J. theor. Biol. (2002) 218, 273–288doi:10.1006/yjtbi.3075, available online at http://www.idealibrary.com on

Interplay between Local Dynamics and Dispersal in Discrete-timeMetapopulation Models

Abdul-AzizYakubu*wz and Carlos Castillo-Chavezyz

wDepartment of Mathematics, Howard University, Washington, DC 20059, U.S.A. zDepartmentof Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca,

New York 14853, U.S.A. and yTheoretical and Applied Mechanics Department, Cornell University,Ithaca, New York 14853, U.S.A.

(Received on 16 August 2001, Accepted in revised form on 3 May 2002)

The effects of synchronous dispersal on discrete-time metapopulation dynamics with local(patch) dynamics of the same (compensatory or overcompensatory) or mixed (compensatoryand overcompensatory) types are explored. Single-species metapopulation models behave assingle-species single-patch models, whenever all local patches are governed by compensatorydynamics. Dispersal gives rise to multiple attractors with complex basin structures, wheneversome local patches are under overcompensatory dynamics. In mixed systems, dispersal iscapable of altering the local dynamics from compensatory to overcompensatory dynamicsand vice versa. Examples are provided of metapopulation models supporting multipleattractors with intermingled basins of attraction.

r 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Efforts to understand the impact of intraspe-cific competition on the life history dynamics ofa local (single) patch population have a longtradition in ecology. Investigations on its role atthe metapopulation level (populations of popu-lations) were initially carried out in genetics byWright (1931), ecology by Levins (1969 a,b) andepidemiology by Ross (1909), and Kermack &Mckendrick (1927). In 1993, Hastings (1993),Gyllenberg et al. (1993), Gonzalez-And !ujar &

*Corresponding author. Mathematical and TheoreticalBiology Institute (MTBI), Department of Biometrics forUndergraduate Research, Cornell University College ofAgriculture and Life Sciences, 432 Warren Hall, Ithaca NY14853-7801, U.S.A.. Tel.: +1-607-255-81-03; fax: +1-607-255-4698.E-mail address: ayakubu@fac.howard.edu ay32@cornell.

edu (A.-A. Yakubu).

0022-5193/02/$35.00/0

Perry (1993) studied a two-patch discrete-timemetapopulation model of coupled logistic differ-ence equations. Their work showed that theinteraction between local dynamics and sym-metric dispersal can lead to the replacement ofchaotic local dynamics by periodic dynamics forsome initial conditions. Doebeli (1995) showedthat this stabilizing effect is enhanced if dispersalis asymmetric.

Ecologists often focus on differences in qua-litative dynamics as an important indication ofthe underlying processes (Hastings 1993). In thispaper, we focus on compensatory (equilibriumdynamics) and overcompensatory (oscillatorydynamics) dynamics, two types of densitydependence intraspecific competition for re-sources (Begon et al. 1996; Brauer & Castillo-Chavez, 2001; Castillo-Chavez & Yakubu, 2001;Hassell, 1976; Hassell et al. 1976; Hastings, 1993,

r 2002 Elsevier Science Ltd. All rights reserved.

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ274

1982; Levins, 1969; May & Oster, 1976; May,1977, 1974; May et al. 1981; Maynard Smith &Slatkin, 1973; Milton & B!elair, 1990; Nicholson,1954; Ricker, 1954; Royama, 1992; Yodzis,1989). Compensatory dynamics is also referredto as pure contest competition because only aconstant number of competitors survive in asingle-patch system. The term was initiallyproposed by Nicholson who contrasted it withpure scramble competition, the most extremeform of overcompensatory dynamics, where allthe competitors may be so adversely affected byeach other that none of them survive (Begonet al. 1996; Hassell, 1976; Henson & Cushing,1996; Nicholson, 1954).

The study of the role of dispersal in contin-uous-time metapopulation models is extensive(Allen, 1987; Gadgil, 1971; Hastings, 1982;Levin, 1974, 1992). Here, we focus on theimpact of dispersal on local populations withdiscrete non-overlapping generations whenmetapopulation dispersal is synchronous (Bestet al. 2001; Castillo-Chavez & Yakubu, 2001;Doebeli, 1995; Gonzalez-And !unjar & Perry,1993; Gyllenberg et al. 1993; Hastings, 1993).The local population dynamics are assumedto be compensatory, overcompensatory or ofmixed types.

In Section 2, we review the impact ofcompensatory and overcompensatory dynamicson ‘‘unstructured’’ single-species, single-patchdiscrete-time models. We use the Beverton–Holt,the bobwhite quail ‘‘hump-with-tail’’, the Ricker,and the Smith–Slatkin models to describe eithercompensatory or overcompensatory or mixeddynamics (Brauer & Castillo-Chavez, 2001;Castillo-Chavez & Yakubu, 2001, 2002 Doebeli,1995; Doebeli & Ruxton, 1997; Earn et al. 2000;Franke & Yakubu, 1996, 1992, 1991; Hassell,1976; Hassell et al., 1976; Hastings, 1993; May &Oster, 1976; May, 1997, 1974; May & Hassell,1981; Maynard Smith & Slatkin, 1973; Milton &B!elair, 1990; Nicholson, 1954; Ricker, 1954;Yodzis, 1989). Only pioneer species, that is,species that thrive at very small populationdensities when left to their own natural devices,are considered (Franke & Yakubu, 1996, 1992,1991).

May et al. (1981) showed that the timing ofdensity effects and parasitism in host–parasite

systems can have a profound impact on thedynamics, and Doebeli (1995) showed thatdifferences in the timing of reproduction anddispersal enhance the stabilizing effect of dis-persal. The basic single-species metapopulationmodel consisting of several pioneer populations(with non-overlapping generations) connectedby post-reproductive dispersal is introduced inSection 3. Dispersal could be multidirectional,bidirectional or unidirectional. Situations inwhich dispersal occurs before reproduction arenot considered here since the results do not seemto differ significantly (Doebeli, 1995; Doebeli &Ruxton, 1997).

In Section 4, we show that metapopulationmodels under the same qualitative local com-pensatory dynamics behave as single-patchsystems, whenever the timing of reproductionand dispersion do not differ from patch to patch.The results of Section 4 are closely related tothose of Hastings and Levin on the stabilizingrole of dispersal on continuous-time metapopu-lation models (Hastings, 1982; Levin & Paine,1974).

In general, dispersal can give rise to multipleattractors with interesting basin structures,whenever the local patch dynamics are over-compensatory (Alexander, et al. 1992; Doebeli,1995; Doebeli & Ruxton, 1997; Hanski, et al.

1995; Hofbauer, et al. 2001; Kan, 1994; Lai et al.1996). In Section 5, several examples areintroduced to show that post-dispersal metapo-pulation models are capable of supportingmultiple attractors, whenever the pre-dispersallocal patch populations are either on identicaln-cycle (non-chaotic) attractors or identicalinterval (chaotic) attractors. Section 6 focuseson the nature of the basins of attraction of thesecoexisting attractors. In particular, it is shownthat post-dispersal metapopulation models arecapable of supporting coexisting attractors withintermingled basins of attraction. That is, dis-persal can make it impossible to determine withabsolute certainty the set of initial populationsizes that lead to each attractor.

In Section 7, the study of two-patch models iscarried out under mixed local dynamics. Severaloutcomes are possible, for example, the post-dispersal metapopulation model is capable ofsustaining an attractor with a period completely

Fig. 1. Bobwhite quail model with ki ¼ 0:65; Ki ¼ 1:15and ni ¼ 3:

DISCRETE-TIME METAPOPULATION MODELS 275

different from those supported by the pre-dispersal local dynamics; dispersal can stabilizeor shift the local dynamics from a stable periodfour to a stable period two or to a stable fixedpoint. That is, period-doubling reversals arepossible. Section 8 discusses some possibleimplications of the results of this manuscript.Relevant mathematical details are collected inthe Appendix.

2. Local Patch Dynamics

The metapopulation approach was ‘‘de-signed’’ to model situations where the landscapeis naturally fragmented (Hanski et al., 1995;Hanski, 1991; Hanski & Gilpin, 1997). Hence,the framework used applies to populations thatcan be modeled under the following set ofassumptions: space is discrete; there is a distinc-tion between habitat patches (suitable for thefocal species) and the rest of the environment;habitat patches have equal areas, and localpopulations in the metapopulation have entirelyindependent dynamics connected by dispersal(Hanski et al., 1995; Hanski, 1991; Hanski &Gilpin, 1997).

The equation for the local dynamics in eachPatch iAf1; 2;y;Ng at generation t after

reproduction but before dispersal is modeled by

xiðtþ 1Þ ¼ xiðtÞgiðxiðtÞÞ ði ¼ 1; 2;y;NÞ; ð1Þ

where xiðtÞ denotes the population size and theper-capita growth functions, gi : ½0;NÞ-ð0;NÞ;are assumed to be strictly decreasing, positiveand twice differentiable (C2 on ½0;NÞ). Typi-cally, these functions must take on values greaterthan 1; otherwise the population will go extinct.It is naturally assumed that limxi-N giðxiÞo1:System (1) is a discrete-time, single-species,population model with N patches. It describesthe uncoupled (hence artificial) population dy-namics of pioneer species, that is, species thatsatisfy the condition gið0Þ41 in each indepen-dent Patch i (Castillo-Chavez & Yakubu, 2001,2002; Franke & Yakubu, 1996, 1992, 1991).

The Patch i pre-dispersal reproduction func-tion, fiðxiÞ ¼ xigiðxiÞ; describes the local dy-namics of the species where xi is the measureof the size of the population in the patch. It isassumed to have a unique positive fixed point

denoted by Xi: Since gi is a strictly decreasingcontinuous function then fiðxiÞ4xi whenever0oxioXi and fiðxiÞoxi whenever xi4Xi: Con-sequently, Ii fið½0;XiÞ is a global attractor,that is, every initial population eventuallyreaches a limit in Ii:

Patch i (local) dynamics is compensatorywhenever fi is orientation preserving and allpositive densities approach the positive equili-brium at Xi monotonically under fi iterations. Iffi increases monotonically from zero with therate of increase slowing down as xi gets largethen the Patch i dynamics is compensatory. TheBeverton–Holt stock recruitment model, fiðxiÞ ¼aixi=ð1þ bixiÞ; portrays compensatory dynamicsin Patch i whenever ai41 and bi40 (Begon et al.,1996; Nicholson, 1954; Yodzis, 1989). Errington(1945) and Milton & B!elair (1990), using thevalues ki ¼ 0:65;Ki ¼ 1:15 and ni ¼ 3; fitted fielddata of the bobwhite quail populations inNorthern Wisconsin to the model fiðxiÞ ¼ xi �ðki þ ðKi=ð1þ xnii ÞÞÞ which has a global attractorat Xi ¼ 1:3173 for these values. Since no initialcondition overshoots Xi the dynamics arecompensatory (see Fig. 1).

Patch i (local) dynamics is overcompensatory

whenever fi is orientation reversing and somepositive population sizes ‘‘overshoot’’ the posi-tive equilibrium at Xi under fi iterations. If fiis an orientation reversing one hump map witha stable (respectively, an unstable) positivefixed point, then the return to the stable fixedpoint takes the form of damped oscillations

Fig. 2. Overcompensatory dynamics (initial conditionsovershoot the positive equilibrium at Xi).

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ276

(respectively, the local behavior near the un-stable fixed point takes the form of divergentoscillations), and Patch i dynamics is over-compensatory (see Fig. 2). Whenever ri41 andfi is given by Ricker’s model, fiðxiÞ ¼ xi expðri �xiÞ; the dynamics in Patch i is overcompensatory(Begon et al., 1996; Best et al. 2001; Brauer &Castillo-Chavez, 2001; Castillo-Chavez & Yaku-bu, 2001, 2002; Doebeli, 1995; Doebeli, & Ruxton,1997; Earn et al., 2000; Errington, 1945; Franke &Yabaku, 1996, 1992, 1991; Gozalez-And!ujar &Perry, 1993; Gyllenberg et al., 1993; Hassel, 1976;Hassel et al., 1976; Hastings, 1993, 1982; May,1974, 1977; May et al., 1981; Maynard Smith &Slatkin, 1973; Milton & B!elair, 1990; Nicholson,1954; Ricker, 1954; Yodzis, 1989).

The bobwhite quail ‘‘hump-with-tail’’ model,fiðxiÞ ¼ xiðki þ ðKi=ð1þ xnii ÞÞÞ; where ki;Ki andni are positive constants ki þ Ki41 and kio1(Errington, 1945; Milton & B!elair, 1990), pro-vides an example of Model (1) where the intrinsicgrowth rate, ki þ Ki; is bigger than 1; and the percapita growth rate, ki þ ðKi=ð1þ xnii ÞÞ; tendsto kio1 as the population tends to infinity. Theparameter ni (a shape parameter) measuresthe level and type of intraspecific competition inthe population. If nioðKi=ðki þ Ki � 1Þð1� kiÞÞthen Patch i dynamics is compensatory, while ifni4ðKi=ðki þ Ki � 1Þð1� kiÞÞ then it is over-

compensatory. The bobwhite quail model has apositive equilibrium at

Xi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiki þ Ki � 1

1� ki

ni

s;

which is locally asymptotically stable whenevernioð2Ki=ðki þ Ki � 1Þð1� kiÞÞ: The fixed pointXi undergoes a period-doubling bifurcation in itsroute to chaos as ni is increased past 2Ki=ððki þKi � 1Þð1� kiÞÞ (Milton & B!elair, 1990).

A more detailed characterization of functionsthat describe compensatory, overcompensatoryor mixed-type dynamics requires the introduc-tion of some definitions. The Patch i reproduc-tion function, fi; is an a-monotone concave map iff 0i ðxiÞ40 and f 00i ðxiÞo0 for each xiA½0; a: Patchi population dynamics are compensatory atpopulation sizes in ½0; a whenever fi is ana-monotone concave map with a unique positivefixed point in the open interval ð0; aÞ. TheSmith–Slatkin model, fiðxiÞ ¼ aixi=ð1þ ðbixiÞ

li Þ;describes both overcompensatory and compen-satory dynamics (depending on parametervalues). If li ¼ 1 and ai41; the Smith–Slatkinmodel reduces to the Beverton–Holt model,an N-monotone concave map [compensatorydynamics Begon et al. (1996), Castillo-Chavez& Yakubu (2002), Doebeli (1995), Nicholson(1954), Yodzis (1989)]

When the reproduction function in Patch i; fi;is an a-monotone concave map and some initialpopulations in the interval ða;NÞ overshoot thepositive fixed point in the closed interval ½0; a;then fi describes situations in which there iscompensatory and overcompensatory dynamicsat lower densities and higher densities, respec-tively. Ricker’s model, fiðxiÞ ¼ xi expðri � xiÞ; isa 1-monotone concave map, since all positiveinitial populations with densities in the interval½0; 1 approach the unique positive fixed pointXi ¼ ri monotonically, but initial populations inð1;NÞ overshoot the positive fixed point when-ever 0orio1: In other words, there is compen-satory dynamics in ½0; 1 and overcompensatorydynamics in ð1;NÞ:

3. Metapopulation Model

The dynamics of N patches, modeled bySystem (1), are coupled via the dispersal of

DISCRETE-TIME METAPOPULATION MODELS 277

individuals at a rate that is proportional to eachlocal density. Reproduction is assumed to occurprior to dispersal within each generation andeach patch. Hence, after reproduction, the fractiondij (assumed constant) of the population dispersesfrom Patch i to j; where iajAf1; 2;y;Ng: Thedynamics of the population sizes in the N patchesare given by the following nonlinear system ofcoupled difference equations:

xiðtþ 1Þ ¼ 1�XNj¼1jai

dij

0BB@

1CCAfiðxiðtÞÞ

þXNj¼1jai

dji fjðxjðtÞÞ ði ¼ 1; 2;y;NÞ ð2Þ

where for each iajAf1; 2;y;Ng; fiðxiÞ ¼ xigiðxiÞ; 0odij ; djio1; 0o

PNj¼1jai

dijo1 and dii ¼ 0:System (2) implicitly assumes that the timing ofreproduction and dispersal do not differ frompatch to patch, that is, it models a multi-patchmetapopulation system with synchronous disper-

sal. Clearly, dij ¼ dji implies that dispersalbetween Patches i and j is symmetric whiledijadji makes it asymmetric. The basic demo-graphic reproductive number in each patchwithout dispersal ðdij ¼ 0Þ is

Rid ¼ gið0Þ:

Rid gives the pre-dispersal average number of

descendants produced by a typically small initialpopulation ðxið0ÞÞ over its lifetime in Patch i: Inthe absence of dispersal, Ri

d41 guarantees thesuccessful invasion and survival of the discretelyreproducing population in Patch i; while Ri

do1guarantees the extinction of the initial populationin the patch. The metapopulation basic demo-graphic reproductive number when all the patchesare connected by dispersal ðdij40 where iajÞ is

Rd ¼min Rid ; 1�

XNj¼1jai

dij þXNj¼1jai

dji

0BB@

1CCA

8>><>>:

� miniAf1;2;y;Ng

Rid

� �):

Rd41 implies that all local populations persist.Hence, a totally connected network ðdij40where iajÞ of pioneer species ðgið0Þ41Þ undercompensatory dynamics gives rise to a metapo-pulation with species persistence. In fact, the fullsystem (metapopulation) supports a uniquepositive globally asymptotically stable equili-brium when Rd41 and all local dynamics arecompensatory. Clearly, Rd41 whenever disper-sal is symmetric. Hence, metapopulations withcompensatory local dynamics are robust when-ever dispersal is symmetric and Ri

d41 for all i:From System (2) one can recover the models

of Hastings (1993), Gyllenberg et al. (1993),Earn et al. (2000), Doebeli (1995), Doebeli &Ruxton (1997) and Castillo-Chavez and Yakubu(2002) through the appropriate selection ofdispersal coefficients. In the absence of dispersal(dij ¼ 0 for all i and jÞ System (2) reduces to theuncoupled System (1).

We end this section with some notation anddefinitions. The vector of population densitiesxðtÞ ¼ ðx1ðtÞ; x2ðtÞ;y;xNðtÞÞ is written as x ¼ðx1;x2;y;xNÞ so that the post-dispersal metapo-

pulation reproduction function F :RNþ-RN

þ maybe written as FðxðtÞÞ ¼ xðtþ 1Þ; where FiðxÞ ¼ð1�

PNj¼1jai

dijÞxigiðxiÞ þPN

j¼1jai

djixjgjðxjÞ for eachiAf1; 2;y;Ng: Ft is the map F composed withitself t times. Ft

j ðxÞ is the j-th component of Ft

evaluated at the point x in RNþ : Therefore, Ft

gives the population densities in generation t:The set of iterates of the map F is equivalent tothe set of all density sequences generated bySystem (2), that is, it is the set of life historytrajectories for a metapopulation with fixeddispersal matrix A ¼ ðdijÞ:

4. Compensatory Local DynamicsConnected Via Dispersal

In this section, we study the effects ofsynchronous dispersal in a metapopulationwhere the local dynamics are assumed to becompensatory. Hence, the species may bemodeled by different local reproduction func-tions, and consequently, capable of supportingdifferent per capita growth rates and differentcarrying capacities in the N patches. In thissetting, synchronous dispersal and Rd41 lead toan overall globally stable positive steady state,

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ278

that is, the qualitative dynamics of the full N-patch system with symmetric or asymmetricdispersal between patches, is qualitatively equiva-lent to those of each of the local single patches(prior to dispersal) (Theorem 1 and Corollary 1).

Theorem 1. Let the local dynamics be modeled byfi; an a-monotone concave map with ð1�PN

j¼1jak

dkj þPN

j¼1jak

djkÞfiðaÞoa and XiAð0; aÞ; for

each Patch i; kAf1; 2;y;Ng: Then, System (2)supports a unique positive fixed point, q¼ðq1; q2;y; qNÞARN

þsuch that FnðxÞ-q as n-N

for every point xA½0; p � f0g; provided thatRd41 and p ¼ ða; a;y; aÞA intðRN

þÞ:

The proof of Theorem 1 is in the Appendix.The following result is an immediate conse-quence of the theorem.

Corollary 1. System (2), the pioneer species multi-

patch metapopulation model with synchronoussymmetric dispersal, has a globally stable positive

fixed point whenever each pre-dispersal localpatch dynamics is compensatory.

The quail, the Ricker and the Smith–Slatkinmodels describe either compensatory or over-compensatory dynamics as parameters are var-ied. Theorem 1 and Corollary 1 guarantee thatsynchronous dispersal (symmetrical and asym-metrical) plays no critical role whenever the localpatch dynamics (without dispersal) is compensa-tory, that is, synchronous dispersal has noqualitative impact on the metapopulation (global)qualitative dynamics. In fact, the system behavesqualitatively like a single population model for apioneer species under compensatory dynamics.Hastings and Levin obtained similar resultsin continuous-time metapopulation models withdispersal between several patches (Hastings, 1982;Levin, 1994, 1992; Levin & Paine, 1974).

Example 1. Consider System (2) with N ¼ 2 and

with

fiðxiÞ ¼ xi ki þKi

1þ xnii

� �

for each iAf1; 2g. Set the following parametervalues:

ki ¼ 0:65; Ki ¼ 1:15 and ni ¼ 3

[parameters from quail field data Errington (1945)and Milton & B!elair (1990)].

In Example 1, l ¼ rðDFð0ÞÞ41 and Xi ¼1:3173 whenever Rd41: Corollary 1 implies thatSystem (2) with quail data has a globally stablepositive equilibrium population for symmetric

or asymmetric dispersal rates wheneverPNj¼1jak

djkpPN

j¼1jak

dkj : Hence, symmetric or asym-

metric dispersal has a stabilizing effect onmetapopulations that are totally connectedwhere the local dynamics are compensatoryand the process of dispersal is synchronized[individuals disperse at the same time in allpatches (Castillo-Chavez & Yakubu (2002),Doebeli (1995), Hastings (1982), Levin & Paine(1974))].

The dynamics of deterministic discrete-timemodels can be greatly modified by the introduc-tion of random fluctuations (Crutchfield et al.,1982; Lasota & Mackey, 1986; Mayer-Kress &Haken, 1981; Smith, 1986; Smith & Waltman,1993). System (2), a deterministic model, enjoysa uniform dissipative condition characterized bya compact trapping region that attracts allpopulation sizes (Corollary A.3 in the Appen-dix). Consequently, the results of Smith andWaltman on perturbation of a globally stablefixed point imply that, Theorem 1 or CorollaryA.3 is an open condition in the parameters(Smith & Waltman 1993). That is, the globallyattracting fixed point is stable under smallperturbations and the resulting perturbed fixedpoint is also globally attracting. Hence, slightchanges in the model parameters would tend topreserve the qualitative properties of the com-pensatory local patch dynamics. The results ofSmith and Waltman imply that compensatorydynamics is robust, and hence, less likely to besusceptible to the impact of ‘‘mild’’ stochasticevents (Smith & Waltman, 1993).

5. Overcompensatory Local DynamicsConnected Via Dispersal

Dispersal is not the only mechanism thatgenerates multiple life histories (multiple attrac-tors). Single-patch models that incorporate apopulation’s structure can generate multipleattractors. For example, the single-patch LPA

Fig. 3. Three attractors: symmetric 4-cycle, asymmetric4-cycle and a period-2 limit cycle, where d12 ¼ d21 ¼ 0:01;ki ¼ 0:5; Ki ¼ 3 and ni ¼ 7:

Fig. 4. The symmetric dispersal rate impacts the natureof the dynamical attractor in the set of asymmetric initialconditions, where ki ¼ 0:5; Ki ¼ 3:1 and ni ¼ 9: On thehorizontal axis, 0pd12 ¼ d21p0:3; and on the vertical axis,0px1p4Þ:

DISCRETE-TIME METAPOPULATION MODELS 279

age-structured model (tied to experimental datafrom the Tribolium, flour beetle, experiment)supports multiple attracting 2-cycles (Hensonet al., 1999). Single-patch plant-populationmodels with age–structure (seeds and plants),capable of maintaining multiple attractors, havealso been studied (Best et al., 2001). Hastings(1993) used identical logistic difference equationswith parameters in the chaotic regime and a setof asymmetric (out-of-phase) initial conditionsin his two-patch model. He found that thequalitative dynamics of the system could beradically different from those on each uncon-nected patch. He showed that out-of-phaseinitial conditions in a two-patch system withidentical local reproduction functions can sup-port multiple attractors with fractal basinboundaries for parameter values in the chaoticregime (overcompensatory dynamics). In thissection, we study the impact of synchronoussymmetric dispersal on System (2) where thelocal reproduction functions are identical ð f1 ¼f2 ¼ ? ¼ fNÞ and portray overcompensatorydynamics. We focus on situations where thelocal populations (in the absence of dispersal)live on either a pre-selected n-cycle or an interval(chaotic) attractor. The pre-selected symmetric(chaotic or non-chaotic) attractor can coexistwith one or more ‘‘new’’ asymmetric attractors(multiple attractors, see Table 1 and Figs 3–10).All positive symmetric initial conditions as wellas some positive asymmetric ones are attractedto the pre-selected (local patch) attractorswhenever dispersal is symmetric. Lemma 1implies that the full system with synchronoussymmetric dispersal and symmetric initial popu-lation sizes behaves as a single patch system

Table 1Pre-dispersal identical local dynamics versus post-

dispersal metapopulation dynamics

Identical local dynamics Metapopulation dynamics

1. 2-cycle attractor Two 2-cycle attractors2. 2-cycle attractor 3 attractors, 2 fixed points

and a 2-cycle3. 4-cycle attractor(Fig. 3).

3 attractors, two 4-cycles andlimit cycles

4. chaotic attractor Two chaotic attractors5. chaotic attractor(Fig. 5)

A chaotic and a 4-cycle at-tractor

whenever the underlying local reproductionfunctions are identical.

Lemma 1. In System (2), if g1 ¼ g2 ¼ ? ¼ gNand dij ¼ dji; then the ‘‘diagonal’’ set of symmetric(in-phase) initial conditions,

D fx ¼ ðx1; x2;y; xNÞ

ARNþ j x1 ¼ x2 ¼ ? ¼ xNg;

is invariant and the dynamics on D are qualita-

tively equivalent to the pre-dispersal local patchdynamics.

The proof of Lemma 1 is in the Appendix.

Fig. 5. A chaotic attractor and a 4-cycle attractor:symmetric interval attractor coexists with a 4-cycle attrac-tor at ð0:9011; 1:1472Þ- ð2:3493; 1:4825Þ- ð1:1472;0:9011Þ- ð1:4825; 2:3493Þ; where d12 ¼ d21 ¼ 0:1; ki ¼0:5; Ki ¼ 3:1 and ni ¼ 9:

Fig. 6. (Fractal basin boundaries) Basin of attraction ofthe chaotic attractor is the white region and that of the 4-cycle is the black region, where the parameter values areexactly as in Fig. 5. On the horizontal axis 0px1p3; and onthe vertical axis 0px2p3:

Fig. 7. Basin of attraction of symmetric 2-cycle is whiteregion and that of the asymmetric 2-cycle is black region,where d12 ¼ d21 ¼ 0:01; ki ¼ 0:5; Ki ¼ 3 and ni ¼ 6: On thehorizontal axis 0px1p4; and on the vertical axis 0px2p4:

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ280

To illustrate the possibility of multiple attrac-tors where identical pre-dispersal local reproduc-tion functions support pre-selected n-cycle orinterval (chaotic) attractors, we consider System(2) with N ¼ 2 and fiðxiÞ ¼ xiðki þ Ki=ð1þ xnii ÞÞfor each iAf1; 2g: We choose the parameters, sothat the pre-dispersal local dynamics and post-

dispersal metapopulation dynamics are as listedin Table 1.

We choose the parameters, so that in theabsence of dispersal each local reproductionfunction, fi; supports a stable 2-cycle (over-compensatory dynamics). Then the fulltwo-patch system supports multiple attractors,a stable positive asymmetric 2-cycle atð1:0646; 1:8251Þ-ð1:8251; 1:0646Þ coexisting withthe symmetric 2-cycle at ð1:0595; 1:0595Þ-ð1:8461; 1:8461Þ; where d12 ¼ d21 ¼ 0:01; ki ¼0:5; Ki ¼ 3 and ni ¼ 6 (see Lemma 1).

The impact of increasing the symmetricdispersal coefficient on the structure of theattractors is studied. The parameters ki; Ki andni are kept fixed at ki ¼ 0:5; Ki ¼ 3 and ni ¼ 6;while the symmetric dispersal rates d12 ¼ d21 areincreased past 0:01: At d12 ¼ d21 ¼ 0:9; thesystem supports three attractors, two out-of-phase fixed points at ð1:1275; 1:6108Þ andð1:6108; 1:1275Þ coexisting with the in-phase2-cycle at ð1:0595; 1:0595Þ-ð1:8461; 1:8461Þ:

Identical pre-dispersal local reproductionfunctions with non-chaotic attracting cycles(overcompensatory dynamics) connected bysymmetric dispersal can generate multiple at-tractors with attracting limit cycles. To illustrate

Fig. 8. Basin of attraction of symmetric 2-cycle (greenregion) and that of two asymmetric fixed points (white andred regions), where d12 ¼ d21 ¼ 0:9; ki ¼ 0:5; Ki ¼ 3 andni ¼ 6: On the horizontal axis 0px1p3; and on the verticalaxis 0px2p3:

Fig. 10. Two coexisting attractors with intermingled basins of attraction. Basin of attraction of symmetric chaoticattractor is red region and that of the asymmetric 4-piece chaotic attractor is green region, where d12 ¼ d21 ¼ 0:08; ki ¼ 0:5;Ki ¼ 3:1 and ni ¼ 9: On the horizontal axis 0px1p3; and on the vertical axis 0px2p3:

Fig. 9. Basins of attraction of the three attractors,where the parameter values are exactly as in Fig. 3. On thehorizontal axis 0px1p3; and on the vertical axis 0px2p3:

doi.10.1006 yjtbi.3075

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ

DISCRETE-TIME METAPOPULATION MODELS 281

this, the parameters ki and Ki are fixed at ki ¼0:5 and Ki ¼ 3; while ni and d12 ¼ d21 are reset atni ¼ 7 and d12 ¼ d21 ¼ 0:9: With this choiceof parameters, the identical pre-dispersal localreproduction functions support a stable in-phase4-cycle at ð1:7648; 1:7648Þ-ð0:9799; 0:9799Þ-ð2:0642; 2:0642Þ-ð1:0706; 1:0706Þ coexisting withtwo out-of-phase post-dispersal attractors, a4-cycle at ð1:8212; 2:0230Þ-ð0:9922; 1:0543Þ-ð2:0230; 1:8212Þ-ð1:0543; 0:9922Þ and a period-2limit cycle (see Fig. 3).

When the pre-selected attractor is a chaotic(interval) attractor with no stable cycles (over-compensatory dynamics), Fig. 10 shows that thefull two-patch metapopulation model is capableof sustaining two chaotic attractors, the chaotic(interval) attractor (line segment in Fig. 10) inthe ‘‘diagonal’’ set of symmetric initial condi-tions and a four piece ‘‘strange’’ attractor in theset of asymmetric initial conditions.

The dependence of the chaotic attractor (inthe set of asymmetric initial conditions) on thedispersal parameters is studied by keeping ki; Ki

and ni fixed at ki ¼ 0:5; Ki ¼ 3:1 and ni ¼ 9;while varying the symmetric dispersal coefficientsd12 ¼ d21 between 0 and 1: For a wide range ofvalues of the dispersal parameters, the systemsustains a chaotic attractor in the set ofasymmetric initial conditions, while for a smallrange of the dispersal parameters the systemsupports an asymmetric 4-cycle attractor (seeFigs 4 and 5). These last examples illustrate thedependence of the dynamical attractor onsymmetric dispersal parameters (for asymmetricinitial conditions).

Metapopulation models with overcompensa-tory local patch dynamics and dispersal generatemultiple attractors. Most of these attractorsemerge from saddle-node bifurcations (the birthof a stable n-cycle accompanied by an unstablen-cycle), discrete Hopf bifurcations (the transi-tion from a stable n-cycle to an unstable n-cycle

accompanied by the emergence of a stable set ofinvariant closed curves), and period-doubling

bifurcations (the transition from a stable n-cycleto an unstable n-cycle accompanied by the birthof a stable 2n-cycle) (Alexander et al., 1992). Thenature and number of the attractors are theresult of a complex interaction between dispersaland pre-dispersal local patch dynamics. The

basins of attraction, the set of all populationsizes that eventually settle into an attractorunder iteration, may provide critical informationon a variety of issues including the ‘‘likelihood’’that a particular attractor is observed. In thenext section, the nature of the basins ofattraction associated with dispersal inducedmultiple attractors is studied.

6. Nature of Basins of Attraction

The qualitative asymptotic behavior of sys-tems that support multiple attractors is deter-mined by the nature of the attractors and thestructure of their basins of attraction. Forexample, when one of the coexisting attractorsis chaotic then the basins of attraction aretypically thin and the basin boundaries arefractal in nature. In this case, long-term predic-tions become difficult because small variations ininitial conditions may induce a switch in thelong-term dynamics (Alexander et al., 1992;Grebogi et al., 1987; Hastings, 1993; Hofbaueret al., 2001; Kan, 1994). In the absence of achaotic attractor, it is possible for the basins ofthe coexisting attractors to be fat, that is, long-term predictions for most initial conditions arepossible. However, the basins of attractionbecome thinner and their boundaries exhibitincreasing fractal structures as the number ofnon-chaotic attractors increases or as the periodof the attractors increases. Figures 6–10 illus-trate the possibility that as the number ofattractors and their periods increase, predictionsbecome less and less reliable.

Typically, a basin of attraction is an open setof population sizes, and in particular it containsdisks (open sets). There is a growing evidence ofriddled basins in population models with multipleattractors (riddled basins contain no disks)(Alexander et al., 1992; Hofbauer et al., 2001;Kan, 1994; Lai et al., 1996). The basin ofattraction of an attractor is riddled if itscomplement intersects every disk in a set ofpositive measure. That is, an initial populationsize chosen at random from any disk in a riddledbasin has positive probabilities (measure-theo-retic) of being in the riddled basin and not beingin it (Alexander et al., 1992; Lai et al., 1996).A more extreme type of basin structure,

Fig. 11. Depending on dispersal coefficients, the fullsystem inherits either Patch 1 or Patch 2 local dynamics,where f1ðx1Þ ¼ x1 expðr1 � x1Þ; f2ðx2Þ ¼ x2ðk2 þ K2=ð1þxn22 ÞÞ; r1 ¼ 2:1; k2 ¼ 0:65;K2 ¼ 1:15 and n2 ¼ 3: In each ofthe four windows, on the horizontal axis the dispersal rateis between 0 and 1; and on the vertical axis 0px1p4:

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ282

intermingled basin, was introduced by Alexanderet al. (1992). In intermingled basins of attraction,two or more basins of attraction are riddled(Alexander et al., 1992; Hofbauer et al., 2001;Kan, 1994). System (2) is capable of sustainingriddled basins and intermingled basins of attrac-tion (see Figs 6 and 10). Very general theoreticalresults on riddled basins and intermingled basinshave been established by Alexander et al. (1992),Hofbauer et al. (2001) and Kan (1994). Inmetapopulation models supporting intermingledbasins of attraction, for almost all initialpopulation sizes accurate predictions of theirlong-term dynamical outcome is impossible. Inother words, the dynamics are ‘‘similar’’ to thoseof stochastic models with a fixed number ofabsorbing states.

7. Mixed Compensatory and OvercompensatoryLocal Dynamics with Dispersal

The potential impact of asymmetric synchro-nous dispersal on two-patch systems undermixed (overcompensatory–compensatory) localdynamics are explored in this section. It isassumed that the Patch 1 dynamics is over-compensatory while that of Patch 2 is compen-satory. Patch 1 local reproduction function,f1ðx1Þ ¼ x1 expðr1 � x1Þ; has a pre-selected at-tractor (for example, an attracting 2-cycle whenr1 ¼ 2:1; see Table 2 and Fig. 11 or a 4-cyclewhen r1 ¼ 2:6; see Table 3 and Fig. 12), and thatof Patch 2; f2ðx2Þ ¼ x2ðk2 þ K2=ð1þ xn22 ÞÞ; has aglobally stable fixed point (for example,whenk2 ¼ 0:65;K2 ¼ 1:15 and n2 ¼ 3). The twopatches are connected via dispersal and theparameters are chosen so that pre-dispersal localdynamics and post-dispersal metapopulationdynamics are as listed in Tables 2 and 3.

When the dispersal coefficient from Patch 1 to2, d12; is equal to 0:5; then the full system has a

Tabl

Pre-dispersal local dynamics vs. post-dispe

Pre-specified stable local dynamics

1. 2-cycle in Patch 1, Positive fixed point in Patch2. 2-cycle in Patch 1, Positive fixed point in Patch3. 2-cycle in Patch 1, Positive fixed point in Patch4. 2-cycle in Patch 1, Positive fixed point in Patch

stable fixed point for all values of the dispersalcoefficient from Patch 2 to 1, d21: However, whend12 is equal to 0:01 then the full system has astable 2-cycle for all values of d21: A shift from astable 2-cycle to a fixed point (period-doublingbifurcation reversal) occurs when d21 equals 0:01or 0:5: To illustrate the nature of this bifurcationand the importance of asymmetric dispersal, weincrease the dispersal coefficient d12 beginningat 0:01 and observe how the aggregate systemchanges from a stable 2-cycle (Patch 1 localdynamics, see Fig. 11 and Table 2) to a stablefixed point (Patch 2 local dynamics, see Fig. 11and Table 2). Hence, depending on the dispersalparameters, the metapopulation model inheritseither the overcompensatory Patch 1 local

e 2rsal metapopulation dynamics in Fig. 11

Stable metapopulation dynamics

2 Fixed point if d12 ¼ 0:5 and d21Að0; 1Þ2 2-cycle if d12 ¼ 0:01 and d21Að0; 1Þ2 2-cycle if d21 ¼ 0:01 and d12 is small2 Fixed point if d21 ¼ 0:01 and d1240:1

Table 3Pre-dispersal local dynamics vs. post-dispersal metapopulation dynamics in Fig. 12

Pre-specified stable local dynamics Stable metapopulation dynamics

1. 4-cycle in Patch 1, fixed point in Patch 2 Positive fixed point if d12 ¼ 0:5 and d21Að0; 1Þ2. 4-cycle in Patch 1, fixed point in Patch 2 4-cycle if d12 and d21 are small3. 4-cycle in Patch 1, fixed point in Patch 2 2-cycle if d12 ¼ 0:01 and d21 Að0; 1Þ4. 4-cycle in Patch 1, fixed point in Patch 2 2-cycle if d21 ¼ 0:01 and intermediate d12 values5. 4-cycle in Patch 1, fixed point in Patch 2 Positive fixed point if d21 ¼ 0:01 and d12 is large

Fig. 12. Depending on dispersal coefficients, the periodof the metapopulation dynamics differs from (local)dynamics, where f1ðx1Þ ¼ x1 expðr1 � x1Þ; f2ðx2Þ ¼ x2ðk2 þK2=ð1þ xn22 ÞÞ; r1 ¼ 2:6; k2 ¼ 0:65;K2 ¼ 1:15 and n2 ¼ 3: Ineach of the four windows, on the horizontal axis thedispersal rate is between 0 and 1; and on the vertical axis0px1p6:

DISCRETE-TIME METAPOPULATION MODELS 283

dynamics or the compensatory local dynamics inPatch 2.

The qualitative dynamics of the aggregatedispersal-linked system may be different fromthe pre-dispersal local patch dynamics. Keepingk2;K2 and n2 fixed at k2 ¼ 0:65; K2 ¼ 1:15 andn2 ¼ 3 and re-setting the parameter r1 so that,r1 ¼ 2:6; sets Patch 1 population on a stable4-cycle (overcompensatory dynamics), Patch 2population on a stable fixed point (compensa-tory dynamics) and the metapopulation linkedby dispersal dynamics on a stable 4-cycle (over-compensatory dynamics). The metapopulationdynamics follows the pre-set pre-dispersal Patch

1 compensatory dynamics whenever all dispersalcoefficients are small. However, the metapopula-tion dynamics follows the pre-set pre-dispersalPatch 2 overcompensatory dynamics wheneverthe dispersal coefficient from Patch 1 to 2; d12; islarge. For intermediate values of d12 and d21; themetapopulation dynamics may differ from bothpre-dispersal local patch dynamics.

Figure 12 and Table 3 illustrate a post-dispersal metapopulation on a stable 2-cyclewhere Patch 1 population is on a stable 4-cycle

while that of Patch 2 is on a stable fixed point (inthe absence of dispersal). Increasing d12 valueswith d21 fixed at d21 ¼ 0:01 or 0:5 cause a shift inthe metapopulation dynamics from a stable 4-cycle (Patch 1 overcompensatory dynamics) to astable 2-cycle (overcompensatory dynamics witha different period), and then to a stable positivefixed point (Patch 2 compensatory dynamics),while increasing d21 values with d12 fixed at d12 ¼0:01 cause a shift in the metapopulationdynamics from a stable 4-cycle (Patch 1dynamics) to a stable 2-cycle (a period differentfrom the pre-specified Patch 1 and 2 periods).The metapopulation remains on a stable positivefixed point (Patch 2 dynamics) for all values ofd21 whenever d12 is fixed at 0:5 (see Fig. 12 andTable 3).

Metapopulation models sustaining mixedcompensatory and overcompensatory local dy-namics and asymmetric synchronous dispersalappear to be as rich dynamically as thoseexperiencing overcompensatory local dynamicsexclusively albeit their biological interpretationshave to be tied to the impact of dispersal. Insome instances, the local dynamics of one patchseems to control or drive the qualitative dynamicoutcome of the full system for some asymmetricdispersal rates. However, it is also possible for

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ284

the metapopulation dynamics to live on anattractor with a period completely different fromthose pre-specified by local pre-dispersal dy-namics. Dispersal can also stabilize or shift thelocal dynamics from a stable period four to astable period two or to a stable fixed point.Clearly, we have only highlighted a few possibi-lities with the selected examples.

8. Conclusions

This paper focuses on expanding the conclu-sions of Hastings (1993,1982), Gyllenberg et al.(1993), Doebeli (1995) and Doebeli & Ruxton(1997) and Castillo-Chavez and Yakubu (2002)on discrete-time metapopulation models con-nected by dispersal. The effects of compensatory,overcompensatory, and mixed-type localdynamics in metapopulation systems connectedby symmetric and asymmetric dispersal areexplored.

The results of Levin and Hastings on con-tinuous-time metapopulation models predictthat an equilibrium population is stable only ifit corresponds to a stable equilibrium withineach patch (Hastings, 1982; Levin & Paine,1974). Our results support this prediction formetapopulations with discrete non-overlappinggenerations connected by symmetric and asym-metric dispersal rates as long as dispersalbetween patches is synchronized.

Stable environments are more likely to sup-port single (‘‘optimal’’) life-history strategieswhile ‘‘harsh’’ environments are likely to favora diversity of strategies (Begon, 1996; Cohen &Levin, 1985; Hairston, 1993). Local unstructuredpopulations under compensatory or overcom-pensatory dynamics tend to support singleattractors, that is, the population has a singlefate. In metapopulation models with overcom-pensatory local dynamics, dispersal can fracturethe basins of attraction through its support ofmultiple attractors. Studies on the role ofdispersal in metapopulations living in highlyfragmented harsh environments would be wel-come. These results may support the need tomaintain dispersal corridors in harsh habitatshousing populations with local overcompensa-tory dynamics in an environment fragmented bydevelopment. The use of dispersal corridors as a

mechanism to increase the number of attractorsmay increase the likelihood of species survival inharsh environments (Doebeli, 1995; Hastings,1993).

The interactions via dispersal of variousforms of intraspecific competition have not onlyled to the generation of a dynamical landscapecapable of supporting multiple attractors, butalso has aided our understanding of the role thatinitial population sizes play in the ultimate fate(life history) of a metapopulation. As thecomplexity of the local dynamics increases, theoverall fate of a metapopulation becomes lesspredictable to the point that it may be impossibleto determine, with any degree of certainty,the fate of such metapopulation. In other words,riddled and intermingled basins of attractiongenerated by the interactions between localintraspecific competition and dispersal bring alarge degree of uncertainty to the notionof prediction in deterministic metapopulationmodels (Castillo-Chavez & Yakubu, 2002, Earnet al., 2000; Gadgil, 1971; Hanski et al., 1995).

Metapopulation models with mixed localdynamics and either symmetric or asymmetricsynchronous dispersal appear to be as richdynamically as those with only overcompensa-tory local dynamics. It is possible for theaggregate (metapopulation) dynamics to differcompletely from the local patch dynamics in thismixed setup. These observations call into ques-tion the reliability of data (time series) collectedon the dynamics of metapopulations with over-compensatory local dynamics or mixed-typelocal dynamics. Almost identical ‘‘experimental’’systems are capable of generating ‘‘uncorre-lated’’ time series and totally different predic-tions for systems with intermingled basins ofattraction.

The authors thank the anonymous referees, thehandling editor and Dr. John Tyson for helpfulsuggestions. The authors also thank Janet Best,James Cushing, Alan Hastings and Josef Hofbauerfor helpful discussions. We thank Gerardo Chowell-Puente for the matlab plots of Figs 3 and 5. Thisstudy was partially supported by grants from theNational Science Foundation and the NationalSecurity Agency to Carlos Castillo-Chavez. Abdul-Aziz Yakubu was partially supported by grants fromNational Science Foundation and Cornell University

DISCRETE-TIME METAPOPULATION MODELS 285

IGERT program. We also thank the office of theprovost of Cornell University for its support.

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APPENDIX

To prove Theorem 1, some general resultscharacterizing the dynamics of System (2) arerecalled while some notation is introduced.

If x ¼ ðx1;x2;y;xNÞ and y ¼ ðy1; y2;y; yNÞARN

þ then xoy and xpy in the correspon-ding inequality holds componentwise. Thenotation x9y is used to denote that xpy; butxay: If xpy; ½x; y ¼ fzARN

þjxpzpyg andðx; yÞ ¼fzARN

þjxozoyg:

Lemma A.1. If for each Patch iAf1; 2;y;Ng;the local reproduction functions fi are modeled by

a-monotone concave maps in ½0; aCRþ then

ðiÞ DFðxÞ40 if 0oxop;

and

ðiiÞ DFðyÞ9DFðxÞ if 0oxoyop;

where p ¼ ða; a;y; aÞAintðRNþÞ and DFðxÞ

is the n� n Jacobian matrix of F evaluatedat x:

Proof.

DFðxÞ ¼

@F1ðxÞ@x1

@F1ðxÞ@x2

y@F1ðxÞ@xN

@F2ðxÞ@x1

@F2ðxÞ@x2

y@F2ðxÞ@xN

: : : :

@FNðxÞ@x1

@FNðxÞ@x2

y@FNðxÞ@xN

26666666664

37777777775

¼

ð1�PN

j ¼ 1

ja1

d1jÞf 01ðx1Þ d21f02ðx2Þ y dN1f

0NðxNÞ

d12f01ðx1Þ ð1�

PN

j ¼ 1

ja2

d2jÞf 02ðx2Þ y dN2f0NðxNÞ

: : : :

d1Nf01ðx1Þ d2Nf

02ðx2Þ y ð1�

PN

j ¼ 1

jaN

dNjÞf 0NðxNÞ

26666666666666664

37777777777777775

:

DISCRETE-TIME METAPOPULATION MODELS 287

Recall that 0odijo1; 0oPN

j¼1jaN

dijo1: Hence,f 0i ðxiÞ40 for each xiA½0;NÞ imply that DFðxÞ40 if x40: In addition, f 00i ðxiÞo0 for eachxiA½0;NÞ imply that DFðyÞ9DFðxÞ if 0oxoy:This establishes Lemma 1.

Corollary A.1. In System (2), limx-0þ; x40 DFðxÞ¼ DFð0Þ:

The proof of Corollary 1 follows from thecontinuity of DFðxÞ and is omitted.

In System (2) the origin is an unstable fixedpoint whenever Rd41: Therefore, by standardlinearization techniques the following result isimmediate:

Corollary A.2. In System (2), l ¼ rðDFð0ÞÞ41where Rd41 and rðDFð0ÞÞ denotes the spectralradius of the Jacobian matrix, DFð0Þ:

Hence, whenever the assumptions behindCorollary A.2 are met, we have a metapopula-tion model with no catastrophic extinctionof the species in all patches. It is easy toshow that all solutions of System (2) arebounded, that is, the metapopulation does notbecome arbitrarily large (Lemma A.2 andCorollary A.3).

Lemma A.2. In System (2), no point has an

unbounded orbit.

Proof. We need to show that for each iAf1; 2;y;Ng; the sequence fFn

i ðxÞgnX0 is bounded,where x¼ðx1; x2;y; xNÞARN

þ is an arbitrary point.

For each i; jAf1; 2;y;Ng; FiðxÞ ¼ð1�PN

j¼1jai

dijÞxigiðxiÞ þ

PNj¼1jai

djixjgjðxjÞ: Thus, if xigiðxiÞXxjgjðxjÞ then FiðxÞpð1�

PNj¼1jai

dij þPN

j¼1jai

djiÞxigiðxiÞ: However, if xigiðxiÞoxjgjðxjÞ then FiðxÞoð1�

PNj¼1jai

dij þPN

j¼1jai

djiÞxjgjðxjÞ: Hence, for

each iAf1; 2;y;Ng;

FiðxÞp 1�XNj¼1jai

dij þXNj¼1jai

dji

0BB@

1CCAðmaxffiðxiÞ

¼ xigiðxiÞ; fjðxjÞ ¼ xjgjðxjÞgÞ:

Recall that Ii fið½0;XiÞ and if xipmax Iithen fiðxiÞ ¼ xigiðxiÞpmax Ii but if xi4max Iithen fiðxiÞ ¼ xigiðxiÞoxi: Now, for each i; kA

f1; 2;y;Ng let *fk ¼ ð1�PN

j¼1jai

dij þPN

j¼1jai

djiÞfk:If

gkð0Þ41

1�PN

j¼1jai

dij þPN

j¼1jai

dji

and *fk has a unique positive fixed point at *Xk:Let *Ik *fkð½0; *XkÞ: Also, if xkpmax *Ik then *fkðxkÞpmax *Ik but if xk4max *Ik then *fkðxkÞoxk: If

gkð0Þp1

1�PN

j¼1jai

dij þPN

j¼1jai

dji

then zero is globally attracting under *fkiterations. Therefore, for each iAf1; 2;y;Ngand each point x ¼ ðx1; x2;y; xNÞARN

þ we haveFiðxÞpmaxiAf1;2;y;Ngfxi;max *Iig: Hence, each

sequence fFni ðxÞgnX0 is bounded and, conse-

quently, no point in System (2) has an un-bounded orbit.

From the proof of Lemma A.2, the followingresult is immediate:

Corollary A.3. In System (2), the o-limitset of every point in RN

þ is a non-empty subset

of the compact invariant set S fx ¼ðx1;x2;y;xNÞARN

þj for each iAf1; 2;y;Ng;xipmax %Ig; where max %I ¼ maxiAf1;2;y;Ngfmax *Iig:

The following results of Hal Smith (1986) formonotone systems will be useful in the proof ofTheorem 1.

Theorem A.1. (Smith, 1986) Let pAintðRNþÞ and

T : ½0; p-½0; p be continuous, C1 in ð0; pÞ andsuppose DTð0Þ exists with limx-0;x40 DTðxÞ ¼DTð0Þ: In addition, assume DTðxÞ40 and DTðyÞpDTðxÞ for 0oxoyop and that Tpop:

If Tð0Þ ¼ 0; let l ¼ rðDTð0ÞÞ: If lp1; then for

every xA½0; p; TnðxÞ-0 as n-N: If l41; thenT has a unique non-zero fixed point q: Moreover,qAð0; pÞ and TnðxÞ-q as n-N for everyxA½0; p � f0g:

If Tð0Þ90; then T has a unique fixed pointqA½0; p: Moreover, qAð0; pÞ and TnðxÞ-q as

n-N for every xA½0; p:

Proof of Theorem 1. All the hypotheses inTheorem A.1 (in the Appendix) are satisfied. Letp ¼ ða; a;y; aÞAintðRN

þÞ; and note that FiðpÞpð1�

PNj¼1jai

dijþPN

j¼1jai

djiÞmaxffiðaÞ ¼ agiðaÞ; fjðaÞ ¼agjðaÞgoa: Hence, F : ½0; p-½0; p: Lemmas A.2,

A.-A. YAKUBU AND C. CASTILLO-CHAVEZ288

A.3, Corollaries A.1, A.2 and A.3 imply that allthe hypotheses of Theorem A.1 are satisfied.Hence, Theorem A.1 (in the Appendix) guaran-tees the existence of a unique positive fixed pointq ¼ ðq1; q2;y; qNÞARN

þ satisfying limn-N FnðxÞ¼ q for all points x¼ðx1;x2;y;xNÞ in theinterior of RN

þ :

Proof of Lemma 1. Let the point ðx1;x2;y;xNÞARN

þ : Since g1ðx1Þ ¼ g2ðx2Þ ¼? ¼ gNðxNÞ anddij ¼ dji; we have that F1ðxÞ ¼ f1ðx1Þ ¼F2ðxÞ ¼ f2ðx2Þ ¼ ? ¼ FNðxÞ ¼ fNðxNÞ and, con-sequently, the set D is F-invariant with FiðxÞ ¼fiðxiÞ for each iAf1; 2;y;Ng: