controlling chaos in ecology: from deterministic to individual-based models

Available online at http://www.idealibrary.com onArticle No. bulm.1999.0141Bulletin of Mathematical Biology(1999)61, 1187–1207

Controlling Chaos in Ecology: From Deterministic toIndividual-based Models

RICARD V. SOLEComplex Systems Research Group,Department of Physics FEN, Universitat Politecnica de Catalunya,Campus Nord B5, 08034 Barcelona,SpainE-mail: [email protected]

JAVIER G. P. GAMARRAComplex Systems Research Group,Department of Physics FEN, Universitat Politecnica de Catalunya,Campus Nord B5, 08034 Barcelona,Spainand Forest Technology Center of Catalonia,Pujada del Seminari s/n,25280 Solsona,SpainE-mail: [email protected]

MARTA GINOVART AND DANIEL L OPEZEscola Superior d’Agricultura de Barcelona,Urgell 187, 08036 Barcelona,Spain

The possibility of chaos control in biological systems has been stimulated by recentadvances in the study of heart and brain tissue dynamics. More recently, some au-thors have conjectured that such a method might be applied to population dynamicsand even play a nontrivial evolutionary role in ecology. In this paper we explorethis idea by means of both mathematical and individual-based simulation models.Because of the intrinsic noise linked to individual behavior, controlling a noisy sys-tem becomes more difficult but, as shown here, it is a feasible task allowed to beexperimentally tested.

c© 1999 Society for Mathematical Biology

1. INTRODUCTION

Complex dynamical patterns are an inherent part of both real and model ecosys-tems and have been a matter of extensive studies in recent years [Hassellet al.(1976); Godfray and Blythe (1990); Tilman and Wedin (1991); Bascompte andSole (1995) and references therein]. Chaos is a particularly relevant example of

0092-8240/99/061187 + 21 $30.00/0 c© 1999 Society for Mathematical Biology

1188 R. V. Soleet al.

these patterns and it can be obtained even from very simple nonlinear models asa result of intrinsic nonlinearities (May, 1974, 1976; May and Oster, 1976; Schaf-fer and Kot, 1986). As an example, let us consider the following one-dimensionalmap:

Xt+1 = Xt exp

[r

(1−

Xt

K

)](1)

describing single-species population dynamics with nonoverlapped generationswherer is a constant representing the population growth rate (under no shortanceof resources) andK is the carrying capacity of the population (maximum popula-tion attainable by the system). This is the well-known Ricker model (Ricker, 1954)but other uniparametric, smooth maps with a single maximum on their interval ofdefinition display similar features (in particular a period-doubling scenario leadingto chaos).

Although the evidence for this type of dynamics in nature is still controversial[seeBerryman and Millstein (1989); Lomnicki (1989); Nisbetet al. (1989); Rohaniand Earn (1997)], recent experimental studies involving populations of flour beetleTribolium castaneum(Constantinoet al., 1995, 1997; Denniset al., 1997) haveconfirmed the theoretical predictions even when stochastic terms were includedin the model. By means of changes in demographic parameters, these authorsobtained a broad variation in the nature of population fluctuations by showing thatexperimental data were consistent with the expected model predictions (includingsteady points, oscillations and chaos).

A different possibility for showing the existence of chaos in real populations isprovided through the theory of chaos control (Ott et al., 1990; Ditto et al., 1995). Ithas been shown that, under some constraints, a chaotic system may be ‘controlled’:unstable periodic orbits can be stabilized by means of small, periodic perturbationsof the system parameters (Ott et al., 1990) or states (Guemez and Matıas, 1993;Parthasarathy and Sinha, 1995). In fact, the very nature of chaotic dynamics, withits sensitive dependence on initial conditions and an infinite number of unstableperiodic orbits simultaneously embedded in phase space, makes feasible the pos-sibility of chaos control by moving and keeping the system’s trajectory close toone of the (unstable) orbits, artificially stabilizing one of them. The flexibility ofthis dynamics is opposed to the case where the fixed point is stable. In such acase, moving to another periodic orbit needs a very large alteration of the system.Further work explored the possibility of chaos control in spatiotemporal dynamics,both in physics and biology (Astakhovet al., 1995; Doebeli and Ruxton, 1997).Biological applications include the stabilization of chaos in neural systems (Schiffet al., 1994; Sole and Menendez de la Prida, 1995), chemical reactions (Petrovetal., 1993) and cardiac tissue (Garfinkelet al., 1992).

Some ecological and evolutionary consequences have already been pointed out.Doebeli (1993) hypothesized about the possibility of natural self-control in a pop-ulation under unstable Ricker dynamics, as a way to keep the population at equi-librium in a form ofK selection. Besides,Doebeli and Ruxton (1997) have argued

Controlling Chaos in Ecology 1189

about the possibility of chaos avoidance in some populations located in areas whereexternal factors such as climate may have seasonal changes that drive the dynamicsfrom a chaotic situation to more stable dynamics with small fluctuations, definingthis external seasonality as a natural way in which populations can be stabilized.In this sense some studies on mammal populations, such as the classical ones onlynx fur returns or vole populations in zones closer to the poles, whose dynamicsis clearly cyclical, could be affected by this kind of control.

Deterministic models of populations with chaos control are well known in theliterature [see, e.g.,McCallum (1992); Stone (1993)]. However, it is fairly wellknown that the dynamics in real ecosystems may be seriously affected when noiseis at work. In this context,Allen et al. (1993) showed that local population noiseis amplified in the presence of chaos, due to the resulting asynchronous behavioramong populations [see alsoSole and Gamarra (1998)]. However, there remainsan open question: might a population under a chaotic regime be stabilized in thepresence of internal noise? If so, which are the ecological consequences for realecosystems? Our initial hypothesis is that populations with global mixing (popula-tions located in a sufficiently small area to allow dispersal to every point in the areawith the same probability, or, inversely, populations with large enough dispersalcapacity to all points) are not affected in the presence of noise by asynchronici-ties large enough to avoid the correlating effect of control in the population. Thusthe techniques for controlling chaotic dynamics would be able to overcome theproblem of noise and consequently, the robustness of control would be a furtherargument supporting the existence of instabilities due to the nonlinear nature ofecosystems.

Here we will consider the applicability of some of the techniques for control-ling chaos in order to test the existence of deterministic chaos in ecosystems andthe possibilities for new experimental manipulations of laboratory populations willbe discussed. The presence of noise in the dynamics and two individual-orientedapproaches will shed light onto the robustness of control under stochastic environ-ments (as those found in microecosystems).

2. METHODS OF STABILIZATION IN DISCRETE SYSTEMS

2.1. Parametric (internal) perturbations. Pioneering work on controlling un-stable orbits was early made byOtt et al. (1990) (OGY method). The methodologyof control needs a previous knowledge of the trajectory of the orbit to stabilize, orat least the location of the fixed points in a Poincare section. The method has beensuccessfully applied in physics (i.e., chaotic oscillators) (Ditto et al., 1990), but it istoo difficult to implement in ecological systems: the need for previous knowledgeof the system’s trajectory implies the use of very long time series and ecologicaldata sets available are still scarce and short.

A more plausible protocol involves the use of periodic perturbations on the popu-


lation growth rate [seeDoebeli (1993); Stone (1993)]. Let us start with the single-species population model described by (1), whereXt+1 = Xt fµ(Xt), such thatfµ(X) = exp[r (1− X)] and thus limX→∞ fµ(X) = 0. Let us assume thatr > rc,so we are at the chaotic domain of the parameter space. This state can be charac-terized through the Lyapunov exponent, defined asλL = 〈ln |∂x fµ(xi )|〉, where theaverage is taken over the entire system’s dynamics, i.e.,:{x1, . . . , xn}. Since weare in the chaotic domain, a fast and intuitive method of control is to perturb theparameterr so that we move tor < rc. That is, if we consider model (1) involvingcontrol eachp generations:

Xt+1 = f ′µ(X′

t)

whereX′t = Xt−(p−1) andµ′ = r + γ whereγ < 0 is a constant rate accountingfor seasonal internal causes of population decline due to density dependence. Thisprocedure, illustrated in Fig.1(a) and (b), where the dynamics is biperiodic forthe Ricker’s map, ensures that the slope of the curve| f ′(µ, x∗t )| < 1, satisfyingthe conditions for stability [seeMay (1976); Stone (1993)], wherex∗t is the fixedpoint, i.e., the intersection of thext+1 = xt line and the functionf (µ, xt), then aperiod-doubling reversal appears as a direct consequence of moving far belowrc

(Stone, 1993). In case we apply a perturbation every two generations, the Ricker’smap must be represented in axt+2 − xt graph (second-iterate map), wherext+2 =

fµ( f ′µ(xt)), and the minimum number of orbits attainable is1n = p = 2, due tothe existence of two stable fixed pointsX∗1 andX∗2.

2.2. State (external) perturbations.Two different approaches may be treated.Both add an external feedback to the population everyp iterations and both arevery easy to apply and have a more meaningful ecological understanding to ac-knowledge the role that dispersal might play among metapopulations in order toavoid chaos and consequently the probability of extinction.

The first one, developed byGuemez and Matıas (1993) (GM or proportionalfeedback method) involves stabilizing one orbit of the many unstable periodic onesof a chaotic system by perturbing the system with periodic pulses, which are pro-portional to the state the system presents. For some values of intensity and pe-riodicity of the perturbation the orbit is stabilized. The protocol should be usedto control systems where we do not know the parameters at work, but where wecan take measurements over the variables, i.e., the population size of biomass of aspecies.

Control is exerted through the application of the feedback to the populationxevery p iterations:

Xt + 1= fµ(X′

t) ∗ (1+ γ )

where, again,X′t = Xt−(p−1), the parameterµ is kept constant andγ representsthe strength of the pulse. This type of control may be interpreted as a seasonalor periodic density-dependent immigration(γ > 0) or emigration(γ < 0) term


00

1

2

N(t

)N

(t)

N(t

)

N (t – 1)

N (t – 1)

N (t – 1)

3

1 2 3 00

1

2

γ = 0.15

γ = 0

γ = 1

γ = 1

3

4

000

1

2

3

1 2 30

1

2

N(t

)

3

1

N (t – 1)

2 3

(c)

(a )

(d)

(b)

1 2 3 4

Figure 1. Graphical interpretation of some of the techniques of control,where some perturbationγ is added to either a parameter or a state variableof the system in the Ricker map, such thatr ′ = r (1− γ ). The graph isshown forr = 3 andTcont = 1 in all cases. AsY increases, the slope(derivative) of the map at the fixed point, i.e., that point where the curvecrosses theN(t) = N(t − 1) line, moves towards the curve, getting values∣∣d F(N)

d N

∣∣ < 1, whereF(N) is the corresponding Ricker map. (a) Chaoticregime, no control. (b) Parametric control:r ′ = r (1− γ ). (c) and (d) areboth external methods GM and PS respectively.

when individuals tend to be clumped or territorial, respectively, thus adding a mul-tiplicative pulse to the population size. Figure1(c) represents the effect of theGM control method on the first-iterate map for a period of controlp = 1 withstrengthγ = 0.15. We can represent ther − γ phase space showing the con-tour lines separating the areas corresponding to different stable periodic orbits, asin Fig. 2(a). In the first return map the sensitivity of the system to the controlfeedback is stronger for negative values ofγ , giving raise to smaller populationsizes by smoothing the single hump of the map return. Forγ > 0, the fixed pointgets higher values, shifts towards the rightmost part of the return map and, under


some values, stabilizes the orbit for larger population sizes. A similar protocolrooting in the target of the perturbation was first developed byMcCallum (1992).The idea grew from assuming a chaotic population following a Ricker’s dynamicsand adding an immigrant population as a structural perturbation every time stepto observe that unstable orbits became periodic under certain values of immigra-tion γ . Stone (1993) further extended this work to demonstrate that immigrationopened the possibility of period-doubling reversals. This method was later statedas the method of constant feedback control (Parthasarathy and Sinha, 1995) (PShereafter). Such state perturbation consists of applying a constant pulseγ every pgenerations in the form

Xt+1 = fµ(X′

t)+ γ

with X′t = Xt−(p−1) and the parameterµ kept constant. Let us consider againthat we haver ≈ rc. The return map in Fig.1(d) shows that the addition of apositiveγ moves up the map and increases the value of the nontrivial fixed pointby displacing it towards the plateau region, therefore changing the value of the firstderivative such that| f ′µ(x)| < 1. The phase-space diagram in Fig.2(b) shows thatif γ < 0 then there will be only a few possible values under which the orbit visitperiodic windows. Control may appear in this method with very small values in1γ , so the nature of the orbits appearing in this system seems to be more sensibleto perturbations at highr values. There is also an area stated as theescaperegion,generally under negative values of gamma, where emigration is so strong, that theresulting return map presents a displacement of the unstable trivial fixed point fromzero toX∗t → 0, so when the system visits values ofXt below this fixed point, thepopulation infallibly becomes extinct.

3. STABILIZATION IN CONTINUOUS T IME

Perturbation of unstable orbits in order to stabilize the dynamics is also possiblefor the case of populations following continuous kinetics and systems with severaldimensions. The extension of the previous perturbation methods to a continuousmodel is straightforward and has been shown to work successfully. In this case,since the nonlinear equations are integrated by means of a characteristicδt-timediscretization, control will be applied eachpδt steps in the simulation.

We have addressed the problem of controlling chaos in a typical ecological sys-tem, such as a model plankton community, where apparently internally driven ir-regular oscillations may occur, as has been observed in some experimental mi-croecosystems (Ringelberg, 1977). This realistic model, a three-dimensional ODEmodel of plankton dynamics, showing a single food web with a series of interac-tions able to produce chaotic dynamics in some of the trophic levels, was made byScheffer (1991). He developed a model with three trophic levels that involvedplanktonic algae(A), herbivorous zooplankton(Z) and carnivorous zooplank-


0–1

–0.25

0.5

Stre

ngth

of

cont

rol

Stre

ngth

of

cont

rol

1.25

2

2.75

–1

–0.75

–0.5

>4

>4 >4

1

1

1

2 4 >4

escapeescape

2

4

–0.25

0

0.25

1 2

Growth rate

3 4

0 1 2 3 4

Figure 2. r − γ phase-space diagrams for (a) GM and (b) PS controlmethods characterizing regions ofp = 1,2,4,> 4 periodic orbits and anescape region, characterized numerically. In both,Tcont= 1.

ton (C) and the differential equations describing the dynamics are

d A

dt= r A − cA2

− gZ

d Z

dt= gZce− zmortZ − pc

dC

dt= pCce− cmortC (2)

wherer > 0 is the algal relative growth rate,c > 0 is a competition coefficientfor algae,g ≥ 0 and p ≥ 0 are respectively a grazing and a predation rate bothbeing Monod saturated functions and thus identifying a type II functional responsefor herbivory and predation.ce ∈ (0,1) is the conversion efficiency from food tobiomass andzmort > 0 andcmort > 0 are mortality rates. In this three-trophic levelmodel, strange attractors may appear as a consequence of increasingr by a slightamount.


In order to test whether control methods may work we ran simulations underthe same parameter values thatScheffer (1991) set to get strange attractors in hismodel. We used the proportional (GM) and constant feedback perturbations ineach of the three trophic levels appearing in equation (2). Graphical results appearin Fig. 3. Through several values ofp and γ , control may appear in the formof a wide variety of dynamical behaviors, from stable nodes [Fig.3(a)], to stablefocus [Fig.3(d)] and limit cycles [Fig.3(b), (c), (e), (f)]. There is a special case[Fig. 3(e)], where using the GM control method the orbit wraps around the strangeattractor. In such case, the regular dynamics continues several iterations after thecontrol routine was stopped, maintaining the system in the stable area.

The control induced in this system also contributes to produce a period-doublingreversal [sensuStone (1993)]. The main point for the control of chaos in thesen-dimensional continuous systems is that a small perturbation applied on one ofthe trophic levels, is able to propagate very rapidly over the whole system by itseffect on the interactions. As far as chaotic dynamics is flexible enough, the wholeecosystem changes the trajectories of its components towards more regular ones.

Now the obvious question is whether or not these control methods will be suc-cessful in controlling real populations. A first criticism comes from the lack ofnoise in our models. Noise can strongly modify our previous expectations. Sincethe standard control techniques require deterministic dynamics in order to guaran-tee that the orbit will visit the desired domains where control is applied, the intro-duction of noise can prevent the dynamics from reaching such domains. Addition-ally, some authors have suggested that the effect of individual, microscopic fluc-tuations might introduce fundamental effects eventually breaking down a macro-scopic description [Fox and Keitzer, 1990; see howeverPeeters and Nicolis, 1992].An additional point concerns the translation of these theoretical models to realis-tic, small-sized and intrinsically noisy laboratory populations. Since the controlmethod is expected to be performed by adding given amounts of individuals to thepopulation, and because of the unavoidable random factors arising from individualcomplexities, there is in principle no guarantee that an efficient control is going tobe reached. In the next two sections we explore this problem by using two differ-ent individual-based models of both discrete and continuous dynamics involvingreasonable population sizes in the range of those used in experimental systems.

4. CONTROL IN A DISCRETE STOCHASTIC IBM

The study of the dynamics of nonoverlaping generations (i.e., generations wherereproductive individuals and offspring do not overlap in time) has already beenwidely carried mainly in terms of one-dimensional nonlinear deterministic models(May, 1974; Hassellet al., 1976). These are able to generate a wide spectrumof behaviors (from stable to chaotic), by simply changing one parameter in theunderlying map (usually the growth rate).


0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0

0.0 2.0 4.0 6.00.0 2.0 4.0 6.0

0.00.0

0.5

1.0ZC

C

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

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0.6

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0.6

0.8

2.0 4.0 6.00.0 2.0A A

A A

Z

GM PS

Z

4.0 6.0

(a) (b)

(e) (f)

(d)(c)

Figure 3. Attractors reconstructed for the planktonic system. Both GMand PS control methods have been used. Control is exerted on algae [(a)and (b)], herbivorous zooplankton [(c) and (d)] and carnivorous zooplank-ton [(e) and (f)]. Dashed lines represent the strange attractor and solid linesrepresent the trajectory under control. Parameters used are: (a)p = 1,γ = −0.1; (b) p = 3, γ = 0.3; (c) p = 3, γ = −0.05; (d) p = 3,γ = 0.01; (e) p = 3, γ = −0.2; (f) p = 3, γ = 0.001. The rest ofparameters are the same as inScheffer (1991).


Some problems may appear in the use of these models, such as the fact that thegrowth rates are considered as fixed, and the interactions taking place among theindividuals. In this section a more realistic simulation includes the presence ofnoise by considering the effects of population size and random variability in thegrowth and mortality rates. As far as we know, this is the first fully individual-based counterpart of the standard mathematical models involving single-speciesdiscrete dynamics. In this IBM model, a detailed description of single individualsis introduced, necessary for a complete description of the within-generation dy-namics. The overall dynamics as described by the population values from genera-tion to generation displays the expected pattern observed from one-hump modelsof nonoverlapped generations.

4.1. Model description. The basic methodology of our system comprises severalcharacteristics to have in mind:

1. Discrete space and time.2. A group of N individuals each with its own physiological properties, such

as biomass or age, all of them dependent on time.3. A group of cells in a lattice, with a certain quantity of energy available on

each cell which may change with time and interact with the individuals.4. Periodic boundary conditions.

In our system we consider individuals moving, feeding on energy, reproducingand dying. For computational purposes, individuals and energy are placed in atwo-dimensionalL × L lattice, although the global mixing makes the spatial di-mension irrelevant. The basic temporal scale under these interactions correspondsto the within-generation dynamics. At the end of each generation each individ-ual reproduces and dies and the next generation starts with the resulting offspringfeeding on a newly updated energy population, i.e., the system behaves as a forcedchemostat. Thus, new generations of individuals (grazers) are conditioned by thefinal results of the within-generations evolution (the full flow diagram is providedin Fig. 4). This protocol is repeated a certain number of iterations.

Again, our system develops as a well-mixed system, where the only local inter-actions account for competition for resources among individuals in a cell (whichmay lay the basic rules for density dependence), but there is global movement ofindividuals and offspring.

At the beginning of each generation there is a quantityE of energy particles ran-domly distributed over the whole lattice. Within each generation, there is neitherenergy input nor energy movement across the system. During the evolution of thegeneration, each individual (level 1 in Fig.4) consumes a random fraction of quan-tity E(µ, σ ) of the energy available in the cell with certain dissipation rate, theindividuals grow and randomly move to other cells looking for more energy, untilthe end of the generation is accomplished (i.e., when individuals have exploitedall the energy in the system). When this happens (level 2 in Fig.4), we have to


START

DESCRIPTION OF INDIVIDUALSAND ENVIRONMENT

INITIAL CONFIGURATION(INDIVIDUALS AND ENVIRONMENTAL

CONDITIONS)

ITERATION STEP

EVALUATION OF CONDITIONSFOR THE END OF GENERATION

LIST OF ACTIVEINDIVIDUALS

FOR EACH INDIVIDUAL -MOVEMENT -NOURISHMENT SEARCH AND CONSUMPTION

-METABOLISM GROWTH AND DISSIPATION

EMPTY LIST?

END OFGENERATION?

-REPRODUCTION-NEW GENERATION-DEATH

VARIABLESUPDATING

DATA ANALYSIS

END

YES

YES

YES

NO

NO

NO

LEVEL 1

MAIN LOOP

INITIAL BLOCK

NUTRIENTRE-INITIALIZATION

FINALCONDITIONS?

LEVEL 2

Figure 4. Schematic representation of the processes taking place in thestochastic Ricker IBM.


compare the individual biomass, which increment is proportional to the quantity ofresource consumed, with the fixed minimum mass for reproduction. The grazersthat achieve this biomass reproduce some fixed quantity of offspring and die andthe resulting offspring is redistributed over the whole. The newborns are the onesthat start the next generation. Environmental conditions such as the quantity of en-ergy available only will be modified by direct consumption during the generationtime. A fixed amount of energy is then updated at the beginning of the next gener-ation. The system is closed, since there is no energy input in the within-generationtime. The list of parameters used in the model is shown in Table1. Using thismodel, we have found all the basic features known from the analysis of the Rickermap equation, from periodic doublings to intermittency. The analysis of the cor-relation dimension for the dynamics shows that it is one-dimensional with fractaldimension less than one for chaotic attractors.

Table 1. Main parameters used in the stochastic Ricker-map IBM.

Parameter Definition Default value Effect when increased

E Energy input 2× 106 Larger amplitudeTcont Control period — PDBa

γ Perturbation size — Larger amplitudeω Number of offsprings/individual 5 Larger amplitude, PDBa

δ Dissipation rate at consumption 0.3 PHBb

Mb Mass at birth 250 PHBb

Mrep Minimum mass of reproduction 103 PDBa

a PDB: Period-doubling bifurcations.b PHB: Period-halving bifurcations.

Again we introduce additive external perturbations in order to stabilize the dy-namics in our Ricker map. Some results are given in Fig.5. From top to bottom inthis figure we show: (a) the system with no control, at the chaotic regime, (b) two-periodic controlled orbit (control is applied from generationn = 300 ton = 700)and (c) single-point controlled attractor. In all these examples we use a systemwith a maximum sizeN = 2000 with default values as those indicated in Table1.The two sets of control values areγ = 150 andTcont = 2 for the periodic con-trolled orbit andγ = 400 andTcont = 1 for the steady state. Extensive numericalsimulations show that this control method is highly effective and in fact we canplot a bifurcation diagram usingγ as a parameter and show that a period-halvingscenario is obtained.

Once again, the system may be controlled under certain regimes of perturbation.In our discrete system, the period of control is not affected by the generation-time, but control is more effective asTcont becomes smaller. The perturbation sizedoes not need to be large (i.e.,γ ≈ L×L

10 ), notwithstanding that the system has toovercome the stochastic fluctuations affecting the dynamics.

4.2. Lotka–Volterra predator–prey individual-based model.In this section weintroduce the basic rules describing a Lotka–Volterra (LV) model with microscopic


1500

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00 500

Generations N (t)

1000 0 300 600 900 1200 1500

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300

1500

1200

900N

(t+

1)

Popu

latio

n si

ze

600

300

1500

1200

900

600

300

0

Figure 5. First iterate map of the output of a simulation on the stochasticRicker IBM with additive (PS) control with control periodsTcont = 1,2.Empty and full circles represent the chaotic and control regimes, respec-tively. In both simulationsdd = 2 andγ = 25. The other parameters aredefined in Table1.

interactions among its elements. Again, one of the reasons to use an individual-based model stems from the implicit randomness of its individual interactions, nowamong different species, setting explicitly the stochastic nature of the system (Jud-son, 1994; Wilson, 1998). A similar LV model was introduced by Ricker involvinga rather sophisticated set of individual properties (Sole and Valls, 1992). Here werestrict ourselves to a much more simple set of interactions. Let us note, however,that the basic properties of the bifurcations are the same as those observed in the


original model. The continuous nature of the model roots from the time-couplingof reproduction among its individuals. The exploration of this model is justifiedsince a single-species population as the one described in the previous section isunlikely to be available. Two-species microecosystems are well known from Huf-faker’s classic studies.

Due to the fact that we are not intending to explore the spatial correlations, butinstead the stochastic nature of the system and the posibilities for control under thepresence of noise, we do not explore the spatially extended version of the modelbut a well-mixed model where interactions among individuals occur as a periodi-cally forced chemostat, where the concentration of substrate or energy enters thesystem periodically or, more specifically, sinusoidally, causing the whole systemto oscillate repeteadly with some characteristic period and amplitude, controlledprimarily by the period and amplitude of the input energy, and thus creating a timedependency in the energy input (Pimm, 1991; Kot et al., 1992; Rinaldiet al., 1993).This forced predator–prey model is much more easy to display chaotic dynamicsand remain stabilized. If a three-species ecosystem is used then the top predator islikely to become extinct, unless very high populations are used.

For simplicity, we consider our physical system as formed by a number ofNnoninteracting sites (where no spatial structure exists). This obviously introducesa limitation into the prey population, which, because of the finiteness of the systemwill grow logistically. Let N1(t), N2(t) be the number of prey and predators at agiven stept . Below, we describe the set of rules used.

Energy release.Prey feed on some type of substrate which we will callenergy,which is randomly introduced into the cells. LetEi (t) be the amount of energyat thei th site(i = 1, . . . , N). In our modelEi (t) ∈ 0,1 depending on whetherthe site is energy-free (0) or not (1). In a first step in which each cell can receiveenergy with a probability

p(t) =1

2

(1+ sin

t

T

)∈ [0,1]

whereT is a parameter introducing the period of the energy fluctuations, i.e.,P =T2π . As expected, high values ofT approach the behavior towards an unforcedsystem, where the system behaves as a typical predator–prey LV system, givingrise to a simple limit cycle. If a site is such thatEi (t) = 0, it becomesEi (t+1) = 1with probability p(t), independently of the prey and predator states in the cell.

Interactions. The following step involves interactions among preys and energyand among predators and preys. LetS1

i (t) andS2i (t) be thestatesof the i th site for

prey and predator, respectively, i.e.,

Sji (t) =

{1 i -site occupied0 i -site free of j


for both prey( j = 1) and predator( j = 2). Then the following rules are appliedn = N times (this defines our timescale: for eacht there areN updatings). Wechoose a site 1≤ i ≤ N at random.

1. If S1i (t) = 0 andS2

i (t) = 0 then we choose the next site.

2. Death. Prey and predators die with certain probabilitym1≤ m2. If there is

no death then:

3. If S1i (t) = 1 andS2

i (t) = 0 (only prey present) two possible outcomes arepossible:

(a) If Ei (t) = 0, then no reproduction is allowed to occur and the preymoves to a new free sitek (such thatS1

k(t) = 0) which is chosen atrandom.

(b) Feeding and reproduction of preys. If Ei (t) = 1 then feeding occurs(Ei (t) = 0) and reproduction is possible with probabilityr 1

∈ [0,1].The parent individual remains in the same position, whilst the offspringmoves to another randomly chosen site.

4. If S1i (t) = 0 andS2

i (t) = 1 (only predator present) then the predator movesto another randomly chosen site looking for a prey [as in rule 3(a)].

5. If S1i (t) − 1 andS2

i (t) = 1 (prey and predator present). Movement, feedingand reproduction occur as in the former case, but the predator feeds on preybefore energy is offset from the cell.

Table2 shows the parameter set used in the model. The onset of control in thischaotic three-species system has been attached through the addition of externalperturbations by addingγ preys eachTcont time (Fig.6). Control is also possibleby adding predators to the system, although the parameter set is somewhat differ-ent. Hastings and Powell (1991) andMcCannet al. (1998) state that chaos mayarise whenever the period of one oscillation (i.e., the cycles followed by the preda-tor population, which are longer than that of the prey due to the longer lifetimesof predators) is not some multiple of the other frequency (i.e., that of the prey).Interestingly, our results suggest that, in order to achieve better control, such aslimit cycles or even stable points with some random fluctuactions, it is necessaryfor the system to have a control period that is some multiple of the period of energyfluctuation, as may be seen in Fig.6. Our conjecture is that by doing this, we get tocouple the oscillations in all three systems, so in some way we are able to ‘unforce’the system, avoiding asynchronization and chaos in all trophic levels. By addinglarge-sized perturbations, we accelerate the dynamics of the system by rapidly in-creasing the prey population to its upper bound in the attractor. As a result theattractor is wrapped around by the limit cycle (see Fig.6).


Table 2. Parameters and default values used in the LV IBM.

Parameter Definition Default value Effect when increased

L × L System size 30× 30 Improves controlT Period of energy sine wave 40 Larger cycles

Tcont Control period 20 Large amplitude, PDBa

γ Perturbation size — Large amplitudem1 Death rate of prey 0.05 Extinction likelym2 Death rate of predator 0.2 Fixed pointr 1 Feeding-reproduction rate of prey 0.95 Short-term correlationsr 2 Feeding-reproduction rate of predator 0.85 Extinction likelyd1 Interspecific DDb factor of prey 40 Improves controld2 Interspecific DDb factor of predator 40 Improves control

a PDB: Period-doubling bifurcations.b DD: Density-dependence.

5. DISCUSSION

Although not fully demonstrated yet, the existence of chaotic dynamics in realecosystems has been a matter of extensive studies over recent years. It opensnew daring questions in biology, due to its intrinsically unpredictable behavior andcounterintuitive results. Even more important, it is a useful platform from whichnew and fascinating testable designs may arise. One step ahead, the possibilitiesfor chaos control help us to unfold the apparently hidden nature of chaos, pullingout the reversibility of one of the main characteristics of chaos: the presence ofperiod-doubling bifurcations (Stone, 1993).

Techniques for stabilizing the infinity of unstable periodic orbits appearing in thepresence of chaos are based mainly in two categories: the addition of perturbationsto the parameters acting over the dynamics (Ott et al., 1990) and to the dynamicvariables at play (Guemez and Matıas, 1993; Parthasarathy and Sinha, 1995). Thefirst one, more difficult to apply in real systems, needs some previous knowledge ofthe dynamics of the system and gets less effective as the system gets more complex.The second one is simpler, although without some knowledge of the system needssome trials before being sure about the effect of positive or negative perturbationsare needed.

The construction of individual-based population models may help us to under-stand the role of noise in the characterization of the attractors and the effectivenessof control techniques in the presence of stochastic fluctuations. So far as we havenot used locally interacting individuals, but global mixing of populations, we haveassumed that each cell of the lattice is independent of another one, where each cellshares the same probabilities of dispersal to every cell of the lattice. Application ofcontrol works well for both systems studied, the LV predator–prey forced chemo-stat and the Ricker individual-based model. For the constant feedback method,


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Figure 6. (a) Numerical output of several simulations of the stochastic LV3D IBM. Attractors defined by the control PS method are represented bysolid lines. (A)T = 10, Tcont = 20, γ = 120. (B)T = 40, Tcont = 20,γ = 200. (C)T = 5, Tcont = 5, γ = 180. (D)T = 5, Tcont = 5, γ = 80.The other parameters are defined in Table2. (b) The same as in Fig.6(a),but representing the evolution of preys in time.


used for both IBMs, the perturbation consists of the addition of individuals in ran-dom sites chosen from a uniform distribution, which have direct effects on the intra-and interspecific density dependence of the populations. Thus, under unstable dy-namics, the system tends to visit extreme values. The nature of the fluctuationsrelies on the following description: if a population has depleted the resources dueto its initially high density, its intraspecific competition for resources available, willhave been strong enough to increase its mortality, so the population will achievevery low values at the end and the next step, when the available resource has beenrenewed, very low initial populations will start to consume it. This process, if fastenough, will be able to decouple the resource and the population dynamics in con-tinuous systems. If, when resources have been depleted, we add some individuals(i.e., we add an structural perturbation) to the already scarce population, higheroffspring individuals will start consuming the available resource when renewed, sobasically the effect of the perturbation has been to increase the number of visitsover less extreme values and, in some cases, to achieve an artificially created pointequilibrium or will couple both subsystems if the requirements of conmensurabil-ity in continuous systems (such as the planktonic or the LV IBM described) arefulfilled. This requirement for control is that the period applied for control shouldbe some multiple of the period of the chemostat, as some authors have previouslysuggested (Hastings and Powell, 1991; McCannet al., 1998). Moreover, its ef-fectiveness depends on the size and frequency of the perturbation applied, as indiscrete systems. For planktonic microecosystems, simple models of competitionamong algae have defined strange attractors, though the set of parameters neededfor it is highly restricted to certain values. Outside this range (i.e., some parametersare modified) the system behaves more or less periodically, so the appearance ofchaos in the real system does not seem likely. However, if the model is able togenerate chaos for some set of parameters, deterministic chaotic dynamics mightin fact be implicit in the basis of the dynamics, and that shift would be certainlydefining a parametric perturbation to leave the chaotic regime.

The relationship with real systems does not come up from a direct interpretationof the results, surely because of the complexity inherent in such systems. Howeversome basic assumptions can be applied from our models. Basically, in populationsundergoing global mixing, i.e., small populations where the size of its geographicrange is closed and sufficiently small to ensure that dispersal is uniform over thewhole range, a systems manager would need to add some individuals [asMcCal-lum (1992) suggested] or remove them (which is equivalent to the assumption ofnegative values ofγ ) from the system. In biological control, if some nonlineardynamics is at play, and a given population needs to be over a certain threshold inorder to produce considerable damage on crops, control techniques may be usedto maintain the population under this threshold, even in the presence of stochasticexternal events, such as short episodes of high temperatures.

The presence of spatial correlations and/or pink or blue noise would increasetemporal correlations and would need some further study. The presence of chaos


in some metapopulation models has already been observed to contribute to spatialasynchrony among the populations by increasing the noise of its local components(Allen et al., 1993; Sole and Gamarra, 1998). Thus, spatial synchronization (bymeans of periodic pulses) seems to be a possible alternative for controlling the pres-ence of chaos and outbreaks in some populations. Currently an artificial change inthe parameters of the systems (such as the growth rate) seems a fairly impossibletask, techniques of control based on perturbations over the dynamic variables seemto be the only plausible alternative.

Some final remarks need to be emphasized. The characterization of an individual-based model following a Ricker map is, as far as we know, the first trial to model asimple system of nonoverlapping generations by means of the coupling of contin-uous and discrete events working at different timescales, giving rise to a perfectlydrawn Ricker dynamics with white noise. The application of control over this sys-tem seems sufficiently plausible to have in mind that chaos control is effectivelyallowed in some real ecosystems.

ACKNOWLEDGEMENTS

The authors thank D. Alonso, J. Bascompte, B. Luque and I. Salazar-Ciudad forinteresting discussions and suggestions. This work has been partially supported bygrant DGYCIT PB97-0693.

REFERENCES

Allen, J. C., W. M. Schaffer and D. Rosko (1993). Chaos reduces species extinction byamplifying local population noise.Nature364, 229–232.

Astakhov, V. V., V. S. Ansihehenko and A. V. Shabunin (1995). Controlling spatiotemporalchaos in a chain of coupled logistic maps.IEEE Trans. Circuits Syst. 142, 352–357.

Bascompte, J. and R. V. Sole (1995). Rethinking complexity: modelling spatiotemporaldynamics in ecology.Trends Ecol. Evol.10, 361–366.

Berryman, A. A. and J. A. Millstein (1989). Are ecological systems chaotic—and if not,why not?Trends Ecol. Evol.4, 26–28.

Constantino, R. F., J. M. Cushing, B. Dennis and R. A. Desharnais (1995). Experimentallyinduced transitions in the dynamic behaviour of insect populations.Nature375, 227–230.

Constantino, R. F., R. A. Desharnais, J. M. Cushing and B. Dennis (1997). Chaotic dynam-ics in an insect populations.Science275, 389–391.

Dennis, B., R. A. Desharnais, J. M. Cushing and R. F. Constantino (1997). Transitions inpopulation dynamics: equilibria to periodic cycles to aperiodic cycles.J. Anim. Ecol.66, 704–729.

Ditto, W. L., S. N. Rauseo and M. L. Spano (1990). Experimental control of chaos.Phys.Rev. Lett.65, 3211–3214.


Ditto, W. L., M. L. Spano and J. F. Lindner (1995). Techniques for the control of chaos.Physica D86, 198–211.

Doebeli, M. (1993). The evolutionary advantage of controlled chaos.Proc. R. Soc. Lond.B 254, 281–285.

Doebeli, M. and G. D. Ruxton (1997). Controlling spatiotemporal chaos in metapopula-tions with long-range dispersal.Bull. Math. Biol.59, 497–515.

Fox, R. F. and J. E. Keitzer (1990). Effect of molecular fluctuations on the description ofchaos by microvariable equations.Phys. Rev. Lett.64, 249–251.

Garfinkel, A., M. L. Spano, W. L. Ditto and J. N. Weiss (1992). Controlling cardiac chaos.Science257, 1230–1235.

Godfray, H. C. J. and S. P. Blythe (1990). Complex dynamics in multispecies communities.Phil. Trans. R. Soc. Lond. B330, 221–233.

Guemez, J. and M. A. Matıas (1993). Control of chaos in uni-dimensional maps.Phys.Lett. A181, 29–32.

Hassell, M. P., J. H. Lawton and R. M. May (1976). Patterns of dynamical behavior insingle-species populations.J. Anim. Ecol.45, 471–486.

Hastings, A. (1993). Complex interactions between dispersal and dynamics, lessons fromcoupled logistic equations.Ecology74, 1362–1372.

Hastings, A. and T. Powell (1991). Chaos in a three-species food chain.Ecology72, 896–903.

Judson, O. P. (1994). The rise of the individual-based model in ecology.Trends Ecol. Evol.9, 9–14.

Kot, M., G. S. Sayler and T. W. Schultz (1992). Complex dynamics in a model microbialsystem.Bull. Math. Biol.54, 619–648.

Lomnicki, A. (1989). Avoiding chaos.Trends Ecol. Evol.4, 239.May, R. M. (1974). Biological populations with nonoverlapping generations: stable points,

stable cycles and chaos.Science186, 645–647.May, R. M. (1976). Simple mathematical models with very complicated dynamics.Nature

261, 459–467.May, R. M. and G. F. Oster (1976). Bifurcations and dynamic complexity in simple eco-

logical models.Am. Nat.110, 573–599.McCallum, H. I. (1992). Effects of inmigration on chaotic population dynamics.J. Theor.

Biol. 154, 277–284.McCann, K., A. Hastings and G. R. Huxel (1998). Weak trophic interactions and the bal-

ance of nature.Nature395, 794–798.Nisbet, R., S. Blythe, B. Gurney, H. Metz and K. Stokes (1989). Avoiding chaos.Trends

Ecol. Evol.4, 238–239.Ott, E., C. Grebogi and J. A. Yorke (1990). Controlling chaos.Phys. Rev. Lett.64, 1196–

1199.Parthasarathy, S. and S. Sinha (1995). Controlling chaos in unidimensional maps using

constant feedback.Phys. Rev. E51, 6239–6242.Peeters, P. and G. Nicolis (1992). Intrinsic fluctuations in chaotic dynamics.Physica A188,

426–435.


Petrov, V., V. Gaspar, J. Masere and K. Showalter (1993). Controlling chaos in theBelousov–Zhabotinsky reaction.Nature361, 240–243.

Pimm, S. L. (1991).The Balance of Nature?: Ecological Issues in the Conservation ofSpecies and Communities, Chicago: University of Chicago Press.

Ricker, W. E. (1954). Stock and recruitment.Fish. Res. Board. Canada11, 559–623.Rinaldi, S., S. Muratori and Y. Kuznetsov (1993). Multiple attractors, catastrophes and

chaos in seasonally perturbed predator–prey communities.Bull. Math. Biol.55, 15–35.Ringelberg, J. (1977). Properties of an aquatic microecosystem. II. Steady-state phenom-

ena in the autotrophic subsystem.Helgolander Meeresunters30, 134–143.Rohani, P. and D. J. D. Earn (1997). Chaos in a cup of flour.Trends Ecol. Evol.12, 171.Schaffer, W. M. (1984). Stretching and folding in lynx for returns: evidence for a strange

attractor in nature?Am. Nat.124, 798–820.Schaffer, W. M. and M. Kot (1986). Chaos in ecological systems: the coals that Newcas-

tle forgot.Trends Ecol. Evol.1, 58–63.Scheffer, M. (1991). Should we expect strange attractors behind plankton dynamics-and if

so, should we bother?J. Plankton. Res.13, 1291–1305.Schiff, S. J., K. Jerger, D. H. Duong, T. Chang, M. L. Spano and W. L. Ditto (1994).

Controlling chaos in the brain.Nature370, 615.Sole, R. V. and J. G. P. Gamarra (1998). Chaos, dispersal and extinction in coupled ecosys-

tems.J. Theor. Biol.193, 539–541.Sole, R. V. and L. Menendez de la Prida (1995). Controlling chaos in discrete neural net-

works.Phys. Lett. A199, 65–69.Sole, R. V. and J. Valls (1992). Nonlinear phenomena and chaos in a Monte Carlo simu-

lated microbial ecosystem.Bull. Math. Biol.54, 939–955.Stone, L. (1993). Period-doubling reversals and chaos in simple ecological models.Nature

365, 617–620.Tilman, D. and D. Wedin (1991). Oscillations and chaos in the dynamics of a peren-

nial grass.Nature353, 653–655.Wilson, W. G. (1998). Resolving discrepancies between deterministic population models

and individual-based simulations.Am. Nat.151, 116–134.

Received 2 May 1999 and accepted 12 July 1999

controlling chaos in ecology: from deterministic to individual-based models

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