interactions between local dynamics and dispersal: insights from single species models

File: DISTIL 134001 . By:CV . Date:17:02:98 . Time:11:34 LOP8M. V8.B. Page 01:01Codes: 5397 Signs: 3575 . Length: 60 pic 4 pts, 254 mm

Theoretical Population Biology�TP1340

Theoretical Population Biology 53, 44�59 (1998)

Interactions between Local Dynamics andDispersal: Insights from Single Species Models

Priyanga AmarasekareDepartment of Ecology and Evolutionary Biology, University of California, Irvine,Irvine, California 92697

Received July 23, 1996

Population persistence in patchy environments is expected to result from an interactionbetween local density dependence, dispersal, and spatial heterogeneity. Using two-patchmodels of single species, I explore two aspects of this interaction that have hitherto receivedlittle attention. First, how is the interaction affected when local density dependence changesfrom negative (logistic) to positive (Allee)? Second, how does dispersal mortality influencepersistence? When local dynamics are logistic, dispersal changes the strength of negative den-sity dependence within patches without much between-patch effect. For example, dispersalmortality reduces the population growth rate and counteracts the tendency towards complexdynamics. Density-dependent disperal amplifies the nonlinearity in the growth rate, thusopposing the stabilizing effects of dispersal mortality. Spatial heterogeneity has little or noeffect on stability. In contrast to the logistic, dispersal under Allee dynamics creates between-patch effects that bring about a qualitative change. Patches that fall below the Allee thresholdare rescued from extinction by immigrants from patches that are above the threshold. Density-dependent dispersal enhances this rescue effect. Persistence below the extinction threshold iscontingent on spatial heterogeneity. Dispersal mortality is of little or no consequence. � 1998

Academic Press

Key Words: dispersal mortality, logistic growth, Allee effects, density-dependent dispersal,population persistence, spatial heterogeneity.

INTRODUCTION

The interaction between local dynamics (changes overtime in births and deaths within a population) and dis-persal (emigration from, and immigration into a localpopulation) has important consequences for populationpersistence. For example, local populations of the but-terfly Melitaea cinixa that are linked by dispersal aremore abundant and hence less susceptible to extinctionthan isolated populations (Hanski et al., 1994). Differen-tial dispersal abilities of clover-feeding insects and theirparasitoid natural enemies cause isolated herbivorepopulations to be released from natural enemycontrol (Kreuss and Tscharntke, 1994). In the Americansouthwest, acorn woodpeckers live in semi-isolated

populations that cannot replace themselves in theabsence of immigration (Stacey and Taper, 1992).

Theory tells us that population persistence in patchyenvironments results from an interaction between localdensity-dependence, dispersal and spatial heterogeneity(Levin, 1974; Chesson, 1981; Kareiva, 1990). Two typesof local density-dependence are likely to be important innature. Negative density-dependence (population growthdecreases with increasing density) causes populations toincrease when rare. Logistic population growth is an exam-ple. Positive density-dependence (population growth de-creases with decreasing density, becoming negative belowa threshold density.) causes populations to go extinct whenrare. Allee population growth exemplifies this situation.

Most theoretical explorations of population dynamicsin patchy environments have focussed on local dynamics

Article No. TP971340

440040-5809�98 K25.00

Copyright ] 1998 by Academic PressAll rights of reproduction in any form reserved.


that are negatively density-dependent and dispersal thatis density-independent (Levins, 1969, 1970; Levin, 1974;Holt, 1985; Hastings, 1993; Doebeli, 1995). An inter-esting question that has not received much attention ishow the interplay between local dynamics, dispersal andspatial heterogeneity is altered when the nature of localdensity-dependence changes from negative to positive.Previous models have also ignored one crucial conse-quence of living in a patchy environment: mortalityduring dispersal.

Here I develop single species models that incorporateboth negative (logistic) and positive (Allee) density-dependence. I examine how dispersal (density-independ-ent and density-dependent) influences local dynamicswhen the environment is spatially heterogeneous andimposes a mortality cost on dispersers.

THE MODEL

Consider a single species occupying a patchy land-scape. The simplest representation of such a situation isa two-patch metapopulation. The dynamics of the i thpatch (i=1, 2) are given by

2Ni=[ f (Ni) Ni&e(Ni) Ni+I] 2t. (1)

The function f (Ni) describes local dynamics withinpatch i. e(Ni) is the fraction emigrating from and I is thenumber immigrating into patch i.

For logistic dynamics, f (Ni)=#i (1&(Ni�K)) where #i

is the per capita growth rate in patch i and K is theuniform carrying capacity of both patches. When #i arenon-negative, populations starting near zero convergeto K.

For Allee dynamics,

f (Ni)=#i \1&Ni

K +\Ni

K&

AK+ ,

where A is the threshold below which population growthin a patch becomes negative. This type of Allee effect(Allee, 1931) can occur due to scarcity of mates, reducedanti-predator defence, inbreeding depression etc. (Lewisand Kareiva, 1993).

The fraction emigrating is given by e(Ni)=:i (Ni�K)s

where :i is the per capita emigration rate from popula-tion i when the population is at carrying capacity, and smeasures the strength of density-dependence in emigra-tion. When s=0, emigration is density-independent andoccurs at rate :.

The immigration term is defined as

I=de(N1) N1+e(N2) N2

2.

A fraction 1&d of the emigrants suffer dispersal mor-tality in transit, and the survivors are distributed equallyover the two patches. Hence, the number of individualsimmigrating into a patch is not directly related to thedensity of that patch.

In this model, local dynamics and dispersal occur con-currently over the time step 2t. When the model isgoverned by discrete-time dynamics, 2t is the time intervalbetween successive generations of an annual species, orthat between successive breeding seasons of a perennialspecies. As 2t � 0, birth, death, emigration and immigra-tion lead to continuous changes in population density,allowing studies of continuous-time dynamics.

The term stability is typically used in the mathematicalsense of attracting nearby initial conditions. Stability canalso be used in a biological sense, to imply a transitionfrom complex to more regular and simple dynamics (e.g.Hastings, 1993; Doebeli, 1995). Throughout, I use theterm to mean biological stability. I will use local stabilityto refer to the former situation.

Interactions between Logistic Local Dynamicsand Dispersal

The dynamics of a two-patch system with logistic localdynamics and dispersal are given by:

2N1=_#N1 \1&N1

K +&:1 \N1

K +s

N1

+d(:1(N1�K)s N1+:2(N2 �K)s N2)

2 & 2t

(2)

2N2=_#N2 \1&N2

K +&:2 \N2

K +s

N2

+d(:1(N1�K)s N1+:2(N2 �K)s N2)

2 & 2t.

Dispersal is density-independent when s=0 and den-sity-dependent when s>0. Spatial heterogeneity isexpressed as differences in dispersal rates betweenpatches.

The behavior of (2) is easier to understand when writ-ten in non-dimensional form (Murray, 1989). Non-dimensionalization reduces the system to a minimal set ofparameters that also highlight the scaling relations

45Local Dynamics and Dispersal


among the various processes underlying the dynamics ofthe system.

The substitutions xi=Ni�K, ;i=:i�#, {=#2t yield thenon-dimensional system

2x1=_x1(1&x1)&;1 xs+11 +

d(;1xs+11 +;2xs+1

2 )2 & {

2x2=_x2(1&x2)&;2xs+12 +

d(;1xs+11 +;2xs+1

2 )2 & {.

(3)

The quantity xi is the fraction of the carrying capacityoccupied by population i. { is the new time metric forupdating x, a composite of the per capita growth rate, #,and the time step over which dynamics occur, 2t. As{ � 0, the system is governed by continuous-timedynamics and can be analyzed by the correspondingdifferential equation system.

The quantity ;i is the ratio of per capita emigrationrate to the per capita growth rate for patch i. When ;i issmall growth predominates emigration, and vice versa.For a population that does not receive any immigrants,persistence requires ;i<1. When immigration is high (i.e.high d ), populations can persist even when ;>1.

Density-independent dispersal. Interactions betweenlogistic dynamics and density-independent dispersalhave been studied extensively (McCallum, 1992; Allen etal., 1993; Gyllenberg et al., 1993; Hastings, 1993; Ruxton,1993, 1994; Stone, 1993; Doebeli, 1995). When the percapita growth rate is non-negative, isolated populationsconverge to the carrying capacity. Increase in the growthrate leads to destabilization of this point equilibrium andtransition to periodic, and eventually chaotic, dynamics.In single patch models, emigration from, and immigra-tion into, the population can convert chaos to simplerdynamics (McCallum, 1992; Ruxton, 1993; Stone, 1993;Doebeli, 1995). In two-patch models, low dispersal rateswith no mortality in transit can convert chaos to locallystable cycles (Hastings, 1993; Gyllenberg et al., 1993;Doebeli, 1995).

Here I illustrate a different stabilizing mechanism fortwo patch models. Mortality during dispersal leads to anet loss of individuals from the system. This reduces theoverall growth rate of the population and hence thetendency to alternatively overshoot and undershoot thecarrying capacity. The effect is most clearly illustratedusing identical patches (;1=;2) with identical densitiesupon which the system defined by (3) converges. Then,2x={[x(1&x&;(1&d ))]. The quantity (1&d ) is themagnitude of dispersal mortality. Higher the mortality,

lower the effective growth rate of the population(1&x&;(1&d )), and lower the equilibrium populationsize x1*=x2*=1& ;(1&d ).

Comparison with an isolated population reveals thestabilizing role of dispersal mortality. For example, aclosed population obeying logistic growth has an equi-librium at 1 which is stable if 0<1<(2�{). With den-sity-independent dispersal, the equilibrium occurs at1&;(1&d ) which is stable if 0<1&;(1&d )<(2�{). Inboth cases, when the equilibrium exceeds 2�{, the systemundergoes a bifurcation to periodic dynamics. However,since the equilibrium population size is lowered due todispersal mortality, a point equilibrium persists at valuesof { that lead to periodic dynamics in the absence of dis-persal. The effect is strongest when there is 1000 disper-sal mortality (d=0).

A similar effect obtains when dispersal is asymmetric(;1{;2). The patch with the higher ;, and hence greaternet loss of dispersers, will be more stable than the patchwith the lower ;. In both symmetric and asymmetricdispersal, stability arises because dispersal mortalityweakens the negative density-dependence within patches.There is little or no effect of spatial heterogeneity.

Density-dependent dispersal. The dynamics of a two-patch system with logistic local dynamics and density-dependent dispersal are given by (3) with s>0. For0<s<1, per capita emigration rate increases at adecelerating rate with increasing density. When s=1, itincreases linearly with increasing density. When s>1,per capita emigration rate increases at an acceleratingrate towards the density-independent rate :.

When dynamics occur in continuous-time, density-dependent dispersal leads to a globally stable point equi-librium as long as ;>0 (Appendix I). Dynamics in dis-crete-time are more complicated. The symmetric disperalcase is again the easiest to analyze (Appendix I). Relativeto the logistic without dispersal, density-independent dis-persal is always stabilizing because dispersal mortalitycounteracts the tendency towards complex dynamics(Table 1). In contrast, density-dependent dispersal whens>1 is always destabilizing with respect to the logistic,i.e., it enhances the tendency towards complex dynamics.This occurs because strong density-dependence inemigration reinforces those in births and deaths, and dis-persal mortality (a linear effect) cannot counteract thisadded non-linearity in the population growth rate. Whens<1, density-dependence in emigration is weaker andhence dispersal mortality has a stronger effect, leading tomore stable dynamics than the logistic (Table 1).

When patches with unequal ; start out in the periodic orchaotic realm, local stability analyses become intractable,

46 Priyanga Amarasekare


TABLE 1

Equilibria and stability conditions for logistic local dynamics and dispersal.

Model Equilibrium Stability criteria

(x*)a { � 0 {>0b

Logistic 1 0<x* 0<x*<2{

Logistic+density-independent dispersal 1&;(1&d ) 0<x* 0<x*<2{

Logistic+density-dependent dispersal1

1+;(1&d )for (s=1) 0<x*+s;(1&d ) x*s 0<x*+s;(1&d ) x*s<

2{

&1+- 1+4;(1&d )2;(1&d )

for (s=2)

a With dispersal, x*<1 because dispersal mortality leads to a net loss of individuals from the system. x* is smallest when dispersal is density-independent (s=0).

b When s=1, x*=1�(1+;(1&d )) and x*+s;(1&d ) x*s=x*+ ;(1&d ) x*=1 which leads to the same local stability condition (0<1<(2�{))as the logistic. When s<1, x*<1�(1+;(1&d )) and s;(1&d )<;(1&d ), making x*+s;(1&d ) x*s<1. Hence, s<1 is stabilizing with respect tothe logistic. When s>1, x*>1�(1+;(1&d )) and s;(1&d )>;(1&d ), making x*+s;(1&d ) x*s>1. Hence, s>1 is destabilizing with respect tothe logistic.

but simulations lead to the same conclusions as above(Fig. 1). As with density-independent dispersal, themajor effect of density-dependent dispersal is to changethe strength of negative density-dependence withinpatches. Spatial heterogeneity is of little consequence.

Interactions between Allee Local Dynamicsand Dispersal

When population growth is logistic, the main effect ofdispersal is to counteract or enhance the tendency forcomplex dynamics. Since Allee population growthinvolves positive density-dependence, dispersal effectsare likely to be different from those on the logistic. Forexample, can dispersal lead to persistence of populationsat densities below the extinction threshold of an isolatedpopulation?

The dynamics of a two-patch system with Allee localdynamics and dispersal are given by

2N1=_#N1 \1&N1

K +\N1

K&

AK+&:1 \N1

K +s

N1

+d(:1(N1 �K)s N1+:2(N2 �K)s N2)

2 & 2t

(4)

2N2=_#N2 \1&N2

K +\N2

K&

AK+&:2 \N2

K +s

N2

+d(:1(N1 �K)s N1+:2(N2 �K)s N2)

2 & 2t,

where A is the population density threshold below whichpopulation growth becomes negative due to Allee effects.Dispersal is density-independent when s=0 and density-dependent when s>0.

Non-dimensionalizing as previously, with the addi-tional parameter a=A�K leads to the discrete system

2x1=_x1(1&x1)(x1&a)&;1 xs+11

+d(;1 xs+1

1 +;2xs+12 )

2 & {

(5)2x2=_x2(1&x2)(x2&a)&;2 xs+1

2

+d(;1 xs+1

1 +;2xs+12 )

2 & {.

When population dynamics occur continuously({ � 0), one obtains the differential-equation systemdxi�d{ with { dropped from the right hand side of (5).

In the absence of dispersal (5) has three steady states:(0, 0), (a, a), and (1, 1). Steady states (0, 0) and (1, 1) arestable, (a, a) is unstable (Appendix II). Hence, isolatedpopulations undergoing Allee dynamics cannot persistbelow the extinction threshold a. As with logisticdynamics, increase in { tends to destabilize the steadystates that are otherwise stable.

I analyze the qualitative dynamics of the continuous-time system by phase plane methods. An excellent intro-duction to this technique is given by Odell (1980). The


File: 653J 134005 . By:XX . Date:10:02:98 . Time:15:06 LOP8M. V8.B. Page 01:01Codes: 1238 Signs: 726 . Length: 54 pic 0 pts, 227 mm

FIG. 1. Effect of asymmetric density-dependent dispersal on a two-patch system under-going logistic local dynamics. Panel (a) depicts a two-patch system in the absence of dispersal. Panels (b) and (c) show dynamics when s�1, and panels (d) and (e), dynamics when s<1. Panel (f ) showsdynamics when dispersal is density-independent (s=0). In all panels, the thick solid line represents patch 1, and the thin solid line, patch 2. Whens�1, dispersal leads to more complex dynamics relative to the logistic, while dispersal when s<1 leads to simpler dynamics. Simulations were con-ducted for 10,000 steps of length {. Dynamics for the last 20 steps are shown. Parameter values are: ;1=0.1, ;2=0.3, d=0.1, and {=2.5.



aim is to show how trajectories for the two patchesbehave in the phase plane. One does this by first plottingthe nullclines, i.e. where dx1=0 and dx2=0, so that thesystem is changing in one direction only. By drawingarrows in the direction of change one can plot the direc-tion of change in x1 and x2 from any given set of initialconditions. The system is at equilibrium only where thedx1=0 and dx2=0 nullclines cross.

Density-independent dispersal. When the two patchesare linked by density-independent dispersal, dynamics ofthe system undergo a qualitative change. Now there areseveral steady states at densities between 0 and K(Fig. 2a). As previously, there are two stable steady statesat 0 and the new carrying capacity determined by the

FIG. 2. Phase portraits for a two-patch system undergoing Allee local dynamics and density-independent dispersal. Panels (a) (;=0.1) and (b)(;=0.2) show symmetric dispersal, and panels (c) (;1=0.05, ;2=0.1) and (d) (;1=0.1, ;2=0.2), asymmetric dispersal. In both cases, low dispersalrates lead to two stable steady states at densities intermediate between 0 and K (panels (a) and (c)). Under high rates of symmetric dispersal boththese equilibria disappear (panel (b)); under high rates of asymmetric dispersal, the equilibrium where the patch with a higher ; has its density abovethe extinction threshold disappears (panel (d)). a=0.4, d=0.8 in all cases.

interaction of local dynamics and dispersal. In addition,there are two (when dispersal is symmetric, Fig. 2a) orone or two (when dispersal is asymmetric, Figs. 2c and d)stable interior equilibria separated by an unstable nodethat determines the extinction threshold. Stability of eachof these equilibria can be investigated analytically (seeAppendix II). The stable interior equilibria representsituations where density of one patch is above the extinc-tion threshold and density of the other patch is below it.When dispersal is asymmetric and ; not too low, there isonly one internal stable steady state which occurs at apoint where density of the patch with the lower ; (i.e. lowemigration relative to local growth) is above the extinc-tion threshold and density of the patch with higher ; isbelow it (Fig. 2d). All stable equilibria are separated fromeach other by saddle points, which together with the



interior unstable node, determine the basins of attractionfor the stable steady states (Fig. 2a).

Thus, when populations undergoing Allee-type localdynamics are linked by density-independent dispersal,they can persist below the extinction threshold for anisolated population. Spatial heterogeneity is a critical

FIG. 3. Effect of dispersal mortality on a two-patch system undergoing Allee dynamics and density-independent dispersal. For both symmetric(panels (a)�(c)) and symmetric (panels (d)�(f )) dispersal, increase in dispersal mortality leads to loss of interior equilibria. At very high dispersal mor-tality there is a net loss of individuals, and both patches go extinct. Parameter values used are: a=0.5, ;=0.1 for symmetric dispersal, a=0.5, ;1=0.1

and ;2=0.2 for asymmetric dispersal.

requirement. For example, if both patches have initialdensities above the extinction threshold they go to carry-ing capacity. If both patches start at densities below thethreshold, extinction results. However, if patches havedifferent initial conditions, immigration from the patchwhose density is above the extinction threshold will



rescue from extinction the patch whose density is belowthe threshold. A similar effect occurs when patches differin growth and dispersal parameters: the patch with thelower ; will rescue the other patch from extinction.

Low levels of emigration and dispersal mortality arenecessary for the existence of the internal stable states. As

FIG. 4. Effect of the Allee threshold on a two-patch system undergoing Allee dynamics and density-independent dispersal. When dispersal is sym-metric, reduction of Allee effects below about 300 of the carrying capacity leads to both populations increasing to their respective carrying capacities.Increase in Allee effects to above 600 of the carrying capacity leads to extinction of both patches. Parameter values used are: d=0.9, and ;=0.1.Similar qualitative results are obtained for asymmetric dispersal as well.

mortality and emigration rates increase these steadystates disappear, and the system collapses to that in theabsence of dispersal, with two stable states at zeroand carrying capacity (Figs. 2 and 3). The extinctionthreshold a has a similar effect (Fig. 4). As a decreases,interior equilibria disappear and the system converges to



the steady state determined by the carrying capacity. Asa increases above 500 of the carrying capacity, bothpatches go extinct.

These results for the Allee model follow directly fromthe perturbation theorem of Levin (1974). Levin used the

FIG. 5. Effect of density-dependent dispersal on a two-patch system undergoing Allee local dynamics. Panels (a) and (b) depict symmetric disper-sal, and panels (c)�(e), asymmetric dispersal. Under density-independent emigration and moderate dispersal mortality (d=0.5), populations cannotpersist below the extinction threshold. Density-dependent dispersal, under the same conditions, makes it possible for both patches to exist below thethreshold. Other parameter values are the same as in Fig. 3.

theorem to show that small amounts of dispersal allowcoexistence of two competing species that wouldotherwise exclude each other. Here, the theoremillustrates how small amounts of dispersal can allow thepersistence of single species populations that would



otherwise go extinct. Briefly, in the absence of dispersalthe system defined by (5) has four stable equilibria:(0, 0), (K1 , 0), (0, K2) and (K1 , K2) where K1 and K2 arethe respective carrying capacities of the two patches.When patches are linked by dispersal the theorempredicts that (K1 , 0) and (0, K2) should move slightly offthe axes to give two new equilibria with one patch abovethe extinction threshold and the other below it. Theseequilibria are stable only for sufficiently small levelsof dispersal. The two homogeneous equilibria (0, 0)and (K1 , K2) remain stable even under high levels ofdispersal.

As has been shown above, these predictions are indeedmet when two patches undergoing Allee dynamics arelinked by dispersal. The same results obtain when {>0,as predited by the discrete version of the perturbationtheorem (Karlin and MacGregor 1972).

Density-dependent dispersal. When dispersal is den-sity-dependent, dynamics of the two-patch system aredescribed by Eq. (5), with s>0.

The general effect of density-dependent emigration isto facilitate the existence of populations below the extinc-tion threshold (Fig. 5). For example, populationsexperiencing symmetric dispersal and moderate dispersalmortality are unable to persist below the extinctionthreshold, but are able to do so when dispersal is density-dependent (Fig. 5). Similarly, asymmetric density-inde-pendent dispersal at moderate ; results in only a singleinternal steady state, while both internal steady statesoccur when dispersal is density-dependent (Fig. 5). Thesequalitative changes occur because when dispersal is den-sity-independent, emigration occurs at a constant ratewhich, if the density falls below the extinction threshold,can lead to extinction. However, when emigration is den-sity-dependent, this constant rate is achieved only at highdensities. Hence, populations are able to persist at alarger range of intermediate densities than when disper-sal is density-independent.

Limitations of Model in Explaining LocalDynamics�Dispersal Interactions

Throughout this analysis I have assumed an islandmode of dispersal where emigrants from all patches, afraction (1&d ) of which suffer dispersal mortality intransit, enter a common pool and are redistributed sym-metrically across all patches. As such, I ignore possibledependence of dispersal on inter-patch distance.Localized dispersal, where individuals move only to thenearest patches, may lead to different conclusions thanmine. Such differences have been observed in multi-patch

analyses of logistic dynamics and density-independentdispersal (Bascompte and Sole, 1994; Hassel et al., 1995).I have considered a situation where local dynamics anddispersal occur simultaneously within a season orgeneration rather than sequentially. While this does notalter the qualitative results I have obtained for interac-tions between dispersal and logistic dynamics, it mayinfluence those obtained for Allee dynamics.

DISCUSSION

Implications of Results

When local dynamics are logistic, the major effect ofdispersal is a modification of existing dynamics. Forexample, dispersal mortality leads to a net loss ofindividuals from the system. This reduces the effectivepopulation growth rate and counteracts the tendencytowards complex dynamics. Density-dependent dispersalopposes the stabilizing effect of dispersal mortality. Itenhances the density-dependence in births and deathsand amplifies the non-linearity in population growthrate, thus enhancing the tendency towards complexdynamics. Dispersal mortality, a linear effect, is not suf-ficient to counteract this added non-linearity. Dispersaleffects arise from changing the strength of negative den-sity-dependence within patches, without much between-patch effect. Spatial heterogeneity has little or noinfluence on stability.

In contrast to the logistic, interactions between Alleedynamics and dispersal create between-patch effects thatlead to a qualitative change in the system. Populationsare now able to persist below the extinction threshold foran isolated population. Patches that are below the extinc-tion threshold are rescued from extinction by immigrantsfrom patches above the threshold. This rescue effectrequires low dispersal mortality and moderate Alleeeffects (30�600 of the carrying capacity). Density-dependent dispersal enhances the possibility of persistencebelow the threshold by making these requirement lessrestrictive. Persistence below the extinction threshold iscontingent on spatial heterogeneity. Dispersal mortalityplays little or no role in generating this outcome.

Although these results for the logistic and Allee modelsare based on two-patch systems, the underlying mecha-nisms are general and should be operative in multi-patchsystems.

The above constrasts are suggestive of important waysin which natural populations may persist. Populationsthat exhibit negative density-dependence (i.e, they



increase when rare) are unlikely to go existinct exceptwhen high per capita growth rates and�or small overlapbetween generations lead to drastic population fluctua-tions. Such fluctuations involve periods of low densitiesduring which stochastic extinction becomes likely (Allenet al., 1993). Under such conditions, dispersal mortalitymay stabilize dynamics and thus reduce the possibility ofextinction. The interesting feature about this mechanismis that is allows populations exhibiting negative density-dependence to persist in spatially homogeneous environ-ments.

For populations experiencing Allee effects in nature,dispersal may reduce the risk of extinction, regardless ofwhether species have overlapping or discrete generations,as long as they inhabit a spatially heterogeneous environ-ment. Random (e.g., unpredictable changes in weather,food sources, or nature enemies) or deterministic (e.g.,permanent differences habitat quality) factors may createspatial heterogeneity in growth and migration rates. Aslong as populations are linked by migration, populationswhose densities fall below the extinction threshold will berescued from extinction. There is abundant evidence toindicate that populations experience a higher risk ofextinction at lower densities due to either stochastic orsystematic factors. Persistence of natural populationsthat do not experience frequent local extinctions andcolonizations may well occur due to an interactionbetween dispersal and Allee dynamics.

Generality of Results

The results obtained for the logistic model show someparallels with two-patch systems of predators and prey.For predator-prey systems with Lotka-Volterra typedynamics, density-independent movement of prey orpredator converts a neutrally stable equilibrium into alocally stable one. A similar conversion occurs whenmovement is spatially density-dependent, but only whenthe degree of density-dependence is weak (Murdoch etal., 1992). These results parallel the effects I have seen forthe logistic model, where density-independent disperalconverts complex dynamics to more simple dynamics,while density-dependent dispersal is able to do this onlywhen s<1 (i.e., weak density-dependence). However,effects of density-independent dispersal occur via two dif-ferent mechanisms. In the single species case, stabilityoccurs because high rates of dispersal mortality reducethe per capita growth rate and therefore the tendency forcomplex dynamics. In the predator-prey models, stabilityoccurs because when dispersal is density-independent,the number of immigrants entering a patch is not relatedto the density of that patch. Hence, the per capita

immigration rate of prey (or predator) declines withincreasing prey (predator) density, thus stabilizingdynamics (Murdoch et al., 1992).

The results for the Allee model are predicted from theperturbation theorem of Levin (1974). Levin has shownthat low amounts of density-independent dispersal canbring about coexistence of two competitors that wouldortherwise exclude each other. My results highlight thegenerality of theorem's peredictions by showing thatsmall amounts of density-independent dispersal canbring about persistence of populations that would goextinct in isolation. This leads to ``coexistence'' of multi-ple populations of the same species.

When dispersal is incorporated to a model of Alleedynamics, popultations can exist at intermediate den-sities as a source-sink system. For example, populationswhose densities are below the extinction threshold(sinks) are rescued from extinction by immigrants frompopulations whose densities are above the threshold(sources). The initial source-sink models were developedfor species with discrete breeding seasons, where breed-ing and dispersal occurred at different times of the year(Pulliam, 1988). Populations increase when rare, andreach an equilibrium determined by the number ofavailable breeding sites in the habitat patch. Patches dif-fer in quality such that populations occupying patcheswith high quality breeding sites have a reproductive sur-plus, while those occupying patches with inferiorqualitybreeding sites have a reproductive deficit. Active disper-sal of the surplus individuals from the source rescues thesink population from extinction. The model I havedeveloped leads to source-sink dynamics regardless ofthe degree of overlap of generations and even when localdynamics and dispersal occur simultaneously, as long aspopulations exhibit at least moderate Allee effects andare linked by dispersal. Dispersal does not have to beactive. As long as there is spatial heterogeneity, density-independent dispersal can lead to source-sink dynamics.These findings suggest that the source-sink concept maybe applicable to a wider variety of natural situations.

APPENDIX I

Stability Analyses for Logistic Dynamics andDensity-Dependent Dispersal

(i) Continuous-Time Dynamics

For symmetric dispersal, the system defined by (3) issimplified to



dxi

d{=_xi (1&xi)&;xs+1

i +d;(xs+1

1 +xs+12 )

2 & , (I.1)

where i=1, 2.Algebraic solutions for the non-trivial equilibrium are

x1*=x2*=1

1+(1&d ) ;for s=1,

and

x1*=x2*=&1+- 1+4;(1&d )

2;(1&d )for s=2.

Local stability of the equilibrium requires (df�dx)x*<0where f is the right hand side of (I.1). Since patches areidentical, f at equilibrium becomes x(1&x*&(1&d );x*s). Then

\dfdx+x*

=(1&x*&(1&d ) ;x*s)

+x*(&1&s;(1&d ) x*(s&1))<0.

The first term vanishes and hence the condition for localstability is x*+s;(1&d ) x*s>0.

This condition is always satisfied, since s, ;, d, and x*are all >0 and d = [0, 1). Hence, when populationdynamics occur continuously, symmetric density-depen-dent dispersal leads to a globally stable equilibrium atcarrying capacity (Table 1).

When dispersal is asymmetric, the non-trivial equi-librium (x1* , x2*) can only be computed numerically.However, local stability of this equilibrium can beinvestigated using phase plane methods. First, nullclines((dxi�d{)=0) are plotted in the phase plane defined byx1 and x2 (Fig. 6a). Equilibrium points occur wheneverthe two nullclines cross each other. Then, tangent linesare drawn to each nullcline at the equilibrium point(Fig. 6b). The configuration of the two tangents providea complete understanding of the system's behavior in theneighborhood of the equilibrium. An excellent introduc-tion to this technique is given in Vandermeer (1972) andBulmer (1976). A more general description of phaseplane methods is given in Odell (1980).

The elements of the jacobian matrix are

a11=\ �f�x1+x1*

=1&2x1*&;1 \1&d2+ x1*

s(1+s)

a12=\ �f�x2+x2*

=(s+1) ;2

d2

x2*s

a21=\ �g�x1+x1*

=(s+1) ;1

d2

x1*s

a22=\ �g�x2+x2*

=1&2x2*&;2 \1&d2+ x2*

s(1+s), (I.2)

where f is the right hand side of (3) for patch 1 and g isthat for patch 2 with { dropped from the equations.

The eigenvalues of (I.2) are a11+a22\((a11&a22)2+4a12a21)1�2 (Vandermeer 1972; Freedman 1987).

a12 and a21 are both positive since s, ;, d, x1* , and x2*are all >0. The signs of a11 and a22 can be determined byexamining f and g on either side of the respectivenullclines (Fig. 6a). As x1 increases from 0, f goes fromf >0 (to the left of x1 nullcline) to f <0 (to the right ofthe x1 nullcline). Hence a11<0. Performing the sameoperation for the x2 nullcline shows that a22<0. Thesesings have intuitive biological meaning. For example, aii

represent the effect of each population on itself. They arenegative because both populations experience a negativefeedback effect of density. aij represent the effect of pop-ulation i on population j. They are positive because eachpopulation augments the density of the other throughimmigration.

The first question is whether (x1* , x2*) is saddle point(a11a22<a12a21) or an asymptotic point (a11a22>a12a21). This can be determined by considering thetangents to the nullclines at the equilibrium point(Fig. 6b). The slopes of x1 and x2 tangents with respect tothe x1 axis are given by tan % and tan ?, respectively.Since both nullclines have positive slopes with respect tothe x1 axis, slopes of their tangents are also positive. Thetangent slopes tan % and tan ? can be related to the par-tial derivatives of the jacobian matrix by implicit differen-tiation of the two nullcline functions f (x1 , x2)=0 andg(x1 , x2)=0. For example,

ddx1

( f (x1 , x2))(x1* , x

2*)

=\ �f�x1+x

1*

+\ �f�x2+x

2* \

dx2

dx1+ fx1* , x

2* ,

and

\dx2

dx1+ fx1* , x

2*=

&\ �f�x1+x

1*

\ �f�x2+x

2*

.



FIG. 6. Phase portrait for a two patch system undergoing logistic dynamics and asymmetric density-dependent dispersal. The quantity s is thestrength of density-dependence in emigration. Panel (a) shows the nullclines for the two patches (dx1 �d{= f =0 for patch 1; dx2 �d{=g=0 for patch2) when s=0.1. When patches are isolated, each stabilizes at its carrying capacity (the point at which the f =0 and g=0 nullclines intersect the x1

and x2 axes respectively). If immigration is allowed and the density of one patch is fixed, the other patch will stabilize at a density higher than thecarrying capacity at zero immigration. As one patch is fixed at higher and higher densities, the density at which the other patch stabilizes increases,thus giving a positive slope to each nullcline. Panel (b) shows an enlarged view of the immediate neighborhood of the non-trivial equilibrium. Thesolid lines labelled x1 and x2 are the respective tangents to x1 and x2 nullclines. The dashed lines indicate the horizontal and vertical axes. % is theangle the x1 tangent makes with the x1 axis. ? is the angle the x2 tangent makes with the x1 axis. Panels (c)�(f ) give completed phase portraits afterlocal stability analysis for s=0.5, 1, 2, and 4.



But, (dx2 �dx1) fx1*, x

2* is the slope of the x1 tangent with

respect to the x1 axis. Hence

tan %=

&\ �f�x1+x

1*

\ �f�x2+x

2*

.

Similarly,

\dx2

dx1+ gx1* , x

2*=tan ?=

&\ �g�x1+x

1*

\ �g�x2+x

2*

.

From Fig. 6b, tan %>tan ? (the tangent function ismonotonic nondecreasing in the interval 0 to ?�2), and

&\ �f�x1+x

1*

\ �f�x2+x

2*

>

&\ �g�x1+x

1*

\ �g�x2+x

2*

.

Using the actual signs of the four partial derivatives,

\ �f�x1+x

1*

\ �f�x2+x

2*

>\ �g

�x1+1*

\ �g�x2+x

2*

.

Hence (x1* , x2*) is an asymptotic point.Since both a11 are a22 <0, (x1* , x2*) is a stable

asymptotic point (i.e. nearby initial conditions converge to(x1* , x2*)). The final question is whether convergence tothe equilibrium is monotonic ((a11&a22)2>&4a12a21)or oscillatory ((a11&a22)2<&4a12a21). Since both a12

and a21 have the same sign, approach to the equilibriumis monotonic.

This completes the stability analysis. The completedphase portraits for various values of s are given in Fig. 6.As long as ;>0 and dispersal mortality is not so high asto cause extinction, asymmetric density-dependent dis-persal leads to a globally stable equilibrium with bothpatches at their respective carrying capacities.

(ii) Discrete-Time Dynamics

When dispersal is symmetric, (3) simplifies to

2xi=_xi (1&xi)&;xs+1i +

d;(xs+11 +xs+1

2 )2 & {. (I.3)

Local stability of the non-trivial equilibrium requires|1+(df�dx)x* |<1 (Nisbet and Gurney 1982) where f isthe right hand side of (I. 3). This leads to the condition0<x*+s;(1&d ) x*s<2�{. The equilibrium becomesunstable at high {.

When dispersal is asymmetric, local stability requireseigenvalues of the system defined by (3) to havemagnitude <1 (i.e., |*i |<1). Both eigenvalues will havemagnitude <1 if |||<2 and 1<1 where |=a11+a22 ,and 1=a11 a22&a12a21 . The conditions for local stabi-lity are the same as for { � 0, with an additional upperbound that involves {. I illustrate this added restrictionby contrasting stability criteria for { � 0 and {>0.

Let p1 , q1 , r1 , s1 denote the elements a11 , a22 , a12 , a21

for the continuous system (I.2). When {>0, a11=1+{p1 , a22=1+{q1 , a12={r1 , and a21={s1 . When{ � 0, local stability ensues if (i) p1+q1<0 and (ii)p1 q1&r1s1>0. When {>0, (i) becomes (&4�{)<p1+q1<0 and (ii), ( p1+q1)+{( p1 q1&r1 s1)<0. As {increases (i) becomes more restrictive. As for (ii), p1+q1<0 and p1 q1&r1s1>0 because the stability of thedifference equation implies that of the corresponding dif-ferential equation, and p1+q1<0 and p1q1&r1s1>0are required for local stability of the latter. When p1q1>r1s1 , (ii) is satisfied only for { sufficiently small such that| p1+q1 |>|{( p1q1&r1 s1)|.

This restriction on stability when there is a finite timeinterval between generations or successive periods ofpopulation growth, is well illustrated in May (1974;Appendix II, pp. 200�202).

The above stability analysis is valid only within therange of {, ;, and d that leads to a point equilibrium.When { is sufficiently large, the system undergoes a trans-ition to complex dynamics. Simulations provide a guideto the effects of dispersal in this case (Fig. 1).

APPENDIX II

Stability Analyses for Allee LocalDynamics and Dispersal

In the absence of dispersal, there are three equilibria,x*=0, x*=1, and x*=a. Local stability criteria forx*=a are 1<a<(2�{)(a&1). Since a<1 by definition(note: a is the fraction of the carrying below which thepopulation exhibits negative growth), this equilibrium isalways unstable. 0 is locally stable if 0<a<(2�{) and 1is stable if 1&(2�{)<a<1.

The two-patch system defined by (5) has multiple equi-libria that can only be solved numerically. Local



stability, when dynamics are continuous, requires thatboth eigenvalues be negative. Elements of the 2x2 matrixare

a11=(1&x1)(x1&a)&;1 \1&d2+ xs

1

+x1 \1&2x1+a&s;1 \1&d2+ xs&1

1 +a12=(s+1) ;2

d2

xs2

a21=(s+1) ;1

d2

xs1

a22=(1&x2)(x2&a)&;2 \1&d2+ xs

2

+x2 \1&2x2+a&s;1 \1&d2+ xs&1

2 + . (II.1)

Both eigenvalues will be negative if (i) |<0 and (ii)1>0 where | and 1 are as defined in Appendix I.

(II.1) does not yield any meaningful algebraic condi-tions for local stability. Hence I evaluated the eigenvaluesnumerically, in order to examine the behavior of trajec-tories in the immediate neigbhorhood of the steady states(Odell 1980). Steady states with both eigenvalues <0correspond to stable notes, and those with both eigen-values >0 correspond to unstable nodes. Steady stateswith one eigenvalue <0 and the other >0 correspond tosaddle points. I used this information to draw thedetailed pattern of trajectories near each of the steadystates (Fig. 2). The phase portraits in Figs. 2�5 give acomplete picture of both local and global stability of thesystem.

When {>0, the equilibria will be locally stable if botheigenvalues have magnitude <1. Numerical evaluationof eigenvalues indicates that steady states with botheigenvalues of magnitude <1 correspond to stablenodes, and those with both eigenvalues of magnitude >1correspond to unstable nodes. Steady states with oneeigenvalue with magnitude <1 and the other withmagnitude >1 are saddle points.

Equation (5) represents an autonomous system andhence the equilibria are independent of the magnitude of{. The magnitude of {, however, affects the stability ofthese equilibria. For instance, let u, v, w, x denote the

elements a11 , a22 , a12 , a21 for the continuous system(II.1). When {>0, a11=1+{u, a22=1+{v, a12={w,and a21={x. Local stability ensues if (i) (&4�{)<u+v<0 and (ii), (u+v)+{(uv&wx)<0. As { increases(i) becomes more restrictive. As for (ii), u+v<0 anduv>wx. Hence, (ii) is satisfied only for { sufficientlysmall such that |u+v|>|{(uv&wx)|. For larger {, eachpoint equilibrium becomes unstable and the systemundergoes a bifurcation to periodic dynamics.

Note added in proof. While this paper was in press,Ruxton et al., (1997. J. Anim. Ecol. 66:289�292) publishedan article illustrating the stabilizing effects of dispersalmortality for another single species model.

ACKNOWLEDGMENTS

This research was supported by NSF Grants DEB�9057331 andDEB�9627259 to S. A. Frank. I am indebted to Steve Frank for help informulating the models. I thank Jordi Bascompte, Jennifer Calkins,Peter Chesson, Steve Frank, Bill Murdoch, and three anonymousreviewers for many helpful comments on the manuscript.

REFERENCES

Allee, W. C. 1931. ``Animal Aggregations: a Study in GeneralSociology,'' Univ. of Chicago Press, Chicago.

Allen, J. C., Schaffer, W. M., and Rosko, D. 1993. Chaos reduces speciesextinctions by amplifying local population noise, Nature 364,229�232.

Bascompte, J., and Sole, R. V. 1994. Spatially induced bifurcations insingle-species population dynamics, J. Anim. Ecol. 63, 256�264.

Bulmer, M. G. 1976. Theory of perdator�prey oscillations, Theor.Popul. Biol. 47, 137�150.

Chesson, P. L. 1981. Models for spatially distributed populations: Theeffect of within-patch variability, Theor. Popul. Biol. 19, 288�325.

Doebli, M. 1995. Dispersal and dynamics, Theor. Popul. Biol. 47,82�106.

Freedman, H. I. 1987. ``Deterministic Mathematical Models in Popula-tion Ecology,'' HIFR Consulting, Edmonton, Canada.

Gyllenberg, M., Soderbacka, G., and Ericsson, S. 1993. Does migrationstabilize local population dynamics? Analysis of a discretemetapopulation model, Math. Biosci. 118, 25�49.

Hanski, I., Kuussaari, M., and Nieminen, M. 1994. Metapopulationstructure and migration in the butterfly Melitaea cinixa, Ecology 75,747�762.

Hassell, M. P., Miramontes, O., Rohani, P., and May, R. M. 1995.Appropriate formulations for dispersal in spatially structuredmodels: Comments on Bascompte and Sole, J. Anim. Ecol. 64,662�664.

Hastings, A. 1993. Complex interactions between dispersal anddynamics: Lessons from coupled logistic equations, Ecology 74,1362�1372.

Holt, R. D. 1985. Population dynamics in two-patch environments:Some anomalous consequences of an optimal habitat distribution,Theor. Popul. Biol. 28, 181�208.



Kareiva, P. M. 1990. Population dynamics in spatially complexenvironments: Theory and data, Philos. Trans. R. Soc. London B 330,175�190.

Karlin, S., and MacGregor, J. 1972. Polymorphisms for genetic andecological systems with weak coupling, Theor. Popul. Biol. 3, 210�238.

Kruess, A., and Tscharntke, T. 1994. Habitat fragmentation, speciesloss, and biological control, Science 264, 1581�1584.

Levin, S. A. 1974. Dispersion and population interactions, Am. Nat.108, 207�228.

Levins, R. 1969. Some demographic and genetic consequences ofenvironmental heterogeneity for biological control, Bull. Entomol.Soc. Am. 15, 237�240.

Levins, R. 1979. Extinction, in ``Some Mathematical Problems inBiology'' (M. Gerstenhaber, Ed.), pp. 77�107, Am. Math. Soc.,Providence, RI.

Lewis, M. A., and Kareiva, P. 1993. Allee dynamics and the spread ofinvading organisms, Theor. Popul. Biol. 43, 141�158.

May, R. M. 1974. ``Stability and Complexity in Model Ecosystems,''Princeton Univ. Press, Princeton, NJ.

McCallum, H. I. 1992. Effects of immigration on chaotic populationdynamics, J. Theor. Biol. 154, 227�284.

Murdoch, W. W., Briggs, C. J., Nisbet, R. M., Gurney, W. S. C., and

Stewart-Oaten, A. 1992. Aggregation and stability in metapopula-tion models, Am. Nat. 140, 41�58.

Murray, J. D. 1989. ``Mathematical Biology,'' Springer-Verlag, NewYork.

Nisbet, R. M., and Gurney, W. S. C. 1982. ``Modeling Fluctuating Pop-ulations,'' Wiley, New York.

Odell, G. M. 1980. Qualitative theory of systems of ordinary differentialequations, including phase plane analysis and the use of Hopf bifur-cation theorem, in ``Mathematical Models in Molecular and CellularBiology'' (L. A. Segel, Ed.), pp. 649�727, Cambridge Univ. Press,Cambridge.

Pulliam, H. R. 1988. Sources, sinks and population regulation, Am.Nat. 132, 652�661.

Ruxton, G. 1993. Linked populations can still be chaotic, Oikos 68,347�348.

Ruxton, G. 1994. Low levels of immigration between chaotic popula-tions can reduce system extinctions by inducing asynchronousregular cycles, Proc. R. Soc. London B 256, 189�193.

Stacey, P. B., and Taper, M. 1992. Environmental variation and thepersistence of small populations, Ecol. Appl. 2, 18�29.

Vandermeer, J. H. 1972. Generalized models of two species interac-tions: A graphical analysis, Ecology 54, 809�818.


interactions between local dynamics and dispersal: insights from single species models

Documents