Bifurcations and chaos in a predator–prey system with the ... · MAIN EQUATIONS We consider the...

Received 11 February 2004Accepted 25 February 2004Published online 6 May 2004

Bifurcations and chaos in a predator–prey systemwith the Allee effectAndrew Morozov1,2, Sergei Petrovskii1,2* and Bai-Lian Li1

1Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences,University of California at Riverside, Riverside, CA 92521-0124, USA2Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prospekt 36, Moscow 117218, Russia

It is known from many theoretical studies that ecological chaos may have numerous significant impactson the population and community dynamics. Therefore, identification of the factors potentially enhancingor suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by theAllee effect. More specifically, we show by means of computer simulations that in a time-continuouspredator–prey system with the Allee effect the temporal population oscillations can become chaotic evenwhen the spatial distribution of the species remains regular. By contrast, in a similar system without theAllee effect, regular species distribution corresponds to periodic/quasi-periodic oscillations. We investigatethe routes to chaos and show that in the spatially regular predator–prey system with the Allee effect, chaosappears as a result of series of period-doubling bifurcations. We also show that this system exhibits period-locking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.

Keywords: Allee effect; predator–prey system; patchiness; chaos; period locking

1. INTRODUCTION

The manifestation of chaos in ecosystems dynamics hasbeen an issue of intense discussion during the past threedecades (May 1976; Scheffer 1991; Hastings et al. 1993;Rinaldi & De Feo 1999; Cushing et al. 2003). Manytheoretical studies have demonstrated that ecologicalchaos, if and when it exists, can affect many importantecosystem features such as predictability (Scheffer 1991),species persistence (Allen et al. 1993; Jansen 1995) andbiodiversity (Huisman & Weissing 1999). Another reasonthat makes this subject so controversial is that, in spite ofa growing number of indications (Ellner & Turchin 1995;Perry et al. 2000), conclusive evidence of chaos in ecologi-cal data has never been shown, although there does existsolid evidence of chaos in a real population under labora-tory conditions (Costantino et al. 1997). One problem isoften seen in an insufficient length of available ecologicaltime series; moreover, it is currently not clear whethersuitable time series can be obtained at all because of theimpact of long-term environmental fluctuations (Denniset al. 2001; Hastings 2001). Another serious problem isimmaturity of the theoretical approaches used to arguechaos from field data, especially under the impact of noise(Dennis et al. 2003). In this situation, a better understand-ing of mechanisms responsible for transitions between dif-ferent types of dynamics and the identification of factorsthat can potentially enhance or suppress chaotic dynamicsare becoming the issues of primary importance. One wayto study these issues is through controlled laboratoryexperiments (cf. Costantino et al. 1997; Cushing et al.2003); a useful insight into the problem can also be madeby means of mathematical modelling and computersimulations.

* Author and address for correspondence: Shirshov Institute of Oceanol-ogy, Russian Academy of Science, Nakhimovsky Prospekt 36, Moscow117218, Russia ([email protected]).

Proc. R. Soc. Lond. B (2004) 271, 1407–1414 1407 2004 The Royal SocietyDOI 10.1098/rspb.2004.2733

While early theoretical studies of ecological chaos wereconcerned with the temporal dynamics of spatially homo-geneous systems, the progress made during the past dec-ade has been to a large extent stipulated by theunderstanding of the role of space (De Roos et al. 1991;Rand & Wilson 1995; Sole & Bascompte 1995). In parti-cular, it was shown that the system’s capability to manifestchaotic dynamics is essentially different in the spatiallyhomogeneous and inhomogeneous cases. The reason isthat, in an autonomous time-continuous system, chaos ispossible only when the dimension of the correspondingstate space is greater than 2. In population dynamics thisfact is often interpreted in terms of the number of inter-acting populations: assuming that each population can bedescribed by a single scalar variable (and there is no otherdynamical variable in the system), chaos appears possiblein a community of three or more species (Gilpin 1978;Hastings & Powell 1991; Rinaldi & De Feo 1999). (Notethat time-discrete models can exhibit chaotic dynamicseven in the single-species case; see May 1976.) However,when the condition of spatial homogeneity is relaxed, cf.‘models with explicit space’, chaos was observed alreadyin a two-species predator–prey system both in a space-discrete metapopulation model (Jansen 1995) and in aspace-continuous model (Pascual 1993; Sherratt et al.1995; Pascual & Caswell 1997; Petrovskii & Malchow1999, 2001). In the latter case, transition to chaos wasattributed to the formation of irregular spatial patterns.The patterns are self-organized in the sense that they arenot induced by environmental inhomogeneity or specificinitial conditions. As a result of the pattern formation, thedomain appears to be dynamically split into a number ofvirtually independent subdomains so that the populationdynamics inside each of them is practically unaffected bythe dynamics inside the others (Petrovskii & Malchow2001; Petrovskii et al. 2003). From a dynamical systemsperspective, chaos in that system becomes possiblebecause splitting into several spatial subdomains increases

1408 A. Morozov and others The Allee effect can enhance chaos

the number of the state variables. Remarkably, if thepopulations’ distribution over space is regular, the systemexhibits periodic or quasi-periodic dynamics (Petrovskii &Malchow 1999, 2001; Sherratt 2001).

The question naturally arising here is whether thatprominent spatial irregularity in the population distri-bution is a necessary property of chaos in a few-species(N � 2) system. The above results were obtained for thespace- and time-continuous predator–prey system wherethe prey population exhibits logistic growth. However, inreal ecosystems species growth is often damped by theAllee effect (Allee 1938). The Allee effect was shown tostrongly affect not only local population dynamics (Dennis1989) but also population interactions in space (Lewis &Kareiva 1993; Amarasekare 1998; Gyllenberg et al. 1999;Petrovskii et al. 2002a,b). However, the impact of theAllee effect on the properties of chaos in a population sys-tem has never been studied in detail.

In this paper, we consider the properties of chaos in apredator–prey system with the Allee effect for the prey.We show by means of computer simulations that in thissystem temporal chaotic fluctuations of population sizeand population density can occur even when the speciesdistribution over space remains remarkably regular, havingthe form of two standing symmetric patches. We then con-sider scenarios of transition to chaos in this system byvarying some of the parameters and show that chaosappears through the succession of period-doublingbifurcations. Also, we show that the system exhibitsperiod-locking behaviour (cf. Cushing et al. 2003) so thatparameter domains corresponding to chaos alternate withthe domains corresponding to regular dynamics.

2. MAIN EQUATIONS

We consider the following model of predator–preyinteraction in a homogeneous environment:

∂H(X,T)∂T = D1

∂2H∂X2 � F(H) � f(H,P), (2.1)

∂P(X,T)∂T = D2

∂2P∂X2 � � f(H,P) � MP (2.2)

(cf. Segel & Jackson 1972; Nisbet & Gurney 1982; Murray1989; Holmes et al. 1994; Sherratt 2001). Here, H and Pare the densities of prey and predator, respectively, atmoment T and position X. D1 and D2 are diffusivities and� is the food utilization coefficient. The function F(H )describes prey multiplication, f(H,P) describes predation,and the term MP stands for predator mortality. To keepthe model as simple as possible (see also the end of § 4),we assume that f(H,P) = AHP , where A is the predationrate, which corresponds to the classical Volterra scheme.

Assuming Allee dynamics for the prey population, itsgrowth rate can be parameterized as follows (Lewis & Kar-eiva 1993):

F(H) =4�

(K � H0)2H(H � H0)(K � H), (2.3)

where K is the prey-carrying capacity, � is the maximumper capita growth rate and H0 quantifies the intensity ofthe Allee effect so that it is called ‘strong’ if 0 � H0 � K

Proc. R. Soc. Lond. B (2004)

(when the growth rate becomes negative for H � H0 ) and‘weak’ if –K � H0 � 0 (Owen & Lewis 2001; Wang & Kot2001). For H0 � –K, the Allee effect is absent (Lewis &Kareiva 1993).

Equations (2.1)–(2.3) contain a large number of para-meters, which makes their numerical investigation cum-bersome. However, by choosing appropriate scales for thevariables (Nisbet & Gurney 1982; Murray 1989; for a moregeneral discussion see Barenblatt 1996), the number ofparameters can be lessened. Considering dimensionlessvariables u = H/K, v = P/(�K), t = aT, x = X(a/D1)1/2, wherea = A�K , from equations (2.1) and (2.2), we obtain

∂u(x,t)∂t =

∂2u∂x2 � �u(u � �)(1 � u) � uv, (2.4)

∂v(x,t)∂t =

∂2v∂x2 � uv � u. (2.5)

Equations (2.4) and (2.5) contain four dimensionlessparameters (against eight in the original equations), i.e.� = H0/K , � = 4�K /(A�(K – H0)2), = M/a and = D2/D1.Thus, the behaviour of dimensionless solutions u and vappears to depend on four dimensionless combinations ofthe original parameters rather than on each of them separ-ately.

System (2.4) and (2.5) has recently been considered byMorozov (2003) and Petrovskii et al. (2002a,b, 2004) inconnection with biological invasion. The species spreadwas studied for the following initial conditions:

u(x,0) = u0 for �12�u � x � 1

2�u, otherwise u(x,0) = 0,(2.6)

v(x,0) = v0 for �12�v � x � 1

2�v, otherwise v(x,0) = 0,(2.7)

where u0, v0 are the initial population densities and �u, �v

give the size of initially invaded domain. It was shownthat, depending on the parameter values, in equations(2.4) and (2.5) describes a wide variety of the regimes ofspecies invasion. Some of them appear to be qualitativelysimilar to those observed earlier in the predator–prey sys-tem without the Allee effect, e.g. the propagation of popu-lation fronts with excitation of chaotic spatio-temporaloscillations in the wake of invasion (cf. Sherratt et al.1995). However, a few new regimes of the system dynam-ics that were essentially attributed to the impact of theAllee effect have also been observed (Petrovskii et al.2002a,b; Morozov 2003). One of them, the regime of for-mation of spatially regular standing (quasi)stationarypatches will be considered in § 3 as a paradigm of tran-sition to chaos not linked to spatial irregularity.

3. RESULTS OF COMPUTER SIMULATIONS

System (2.4)–(2.5) with the initial conditions (2.6) and(2.7) was solved numerically in the domain � 1

2L � x �12L by the finite-difference method using the implicitscheme. The steps of the numerical mesh were chosen as�x = 0.2 and �t = 0.001 and it was checked that a furtherdecrease of the step values did not lead to any significantmodification of the results. The ‘no-flux’ condition wasused at the boundaries and the length L of the numerical

The Allee effect can enhance chaos A. Morozov and others 1409

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

0

–50 –30 –10 10 30 50

–50 –30 –10 10 30 50

–50 –30 –10 10 30 50space

space

space

(a)

(b)

(c)

prey

and

pre

dato

r de

nsit

ies

prey

and

pre

dato

r de

nsit

ies

prey

and

pre

dato

r de

nsit

ies

0

0

0

Figure 1. Population density versus space (solid curve forprey, dashed curve for predator) calculated from equations(2.4) and (2.5) at different times for parameters � = 0.34, = 0.6, � = 5, = 1, L = 100 and initial distribution (2.6)–(2.7) with u0 = 1, v0 = 0.1, �u = 19, �v = 4. Note that exceptfor the early stage (t � 100) the position of the patch centresdoes not change in the course of system dynamics. (a)t = 240, (b) t = 710 and (c) t = 1200.

domain was chosen large enough to make the impact ofthe boundaries as small as possible during the simu-lation time.

We want to mention that, owing to the Allee effect, thedynamics of system (2.4)–(2.5) depends on the initial con-ditions (cf. Lewis & Kareiva 1993): if u0 and/or �u aresmall (e.g. u0 � �) or v0 is large, the species go extinct(the impact of �v is less prominent). As consideration ofthis threshold phenomenon is beyond the scope of thispaper, in the computer simulations parameters u0, �u andv0 have always been chosen sufficiently large and suf-ficiently small, respectively.

Figure 1 gives an example of standing quasi-stationarypatches showing the population densities calculated att = 240 (a), t = 710 (b) and t = 1200 (c) for � = 0.34, = 0.6, � = 5, = 1, u0 = 1, v0 = 0.1, �u = 19, �v = 4 and


L = 100. For this regime, an early evolution of the initialspecies distribution leads to the formation at t � 100 oftwo dome-shaped patches, see figure 1a. During the laterstages of the system dynamics, the position of the patchcentres remains constant while the population densities inthe patches profile oscillate with time, as in figure 1a–c.Similar behaviour can be observed for somewhat differentparameter values, e.g. in the case where � and � are fixedas above, for between 0.597 and 0.637. For the valuesof close to the right-hand end of this interval, the oscil-lations in the patches profile cease and the population dis-tribution over space becomes stationary.

As the population density is of an apparently differentorder of magnitude near the patch centres and betweenthem (e.g. for x � 0), to quantify the dynamics of the sys-tem as a whole we consider the population size of preyand predator:

U(t) = � L/2

�L/2

u(x,t)dx, V(t) = � L/2

�L/2

v(x,t)dx. (3.1)

Our main goal is to acquire an insight into the proper-ties of the population oscillations in the patch profile andhow these properties may change with a change in theparameter values. For that purpose, we fix the values of �,� and as above and perform a large number of computersimulations corresponding to different (the effect ofother parameters’ variation will be considered in § 4). Foreach , system (2.4)–(2.5) with the initial conditions(2.6)–(2.7) is solved over sufficiently large time (to allowthe transients to die out) and the properties of the systemdynamics at large time are analysed in terms of populationsizes U and V.

We start with = 0.637 and consider how the systemdynamics changes with a decrease in . For only slightlyless than this value, the population densities relax, in thecourse of time, to a stationary dome-shaped distribution(similar to that shown in figure 1a), which corresponds toa point in the (U,V )-phase plane. We obtain the resultthat stationarity breaks at = 0.6185 and for 0.6036� � 0.6185 the system exhibits periodic oscillations; seefigure 2a(i),b(i). At = 0.6036, a period-doubling bifur-cation takes place; cf. figure 2a(i,ii),b(i,ii). More period-doubling bifurcations take place for smaller values of and, as a result of this cascade, the dynamics of the systembecomes chaotic at = 0.6009; see figure 2a(iii),b(iii).

To prove more rigorously that what is shown in figure2a(iii),b(iii) is really chaos, we estimate the dominant Lya-punov exponent �. For that purpose, we induce a smallperturbation to the patches and then consider the behav-iour of the following value:

�M(t) = {[U2(t) � U1(t)]2 � [V2(t) � V1(t)]2}1/2, (3.2)

where U1 and V1 and U2 and V2 are the values of theprey and predator population sizes in the undisturbed anddisturbed system, respectively. Thus, �M gives the dis-tance between the two initially close systems’ trajectories.The perturbation is applied at time t0 = 1200 to excludepossible impact of the transients, and �M(t0) is variedbetween 10�7 and 10�12.


6

4

2

0

–2

–4

–6

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

5.55.04.54.03.53.02.52.01.5

6

4

2

0

–2

–4

6

4

2

0

–2

–4

6.0

1.0

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

0 0.05 0.10 0.15 0.20 0.25

0 0.05 0.10 0.15 0.20 0.25

0 0.05 0.10 0.15 0.20 0.253 4 5 6 7 8 9

population size of prey time–1

(a) (b)

(i) (i)

(ii) (ii)

(iii) (iii)

pow

erpo

wer

pow

er

popu

lati

on s

ize

of p

reda

tor

popu

lati

on s

ize

of p

reda

tor

popu

lati

on s

ize

of p

reda

tor

Figure 2. A cascade of period-doubling bifurcations leading to the onset of chaotic oscillations: (a) trajectory of the systemdynamics in the phase plane (U,V ) for different values of predator mortality, (i) = 0.607, (ii) = 0.602 and (iii) = 0.6.Other parameters are the same as in figure 1. (b) The corresponding power spectra of the prey population size. Note that, inspite of chaos, the spatial population distribution remains regular; cf. figure 1.

To test the procedure, we first apply it to the case ofperiodic dynamics. We obtain the result that �M(t) neverexceeds 10�7 and the estimated value of � is zero (withinthe computation error) as is predicted by theory. We thenapply the procedure to the case shown in figure2a(iii),b(iii). For this regime, we obtain the result thatthere is an exponential discrepancy between the close tra-jectories, i.e. �M(t) � exp[�(t – t0)] where � is positive andits value is estimated as � = 0.009 ± 0.001. Note that, inspite of chaotic temporal oscillations of the populationsize, the spatial distribution of the populations remainsfairly regular; see figure 1 obtained for exactly the sameparameter values as in figure 2a(iii),b(iii).

It appears, however, that a further decrease in restoresregularity in temporal oscillations. Figure 3a shows the(U,V )-phase plane obtained for = 0.5998. The dynamicsis periodic in this case, although the form of the cycle is


rather complicated. A further decrease in leads toanother period-doubling cascade (cf. figure 3a(i,ii),b(i,ii))and for = 0.5993 the dynamics becomes chaotic againwith the dominant Lyapunov exponent estimated as� = 0.016 ± 0.001. We stress that in this case, again, thepopulation distribution over space is regular and qualitat-ively the same as shown in figure 1.

The above results are summarized in the form of thebifurcation tree shown in figure 4. To obtain the bifur-cation tree, we apply the following procedure. For eachvalue of , we first calculate the mean value of the predatorpopulation size V∗ at the time interval 1000 � t � 4200(a smaller time has not been taken into account to makesure that the transients die out completely). The cross-section points between the horizontal straight line V = V∗

and the trajectory of the system dynamics in the (U,V )-phase plane during that time interval give the set of values


5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

6

4

2

0

–2

–4

6

4

2

0

–2

–4

6

4

2

0

–2

–4

6.0

1.0

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

0 0.05 0.10 0.15 0.20 0.25

0 0.05 0.10 0.15 0.20 0.25

0 0.05 0.10 0.15 0.20 0.253 4 5 6 7 8 9population size of prey time–1

(a) (b)(i) (i)

(ii) (ii)

(iii) (iii)

pow

erpo

wer

pow

er

popu

lati

on s

ize

of p

reda

tor

popu

lati

on s

ize

of p

reda

tor

popu

lati

on s

ize

of p

reda

tor

Figure 3. The next cascade of period-doubling bifurcations after regularity was restored: (a) trajectory of the system dynamicsfor (i) = 0.5998, (ii) = 0.5996 and (iii) = 0.5993. Other parameters are the same as in figures 1 and 2. (b) Thecorresponding power spectra of the prey population size. In all cases the spatial distribution of the populations remainsregular, having the form of two symmetric dome-shaped patches.

Uk, which are then plotted in figure 4. Apparently, in thecase of stationary patches there is only one point corre-sponding to given (see the right-hand side of figure 4),in the case of simple periodic dynamics before the firstperiod-doubling bifurcation there are only two points, andin the case of chaos the points form a ‘dense’ set spreadover a certain interval corresponding to the size of thestrange attractor; cf. figures 2a(iii),b(iii) and 3a(iii),b(iii).

The simulations fulfilled for from 0.597 � � 0.5993show that chaos and regularity may also alternate in thatparameter range; however, the intervals between thevalues of corresponding to chaos and regularity becomeso small there that it makes detailed investigation very dif-ficult. For � 0.597 the regime of standing patches givesway to the regime of species extinction (Petrovskii et al.


2002b; Morozov 2003) and for 0.637 it turns to theregime of patchy spread (Petrovskii et al. 2002a,b).

It should be noted that, as long as the values of u0, v0,�u, �v corresponding to species extinction are excluded(see the comment at the beginning of § 3), the resultspresented above appear to be robust to the choice of initialconditions. The initial conditions affect only the earlystage of the system dynamics but not the later stage whenthe transients die out. Although minor details of the spa-tial pattern (e.g. the position of the patches) do dependon the initial species distribution, the regularity of the pat-tern is not affected by reasonably small variations of theinitial conditions. The system dynamics exhibit essentiallythe same properties; in particular, the shapes of the attrac-tors in figures 2 and 3 and the bifurcation tree in figure 4


8.0

7.5

7.0

6.5

6.0

5.5

5.0

4.5

4.00.5975 0.6000 0.6025 0.6050 0.6075 0.6100 0.6125 0.6150 0.6200

U

δ

Figure 4. The bifurcation diagram of the system (2.4)–(2.5) showing the changes in the system dynamics appearing from adecrease in the predator mortality rate . Other parameters are the same as in figures 1–3.

do not depend on the parameters of initial conditions (2.6)and (2.7).

4. CONCLUSIONS AND DISCUSSION

The Allee effect has been attracting much attentionrecently owing to its strong potential impact on populationdynamics (Amarasekare 1998; Gyllenberg et al. 1999;Owen & Lewis 2001; Wang & Kot 2001; Petrovskii et al.2002a,b). In this paper, we show that the Allee effect sig-nificantly increases the system’s spatio-temporal com-plexity and may enhance chaos in population dynamics;in the predator–prey system with the Allee effect, chaotictemporal population oscillations can appear in the casewhen the spatial distribution of the species remains reg-ular. By contrast, a spatially regular species distributionin the system without the Allee effect can only lead toperiodic/quasi-periodic population oscillations (Petrovskii& Malchow 1999, 2001; Sherratt 2001).

To visualize the system’s dynamics, we used the valuesof population size U and V; see equation (3.1). However,based on the results of our computer simulations we wantto mention that the type of the oscillations in populationsize, i.e. periodic or chaotic, conforms to the type of the(local) oscillations in population density. Particularly,chaos in the dynamics of population size always corre-sponds to local chaotic oscillations, although the magni-tude of the oscillations is very different near the centre ofthe patches and between them.

A question of interest is whether our results are robustto parameter variations. To address this issue, we per-formed extensive computer simulations for different para-meter values (in total, more than 100 parameter sets wereexamined). The results are summarized in figure 5, whichshows a map in the (�, ) parameter plane obtained for� = 5 and = 1. It appears that the parameter range wherespecies introduction leads to the formation of standingstationary/quasi-stationary patches (cf. figure 1) has theform of a stripe. Inside this domain there is a narrowerstripe where chaos and period locking in temporal popu-lation oscillations are observed for a spatially regularpopulation distribution. The parameters from above the


0.40

0.38

0.36

0.34

0.32

0.30

0.280.55 0.650.60

β

δ

borders of quasi-stationarypatches regime

chaos andperiod locking

regimes of invasion overthe whole area

periodicoscillations

Figure 5. The map in the (�, ) parameter plane showingthe domain corresponding to the regime of standingstationary/quasi-stationary patches and the subdomaincorresponding to chaos and period locking.

hatched domain correspond to species extinction, whereasthe parameters from below the hatched domain corre-spond to species invasion. Depending on parameters, theinvasion can take place either through propagation of atravelling population wave or through spreading of separ-ate self-multiplicating patches (cf. Petrovskii et al.2002a,b; Morozov 2003). On the whole, the spatiallyextended predator–prey system with the Allee effect exhib-its very rich dynamics, which will be described in detailelsewhere (Petrovskii et al. 2004).

Regarding the impact of other parameters, we haveobserved that variation of parameter � does not make asignificant impact, and chaos can be easily observed forvalues of � from a wide range. We also emphasize thatthe regime of standing quasi-stationary patches along withchaotic oscillations in population size can also be observedif the diffusivity is different for prey and predator. For


� 1, the position of the chaotic domain in the (�,)plane changes slightly while its size decreases with anincrease of the difference between and 1 so that for � 0.5 and 1.4 chaotic oscillations become very diffi-cult to observe. Note that this condition is not too restric-tive if applied to real ecological communities. Oneexample where the diffusivities of prey and predator arevirtually equal is given by the plankton system, where spa-tial mixing mainly occurs owing to the impact of turbu-lence (Okubo 1980). (Remarkably, plankton systems haveoften been used as a paradigm of chaotic spatio-temporaldynamics; see Pascual 1993; Sherratt 2001; Medvinsky etal. 2002.) In addition, in terrestrial communities, althoughthe individual motion of a predator tends to be faster thanthat of its prey, in many cases they are on the same order;examples may be given by lynx–hare and wolf–deertrophic pairs.

Analysis of the (�,)-plane structure thus leads to a clearunderstanding that the phenomenon as a whole (i.e. tem-poral chaos along with spatial regularity) should be essen-tially attributed to the impact of the strong Allee effectbecause the lower corner of the chaotic domain is alwayssituated at a positive �; see figure 5. For a weak Alleeeffect (–1 � � � 0) or when the Allee effect is absent(� � –1), chaotic oscillations of this type are not observed.

The existence of self-organized irregular spatial patternshas long been considered a ‘fingerprint’ of chaos (Cominset al. 1992; Nowak & May 1992) and was often linked tochaotic temporal fluctuations (Pascual 1993; Sherratt etal. 1995). However, the question remained open to whatextent a spatial structure is necessary to enhance chaoticoscillations. In this paper, we show that the impact of theAllee effect can reduce the degree of required spatial het-erogeneity and invokes temporal chaotic oscillations in asystem where they would otherwise be impossible. Ourresults make a certain contribution towards a betterunderstanding of possible levels of spatio-temporal com-plexity in ecological communities.

It should be noted that, if considered in a somewhatbroader ecological perspective, our results have an intuit-ively clear meaning. There has been a growing under-standing during the past years that, regarding thedynamics of real ecosystems, it is perhaps not chaos itselfthat is important but the transitions between differentdynamical regimes arising as a result of perturbation ofthe system’s parameters (cf. Cushing et al. 2003). Fromthis standpoint, it seems interesting to know how thedynamics varies when the parameters move across the dia-gram shown in figure 5. For simplicity, let us assume thatonly one parameter is changing, either or �, othersremaining fixed. A decrease in the predator mortality changes the regime of species invasion to the regime ofstanding patches and then to species extinction. Appar-ently, from a given species’ perspective, invasion is a posi-tive factor enhancing its regional persistence. A decreasein predator mortality destabilizes the system, hampers andeventually blocks species invasion, and thus potentiallydecreases species persistence. Similar effects are producedby an increase in �. These inferences appear to be in goodagreement with current understanding of the impact ofpredation and the Allee effect on population dynamics.

In conclusion, we mention that the results of this paperwere obtained under the assumption that predation is


described by the bilinear function of the prey and predatordensities (see the lines below equations (2.1) and (2.2)).This assumption, however, is not crucial and has beenmade mainly to avoid additional parameters. Our com-puter experiments indicate that the system exhibits quali-tatively similar behaviour when predator response is ofHolling type II, e.g. f(H,P) = AHP/(H � B), where B isanother parameter, and the regime of standing quasi-stationary patches with chaotic oscillations can also beobserved in this case.

The authors thank R. Arditi, A. Medvinsky and S. Ruan fortheir helpful remarks. This work was partly supported by theRussian Foundation for Basic Research under grant nos. 03-04-48018 and 04-04-49649, by the US National ScienceFoundation under grant nos. DEB-00-80529 and DBI-98-20318, by the University of California Agricultural ExperimentStation and by the UCR Center for Conservation Biology.

REFERENCES

Allee, W. C. 1938 The social life of animals. New York: Nortonand Co.

Allen, J. C., Schaffer, W. M. & Rosko, D. 1993 Chaos reducesspecies extinction by amplifying local population noise.Nature 364, 229–232.

Amarasekare, P. 1998 Interactions between local dynamics anddispersal: insights from single species models. Theor. Popul.Biol. 53, 44–59.

Barenblatt, G. I. 1996 Scaling, self-similarity, and intermediateasymptotics. Cambridge University Press.

Comins, H. N., Hassell, M. P. & May, R. M. 1992 The spatialdynamics of host–parasitoid systems. J. Anim. Ecol. 61,735–748.

Costantino, R. F., Desharnais, R. A., Cushing, J. M. &Dennis, B. 1997 Chaotic dynamics in an insect population.Science 275, 389–391.

Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais,R. A. & Henson, S. M. 2003 Chaos in ecology: experimentalnonlinear dynamics. San Diego: Academic.

Dennis, B. 1989 Allee effects: population growth, critical den-sity, and the chance of extinction. Nat. Res. Model. 3,481–538.

Dennis, B., Desharnais, R. A., Cushing, J. M., Henson,S. M. & Costantino, R. F. 2001 Estimating chaos and com-plex dynamics in an insect population. Ecol. Monogr. 71,277–303.

Dennis, B., Desharnais, R. A., Cushing, J. M., Henson,S. M. & Costantino, R. F. 2003 Can noise induce chaos?Oikos 102, 329–339.

De Roos, A. M., McCauley, E. & Wilson, W. G. 1991Mobility versus density-limited predator–prey dynamics ondifferent spatial scales. Proc. R. Soc. Lond. B 246, 117–122.

Ellner, S. & Turchin, P. 1995 Chaos in a noisy world: newmethods and evidence from time-series analysis. Am. Nat.145, 343–375.

Gilpin, M. E. 1978 Spiral chaos in a predator–prey model. Am.Nat. 112, 306–308.

Gyllenberg, M., Hemminki, J. & Tammaru, T. 1999 Alleeeffects can both conserve and create spatial heterogeneity inpopulation densities. Theor. Popul. Biol. 56, 231–242.

Hastings, A. 2001 Transient dynamics and persistence of eco-logical systems. Ecol. Lett. 4, 215–220.

Hastings, A. & Powell, T. 1991 Chaos in a three-species foodchain. Ecology 72, 896–903.

Hastings, A., Hom, C. L., Ellner, S., Turchin, P. & Godfray,H. C. J. 1993 Chaos in ecology: is mother nature a strangeattractor? A. Rev. Ecol. Syst. 24, 1–33.


Holmes, E. E., Lewis, M. A., Banks, J. E. & Veit, R. R. 1994Partial differential equations in ecology: spatial interactionsand population dynamics. Ecology 75, 17–29.

Huisman, J. & Weissing, F. J. 1999 Biodiversity of planktonby species oscillations and chaos. Nature 402, 407–410.

Jansen, V. A. A. 1995 Regulation of predator–prey systemsthrough spatial interactions: a possible solution to the para-dox of enrichment. Oikos 74, 384–390.

Lewis, M. A. & Kareiva, P. 1993 Allee dynamics and thespread of invading organisms. Theor. Popul. Biol. 43, 141–158.

May, R. M. 1976 Simple mathematical models with very com-plicated dynamics. Nature 261, 459–467.

Medvinsky, A. B., Petrovskii, S. V., Tikhonova, I. A., Mal-chow, H. & Li, B.-L. 2002 Spatiotemporal complexity ofplankton and fish dynamics. SIAM Rev. 44, 311–370.

Morozov, A. Y. 2003 Mathematical modelling of ecological factorsaffecting stability and pattern formation in marine plankton com-munities. PhD thesis. Shirshov Institute of Oceanology,Moscow.

Murray, J. D. 1989 Mathematical biology. Berlin: Springer.Nisbet, R. M. & Gurney, W. S. C. 1982 Modelling fluctuatingpopulations. Chichester: Wiley.

Nowak, M. A. & May, R. M. 1992 Evolutionary games andspatial chaos. Nature 359, 826–829.

Okubo, A. 1980 Diffusion and ecological problems: mathematicalmodels. Berlin: Springer.

Owen, M. R. & Lewis, M. A. 2001 How predation can slow,stop or reverse a prey invasion. Bull. Math. Biol. 63, 655–684.

Pascual, M. 1993 Diffusion-induced chaos in a spatial pred-ator–prey system. Proc. R. Soc. Lond. B 251, 1–7.

Pascual, M. & Caswell, H. 1997 Environmental heterogeneityand biological pattern in a chaotic predator–prey system. J.Theor. Biol. 185, 1–13.

Perry, J. N., Smith, R. H., Woiwod, I. P. & Morse, D. R. (eds)2000 Chaos in real data: the analysis of nonlinear dynamics fromshort ecological time series. Dordrecht, The Netherlands:Kluwer.

Petrovskii, S. V. & Malchow, H. 1999 A minimal model ofpattern formation in a prey–predator system. Math. Comput.Model. 29, 49–63.


Petrovskii, S. V. & Malchow, H. 2001 Wave of chaos: newmechanism of pattern formation in spatio-temporal popu-lation dynamics. Theor. Popul. Biol. 59, 157–174.

Petrovskii, S. V., Morozov, A. Y. & Venturino, E. 2002a Alleeeffect makes possible patchy invasion in a predator–prey sys-tem. Ecol. Lett. 5, 345–352.

Petrovskii, S. V., Vinogradov, M. E. & Morozov, A. Y. 2002bSpatio-temporal horizontal plankton patterns caused by bio-logical invasion in a two-species model of plankton dynamicsallowing for the Allee effect. Oceanology 42, 384–393.

Petrovskii, S. V., Li, B.-L. & Malchow, H. 2003 Quantificationof the spatial aspect of chaotic dynamics in biological andchemical systems. Bull. Math. Biol. 65, 425–446.

Petrovskii, S. V., Morozov, A. Y. & Li, B.-L. 2004 Regimesof biological invasion in a predator–prey system with theAllee effect. Bull. Math. Biol. (Submitted.)

Rand, D. A. & Wilson, H. B. 1995 Using spatio-temporalchaos and intermediate-scale determinism to quantify spati-ally extended ecosystems. Proc. R. Soc. Lond. B 259, 111–117.

Rinaldi, S. & De Feo, O. 1999 Top predator abundance andchaos in tritrophic food chains. Ecol. Lett. 2, 6–10.

Scheffer, M. 1991 Should we expect strange attractors behindplankton dynamics—and if so, should we bother? J. Plankt.Res. 13, 1291–1305.

Segel, L. A. & Jackson, J. L. 1972 Dissipative structure: anexplanation and an ecological example. J. Theor. Biol. 37,545–559.

Sherratt, J. A. 2001 Periodic travelling waves in cyclic pred-ator–prey systems. Ecol. Lett. 4, 30–37.

Sherratt, J. A., Lewis, M. A. & Fowler, A. C. 1995 Ecologicalchaos in the wake of invasion. Proc. Natl Acad. Sci. USA 92,2524–2528.

Sole, R. V. & Bascompte, J. 1995 Measuring chaos from spatialinformation. J. Theor. Biol. 175, 139–147.

Wang, M.-E. & Kot, M. 2001 Speeds of invasion in a modelwith strong or weak Allee effects. Math. Biosci. 171, 83–97.

As this paper exceeds the maximum length normally permitted, theauthors have agreed to contribute to production costs.

Bifurcations and chaos in a predator–prey system with the ... · MAIN EQUATIONS We consider the...

Documents

Transcript of Bifurcations and chaos in a predator–prey system with the ... · MAIN EQUATIONS We consider the...