Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
Secondary article Article Contents . Introduction
Bruce E Kendall, University of California, Santa Barbara, California, USA
. Historical development
Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of nonlinear dynamics, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time. 7
Introduction Throughout much of ecology’s history, most ecologists have believed that ecological systems tend towards an equilibrium – a notion popularized as ‘the balance of nature’ – and that deviations from that equilibrium are caused by external perturbations. By assuming that systems were close to equilibrium, ecologists could describe them with linear models that were mathematically tractable. However, ecology is rife with processes whose rates depend nonlinearly on the state of the system – such as nutrient uptake, density dependence and predation. Advances in computing power and availability in the 1970s led to the discovery that such nonlinearities could cause spontaneous oscillations. Interest initially focused on chaos as a potential explanation for all of the irregular fluctuations in nature. Two decades of investigation suggest that ecological chaos is rare in natural systems – ecology really is ‘noisy’. Nevertheless, the importance of nonlinear processes, and the resulting nonequilibrium dynamics, are now fully accepted at the core of ecology.
Historical development The study of complex population dynamics is nearly as old as population ecology. In the 1920s, Alfred Lotka and Vito Volterra independently developed a simple model of interacting species that still bears their joint names. This was a nearly linear model, but the predator–prey version displayed neutrally stable cycles. Experimentalists such as G. F. Gause soon tried to recreate these cycles in laboratory populations of protozoa. Also working in the laboratory, A. J. Nicholson was trying to demonstrate his ideas about density-dependent population regulation and managed to produce cycles in single-species cultures of blowflies (Figure 1). In the meantime, Charles Elton had become fascinated by cyclic fluctuations in field populations of voles (small rodents; Figure 2) and in the numbers of furs of Canadian lynx and other large mammals traded by the Hudson’s Bay Company. How-
. Nonlinear Attractors: Limit Cycles, Quasiperiodicity and Chaos . Lyapunov Exponents . Environmental Variability and Nonlinear Dynamics . Empirical Investigations of Nonlinear Dynamics . Evolution of Stability or Chaos
ever, it was not until Michael Rosenzweig, as a graduate student with Robert MacArthur in the 1960s, added density dependence and predator behaviour to the Lotka– Volterra equations that an ecological model capable of displaying true nonlinear limit cycles was developed (Nicholson’s blowflies were not adequately modelled until 1980). Chaos was first knowingly observed in a mathematical model by meteorologist Edward Lorenz in the early 1960s. Not only did he observe deterministic aperiodic fluctuations, he also discovered sensitive dependence on initial conditions: restarting a simulation model without bothering to copy all of the significant digits, Lorenz observed a sequence of numbers that soon started to diverge from the original simulation. Within the context of turbulence studies, the mathematical and computational strands of chaos were woven together in the 1970s and 1980s. Mathematicians David Ruelle and Floris Takens coined the evocative term ‘strange attractor’ to describe the complex structures of chaos in 1971. In the early and mid 1970s, physicist Robert May took theoretical ecology by storm with a series of papers demonstrating that ‘complex dynamics’ (cycles and chaos) could be generated by the simplest ecological models (e.g. May 1976), including the ‘logistic map’, an extremely simple model of density-dependent population growth with discrete generations: N(t 1 1) 5 rN(t) [1 2 N(t)/K]. Theoreticians rapidly embraced the new science of nonlinear dynamics, hoping that chaos could explain some of the erratic fluctuations that were hitherto called ‘noise’. Empirical ecologists were rather more sceptical, and never embraced the claims that chaos was important in the real world; but they gradually accepted the idea that nonlinear dynamics in general were important. This acceptance was indicated by a 1990 symposium sponsored by the Ecological Society of America: ‘The shift from an equilibrium to a nonequilibrium paradigm in ecology’.
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Nonlinear Dynamics and Chaos
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Nonlinear Attractors: Limit Cycles, Quasiperiodicity and Chaos The time-series, which displays fluctuations through time, is a familiar representation of dynamic systems (Figures 1 and 2). A less intuitive, but powerful, representation is the phase space. The axes of the phase space are defined by the state variables so that, at a given moment, the state of the system is represented by a point in the phase space. Plotting the system at successive moments in time reveals the trajectory, or the path that the system follows through the phase space. Examples showing the relationship between the phase space and time-series representations are shown in Figure 3. The power of the phase space representation comes about because, as long as the system’s fluctuations are bounded, an infinitely long trajectory inhabits a finite region of the phase space. The trajectory becomes a geometric object. In a deterministic system, the trajectory will eventually conform to another geometric object in the phase space: the attractor. A trajectory that is sufficiently ‘near’ (in the phase space) an attractor will move towards it, on average, and once it is on the attractor it will not spontaneously move away. The simplest form of attractor is a stable equilibrium, which is an isolated point in the phase space. Nonlinear systems can have more complex attractors: limit cycles, quasiperiodicity and chaos (Figure 3). Periodic, or limit cycle, attractors repeat the same sequence of values over and over, in strictly periodic oscillations. In discrete systems the trajectory visits a finite set of points in the phase space, repeating them in theI0006988 same order. In continuous systems the trajectory follows a closed loop in the phase space. 2
Quasiperiodic attractors have two fundamental frequencies whose ratio is an irrational number. In discrete systems one of these frequencies is the time step, so that a quasiperiodic time series appears to be periodic with an irrational period. In the phase space the trajectory fills in all of the points in a closed loop. In continuous systems, the lower frequency provides an envelope that modulates amplitude of the higher frequency oscillations, producing a ‘beat’ phenomenon. In the phase space, the orbit covers the surface of a torus. Chaotic attractors have trajectories that never repeat, and exhibit apparently erratic fluctuations despite being deterministic. However, there is regularity in the dynamics. The attractors are confined to a finite volume of the phase space, and there is fine-scale structure that is fractal – visible at all magnifications. Thus the attractors do not fill the volume in the phase space, but have a noninteger (‘fractal’) dimension. Chaotic attractors also exhibit ‘sensitivity to initial conditions’: a small change in the starting state of the system will grow, on average, exponentially through time (up to the size of the attractor). Periodic and chaotic attractors may be found in discrete systems with only one state variable, but quasiperiodicity requires at least two variables. In contrast, continuous systems require two state variables for a limit cycle, and three variables for quasiperiodicity or chaos. If one or more parameters varies periodically in time (as in seasonal or diurnal effects) then the required number of state variables is reduced by one. If there are time delays (the current dynamics depend on past values of the state variable) then all attractors are possible even with only one state variable. A system may have multiple coexisting attractors, which may be qualitatively different (a stable equilibrium and
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Nonlinear Dynamics and Chaos
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Figure 2 Examples of nonlinear dynamics in field populations. (a) Coffee leaf-miners (Leucoptera spp.) in Tanzania. (b) Larch budworm (Zeiraphera diniana) in Switzerland. (c) Voles (Microtus and Clethrionomys) in Finland. (d) Red grouse (Lagopus lagopus scotius) in Scotland. (Data from Bigger M (1973) Journal of Animal Ecology 42: 417–434; Baltensweiler W and Fischlin A (1988) In: Berryman AA (ed.) Dynamics of Forest Insect Populations, pp. 331–351. New York: Plenum Press; Hanski I et al. (1993) Nature 364: 232–235; Middleton AD (1934) Journal of Animal Ecology 3: 231–249.)
chaos, for example). The set of initial conditions that leads to each attractor is known as the basin of attraction of that attractor. The boundary between the basins of attraction of coexisting attractors often itself has fractal structure.
Bifurcations and routes to chaos The examples shown in Figure 3 illustrate the fact that the same model can produce very different dynamics with different parameter values. A small change in a parameter may have only a small quantitative effect on the dynamics: the amplitude of a limit cycle may increase, for example, or the period of a quasiperiodic cycle may decrease. However, other equally small changes in parameter values may lead to a bifurcation: an abrupt, qualitative change in the dynamics. These changes include shifts among qualitative attractor types, doubling of the period of a
limit cycle, or, in systems with multiple attractors, the complete destabilization of one of the attractors. Bifurcations can be visualized using bifurcation diagrams (Figure 4). There are two archetypal sequences of bifurcations, or ‘routes to chaos’. The first is known as the ‘period-doubling route’ (Figure 4). A stable equilibrium gives way to a simple limit cycle. Further changes in the parameter lead to a doubling of the period. Subsequent period-doublings occur at an accelerating rate, until at the limit of ‘infinite period’ the system becomes chaotic. In contrast, in the ‘torus route’ (Figure 4) the equilibrium is destabilized to a quasiperiodic attractor. For certain ranges of the parameter value, the quasiperiodic attractor may be replaced by a limit cycle. The structure of the quasiperiodic attractor becomes steadily more complex, and eventually becomes chaotic.
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Figure 4 Bifurcation diagrams of (a) the logistic model (illustrating the period-doubling route to chaos) and (b) the host–parasitoid model from Figure 3 (illustrating the torus route to chaos). In each case a model parameter varies along the horizontal axis. Above that parameter value, successive values of one of the state variables are plotted on the vertical axis. A single point represents an equilibrium, two points represent periodic dynamics with period 2, and so on. Chaos and quasiperiodic dynamics result in a dense infilling of points. Notice the ‘windows’ of periodic dynamics within the chaotic and quasiperiodic regimes.
Lyapunov Exponents Lyapunov exponents are properties of attractors, and quantify an important aspect of the dynamics associated with the attractors described above. If a system is perturbed slightly, will it return to the trajectory it would have followed in the absence of the perturbation, or will the effect of the perturbation grow through time? If the largest Lyapunov exponent is negative, then the effects of the perturbation will die out; this is associated with stable
equilibria and limit cycles. If the largest Lyapunov exponent is positive, then perturbations will tend to grow; this is a property of chaotic attractors. Quasiperiodic attractors have a largest Lyapunov exponent of exactly zero: perturbations neither grow nor decay. The Lyapunov exponent is related to the derivatives of the functions describing the dynamics. Consider a onedimensional map, x(t 1 1) 5 f(x(t)). If the absolute value of the derivative of f is less than 1, then |f(x) 2 f(x 1 e)| 5 e, and so the perturbation e shrinks through time. This
Figure 3 Time-series and phase space representations of a discrete-time host–parasitoid model (left panels) and a continuous-time predator–prey model with seasonal variation in the prey carrying capacity (right panels). Top rows: limit cycles. Middle rows: quasiperiodic cycles. Bottom rows: chaos. For the host–parasitoid model the parasitoid attack rate increases from top to bottom. For the predator–prey model the strength of seasonality increases from top to bottom (there is no seasonality in the top pane); note that the period of the predominant oscillation remains longer than one year. (Models from Beddington JR, Free CA and Lawton JH (1976) Nature 225: 58–60; Rinaldi S, Muratori S and Kuznetsov Y (1993) Bulletin of Mathematical Biology 55: 15–35.)
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corresponds to a negative Lyapunov exponent. In contrast, if the derivative of f is greater than 1, then |f(x) 2 f(x 1 e)| 4 e, and so the perturbation e grows through time. This corresponds to a positive Lyapunov exponent. The multidimensional version of the derivative is called the ‘Jacobian’ matrix. If the dynamical system is described by dxi/dt 5 fi(X) (for continuous systems) or xi(t 1 1) 5 fi(X(t)) (for discrete systems), then the Jacobian matrix J has elements Jij 5 @fi/@xj. At each point X(t) on the attractor, the Jacobian matrix J[X(t)] is evaluated. The maximum Lyapunov exponent is given by eqn [1]. l 5 limt!1 (1/t) log | |J[X(t)] J[X(t 2 1)] _ J[X(1)]| | [1] where | |J| | represents a matrix norm (some measure of the ‘size’ of J).
Environmental Variability and Nonlinear Dynamics The above descriptions of nonlinear attractors have all assumed that the system is in a constant, noise-free environment. There are two important ways these assumptions could be violated: periodic variation in parameter values (e.g. seasonality) or ‘noise’ (random perturbations to the parameters or state variables due to environmental variability). Seasonal variation can induce complex dynamics in systems that would be stable in a complex environment. For example, one of the standard epidemiological models of childhood infectious disease has a stable equilibrium in a constant environment. Allowing the contact rate parameter to fluctuate through the year (to simulate the effect of school terms) induces, not surprisingly, a limit cycle with a 1-yr period. However, further increases in the magnitude of seasonality lead to a periodic or chaotic modulation of the basic annual cycle. A small amount of noise merely blurs the attractor. The fractal structure of a chaotic attractor is destroyed, but the basic dynamical characteristics of the attractor, such as the Lyapunov exponent, are unchanged. In contrast, a large noise variance can have surprising effects on the dynamics. If there are multiple coexisting attractor in the system, then the noise can bounce the trajectory among them, leading to episodes of qualitatively different dynamics. In addition, many systems contain complex transient dynamics as the trajectory returns to the attractor from a distant part of the state space, even when the attractor is a limit cycle or equilibrium. A trajectory following such transients can have all the characteristics of chaos – indeed, the transients are often the ghosts of chaotic attractors from other parameter values. Thus a sufficiently large perturbation can lead to an extended chaotic transient. If the perturba6
tions are frequent enough, the trajectory will spend more time on the transients than on the deterministic attractor.
Empirical Investigations of Nonlinear Dynamics Empirical investigations of nonlinear population dynamics have generally proceeded along three directions: statistical analysis of time-series data, experimental manipulations of whole populations, and mechanistic models built on known processes.
Time-series analysis The first step in analysing a time-series for nonlinear dynamics is to determine that the fluctuations are regular and not random. Identifying cyclic time-series ‘by eye’ is notoriously unreliable. Spectral analysis decomposes the time-series into sinusoidal components, and asks whether the most important such component is larger than should be expected by chance. For periodic and quasiperiodic time-series, the power associated with the dominant period should be very high, of course, but even chaotic time series have a strong spectral peak (even though they do not repeat exactly). A recent analysis of 700 long population timeseries revealed that 30% of them displayed statistically significant periodicity (Kendall et al., 1998). All methods for identifying and characterizing chaotic dynamics take advantage of a remarkable mathematical theorem by Floris Takens. If we have observed only one time series, x(t), of the n state variables that make up the full system, X(t), then we can create a synthetic state vector by ‘embedding’ the time series: Y(t) 5 (x(t), x(t 2 L), x(t 2 2L), _ x(t 2 (m 2 1)L)). With appropriate choices of L (the ‘embedding lag’) and m (the ‘embedding dimension’), the attractor defined by the points Y is topologically equivalent to the attractor defined by the full system X. In particular, the Lyapunov exponent and the fractal dimension of the attractor will be the same. The choice of L and m is something of an art, but m needs to be at least as large as n, the dimension of the original system. Early approaches for estimating the Lyapunov exponent and the attractor dimension were developed by physicists for use on long (hundreds of points), noise-free time-series. Olsen and Schaffer (1990) applied these techniques to data on childhood diseases and concluded that chickenpox displayed noisy periodicity whereas measles was chaotic. However, these techniques are unsuited to the short, noisy time-series that characterize most ecological systems. Noise destroys most of the fractal character of the attractor, so the dimension is not a useful statistic. Lyapunov exponent estimators need to distinguish between the effects of the noise and the effects of the underlying deterministic dynamics.
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Nonlinear Dynamics and Chaos
Modern techniques for estimating the Lyapunov exponent attempt to use the power of the model-based methods described above (Ellner and Turchin, 1995). Lacking information about the underlying processes or even about all of the state variables, we must try to estimate the function describing the time evolution of the embedded state variable: x(t 1 1) 5 f|Y(t)| 1 e(t), where e(t) represents the noise component. The model f is some sort of flexible function, such as a generalized polynomial neural network, or thin plate spline; the parameters of f are estimated using nonlinear regression. The Jacobian of f is evaluated at each data point Y(t), and the eigenvectors and eigenvalues are used to calculate the average Lyapunov exponent along the trajectory just as if f were the true model of the system. Thus the Lyapunov exponents are only estimated from the deterministic component of the model (f), but the exponents are evaluated along the noisy trajectory that is actually followed, rather than on the unrealized deterministic attractor. Ellner and Turchin (1995) applied this technique to a variety of laboratory and field population time-series and found evidence for chaos in a few, but not many, of the populations.
Experiments There have been numerous laboratory populations raised with the intention of reproducing the predator–prey cycles predicted by the Lotka–Volterra model. Most of these experiments were performed with interacting species (predators and prey) of protozoa or mites (Figure 1b). Elaborate spatial structure and prey refugia were required to prevent rapid extinction of these populations: in the simple laboratory environment, the predators were simply too efficient! However, these experiments, along with single-species populations such as the blowflies raised by A. J. Nicholson (Figure 1a), demonstrated that population fluctuations could occur in a constant environment. A few studies have modified the dynamics of fluctuating populations by modifying demographic parameters or interaction rates. For example, Constantino et al. (1995) induced transitions between equilibrium and cyclic dynamics in laboratory populations of the flour beetle (Tribolium castaneum) by artificially increasing the adult mortality rate. In contrast, Nicholson stabilized the blowfly cycles by increasing the juvenile death rate. Krebs et al. (1995) added food and reduced predation on snowshoe hares in 1 km2 enclosures during the course of a cycle. Each manipulation increased the peak density (and delayed the peak); when combined, the total increase in density was more than twice the additive effects of the individual factors, suggesting that hare density is controlled by an interaction between food supply and predation pressure. Density manipulations have been used to prolong the cycle peak of red grouse. The results of these manipulations are often initially counterintuitive, as most
of us have intuition based on linear situations. Thus these results can be best understood by making analogous manipulations to a nonlinear model.
Mechanistic models Time-series analysis does not provide any information about the processes that govern nonlinear dynamics. Using verbal models to generate predictions leaves the results of experimental manipulations less than sharp. Both of these empirical approaches greatly benefit from the thoughtful use of mechanistic models. For example, the conclusions about smallpox and measles were strengthened by the analysis of a nonlinear epidemiological model that supported the time-series analysis: with parameters appropriate to smallpox the model produces stable limit cycles, whereas with measles parameters it produces chaos (Olsen and Schaffer, 1990). Kendall et al. (1999) demonstrated techniques whereby alternative mechanistic models could be fitted to a time-series, and then time-series analysis could be used to discriminate which model better reproduced the patterns in the data. The Tribolium study used a mechanistic model to make predictions (which were sometimes confirmed) about the qualitative changes in dynamics as the mortality rate was varied (Constantino et al., 1995). The ideal study to uncover the underlying causes of nonlinear dynamics in a population would start with a set of mechanisms that are consistent with the existing data, in the sense of reproducing the dynamic patterns when entered into models. It would then use these models to find the critical experiments that would conclusively distinguish among the competing mechanisms. Finally, the outcomes of these experiments would be analysed rigorously using time-series statistics. Such an ideal study has never been performed.
Evolution of Stability or Chaos Chaos presents an apparent paradox for the evolution of population dynamics. In simple ecological models, chaos is associated with high population growth rates. All else being equal, high growth rates should be selected for, and hence chaos should be common. But chaos appears to be rare in field populations. Initial efforts to explain this paradox invoked group selection: populations that evolve chaotic dynamics are more likely to go extinct, for the fluctuations frequently bring their densities near zero. More recently, life-history theory has been applied, with the recognition that all is not equal. There is often a trade-off, for example, between the ability to grow rapidly at low densities and the ability to compete effectively at high densities – the classic distinction between ‘r-selection’ and ‘K-selection’. A population made up of K-selected individuals has a stable equilibrium
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Nonlinear Dynamics and Chaos
at high densities, and hence is resistant to invasion by rselected individuals. K-selection is thus an ‘evolutionarily stable strategy’ (ESS). In contrast, a population of rselected individuals fluctuates between states that are inimical to K-selected individuals (low densities) and states that favour them (high densities). Thus it will often be possible for the K-selected genotype to invade. This differential invasibility can occur even when the difference between the phenotypes is small. As new mutants arise that are slightly towards the K-selected strategy, the population steadily evolves towards ever-simpler dynamics. Evolution towards simpler dynamics is commonly observed in laboratory populations. For example, the pronounced cycles in Nicholson’s blowflies started to break down after about a year (Figure 1a). Examining the flies at the end of the experiment, Nicholson found that they had evolved physiological traits that allowed them to reproduce modestly at high densities while sacrificing their ancestors’ ability to be highly fecund at low density. Subsequent analysis of the time-series with a mechanistic model suggested that this change was happening continuously through the duration of the experiment, and the abrupt shift in dynamics thus represents a bifurcation. This process should select against all forms of complex population dynamics, not just chaos. Yet population cycles are relatively common, and can persist for at least hundreds of years. Is there a countervailing selective force in favour of complex dynamics? Recent theory suggests that in a spatially structured environment a chaotic phenotype may be able to coexist with a stable phenotype if there is a trade-off between dispersal ability and competitive ability. Especially in the presence of spatial and temporal variability in the environment, this may be sufficient to allow the population as a whole to exhibit regular oscillations. The above arguments are all based on single-species models. In contrast, many of the population cycles in nature appear to be the result of predator–prey interactions. Understanding the co-evolution between predator and prey in the context of complex population dynamics is rather difficult. However, it appears possible to obtain coevolutionary cycles in the traits that influence the predation rate. In a heterogeneous habitat, optimizing the predator and prey dispersal rates leads to stability. However, if both predator and prey populations are able to adapt their dispersal behaviour to maximize their fitness at each generation (by distributing themselves according to the ‘ideal free distribution’) then chaotic population dynamics always result (van Baalen and Sabelis, 1999). The analyses described above all assumed that the different phenotypes are inherited clonally, and one might
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think that sex and recombination would mute those effects. However, models of Mendelian or quantitative genetics in the context of density- or frequency-dependent selection can also lead to fluctuating polymorphisms. When coupled with complex population dynamics, this destroys the notion of a static ‘fitness surface’ that underlies much of classical evolutionary theory. Thus nonlinear population dynamics present a major challenge to evolutionary theory.
References Constantino RF, Cushing JM, Dennis B and Desharnais RA (1995) Experimentally induced transitions in the dynamic behaviour of insect populations. Nature 375: 227–230. Ellner S and Turchin P (1995) Chaos in a noisy world: new methods and evidence from time-series analysis. American Naturalist 145: 343–375. Kendall BE, Briggs CJ, Murdoch WW et al. (1999) Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80: 1789–1805. Kendall BE, Prendergast J and Bjørnstad ON (1998) The macroecology of population dynamics: taxonomic and biogeographic patterns in population cycles. Ecology Letters 1: 160–164. Krebs CJ, Boutin S, Boonstra R et al. (1995) Impact of food and predation on the snowshoe hare cycle. Science 269: 1112–1115. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261: 459–467. Olsen LF and Schaffer WM (1990) Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. Science 249: 499–504. van Baalen M and Sabelis MW (1999) Nonequilibrium population dynamics of ‘ideal and free’ prey and predators. American Naturalist 154: 69–88.
Further Reading Ferrie`re R and Fox GA (1995) Chaos and evolution. Trends in Ecology and Evolution 10: 480–485. Gleick J (1987) Chaos: Making a New Science. New York: Viking. Hastings A, Hom CL, Ellner S, Turchin P and Godfrey HCJ (1993) Chaos in ecology: is Mother Nature a strange attractor? Annual Review of Ecology and Systematics 24: 1–33. May RM (1973) Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press. May RM (1986) When two and two do not make four: nonlinear phenomena in ecology. Proceedings of the Royal Society of London B 228: 241–266. Pahl-Wostl C (1995) The Dynamic Nature of Ecosystems: Chaos and Order Intertwined. Chichester: Wiley. Royama T (1992) Analytical Population Dynamics. London: Chapman and Hall. Schaffer WM and Kot M (1986) Chaos in ecological systems: the coals that Newcastle forgot. Trends in Ecology and Evolution 1: 58–63. Stone L and Ezrati S (1996) Chaos, cycles and spatiotemporal dynamics in plant ecology. Journal of Ecology 84: 279–291.
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Secondary article Article Contents . Introduction
Bruce E Kendall, University of California, Santa Barbara, California, USA
. Historical development
Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of nonlinear dynamics, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time. 7
Introduction Throughout much of ecology’s history, most ecologists have believed that ecological systems tend towards an equilibrium – a notion popularized as ‘the balance of nature’ – and that deviations from that equilibrium are caused by external perturbations. By assuming that systems were close to equilibrium, ecologists could describe them with linear models that were mathematically tractable. However, ecology is rife with processes whose rates depend nonlinearly on the state of the system – such as nutrient uptake, density dependence and predation. Advances in computing power and availability in the 1970s led to the discovery that such nonlinearities could cause spontaneous oscillations. Interest initially focused on chaos as a potential explanation for all of the irregular fluctuations in nature. Two decades of investigation suggest that ecological chaos is rare in natural systems – ecology really is ‘noisy’. Nevertheless, the importance of nonlinear processes, and the resulting nonequilibrium dynamics, are now fully accepted at the core of ecology.
Historical development The study of complex population dynamics is nearly as old as population ecology. In the 1920s, Alfred Lotka and Vito Volterra independently developed a simple model of interacting species that still bears their joint names. This was a nearly linear model, but the predator–prey version displayed neutrally stable cycles. Experimentalists such as G. F. Gause soon tried to recreate these cycles in laboratory populations of protozoa. Also working in the laboratory, A. J. Nicholson was trying to demonstrate his ideas about density-dependent population regulation and managed to produce cycles in single-species cultures of blowflies (Figure 1). In the meantime, Charles Elton had become fascinated by cyclic fluctuations in field populations of voles (small rodents; Figure 2) and in the numbers of furs of Canadian lynx and other large mammals traded by the Hudson’s Bay Company. How-
. Nonlinear Attractors: Limit Cycles, Quasiperiodicity and Chaos . Lyapunov Exponents . Environmental Variability and Nonlinear Dynamics . Empirical Investigations of Nonlinear Dynamics . Evolution of Stability or Chaos
ever, it was not until Michael Rosenzweig, as a graduate student with Robert MacArthur in the 1960s, added density dependence and predator behaviour to the Lotka– Volterra equations that an ecological model capable of displaying true nonlinear limit cycles was developed (Nicholson’s blowflies were not adequately modelled until 1980). Chaos was first knowingly observed in a mathematical model by meteorologist Edward Lorenz in the early 1960s. Not only did he observe deterministic aperiodic fluctuations, he also discovered sensitive dependence on initial conditions: restarting a simulation model without bothering to copy all of the significant digits, Lorenz observed a sequence of numbers that soon started to diverge from the original simulation. Within the context of turbulence studies, the mathematical and computational strands of chaos were woven together in the 1970s and 1980s. Mathematicians David Ruelle and Floris Takens coined the evocative term ‘strange attractor’ to describe the complex structures of chaos in 1971. In the early and mid 1970s, physicist Robert May took theoretical ecology by storm with a series of papers demonstrating that ‘complex dynamics’ (cycles and chaos) could be generated by the simplest ecological models (e.g. May 1976), including the ‘logistic map’, an extremely simple model of density-dependent population growth with discrete generations: N(t 1 1) 5 rN(t) [1 2 N(t)/K]. Theoreticians rapidly embraced the new science of nonlinear dynamics, hoping that chaos could explain some of the erratic fluctuations that were hitherto called ‘noise’. Empirical ecologists were rather more sceptical, and never embraced the claims that chaos was important in the real world; but they gradually accepted the idea that nonlinear dynamics in general were important. This acceptance was indicated by a 1990 symposium sponsored by the Ecological Society of America: ‘The shift from an equilibrium to a nonequilibrium paradigm in ecology’.
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Figure 1 Examples of nonlinear dynamics in laboratory populations. (a) Blowflies (Lucilia cuprina). (b) The protozoa Didinium nasutum (predator) and Paramecium aurelia (prey). (Data from Nicholson AJ (1957) Cold Spring Harbor Symposia on Quantitative Biology 22: 153–173; Luckinbill LS (1973) Ecology 54: 1320–1327.)
Nonlinear Attractors: Limit Cycles, Quasiperiodicity and Chaos The time-series, which displays fluctuations through time, is a familiar representation of dynamic systems (Figures 1 and 2). A less intuitive, but powerful, representation is the phase space. The axes of the phase space are defined by the state variables so that, at a given moment, the state of the system is represented by a point in the phase space. Plotting the system at successive moments in time reveals the trajectory, or the path that the system follows through the phase space. Examples showing the relationship between the phase space and time-series representations are shown in Figure 3. The power of the phase space representation comes about because, as long as the system’s fluctuations are bounded, an infinitely long trajectory inhabits a finite region of the phase space. The trajectory becomes a geometric object. In a deterministic system, the trajectory will eventually conform to another geometric object in the phase space: the attractor. A trajectory that is sufficiently ‘near’ (in the phase space) an attractor will move towards it, on average, and once it is on the attractor it will not spontaneously move away. The simplest form of attractor is a stable equilibrium, which is an isolated point in the phase space. Nonlinear systems can have more complex attractors: limit cycles, quasiperiodicity and chaos (Figure 3). Periodic, or limit cycle, attractors repeat the same sequence of values over and over, in strictly periodic oscillations. In discrete systems the trajectory visits a finite set of points in the phase space, repeating them in theI0006988 same order. In continuous systems the trajectory follows a closed loop in the phase space. 2
Quasiperiodic attractors have two fundamental frequencies whose ratio is an irrational number. In discrete systems one of these frequencies is the time step, so that a quasiperiodic time series appears to be periodic with an irrational period. In the phase space the trajectory fills in all of the points in a closed loop. In continuous systems, the lower frequency provides an envelope that modulates amplitude of the higher frequency oscillations, producing a ‘beat’ phenomenon. In the phase space, the orbit covers the surface of a torus. Chaotic attractors have trajectories that never repeat, and exhibit apparently erratic fluctuations despite being deterministic. However, there is regularity in the dynamics. The attractors are confined to a finite volume of the phase space, and there is fine-scale structure that is fractal – visible at all magnifications. Thus the attractors do not fill the volume in the phase space, but have a noninteger (‘fractal’) dimension. Chaotic attractors also exhibit ‘sensitivity to initial conditions’: a small change in the starting state of the system will grow, on average, exponentially through time (up to the size of the attractor). Periodic and chaotic attractors may be found in discrete systems with only one state variable, but quasiperiodicity requires at least two variables. In contrast, continuous systems require two state variables for a limit cycle, and three variables for quasiperiodicity or chaos. If one or more parameters varies periodically in time (as in seasonal or diurnal effects) then the required number of state variables is reduced by one. If there are time delays (the current dynamics depend on past values of the state variable) then all attractors are possible even with only one state variable. A system may have multiple coexisting attractors, which may be qualitatively different (a stable equilibrium and
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chaos, for example). The set of initial conditions that leads to each attractor is known as the basin of attraction of that attractor. The boundary between the basins of attraction of coexisting attractors often itself has fractal structure.
Bifurcations and routes to chaos The examples shown in Figure 3 illustrate the fact that the same model can produce very different dynamics with different parameter values. A small change in a parameter may have only a small quantitative effect on the dynamics: the amplitude of a limit cycle may increase, for example, or the period of a quasiperiodic cycle may decrease. However, other equally small changes in parameter values may lead to a bifurcation: an abrupt, qualitative change in the dynamics. These changes include shifts among qualitative attractor types, doubling of the period of a
limit cycle, or, in systems with multiple attractors, the complete destabilization of one of the attractors. Bifurcations can be visualized using bifurcation diagrams (Figure 4). There are two archetypal sequences of bifurcations, or ‘routes to chaos’. The first is known as the ‘period-doubling route’ (Figure 4). A stable equilibrium gives way to a simple limit cycle. Further changes in the parameter lead to a doubling of the period. Subsequent period-doublings occur at an accelerating rate, until at the limit of ‘infinite period’ the system becomes chaotic. In contrast, in the ‘torus route’ (Figure 4) the equilibrium is destabilized to a quasiperiodic attractor. For certain ranges of the parameter value, the quasiperiodic attractor may be replaced by a limit cycle. The structure of the quasiperiodic attractor becomes steadily more complex, and eventually becomes chaotic.
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Figure 4 Bifurcation diagrams of (a) the logistic model (illustrating the period-doubling route to chaos) and (b) the host–parasitoid model from Figure 3 (illustrating the torus route to chaos). In each case a model parameter varies along the horizontal axis. Above that parameter value, successive values of one of the state variables are plotted on the vertical axis. A single point represents an equilibrium, two points represent periodic dynamics with period 2, and so on. Chaos and quasiperiodic dynamics result in a dense infilling of points. Notice the ‘windows’ of periodic dynamics within the chaotic and quasiperiodic regimes.
Lyapunov Exponents Lyapunov exponents are properties of attractors, and quantify an important aspect of the dynamics associated with the attractors described above. If a system is perturbed slightly, will it return to the trajectory it would have followed in the absence of the perturbation, or will the effect of the perturbation grow through time? If the largest Lyapunov exponent is negative, then the effects of the perturbation will die out; this is associated with stable
equilibria and limit cycles. If the largest Lyapunov exponent is positive, then perturbations will tend to grow; this is a property of chaotic attractors. Quasiperiodic attractors have a largest Lyapunov exponent of exactly zero: perturbations neither grow nor decay. The Lyapunov exponent is related to the derivatives of the functions describing the dynamics. Consider a onedimensional map, x(t 1 1) 5 f(x(t)). If the absolute value of the derivative of f is less than 1, then |f(x) 2 f(x 1 e)| 5 e, and so the perturbation e shrinks through time. This
Figure 3 Time-series and phase space representations of a discrete-time host–parasitoid model (left panels) and a continuous-time predator–prey model with seasonal variation in the prey carrying capacity (right panels). Top rows: limit cycles. Middle rows: quasiperiodic cycles. Bottom rows: chaos. For the host–parasitoid model the parasitoid attack rate increases from top to bottom. For the predator–prey model the strength of seasonality increases from top to bottom (there is no seasonality in the top pane); note that the period of the predominant oscillation remains longer than one year. (Models from Beddington JR, Free CA and Lawton JH (1976) Nature 225: 58–60; Rinaldi S, Muratori S and Kuznetsov Y (1993) Bulletin of Mathematical Biology 55: 15–35.)
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corresponds to a negative Lyapunov exponent. In contrast, if the derivative of f is greater than 1, then |f(x) 2 f(x 1 e)| 4 e, and so the perturbation e grows through time. This corresponds to a positive Lyapunov exponent. The multidimensional version of the derivative is called the ‘Jacobian’ matrix. If the dynamical system is described by dxi/dt 5 fi(X) (for continuous systems) or xi(t 1 1) 5 fi(X(t)) (for discrete systems), then the Jacobian matrix J has elements Jij 5 @fi/@xj. At each point X(t) on the attractor, the Jacobian matrix J[X(t)] is evaluated. The maximum Lyapunov exponent is given by eqn [1]. l 5 limt!1 (1/t) log | |J[X(t)] J[X(t 2 1)] _ J[X(1)]| | [1] where | |J| | represents a matrix norm (some measure of the ‘size’ of J).
Environmental Variability and Nonlinear Dynamics The above descriptions of nonlinear attractors have all assumed that the system is in a constant, noise-free environment. There are two important ways these assumptions could be violated: periodic variation in parameter values (e.g. seasonality) or ‘noise’ (random perturbations to the parameters or state variables due to environmental variability). Seasonal variation can induce complex dynamics in systems that would be stable in a complex environment. For example, one of the standard epidemiological models of childhood infectious disease has a stable equilibrium in a constant environment. Allowing the contact rate parameter to fluctuate through the year (to simulate the effect of school terms) induces, not surprisingly, a limit cycle with a 1-yr period. However, further increases in the magnitude of seasonality lead to a periodic or chaotic modulation of the basic annual cycle. A small amount of noise merely blurs the attractor. The fractal structure of a chaotic attractor is destroyed, but the basic dynamical characteristics of the attractor, such as the Lyapunov exponent, are unchanged. In contrast, a large noise variance can have surprising effects on the dynamics. If there are multiple coexisting attractor in the system, then the noise can bounce the trajectory among them, leading to episodes of qualitatively different dynamics. In addition, many systems contain complex transient dynamics as the trajectory returns to the attractor from a distant part of the state space, even when the attractor is a limit cycle or equilibrium. A trajectory following such transients can have all the characteristics of chaos – indeed, the transients are often the ghosts of chaotic attractors from other parameter values. Thus a sufficiently large perturbation can lead to an extended chaotic transient. If the perturba6
tions are frequent enough, the trajectory will spend more time on the transients than on the deterministic attractor.
Empirical Investigations of Nonlinear Dynamics Empirical investigations of nonlinear population dynamics have generally proceeded along three directions: statistical analysis of time-series data, experimental manipulations of whole populations, and mechanistic models built on known processes.
Time-series analysis The first step in analysing a time-series for nonlinear dynamics is to determine that the fluctuations are regular and not random. Identifying cyclic time-series ‘by eye’ is notoriously unreliable. Spectral analysis decomposes the time-series into sinusoidal components, and asks whether the most important such component is larger than should be expected by chance. For periodic and quasiperiodic time-series, the power associated with the dominant period should be very high, of course, but even chaotic time series have a strong spectral peak (even though they do not repeat exactly). A recent analysis of 700 long population timeseries revealed that 30% of them displayed statistically significant periodicity (Kendall et al., 1998). All methods for identifying and characterizing chaotic dynamics take advantage of a remarkable mathematical theorem by Floris Takens. If we have observed only one time series, x(t), of the n state variables that make up the full system, X(t), then we can create a synthetic state vector by ‘embedding’ the time series: Y(t) 5 (x(t), x(t 2 L), x(t 2 2L), _ x(t 2 (m 2 1)L)). With appropriate choices of L (the ‘embedding lag’) and m (the ‘embedding dimension’), the attractor defined by the points Y is topologically equivalent to the attractor defined by the full system X. In particular, the Lyapunov exponent and the fractal dimension of the attractor will be the same. The choice of L and m is something of an art, but m needs to be at least as large as n, the dimension of the original system. Early approaches for estimating the Lyapunov exponent and the attractor dimension were developed by physicists for use on long (hundreds of points), noise-free time-series. Olsen and Schaffer (1990) applied these techniques to data on childhood diseases and concluded that chickenpox displayed noisy periodicity whereas measles was chaotic. However, these techniques are unsuited to the short, noisy time-series that characterize most ecological systems. Noise destroys most of the fractal character of the attractor, so the dimension is not a useful statistic. Lyapunov exponent estimators need to distinguish between the effects of the noise and the effects of the underlying deterministic dynamics.
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Nonlinear Dynamics and Chaos
Modern techniques for estimating the Lyapunov exponent attempt to use the power of the model-based methods described above (Ellner and Turchin, 1995). Lacking information about the underlying processes or even about all of the state variables, we must try to estimate the function describing the time evolution of the embedded state variable: x(t 1 1) 5 f|Y(t)| 1 e(t), where e(t) represents the noise component. The model f is some sort of flexible function, such as a generalized polynomial neural network, or thin plate spline; the parameters of f are estimated using nonlinear regression. The Jacobian of f is evaluated at each data point Y(t), and the eigenvectors and eigenvalues are used to calculate the average Lyapunov exponent along the trajectory just as if f were the true model of the system. Thus the Lyapunov exponents are only estimated from the deterministic component of the model (f), but the exponents are evaluated along the noisy trajectory that is actually followed, rather than on the unrealized deterministic attractor. Ellner and Turchin (1995) applied this technique to a variety of laboratory and field population time-series and found evidence for chaos in a few, but not many, of the populations.
Experiments There have been numerous laboratory populations raised with the intention of reproducing the predator–prey cycles predicted by the Lotka–Volterra model. Most of these experiments were performed with interacting species (predators and prey) of protozoa or mites (Figure 1b). Elaborate spatial structure and prey refugia were required to prevent rapid extinction of these populations: in the simple laboratory environment, the predators were simply too efficient! However, these experiments, along with single-species populations such as the blowflies raised by A. J. Nicholson (Figure 1a), demonstrated that population fluctuations could occur in a constant environment. A few studies have modified the dynamics of fluctuating populations by modifying demographic parameters or interaction rates. For example, Constantino et al. (1995) induced transitions between equilibrium and cyclic dynamics in laboratory populations of the flour beetle (Tribolium castaneum) by artificially increasing the adult mortality rate. In contrast, Nicholson stabilized the blowfly cycles by increasing the juvenile death rate. Krebs et al. (1995) added food and reduced predation on snowshoe hares in 1 km2 enclosures during the course of a cycle. Each manipulation increased the peak density (and delayed the peak); when combined, the total increase in density was more than twice the additive effects of the individual factors, suggesting that hare density is controlled by an interaction between food supply and predation pressure. Density manipulations have been used to prolong the cycle peak of red grouse. The results of these manipulations are often initially counterintuitive, as most
of us have intuition based on linear situations. Thus these results can be best understood by making analogous manipulations to a nonlinear model.
Mechanistic models Time-series analysis does not provide any information about the processes that govern nonlinear dynamics. Using verbal models to generate predictions leaves the results of experimental manipulations less than sharp. Both of these empirical approaches greatly benefit from the thoughtful use of mechanistic models. For example, the conclusions about smallpox and measles were strengthened by the analysis of a nonlinear epidemiological model that supported the time-series analysis: with parameters appropriate to smallpox the model produces stable limit cycles, whereas with measles parameters it produces chaos (Olsen and Schaffer, 1990). Kendall et al. (1999) demonstrated techniques whereby alternative mechanistic models could be fitted to a time-series, and then time-series analysis could be used to discriminate which model better reproduced the patterns in the data. The Tribolium study used a mechanistic model to make predictions (which were sometimes confirmed) about the qualitative changes in dynamics as the mortality rate was varied (Constantino et al., 1995). The ideal study to uncover the underlying causes of nonlinear dynamics in a population would start with a set of mechanisms that are consistent with the existing data, in the sense of reproducing the dynamic patterns when entered into models. It would then use these models to find the critical experiments that would conclusively distinguish among the competing mechanisms. Finally, the outcomes of these experiments would be analysed rigorously using time-series statistics. Such an ideal study has never been performed.
Evolution of Stability or Chaos Chaos presents an apparent paradox for the evolution of population dynamics. In simple ecological models, chaos is associated with high population growth rates. All else being equal, high growth rates should be selected for, and hence chaos should be common. But chaos appears to be rare in field populations. Initial efforts to explain this paradox invoked group selection: populations that evolve chaotic dynamics are more likely to go extinct, for the fluctuations frequently bring their densities near zero. More recently, life-history theory has been applied, with the recognition that all is not equal. There is often a trade-off, for example, between the ability to grow rapidly at low densities and the ability to compete effectively at high densities – the classic distinction between ‘r-selection’ and ‘K-selection’. A population made up of K-selected individuals has a stable equilibrium
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at high densities, and hence is resistant to invasion by rselected individuals. K-selection is thus an ‘evolutionarily stable strategy’ (ESS). In contrast, a population of rselected individuals fluctuates between states that are inimical to K-selected individuals (low densities) and states that favour them (high densities). Thus it will often be possible for the K-selected genotype to invade. This differential invasibility can occur even when the difference between the phenotypes is small. As new mutants arise that are slightly towards the K-selected strategy, the population steadily evolves towards ever-simpler dynamics. Evolution towards simpler dynamics is commonly observed in laboratory populations. For example, the pronounced cycles in Nicholson’s blowflies started to break down after about a year (Figure 1a). Examining the flies at the end of the experiment, Nicholson found that they had evolved physiological traits that allowed them to reproduce modestly at high densities while sacrificing their ancestors’ ability to be highly fecund at low density. Subsequent analysis of the time-series with a mechanistic model suggested that this change was happening continuously through the duration of the experiment, and the abrupt shift in dynamics thus represents a bifurcation. This process should select against all forms of complex population dynamics, not just chaos. Yet population cycles are relatively common, and can persist for at least hundreds of years. Is there a countervailing selective force in favour of complex dynamics? Recent theory suggests that in a spatially structured environment a chaotic phenotype may be able to coexist with a stable phenotype if there is a trade-off between dispersal ability and competitive ability. Especially in the presence of spatial and temporal variability in the environment, this may be sufficient to allow the population as a whole to exhibit regular oscillations. The above arguments are all based on single-species models. In contrast, many of the population cycles in nature appear to be the result of predator–prey interactions. Understanding the co-evolution between predator and prey in the context of complex population dynamics is rather difficult. However, it appears possible to obtain coevolutionary cycles in the traits that influence the predation rate. In a heterogeneous habitat, optimizing the predator and prey dispersal rates leads to stability. However, if both predator and prey populations are able to adapt their dispersal behaviour to maximize their fitness at each generation (by distributing themselves according to the ‘ideal free distribution’) then chaotic population dynamics always result (van Baalen and Sabelis, 1999). The analyses described above all assumed that the different phenotypes are inherited clonally, and one might
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think that sex and recombination would mute those effects. However, models of Mendelian or quantitative genetics in the context of density- or frequency-dependent selection can also lead to fluctuating polymorphisms. When coupled with complex population dynamics, this destroys the notion of a static ‘fitness surface’ that underlies much of classical evolutionary theory. Thus nonlinear population dynamics present a major challenge to evolutionary theory.
References Constantino RF, Cushing JM, Dennis B and Desharnais RA (1995) Experimentally induced transitions in the dynamic behaviour of insect populations. Nature 375: 227–230. Ellner S and Turchin P (1995) Chaos in a noisy world: new methods and evidence from time-series analysis. American Naturalist 145: 343–375. Kendall BE, Briggs CJ, Murdoch WW et al. (1999) Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80: 1789–1805. Kendall BE, Prendergast J and Bjørnstad ON (1998) The macroecology of population dynamics: taxonomic and biogeographic patterns in population cycles. Ecology Letters 1: 160–164. Krebs CJ, Boutin S, Boonstra R et al. (1995) Impact of food and predation on the snowshoe hare cycle. Science 269: 1112–1115. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261: 459–467. Olsen LF and Schaffer WM (1990) Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. Science 249: 499–504. van Baalen M and Sabelis MW (1999) Nonequilibrium population dynamics of ‘ideal and free’ prey and predators. American Naturalist 154: 69–88.
Further Reading Ferrie`re R and Fox GA (1995) Chaos and evolution. Trends in Ecology and Evolution 10: 480–485. Gleick J (1987) Chaos: Making a New Science. New York: Viking. Hastings A, Hom CL, Ellner S, Turchin P and Godfrey HCJ (1993) Chaos in ecology: is Mother Nature a strange attractor? Annual Review of Ecology and Systematics 24: 1–33. May RM (1973) Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press. May RM (1986) When two and two do not make four: nonlinear phenomena in ecology. Proceedings of the Royal Society of London B 228: 241–266. Pahl-Wostl C (1995) The Dynamic Nature of Ecosystems: Chaos and Order Intertwined. Chichester: Wiley. Royama T (1992) Analytical Population Dynamics. London: Chapman and Hall. Schaffer WM and Kot M (1986) Chaos in ecological systems: the coals that Newcastle forgot. Trends in Ecology and Evolution 1: 58–63. Stone L and Ezrati S (1996) Chaos, cycles and spatiotemporal dynamics in plant ecology. Journal of Ecology 84: 279–291.
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