Preface

Chaos in ecology

Vikas Rai a, W.M. Scha�er b

a Structural Biology Unit and Computer Centre, National Institute of Immunology, Aruna Asafali Road, New Delhi 110 067, Indiab Department of Ecology and Evolutionary Biology, Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721, USA

Overview

As noted by Hao Bai-Lin in the preface to his admirable collection [29,30] of in¯uential papers on non-linear dynamics, the discovery [48,55,56] of chaos in ecological di�erence equations, as much as anythingelse, fertilized a ¯owering of interest in this subject some twenty-®ve years ago. Perhaps not surprisingly, itwas in the physical, as opposed to the biological, sciences that ``chaos theory'', as it is often (and inac-curately!) referred to, really took hold. In ecology itself, the ubiquity of chaos and other non-linear phe-nomena in both discrete and continuous models was subsequently con®rmed 1 [1,3,4,9,27,34,46,57,70±72].At the same time, convincing evidence for chaos in natural systems proved harder to come by [17,33,64].For example, in the case of pre-vaccination epidemics of measles in large, ®rst world cities, what was oncejudged [63] to be one of the more likely examples of real-world ecological chaos, is now the subject ofdivergent opinion [16,18].

In as much as ecological systems have all the necessary ingredients: palpable non-linearity, multiplestate variables, etc., for chaotic dynamics, the lack of evidence for chaos in nature may strike some assurprising. The conventional view [64] is that ecological data sets are too short and too corrupted byobservational error and process noise to allow for accurate characterization of the underlying dynamics.This has prompted a search for better methods of extracting the deterministic signal from noisy timeseries [17,19,20,58,80±82]. The inherent di�culty of this task is underscored by two facts: in the ®rstplace, as emphasized by Auerbach and others [7,47], chaotic motion can be viewed as a choreographyinvolving non-stable cycles, with the lead dancer, the evolving orbit, successively favoring di�erentpartners. It follows that chaotic time series will often have strong periodic components rendering itdi�cult to distinguish them from null models [17] consisting of periodic motions in the presence ofnoise. 2

An additional problem results from the fact that the time evolution of non-linear systems is oftenre¯ective not only of the attractors to which trajectories tend in the limit of large time, but also of thepresence of non-stable sets elsewhere in the phase space. 3 For example, in the case of the Lorenzequations, Yorke and Yorke [87] have pointed out some time ago that there exist parameter values for

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Chaos, Solitons and Fractals 12 (2001) 197±203

1 Many of the early observations [2,6,37,75] of complex dynamics in ecological models were published in physics and applied

mathematics journals where they remain, to this day, unknown to most ecologists.2 Such di�culties are exacerbated by what has been called [40,41] ± ``transient periodicity''. In this case, a chaotic attractor contains

well-de®ned semi-attractors on which the motion is semi-periodic with the consequence that episodes of ``noisy periodicity'' [51] are

interspersed with periods in which the dynamics are more obviously aperiodic. In the case of short time series in which there occur only

a few such transitions, the naive observer is likely to suspect some undetected change in environmental conditions as causing the

change in behavior.3 The reality of non-stable invariant sets, and their in¯uence on observable dynamics, have recently been con®rmed in the case of

¯our beetle dynamics in the laboratory [13].

0960-0779/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. All rights reserved.

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which one can observe arbitrarily long chaotic transients even though the dynamics tend asymptoticallyto equilibrium. This phenomenon, which they termed ``metastable chaos'', results from the fact thatcoexisting with a pair of stable ®xed points is a chaotic saddle (Sparrow [78] calls it a ``strange invariantset''.) on which the motion is every bit as chaotic as on a bona ®de strange attractor. When such systemsare subjected to random shocks without the parameter values, there arises the possibility of what Kantzand Grassberger [39] call the ``noise-stabilized chaos'', i.e., the system spends most of its time in thevicinity of the chaotic saddle even though the deterministic behavior is simple. 4 In such cases, e�orts toferret out the asymptotic dynamics miss the essential point i.e., that it is the saddles, not the attractors,that determine the motion.

An alternative point of view is that chaos in nature really isa rarity and that this re¯ects somethingfundamental about the organization of ecological systems. One possibility [65,83] is that the reality ofecological interactions, their non-linear character notwithstanding, necessitates high levels of dissipa-tion which, in turn, preclude the possibility of chaotic motion in the absence of noise. Another [10] isthat chaotically ¯uctuating populations are more prone to extinction than equilibrial populations withthe consequence that group selection acts to eliminate species which would otherwise evolve intochaos. 5

Contributing to the laggardly rate of progress in non-linear ecology is the fact that natural systems arenot readily brought into the laboratory. An exception is the ¯our beetle, Tribolium sp., an agricultural pestthat has infested mankind's granaries since the days of the ancient Egyptians [77] at least. In this case, thelaboratory system (one or more species in a jar containing ¯our) is not unlike the natural one (one or morespecies in a silo). Moreover, the simplicity of the system, and the fact that it is well understood, allows forthe formulation of tractable models whereby quantitative predictions can be tested under a variety ofexperimental conditions. Gratifyingly, many of these predictions, including the observation of chaos, havebeen borne out by the experiments [13±15]. In short, the ¯our beetle system allows ecologists to do whatworkers in other ®elds often take for granted: to write down mechanistic models which predict sequences ofdynamical behaviors that are obtained in response to variations in parameters and then to determinewhether or not the said sequences are actually observed. The results of the beetle experiments, which are inthe time-honored tradition of Gause [25], Nicholson [61], Hu�aker [36], etc., cannot, of course, be applieddirectly to the ®eld. What they can do, however, is to give plausibility to the idea of modeling ecologicalsystems mathematically, a proposition which is not well accepted by ecologists and ®eld biologists.Moreover, the experiments establish in an undeniable way the reality of non-linear phenomena in thecontext of population biology.

The prevailing uncertainty regarding complex dynamics in nature has hardly dampened the enthusiasmof theoreticians intent on elucidating the possible consequences of non-linearity to the natural world. In thisvein, Allen and his associates [5,67] have challenged the assertion that group selection acts to eliminatespecies which ¯uctuate chaotically. Drawing on the observation that most species consist of semi-isolatedpopulations, these authors argued that exponential divergence of nearby trajectories will often reduce thelikelihood of ensemble, i.e., of species, extinction that would otherwise result from the vagaries of a ¯uc-tuating environment. From the perspective of species persistence, chaos in nature, by this reckoning, isgood.

The protective e�ect of chaos observed by Allen et al. is a consequence of sensitivity to initial conditions[69,76], the de®ning characteristic of chaotic motion which, in this case, acts to counter the e�ect of mi-gration which would otherwise act to synchronize spatially disjunct populations. However, as pointed outby Farmer and his associates [21], there are di�erent levels of sensitive dependence. In the case of the

4 Closely related are the supertransients which occur in spatially extensive systems [32,38]. In this case, one also observes extended

episodes of chaotic behavior before the system ®nally settles down to its asymptotic state ± typically a ®xed point. From a

phenomenological perspective, supertransients are distinguishable from transient chaos in spatially homogeneous systems by virtue of

the fact that transient chaos appears to obtain over open intervals of parameter values.5 The argument depends on the observation that in simple models, such as the logistic and Ricker [66] maps, minimum population

sizes decline as one enters, and then moves farther into, the chaotic region. For an earlier, ``pre-chaos'' example of such reasoning, see

Gilpin's [26] monograph on the evolution of predator±prey systems.

198 Preface

R�ossler band [68], for example, time series for two of the three variables are characterized by power spectrain which ``instrumentally sharp'' peaks are superposed on a noisy background. 6;7 Farmer et al. havetermed such a motion as ``phase coherent'', and it is not surprising that coupling two or more phase co-herent systems spatially leads to correlated dynamics in space. Indeed, just such a ``phase synchronization''has been reported by Blasius et al. [11,50]. In this study, a three-level food chain model with R�ossler-likedynamics is assumed to determine the within-patch behavior, and local populations are coupled by mi-gration. Of ecological interest is the fact that the system's behavior is reminiscent of the lynx-hare cycle [23]in that variations in cycle amplitude exceed variations in period. Also of interest is the authors' suggestionthat the high degree of synchrony among lynx populations across North America is re¯ective of phasecoherent chaos 8 in local populations.

Wildlife's ten-year cycle [42] in Canada's north woods is the subject of two other recent investigations.Reviewing lynx fur returns for the Hudsons Bay Company, Gamarra and Sol�e [24] have argued in favor ofan earlier suggestion [70] that the system underwent a bifurcation, possibly in response to increasedtrapping pressure, in the early 19th century. With regard to contemporary cycle dynamics, a realistic (asthese things go) model of snowshoe hare demography was recently proposed [44] which induces 8±12-yearoscillations for parameter values that fall within the ranges of independent estimates deriving from ®eldobservations. 9 Of course, the ®eld-derived estimates are crude, often being no better than the order ofmagnitude, with the consequence that the observed concordance of theory and observation is hardly de-®nitive. Nonetheless, this e�ort marks the ®rst successful attempt to model what is arguably ecology's most-celebrated oscillation in a falsi®able manner.

Of more general import is the fact that the snowshoe hare model can be formulated as a perturbation ofa Hamiltonian limit wherein arise the motions of ecological interest. The proposed restoration [43] ofHamiltonian dynamics [35,49] to a central place in ecological theory is both novel and very much againstthe conventional wisdom [54] which holds that conservative systems [84,85], by virtue of their structuralinstability, are irrelevant to the behavior of realistic models. The ¯aw in this point of view turns out to bethe structural stability criterion itself, which is based on the notion of topological conjugacy. Essentially,many of the orbits of structurally unstable systems ± in the present case, an in®nite number ± can persistunder perturbation over open intervals of parameter values. The surviving motions include periodic orbitsthat give rise to stable cycles in the dissipative regime and probably also to the saddle cycles about whichchaotic attractors are organized [43].

About this volume

The foregoing (and other) developments, the historical importance of ecological theory to non-lineardynamics, and the authors' personal opinion that non-linearity will eventually be accepted as a funda-mental property of ecological systems, motivated the decision to produce a special issue of Chaos, Solitonsand Fractals on chaos in ecology. Each of the papers included addresses an important topic. We now devotea few sentences to each.

6 By ``instrumentally sharp'', we refer to peaks as narrow as the sample size permits. As in the case of R�ossler's equations, the degree

of phase coherence in tri-trophic level ecological models [34] is parameter-dependent. Recently [43], it has been pointed out that the

origins of this dependence lie in the fact that such systems can be formulated as perturbations of one or more Hamiltonian limits [35,49]

from which emanate resonance horns corresponding to cycles of all possible base rotation numbers. With increasing levels of

dissipation, the cycles are destroyed by saddle node bifurcations in a well-de®ned sequence. Phase coherent chaos results when the

cycles, about which a chaotic attractor is organized, are all associated with a single resonance horn. Since all the cycles have the same

base period, it follows that the power spectrum will have sharp peaks and that, from a time series point of view, variations in cycle

amplitude will exceed variations in period.7 Animations illustrating the mixing properties of the R�ossler band and other chaotic attractors may be viewed and downloaded at

http:bill.srnr.arizona.edu. (Click on Cool Chaos Demos.)8 An alternative explanation, of course, is periodicity in the presence of noise.9 Information on the ``ten-year'' cycle generally and the snowshoe hare model, in particular, is available on-line at http://

two.ucdavis.edu/aking/mam99.

Preface 199

The paper by Rai and Upadhyay attempts to understand the general failure in the e�orts to observechaos in natural populations. For two representative models, it is shown that the range of parameter valuescorresponding to chaos is both narrow and biologically unrealistic. It is further observed that for thoseparameter regimes in which chaos does obtain, the generative mechanism involves catastrophic bifurcationsknown as ``crises'' [28].

The contribution of Cushing et al. summarizes the aforementioned studies of laboratory populations ofthe ¯our beetle, Tribolium castaneum. The methodology is both elegant and rigorous. In the ®rst place, adeterministic model is shown capable of inducing time-series which qualitatively match those obtainedexperimentally. Quantitative correspondence is then achieved by replacing the deterministic model with itsstochastic counterpart thereby providing for the e�ects of demographic and environmental noise. As notedabove, this work has yielded the strongest evidence to date for chaos in an ecological system. Together withthe interdisciplinary approach (modeling, experimentation and statistical analysis), it will likely serve as amodel for future investigations of ecological population dynamics.

The third paper by Gamarra et al. focuses on chaos control [22] and techniques which can be used toidentify chaos in non-linear population models. The authors demonstrate that their methods are robust toenvironmental stochasticity and can further be applied to individually oriented studies. Of the variouschaos control strategies currently available, Gamarra et al. identify the one best suited for controlling chaosin realistic models. Further research along these lines may also facilitate the detection of chaos in naturaldata.

The next two contributions focus on periodically forced predator±prey interactions. In the ®rst, Scha�erand co-workers argue that the analysis of Hamiltonian systems can help explain cyclic phenomena in nature(see above). In particular, they review the mechanism by which periodic motions arise in the Hamiltonianlimit and point out that empirical observations [12] of allometric scaling (cycle period versus body size) inmammals result as a natural consequence. The second paper by Vandermeer and colleagues uses circlemaps to describe the dynamics of a periodically forced Lotka±Volterra system. As predicted by earlierinvestigations [43,45], coexisting attractors are observed. Vandermeer and his associates further observethat the system is characterized by fractal basin boundaries [52,53] and that, in some instances, the basins ofattraction additionally manifest the so-called ``Wada'' property [62]. These matters are further discussedwith regard to the model (see above) studied by Blasius et al. [11].

The next two papers concern aquatic systems. The study by Tikhonov et al. discusses the dynamics ofschooling behavior in ®sh using reaction-di�usion equations. Two types of behaviors are distinguished:motions of low-persistence and behaviors which are highly persistent. In the former instance, fractals on alltime scales are obtained. In the latter, multi-fractals are observed for large-scale displacements. Themovement patterns of ®sh are of obvious consequence to the organisms they eat which, depending onthe size of the ®sh in question, are either smaller ®sh or zooplankton. These consequences are the subject ofthe paper by Edwards and Bees. The latter authors consider the dynamics of tri-trophic plankton models 10

which utilize ``closure terms'', [79] of the form ÿdZm, to model the e�ects of ®sh predation on the zoo-plankton concentration, Z. This device allows for the omission of ®sh population density as an explicitvariable. Historically, linear closure terms have been used to model ``passive'' predators, for example, ®lterfeeders, and quadratic terms, to model predators which are attracted to the regions of high prey concen-tration. Arguing that the fractions of predators employing alternative feeding strategies vary with envi-ronmental conditions such as turbulence, Edwards and Bees consider non-integer values of the ``index ofclosure'', m. Among other things, they ®nd that dynamical complexity observed for the special case, m � 2,is persistent over a range of values in the interval, m � 1; 2.

Ecological exploitation is also the subject of the paper by Bernard Cazelles who establishes that it ispossible to construct simple predator±prey models with multiple attractors and riddled basin boundaries[86]. The practical signi®cance of this special type of boundary is that even qualitative predictions regardinga system's time evolution are impossible.

10 The state variables are concentrations of nutrients, phytoplankton and zooplankton.

200 Preface

The paper by Rinaldi et al. addresses the problem of forecasting successive maxima in ecological timeseries. As foresters and their entomological colleagues will readily attest, this is a matter of practical, as wellas theoretical importance. 11 The authors focus on the next amplitude maps which, when the dynamics areon a two-torus or a low-dimensional strange attractor, a�ord a straightforward device for predicting thetiming and amplitude of the next irruption. The utility of this scheme, which has the practical advantage ofrequiring limited information about the previous outbreaks, depends on the prevalence of low dimensionalecological motion. In essence, this work takes us back to earlier attempts [73,74] to use such maps todistinguish chaos from random ¯uctuations.

Not unconditionally, the problem of distinguishing deterministic from stochastic dynamics is the subjectof the paper that follows. In this contribution, Kendall considers data generated by a discrete host-parasitoid model in the presence of noise. Kendall compares the ability of a non-parametric model todescribe the data with that of the original equations. He concludes that the original model outperforms thenon-parametric scheme which, in turn, outperforms a second incorrect model chosen for comparison.Kendall also observes that oscillatory transients provide su�cient structure for parameter estimation andidenti®cation of the correct model and that the non-parametric model is most readily rejected when thedynamics are chaotic. This analysis, which is in the tradition of Morris [59] and other early investigators[73], by virtue of its situation-speci®c nature, underscores the di�culties which attempt to formulate ageneral methodology for detecting complex determinism in the absence of the ability to conduct controlledexperiments.

The ®nal paper is by Jon Allen and his associates. The model considered here belongs to a distinct classknown as ``discrete spatial convolution'' models, whereby the e�ects of migration and dynamics at a pointare studied in the context of spatially extensive systems. Speci®cally, the authors con®rm the stabilizinge�ects of spatial structure for a host±parasitoid system under various assumptions. In addition, the analysisof spatial convolution models via the frequency response of the system transfer function is illustrated. Real-world systems are, of course, distributed over the surface of a two- (and sometimes three-) dimensionalplanet, with the consequence that the ``well-stirred'' dynamics of discrete maps and ordinary di�erentialequations are always an approximation. Often, it is also a poor one ± for example, when the spatiallyextensive generalization of a non-spatial model is characterized by spatial structures [31] or supertransients[32,38]. It is therefore ®tting that the concluding contribution to this volume takes up the problem ofecological dynamics in space, as well as time.

Acknowledgements

We thank Tatiana Bronnikogva, Bob Costantino and Aaron King for their criticism and comments. Themistakes, of course, remain our own.

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