsforthesystemwould become
91n (7)
dxiP/dt- Fi(X, D) - fill(xA)+ JjA= f4A -
(xA) + .Dt"(xiP -
1

where i=1, .. ., n and , ..., ,and where

= (X', x2 Use)

and
D _(Djlll .. . .. I Dltmp..
Ditnl . . .,I D11"m DnMM).
Thus, the formof the equations denotesthe dependenceof the growthrates
not only on the species densitiesbut on the parametersDvAlas well. The
analogous discrete versions were considered by Karlin and McGregor
(1972) ; resultsobtainedbelow are continuousanalogues of theirresults.
More generally,the governingequations of the systemare given com-
pactly by

dxjI/dt= Age(X, D ), (8)

wherePFg'(X,0) fin(xg). Here 0 is the zero vectorin the rnM2
parameter
space. Note furtherthat the assumptionspreviouslymade on the functions
figand thenonnegativityof theelementsofD guaranteethat
F,(X, D) > O if x O and X e X, (9)
and the Fig are continuouslydifferentiable.
Equations (8) and (9) summarizethe' essential propertiesof the dy-
namics. Moreover,as indicated earlier, (8) and (9) describe a touch more

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 DISPERSION AND POPULATION INTERACTIONS 213

general frameworknot tied, for example, to the colonizationassumptions

underlying(6).
Using this framework,two questions of biological interestare assessed
in succeedingsections: (i) To what extent does the spatial componentof
the environmentlead to the coexistenceof species which could not coexist
within a single patch? (ii) To what extentis the behavior of the system
affectedby migrationrates?

COMPETITION IN PATCHY ENVIRONMENTS

The outcomeof interspeciesinteractionsmay not be completelydeter-

minatebut may depend in an essentialway on initial densities(Park 1962;
Slobodkin 1961). Species which are able to reach and colonize areas first
may be able to establishthemselvesand therebyresist invasionsby species
that mightotherwiseexclude them.Indeed, the possibilityof such a situa-
tion is inherentin even the simplestof competitionmodels,provided only
that parametersare properlychosen. It occurs,for example,in the Lotka-
Volterracompetitionequations,rewrittenas
dxldt = x (B ax by),
dy/dt=y(S- cx-dy),
providedinterspecific
competitionoutweighsintraspecific:
a/c < R/S < b/d. (10)
Here x and y representspecies densities.Note that in this formulation,
the "saturation values" (Slobodkin 1961) are R/a and S/d respectively,
and the "coefficientsof competition" are respectivelyb/a and c/d. The
situation is also present in much more general models and, indeed, will
occurwheneverthedefiningequationsare of theform
dxi/dt=fi(x), i =n (11)
and possess more than one stable equilibrium (or, more generally,more
than one stable attractor).
Considerthena networkof m initiallyidenticalpatches,in each of which
consideredalone one or more of the n species would becomeextinctbut in
each of which every species has the potentialto survive given a sufficient
lead. Then, for any specifiedcollectionof species,thereexists a numberN
such that,if thenumberof patchesm exceedsN and if thereis no migration
betweenpatches,a stable configuration is possible with all of those species
present.(B elow I show that the "no migration" conditioncan be relaxed.)
In particular,if m > N, then'a stable arrangementwould be possiblewith,
say, species 1 (and perhaps other species) established in patch 1 (and
perhaps elsewhere),,species 2 in patch 2, etc.
If some migrationis now allowed betweenpatches (D74 0 but D "close
tot"0), a perturbationtheorem(Appendix 1) applies and a new equilibrium
results with all species present. In this case, the equilibriumnumber of

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 214 THE AMERICAN NATURALIST

species in the mosaic is dependenton initial densitiesand the connected-

ness D of thenetwork.In particular,if the matrixAi = (de") is irreducible
(Gantmakher1959), then species i will at equilibriumbe representedin
every patch. In general, in a patchy environment,individual patches (or
islands) may be expected to vary substantially in their species lists
(Simberloffand Wilson 1969, 1970; Root 1973; Paine, personalcommunica-
tion).
The above results correspondto similar results derived in the discrete
time systemby Karlin and McGregor (1972).
Assuming that extinctionsof local populations due to external factors
occur on a slowertimescale than that of intrapatchdynamics,the individ-
ual patches may be reasonably regarded as "equilibrium patches," in
equilibria which track the changes in externalfactors,and a colonization-
extinctionapproach to the expectedequilibriumnumberof species becomes
relevant.The theorypresentedin this sectionmay be consideredas directed
to the fast time scale and would be complementedby a colonization-
extinctionapproach to the colonization-extinction equilibrium. Such an
approach in general would have to considerthe distributionsin sizes and
relativegeographiclocations(throughD) of thevarious equilibriumpatches
and would seek the correlationbetweencommunitiesas a functionof dis-
tance, as did Kimura and Weiss (1964), Weiss and Kimura (1965), and
othersin consideringstepping-stonemodels of geneticcorrelation.
In summary,coexistenceis made possible in a patchy environmentbe-
cause of a scenario wherebyinitially identical patches subject to random
colonizationdiverge in species lists, througha kind of founderprinciple,
culminatingin. a "linked " joint equilibrium in which a much higher
diversityresultsthan would be possible in a single patch. Diversity (num-
ber of species) generallyincreasesas the numberof patches increasesuntil
it reachesits maximumpossiblevalue. However,since in some sense perfect
mixing (equivalently,a single homogeneouspatch) is the limitingcase as
the interpatchelementsof P become infinite,diversityeventuallymay be
expectedto decrease as migrationincreasesbeyond some critical threshold.
This result needs to be made more precise,however,and computerstudies
of the relation of the equilibriumnumberof species to D would be very
enlightening.
As pointed out elsewhere (Levin 1970), stable coexistenceis only pos-
sible provided the effectivenumber of limitingfactors is as great as the
numberof species. When, for example,two species are limitedby a single
resource,no stable equilibriumexists.A patchy environment may, however,
increase the numberof limitingfactors,say by increasingthe number of
resource or prey species, and furtherby making relevant local densities
withinpatches ratherthan overall densities.Though the environmentmay
have been uniform initially, it becomes heterogeneous(in the sense of
Smith [1972]) as a result of what may be random initial disturbances.
Obviously,this scenario depends neither6n the initial uniformitynor on
the randomnessof disturbances.What is strikingis that it can occur even
in the face of these.
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 DISPERSION AND POPULATION INTERACTIONS 215

Note that in this example coexistenceoccurs because of the spatial com-

ponentof the environment but is not dependenton fluctuationsin local den-
sities. Further,highermigrationfates tend to homogenizethe systemand
to reduce the potentialfor coexistence.This contrastswith coexistencedue
to fugitivestrategies,wherehigh migrationrates do not homogenize(quite
the contrary) and coexistenceis the result of sustained fluctuationsin the
environment.Patchiness does not require the existenceof multiple types
of patches. The above resultswere derived under the assumptionthat all
patcheswereidentical,preciselyto emphasizethispoint.If, however,patches
are of a variety of types, the perturbationtheoremstill applies; that is,
small amounts of migrationdo not destabilize an arrangementwhich is
stablewithoutmigrations.To quote Karlin and McGregor(1972), "complex
systemswhen combinedwith slight migrationbetweenthem produce even
more complexsystemswith more possibilitiesand representationsof stable
polymorphisms. "

COMPETITION BETWEEN TWO SPECIES OVER TWO PATCHES

To exemplifythe general results just given, this section deals with the
simplestcase, two species in competitionin two patches which differonly
with regard to the species densities. For simplicityof illustration,the
dynamicsintroducedearlier,
dx/dt=x(R-ax-by),
(12)
dy/dt_ y(S -x-dy),
are assumed within each patch. (Again, the qualitative results do not
depend on such oversimplified equations.) Equivalence of the two patches
means that the same parametersR, S, a, b, c, and d apply in each patch.
Assume furtherthat a/c < R/S < b/d, condition (10). This means that
coexistenceis not possible withina single patch but that eitherspecies can
establishitselfgiven a sufficient
lead.
As is well known,the system (12) has four equilibria (fig. 1), stable
ones at (R/a, 0) and (0, S/d) and unstableones at 0 = (O, 0) and
(Rd-Sb -Rc+Sa
U ad-bc' ad-beb
When the two patches are consideredtogether,and migrationis passive,
thecorrespondingequationstake the form
dxi/dt= xi(R - axi - byi) + D(xj -x),
(13)
dyildt= yj (S - xi dyi) + Dy(yj - y), i, j=1, 2; i:& j
whereD, and D1,are the species-specific
migrationrates betweenpatches.
In the completelysymmetriccase, when the species behave identically
except for the (symmetric) effectsof interspecificcompetition,R = $,
a - d, b = c, and D, = Dv. Congenericspecies might reasonably be ex-
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216 THE AMERICAN NATURALIST

S/d

\OU

C >
R/a
FIG. 1.-The fourequilibriaforthesystem(12)

pected to approximatethis behavior,if sufficiently close, but the choice is

made heresimplyforease of illustration.In thiscase,thesystembecomes
dx/dt = xi(R - ax -bye) + D(xj - ),
(14)
dyi/dt= y(R- bxi -ay)+D(yj- y), i, j=l,2; i =/j,
wherethesubscripton D has been dropped.Equation (10) becomessimply
a < b. (15)
For small D, the qualitativebehaviorof (14) is the same as that for (13),
but the symmetry makesarithmeticcalculationa great deal simpler.Hence,
(14) will be analyzedin closerdetail.
Note firstthattheuncoupledequations
dx/dt=x(R-ax-by)
dy/dt= y(R - bx-ay)
have fourpossibleequilibria (see fig.2):
I: x-O. y O0 (0)
II: x 0, y R/a,
III: x =R/a, y =0,
IV: x-R/(a + b), y= R/(a + b). (U)
In lightof (15), only II and III are stable,and correspondto the exclusion
of one or the other species. Moreover,almost every solution tends either
to state II or to III, dependingon whichspecies is initiallymoreabundant.

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 DISPERSION AND POPULATION INTERACTIONS 217

(OR/o)
Basin of
attraction
for :

Basin of
/(Ra+bR/a+b) attraction
for m

(0,0) E
( R/o,Q)
FIG. 2.-The four possible equilibrium states for the system x = -ax - by),
p p i-( bx - ay), with basins of attractionshown.

Only the thin ridge x - y of initial conditionsdoes not lead to II or III

beingperchedprecariouslybetweenthetwobasins of attraction.
When D - 0, the system(14) clearly has 16 possible equilibriumstates,
since it is still uncoupled and each patch has four possible equilibria. How-
ever,onlyfourof these,denoted(II, II), (II, III), (III, II), and (III, III),
are stable. When D > 0, the four homogeneousequilibriumstates (I, I),
(II, II), (IIT, III), and (IV, IV) remainand withthe same stabilitychar-
acteristicsas previously,respectivelyunstable,stable, stable, and unstable.
The coupling destroys10 of the other equilibria not under the protection
of the perturbationtheorem,since they were initially unstable, but the
remainingtwo, (II, III) and (III, II), are the mostinterestingof all. Due
to the symmetry, it suffices
to considerjust (II, III).
When thereis no coupling,the system(14) as stated permitsan equilib-
riumat (II, III), thatis, at

X1 , Y1 =- Ra, X2 = R/a, Y2 = 0.

This equilibriumis shown in figure3a, where 1 denotes the equilibrium

in patch 1, 2 the equilibriumin patch 2; and again in figure3b. In short,
the equilibriumdiscussedis one in whichspecies 2 alone prevailsin patch 1,
and species1 in patch 2.
When D > 0 and is small, the perturbationtheorem (Appendix 1)
guarantees that the equilibrium (II, III) does not disappear. Rather, it
movesslightlyofftheaxes. To be precise,for

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218 THE AMERICAN NATURALIST

00 000 0K 0
0 000 0
00 0
0 0~ 0
000 0 0
0
2/ ~~~0 0 0

PATCH I PATCH 2
a b
FIG. 3.-a, The equilibrium (II, III) for the system (14) when D = 0. b, An alternate
representationof the equilibriumin a. Circles indicate individuals of species 2; diamonds,
individuals of species 1.

-
0 D <:, 2 * b+ a(16)
2 b+a'
(14) has an equilibriumat
R 2D 1 If b+a
Xi Y2 - -l R-2DJIR- 2D __
2a 2a a/

X2 Y
R-
2a
2D
-+
1
2a
B- D -2D
b+a
- a/
V (17)

It is easy to check, using (15) and (16), that these values are real and
(for D > 0) positive.Furthermore,as D tends to 0, the equilibriumclearly
tends to (II, III), that is, to x1 Y2 0 and X2 - 1 - R/a, as pre- -

dicted by the perturbationtheorem. Finally, note that as D tends to

(R/2) * (b -a)/(b +a), the equilibrium tends to the homogeneousone
(IV, IV), that is, to
X1 =Y2 = X2y R/(b + a),
whichexplains condition(16).
The perturbationtheoremdoes not guarantee that (17) remains stable
but only that it does so forD small enough.Condition(16) is not sufficient
to guarantee this, and in fact it may be shown (Appendix 2) that the
equilibriumis stable for
R b-a
2 2b +a (18)

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 DISPERSION AND POPULATION INTERACTIONS 219

but unstablefor
R
b a<D b-a (19)
2 2b+a 2 b+a
The entiresituationis summedup in figure4. In figure4a, as D increases
from 0 toward the thresholdvalue (R/2) * (b -a)/ (b +a), the points
(xi, yi) and (X2, Y2), representingthe equilibriumdensitieswithinthe two
patches,movesymmetrically towardeach otheralong thehyperbola
ax2 + 2bxy+ ay2 - Rx Ry =0,
finally coalescing when D hits the value (R/2) (b - a)/(b + a). The
joint equilibriumis stable only until D reachesthe value (R/2) * (b -a)/
(2b + a), whichcorrespondsto the equilibrium
R b?2a 1? 72A1

2a 2b+a 2a(2b + a) -p,

R b + 2a + R
X2 1 V b+ 2ab.
2a 2b+a 2a(2b+a)
In summary,when there are at least two patches in the environment,
coexistenceof two species that would otherwiseexclude each other is pos-
sible. Each establishesitselfin one patch sufficientlyto withstandinvasion,
and each is found in the otherpatch due to sustained migrationfromthe
favored territory.When the migrationis too high (condition 19), mixing
is rapid enoughthat thereis effectively only a single patch, and coexistence
is no longerpossible.
Global analysis for this case has been performedby the author and
L. E. Payne and will be publishedseparately.

, O

R/be 0R/ab2Rab

X ,,, O o o0 o

b R/0 PATCH I PATCH 2

a b
FIG. 4.-a, The nonhomogeneous competitivecoexistenceequilibriumfor the system
(14) whenD > 0 and is small. b, An alternaterepresentation of the equilibriumin a.
individualsof species1.
Circlesindicateindividualsof species2; diamonds,

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 220 THE AMERICAN NATURALIST

THE DEVELOPMENT OF SPATIAL AND SPATIO-TEMPORAL PATTERNS

Opportunitiesfor movementand habitat diversification provided by the

of
spatial aspect environment make possible in a variety ways coexistence
of
of species which could not otherwisesurvive together.If the environment
is heterogeneous,differentcombinationsof species are likely to be favored
in the various local regions and maintained elsewhereprincipally by dis-
persal frommore favored regions,and this will act to increase the overall
speciesrichness.This spatial heterogeneity in environment may be externally
imposed, consistingprincipally of variation in weather,climate, edaphic
factors,etc., or, as shown earlier, it may arise as a result of a divergence
betweensubregionsor patches due to an essentiallyrandom variation with
respectto colonization.In any case, such spatial patternsare steady state,
and are not relatedto temporalfluctuations in local densities.
When species life historypatterns are such that local populations have
periodic dispersal episodes, or when the environmentvaries in time, a
premiumis placed on how fast species can get to and utilize choice areas.
In this case, opportunityexists for species which would otherwisebecome
extinct to survive as fugitives,distributedaccording to spatio-temporal
patterns involving consistentlyfluctuatinglocal population densities. As
with the steady-statestrategy,the fugitivestrategycan arise in an initially
spatially uniform environment,since the spatio-temporalfluctuationsin
environmentmay be stronglycoupled in a feedback relationshipto the
fluctuationsin species densities.Such situationsmay be studied by seeking
stable nonconstantsolutionsto (7) or continuousanalogues of it, typically
of theform
ax, = f4(X,z) +V (DjVxi),
at
where V denotesthe gradientwith respectto the spatial variables,Z. Such
models,discreteor continuous,provide as well the startingpoint for the
computation of the spatially nonuniformbut steady-statedistributions
describedpreviously.This computationis a difficult mathematicalproblem,
but an elegant beginning to its solution may be found in Othmer and
Scriven (1973).
Such spatial patterns,both steady state and temporallyvarying,may
arise in a varietyof ways. They may be, as statedearlier,determinedby the
patternsof initial colonizationin newly available areas. Alternatively,they
may arise simplydue to invasionepisodes of sufficient intensityto allow the
invader to becomeestablishedin some locale or througha fugitivestrategy.
Finally, theymay arise througha change in eitherthe species densitiesor
in the basic interactionsbetween species, when the change is sufficient to
destabilize a previously uniform coexistence pattern. One particularly
strikingexample,owing in motivationto Turing (1952); is that in which
an increase in the ability to migrate by some species can destabilize the
system.This phenomenonmay be termeddiffusiveor dissipativeinstability
(Segel and Jackson 1972). Many results discussed below parallel results

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 DISPERSION AND POPULATION INTERACTIONS 221

discoveredin the spatially continuouscase by Segel and Jackson (1972). As

with fugitivespecies, the lack of perfectmixing (over the spatial region)
is responsible,since this effectivelyintroducesa destabilizingtime lag.
Basically, diffusiveinstabilitiesarise when diffusionor migration de-
stabilizesan otherwisestable situation,in contrastto one's usual intuitions
concerningthe effectsof diffusion.The theoryhas been exploredby Turing
(1952), Othmerand Scriven (1971, 1973), Keller and Segel (1970), Segel
and Jackson (1972), and others,principally in developmentalcontexts.
The effectof such instabilitiesis that,for example,a predator-preyrelation-
ship which permitsstable coexistencewhen migrationrates are low might
become destabilizedwhen the ability to migrateincreases. The two species
mightstill coexist,but in an oscillatorypatternsimilar to that caused by
the fugitivespecies or in a steady-statebut spatially nonuniformdistribu-
tion such as discussedabove. Althoughthe prey could persisteven without
oscillations,thereis clear advantage to individual prey to migrateto escape
predation,and this may lead to evolutionarypressurestoward destabiliza-
tion. From the "viewpoint" of the predatorpopulation,increasedmigration
becomesa necessityto stabilize an otherwiseunstable situation and allow
survival.
Diffusiveinstabilitiesdo not arise in purelycompetitivesystems.Further,
they only arise in predator-preysystemswhen one species is rare enough
thatan Allee effectis applicable; thatis, theyarise onlywhenthe per capita
rate of growthof the rare species is an increasingfunctionof densitydue,
say, to an increased ability to find mates. The Allee effectis inconsistent
withstabilityin thepurelycompetitivecase.
By way of example,considerthe interactionbetweenan abundant prey
and a rare predator,describednear equilibriumby modifiedLotka-Volterra
equations:
dx/dt =x(K - ax-by),
dy/dt=y(-L+cx+dy).
These differfromthe usual Lotka-Volterrapredator-preyequations only in
thesign of d, whichreflectstheAllee effect.
Theseequationspermita stableequilibriumat
Lb- Kd Ko-La
bc ad' Y be-ad'
providedad < be and
a K a b+d= a a bec-ad
<
L
< = + * ~~~~~~~~(20)
c d a+c c ed a+c
The correspondingsystem over two patches, with differentialmigration
,u,v included,
coefficients
dx/dt= x(K - axm- bye) + I(x - ),
(21)
dyi/dt= y,(-L + cxi+ dye)+ v(y - yj), j = i,

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222 THE AMERICAN NATURALIST

obviouslypermitsthe equilibrium
Lb-Kd _ _ Kc -La
xl $2 -
be - d
bc-ad' 1= Y2 be- a
bc-adt
but (Appendix 3) this equilibriumis unstablefor large ,uand small v, that
is, for

2F+-)
(v- 2 ) < 4
This inequalitydefinesa region in (pa, v) -space with boundary the hyper-
bola

+ ax, dyl bcZ191

2 2 ~~~~4
The regionsof stabilityand instabilityare illustratedin figure5.
Thus,an increasein mobilityof theprey,expressedas increasedmigration
,u,will lead to a destabilizationof the balance betweenpredatorand prey.
However no rate of migrationby the prey.is too great for the predatorto
overcomethroughincreasedabilityto migrateon its ownpart. In particular,
if v > d51/2,thesystemis always stable.
This linear approach to the destabilizationof the predator-preysystem
does not distinguishbetweenthe alternativeswhich result: (i) attainment
of a new steady state,very possibly spatially nonuniform,(ii) limit cycle
oscillations,in which predator and prey undergo their familiar coupled
temporaloscillations,and (iii) extinctionof the system.Moreover,it is not
clear how commonthis phenomenonis in nature. It is, however,an in-
triguingpossibilitywhichlends itselfto laboratorytesting.
Finally, cooperativeeffectssimilarin effectto the Allee effectmay be more
in
common prey than in predator,as increasingprey population size may
decrease the per capita predationload (Rosenzweigand MacArthur 1963;
Rosenzweig1969). Resultssimilarto thosediscussedin thissectionare found
for systemsin whichtheseeffectsoccur (Segel and Jackson1972).

Region of
stability

Region of instability

FIG. 5.-Stability diagram for the homogeneousequilibriumin the predator-prey

system(21).

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 DISPERSION AND POPULATION INTERACTIONS 223

SUMMARY

The spatial componentof environment,oftenneglected in modeling of

ecologicalinteractions,in generaloperatesto increasespecies diversity.This
arises due to the heterogeneity of the environment, but such heterogeneity
can arise in an initially homogeneousenvironmentdue to what may be
random initial events (e.g., colonizationpatterns), effectsof which are
magnifiedby species interactions.In this way, homogeneousenvironments
may becomeheterogeneousand heterogeneousenvironmentseven more so.
In patchy environments, distinctpatches are likelyto be colonizedinitially
by different species, and therebya kind of foundereffectresults whereby
individual patches evolve along differentpaths simply as a consequence
of initial colonizationpatterns. Species which would be unable to invade
may neverthelesssurviveby establishingthemselvesearly and will moreover
be found in lower densitiesin other areas as overflowfromtheir "safe"
areas. Spatially continuous environmentsmay evolve toward essentially
patchy ones by this kind of process.Overall species richnessis expectedto
be higherin patchy environments but to decreaseas the abilityof species to
migratebecomeslarge. These resultsare due to patchinessper se and do not
depend on the existenceof several kinds of patches,a situationwhich will
tend to reinforcethese effects.
Diversityis also increased in such environments with spatial extentdue
to the opportunitiesfor fugitive-typespatio-temporalstrategies.In these,
local population oscillationsprovide the salvation for species which are for
example competitivelyinferioror easy victimsto predationbut which can
surviveby superiormigratoryability and (in patchy environments)talent
forrecolonization.Again, dependenceis on spatial heterogeneity, in addition
to temporalheterogeneity;again, this may be externallyimposed or the
resultlargelyof internalprocesses.
Some gross statisticsfor these processes,principally patch occupancy
fractions,may prove useful for a simplifiedtreatmentof colonization-
extinctionequilibria, as in the approaches of Cohen (1970), Levins and
Culver (1971), Horn and MacArthur (1972), and Slatkin (in preparation).
For such considerations,however,one cannot assume independenceof dis-
tributions;and the approach of Cohen (1970) and Slatkin (in preparation),
whichallows for considerationof covariance,is favored.

ACKNOWLEDGMENTS

I gratefullyacknowledge support under NSF grant GP33031X and

numeroushelpfulcommentsby and discussionswith Peter Brussard, Jamie
Cromartie,Hugh Gauch, Charles Hall, Alan Hastings, Carole Levin' L. E.
Payne, David Pimentel,Richard B. Root, Michael Rosenzweig,Lee Segel,
MontgomerySlatkin, J. Daniel Udovic, George White, and Robert H.
Whittaker.

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224 THE AMERICANNATURALIST

APPENDIX 1
PERTURBATIONTHEOREM
The theorem quiteintuitive
belowis mathematically and is an analogueof one
provedin Karlin and McGregor(1972) in the discretetimecase. As theypoint
forpopulationdynamics
out,itsimplications are substantial.
Theorem:Assume that FsiL(X,D) > 0 when xl, 0 and X >, 0, and that the -

system dx~i'/dt= FPt(X, Do) has a stable equilibrium (the eigenvalues of the
Jacobian matrixhave negativereal parts) at X X0 > 0. Then for D sufficiently
close to Do, the system dxe"/dt Fe (X, D) has a stable equilibrium at some
point XD , 0, whereXD tends to X0 as D tends to Do.
The notationX) 0 meansthat all components of X are nonnegative. The
quantityFig is assumedto be continuously in X and D.
differentiable
The proofof theperturbation theoremis givenin Levin (in preparation)and
restson theconditionFg(X, D) > 0 whenxe = 0, whichassuresthattrajectories
do notcrossthecoordinate planeintothenegativeregion.

APPENDIX 2
STABILITY ANALYSIS FOR (17)
Here I provethat(17) is stableif D < (R/2) *(b - a)/(2b + a) and unstable
if D > (R/2) (b - a)/(2b + a). For clarity,
thebar notationdenotesequilibrium
valuesto freex1,Yl,x2,and y2 formoregeneralusage.Thus,
R 2D .1 II' b+a
Y1=112=2 2
2a - UR -2D BR- 2D~b- a)'
2aJ
BR- 2D 1 I \ a
+-JI -2DJIR-- 2D___
$2 =Y1- 2a + 2a - b -a
Setting

Y(1-1 \
U = I
Y2- J2
(X2 -
2
one obtainsthe systemof equations
and linearizing(14) about the equilibrium,

d-^- (12 T)J+ DK (s) K) U + ...............(22)

dt
Here

12-(0 1) K_ (1 0)
} -R 201 - by, D- xA
\ byl R -2aV, - DJ
bxl-~,
(R-b2 -2a2-D ? -bV2 D
-by2 )r - 2aZ2 - b2 DJ
and ? denotesthetensorproductof twomatrices.(For an excellentintroduction
to thisnotationand themethods used in thisAppendix,see Othmerand Scriven
withcommon
[1971].) Sincethematrices12 and K are self-adjoint spectralrepre-

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 DISPERSION AND POPULATION INTERACTIONS 225

a theorem
sentation, of Friedman(1956) applies.The stabilityof (22) is deter-
minedby theeigenvaluesof 12 ? J+ DK ? K, whichare theeigenvalues of the
matricesJ ? DK. But
J~~~~
-D = 2( by, -D +D- bXI

The foureigenvalues
of thesetwomatrices
will all havenegativereal partsif and
onlyif the (common)traceT of thesematricesis negativeand the determinants
positive.But
T = 2R - (2a + b) (Y, + Yg) - 2D

212R?- (2a+b) -2D

a
b 2D
a a
2(b+a) _ b
a 2 b+a
Thus,by (16), T is alwaysnegative.
Finally,thedeterminants willbe positiveif and onlyif
(R- 2a51-b1 -D) (R - bi - 2ag1 -D) > (D + by1)(D + byi),
thatis, if and onlyif
[(R-2ax1) - (by,+?D) [(R-2a-y) - (b7I+D)]
> (D + bY,) (D + by,).
Thiscondition to
simplifies
(R - 2aX1)(R -2a1) > (b1 +?D) (R-axl) + (by,+ D) (R-2ady),
and further
to
R2 - 2aR(x1+ Y,) + 4a2x1g1 > 2DR +
(Rb - 2aD) (x1 + Y1) - 2ab(X12+ V72).

Simplifying oneobtains
further,
R(R -2D) + [2aD- R(b+2a)](x1+l1)
>- 2ab (Y. + pl)2 + 4a(b -a) Yl.
Substituting
R - 2D D(R -2D)
XI + Y1= -, xy
a a(b- a)
one gets

R (R-2D) + [2aD-R (b +2a) ] R2D

> -2ab (R-2D) + 4D(R-2D)

a2

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226 THE AMERICAN NATURALIST

Since R - 2D > 0, this becomes

R+-[2aD-- B(b+2a)] >-2-(R-2D) +4D,

a a
that is,

R(b ) > D (? + -)
aa
Thus, the equilibriumis stable if
<R b-a
D<-.
2 a+2b
and unstableif
R b -a
2 a+2b

APPENDIX 3
EQUATIONS
STABILITYANALYSISFORPREDATOR-PREY
The stabilityof the homogeneousequilibrium
Lb-Kd _ -
-La
X.,=X =I
=be - bc-ad
ad ~be-ad
b -ad23
for the system(21) depends on the eigenvalueshaving negative real parts for the
equilibriumJacobian matrix:

- 0 V
Mz: = | iCY dyg 0
tt 0 -ay., -t - bx I
0 V C~l dy, - v

- I
ax, 1
- 2 ( -bxl + ( (t ,

Using the theoremof Friedman (1956) employedin Appendix 2, one findsthat the
of thematrices
ofM are theeigenvalues
eigenvalues
-a1 -bXj \ ( -a 2[t -bY1
- 2,u
\ cy dy1 kn cyl dy1- 2v/
By the stabilityof the equilibriumwhen A - v 0 the criterionwhen iz, v 7L 0,
Oreducesto
as was t dylo
-brv
tt + 2Jv 2 >
as was to be proved.

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