25SINGLE-SPECIES DYNAMICS(a)(b)eno+enPopulation density timetPopulation densitytimef(c)+aePopulation density time tFigure 3.7 Increasing the nonlinearity of the response of population growth to density can lead from the monotonic approach toequilibrium in Fiqure 3.3 to an oscillatory approach (a) and then a two-point cycle (b) and finally chaos (c). Examples of cobwebbing; theunderlying population model is the Ricker equation, xi+1 = x: exp(r(1 x:) where x is population density (scaled to be 1 at carrying capacity)and r is fecundity.thentoconductastabilityanalysistodeterminethearetheirmultidimensionalequivalentscanbeequilibrium's properties. In some cases it is pos-derived (May,1972).In this section we have focused on a simple,sibleto showthat an equilibrium such as thatinFigure3.3is globally stable, but such analyses areunstructured population in discrete time. Wemathematically challenging and often there is nocould alternativelyhave studied a population inglobal equilibrium. Instead, a local stability ana-continuoustimewhosedynamicsaredescribedbylysis is performed.For populations described bythe equationthe type of growth curve shown in Figures 3.2-3.7dN(3.6)we have beenable toderive the local stability=[(N)-d(N)N =r(N)criteria using intuition helped by a little cobweb-bing.Moving to more complex structuredorwhere birth rate (b), death rate (d), and net popu-multispecies communities these simple geome-lationgrowth (r)arenowall functionsofpopula-trical insightsarelost,butalgebraicconditionsthattion size N.Plotting r(N)against N would similarly
then to conduct a stability analysis to determine the equilibrium’s properties. In some cases it is possible to show that an equilibrium such as that in Figure 3.3 is globally stable, but such analyses are mathematically challenging and often there is no global equilibrium. Instead, a local stability analysis is performed. For populations described by the type of growth curve shown in Figures 3.2–3.7 we have been able to derive the local stability criteria using intuition helped by a little cobwebbing. Moving to more complex structured or multispecies communities these simple geometrical insights are lost, but algebraic conditions that are their multidimensional equivalents can be derived (May, 1972). In this section we have focused on a simple, unstructured population in discrete time. We could alternatively have studied a population in continuous time whose dynamics are described by the equation dN dt ¼ ½ bðNÞ dðNÞ N ¼ rðNÞN ð3:6Þ where birth rate (b), death rate (d), and net population growth (r) are now all functions of population size N. Plotting r(N) against N would similarly Population density time t+1 Population density time t (a) Population density time t Population density time t+1 (b) Population density time t+1 Population density time t (c) Figure 3.7 Increasing the nonlinearity of the response of population growth to density can lead from the monotonic approach to equilibrium in Figure 3.3 to an oscillatory approach (a) and then a two-point cycle (b) and finally chaos (c). Examples of cobwebbing; the underlying population model is the Ricker equation, xtþ1 ¼ xt expðrð1 xtÞÞ where x is population density (scaled to be 1 at carrying capacity) and r is fecundity. SINGLE-SPECIES DYNAMICS 25
26THEORETICAL ECOLOGYtell us much about the dynamics of the system.generations. Many of these systems show popula-There is, however, a difference between popula-tion cycles with periods shorter than those pre-tions described by eqn 3.6 and those that we stu-dicted by unstructured models (e.g. Figure 3.7).died in Figures 3.1-3.7. In eqn 3.6 the effects ofThe details differ with the natural history of thepopulation density act instantaneously on popu-different systems but a common pattern is for anlation growth rate.In our discrete-time examplesolder cohort of individuals to reduce the numbersthere is an implicittime lag:thelevel of competi-in a younger cohort by out-competing them fortion experienced by individuals in the currentfood or through cannibalism. When the depletedgeneration is set by interactions that occurred inyounger cohort grow old enough to be dominantthe last generation.Such time lags tend to becompetitors or cannibalsthemselvestherearenotdestabilizingastheydelaythe onset ofa reductionenough of them to reduce significantly the nextinpopulationgrowthrates asdensitiesclimb,andcohort coming through.This means that the nextmake it more likely that any equilibrium is over-group of individuals to mature into the oldershot.Indeed,it is mathematically impossible for acohort are very numerous and decimate the cur-population governed byeqn 3.6 to show chaos.Werent younger cohort, and the cycle begins again.shouldstressthatitisnotthedifferencebetweencontinuous-and discrete-timeformalisms that lies3.2.1 Chaosbehind the contrasting stability,but the presence ofThe pioneers of modern mathematical dynamics,the timelag.Indeed, in continuous time we can getexactly the same dynamics by explicitlymakingparticularly Poincare at the end of the nineteenthcentury, realized that the behaviour of highlynet populationgrowthrates a function of previouspopulationdensities,nonlinear systems could be very odd, but in theabsence of computers to help visualize theirdynamics, progress on understanding what wasdN=(Ni)N(3.7)happening was very slow.When computersbegandttobecomeavailableinthe1960sworkersinfieldswhere t is a time lag of approximately one gen-such as meteorology and ecology were able to seeeration.the complex dynamics produced by beguilinglyStructured population models with densitystraightforward equations, and this led to a burstdependence can be built using the same matrix,of interest inboth pure and applied mathematicsthat laid the foundations of the modern field ofintegral equation, or partial differential equationapproaches discussed in the section on densitychaotic dynamics. In population ecology,theclas-independence. Naturally they are more complex,sic paper is May's Simple mathematical models withand often with a greater potential for destabilizingvery complicated dynamics (May,1976a),which nottime lags. Relaxing the assumption that all indi-only introduced the notion ofchaosto thefield butviduals are equal also leads to the possibility ofshowed that lurking underneath the seemingmore complicated types of interaction than areunpredictability of chaotic dynamics was con-possible for unstructured populations.Competi-siderable order and pattern. We shall now exploretion may be asymmetric, typically with smallera population model of exactly the type that Mayindividuals suffering disproportionately at theanalysed.hands (or roots) of larger individuals.Moreover,Chaos has already been encountered in thiscannibalism ismuchmorecommon intheanimalchapter as the dynamics that emerge in a simplekingdom than often realized, and when it occurs itdiscrete-timepopulationmodel asthepopulationis nearly always size-related, with older largergrowth curve (ormap)becomes sufficientlynon-individualsconsuming their smaller conspecificslinear(the‘humpiness'ofthecurvesinFigure3.7).Such age-specific interactions have been studiedLet us now specify a family of curves that can giveindetail,particularlvin insectsvstemsthatcanrisetothemaps inFigures3.3and3.7.Forreasonsbe maintainedin the laboratoryformultiplethat will be explained in a few paragraphs it does
tell us much about the dynamics of the system. There is, however, a difference between populations described by eqn 3.6 and those that we studied in Figures 3.1–3.7. In eqn 3.6 the effects of population density act instantaneously on population growth rate. In our discrete-time examples there is an implicit time lag: the level of competition experienced by individuals in the current generation is set by interactions that occurred in the last generation. Such time lags tend to be destabilizing as they delay the onset of a reduction in population growth rates as densities climb, and make it more likely that any equilibrium is overshot. Indeed, it is mathematically impossible for a population governed by eqn 3.6 to show chaos. We should stress that it is not the difference between continuous- and discrete-time formalisms that lies behind the contrasting stability, but the presence of the time lag. Indeed, in continuous time we can get exactly the same dynamics by explicitly making net population growth rates a function of previous population densities, dNt dt ¼ rðNttÞNt ð3:7Þ where t is a time lag of approximately one generation. Structured population models with density dependence can be built using the same matrix, integral equation, or partial differential equation approaches discussed in the section on density independence. Naturally they are more complex, and often with a greater potential for destabilizing time lags. Relaxing the assumption that all individuals are equal also leads to the possibility of more complicated types of interaction than are possible for unstructured populations. Competition may be asymmetric, typically with smaller individuals suffering disproportionately at the hands (or roots) of larger individuals. Moreover, cannibalism is much more common in the animal kingdom than often realized, and when it occurs it is nearly always size-related, with older larger individuals consuming their smaller conspecifics. Such age-specific interactions have been studied in detail, particularly in insect systems that can be maintained in the laboratory for multiple generations. Many of these systems show population cycles with periods shorter than those predicted by unstructured models (e.g. Figure 3.7). The details differ with the natural history of the different systems but a common pattern is for an older cohort of individuals to reduce the numbers in a younger cohort by out-competing them for food or through cannibalism. When the depleted younger cohort grow old enough to be dominant competitors or cannibals themselves there are not enough of them to reduce significantly the next cohort coming through. This means that the next group of individuals to mature into the older cohort are very numerous and decimate the current younger cohort, and the cycle begins again. 3.2.1 Chaos The pioneers of modern mathematical dynamics, particularly Poincare´ at the end of the nineteenth century, realized that the behaviour of highly nonlinear systems could be very odd, but in the absence of computers to help visualize their dynamics, progress on understanding what was happening was very slow. When computers began to become available in the 1960s workers in fields such as meteorology and ecology were able to see the complex dynamics produced by beguilingly straightforward equations, and this led to a burst of interest in both pure and applied mathematics that laid the foundations of the modern field of chaotic dynamics. In population ecology, the classic paper is May’s Simple mathematical models with very complicated dynamics (May, 1976a), which not only introduced the notion of chaos to the field but showed that lurking underneath the seeming unpredictability of chaotic dynamics was considerable order and pattern. We shall now explore a population model of exactly the type that May analysed. Chaos has already been encountered in this chapter as the dynamics that emerge in a simple discrete-time population model as the population growth curve (or map) becomes sufficiently nonlinear (the ‘humpiness’ of the curves in Figure 3.7). Let us now specify a family of curves that can give rise to the maps in Figures 3.3 and 3.7. For reasons that will be explained in a few paragraphs it does 26 THEORETICAL ECOLOGY
SINGLE-SPECIES DYNAMICS27not particularlymatter which familywe chose,andcycle:the population oscillates for ever betweenwe plump for the Ricker equation as it is commonlytwo densities, one greater and one less than thenowunstableequilibrium n=1.InFigure3.8thisused in applied population biology,particularlyfor fisheries (Ricker, 1954).appears as two points. The value of this repre-sentation now becomes clear because instead of(3.8)nt+1 = ntexp[r(1 - n)]having to tryto comparea large number ofcobwebHere n, is population density (scaled to equal 1 atdiagrams we can see at one glance how theequilibrium).When rare thepopulation increasescycles appear at r=2 and then increase in ampli-each generation by a factor explr but as densitiestude as r gets bigger. The change of behaviour atapproach1the increaseslowsandabove1itr=2 is called for obvious reasons a bifurcation,reverses.If r is high there is the potential for theand the representation itself isa bifurcationdia-population to overshoot the equilibrium.gram.Wecanalsoseethatatr=2.5asecondWe want to picture the dynamics of the wholebifurcation occurs to givea four-point cycle,andsystem for different values of the sole adjustablethen further bifurcations at increasingly smallerparameter r. To do this, imagine iterating theintervals of r until a limit is reached."WhathappensequationbycobwebbingasinFigures3.3and3.7at the point of accumulation [the limit]?" is what Mayand then throwing away allthetransientdynamics,scrawled on a blackboard in theTheoretical Phy-perhaps the first 50generations.For Figure 3.3sicsDepartment at SydneyUniversityin the early(correspondingtoavalueof r=1)thenon-transient1970s.dynamics would not be very interesting: it wouldMay showed that what happens is chaos.As thesimplybeapopulationatstableequilibrium,inthisthird panel in Figure 3.7 illustrates, the trajectorycase n=1. In Figure3.8 weplot ralong the xaxisnever converges on a simple cycle but fluctuatesand thenon-transient dynamics on theyaxis; foraperiodically around very many values of n, neverr=1 there is a single point at n=1. The dynamicsrepeating itself.This is represented in Figure 3.8 byof the population described by the first panel ina vertical line containing numerous, in fact anFigure 3.7 (r=1.9) differ only in their transientinfinitenumberof,points.Cobwebbingcanalsobebehaviour and so it too would be represented by aused todemonstratea cardinal property of chaos:single point at n =1. Indeed, for the Ricker equa-namely sensitivity to initial conditions.Start twotion a stable equilibrium occurs for all persistenttrajectories very close together and sooner or laterpopulations withr<2,whichgivesthe straight linethey will diverge. This is not due to a lack ofat n =1 in the left-hand part of Figure 3.8.computingpower:nomatterhowclosethetwoThe persistent dynamics depicted by the middleinitial values they will come to diverge. Morepanel of Figure 3.7 (r=2.3) are a two-point limitaccurate estimationof initial values,sothatthe5isuaonenod0.5C1.522.53Figure3.8 The bifurcation diagram for the RickerFecundity parameter rpopulation model (see the legend of Figure 3.7)
not particularly matter which family we chose, and we plump for the Ricker equation as it is commonly used in applied population biology, particularly for fisheries (Ricker, 1954). ntþ1 ¼ nt exp½rð1 ntÞ ð3:8Þ Here nt is population density (scaled to equal 1 at equilibrium). When rare the population increases each generation by a factor exp[r] but as densities approach 1 the increase slows and above 1 it reverses. If r is high there is the potential for the population to overshoot the equilibrium. We want to picture the dynamics of the whole system for different values of the sole adjustable parameter r. To do this, imagine iterating the equation by cobwebbing as in Figures 3.3 and 3.7 and then throwing away all the transient dynamics, perhaps the first 50 generations. For Figure 3.3 (corresponding to a value of r ¼ 1) the non-transient dynamics would not be very interesting: it would simply be a population at stable equilibrium, in this case n ¼ 1. In Figure 3.8 we plot r along the x axis and the non-transient dynamics on the y axis; for r ¼ 1 there is a single point at n ¼ 1. The dynamics of the population described by the first panel in Figure 3.7 (r ¼ 1.9) differ only in their transient behaviour and so it too would be represented by a single point at n ¼ 1. Indeed, for the Ricker equation a stable equilibrium occurs for all persistent populations with r < 2, which gives the straight line at n ¼ 1 in the left-hand part of Figure 3.8. The persistent dynamics depicted by the middle panel of Figure 3.7 (r ¼ 2.3) are a two-point limit cycle: the population oscillates for ever between two densities, one greater and one less than the now unstable equilibrium n ¼ 1. In Figure 3.8 this appears as two points. The value of this representation now becomes clear because instead of having to try to compare a large number of cobweb diagrams we can see at one glance how the cycles appear at r ¼ 2 and then increase in amplitude as r gets bigger. The change of behaviour at r ¼ 2 is called for obvious reasons a bifurcation, and the representation itself is a bifurcation diagram. We can also see that at r ¼ 2.5 a second bifurcation occurs to give a four-point cycle, and then further bifurcations at increasingly smaller intervals of r until a limit is reached. ‘‘What happens at the point of accumulation [the limit]?’’ is what May scrawled on a blackboard in the Theoretical Physics Department at Sydney University in the early 1970s. May showed that what happens is chaos. As the third panel in Figure 3.7 illustrates, the trajectory never converges on a simple cycle but fluctuates aperiodically around very many values of n, never repeating itself. This is represented in Figure 3.8 by a vertical line containing numerous, in fact an infinite number of, points. Cobwebbing can also be used to demonstrate a cardinal property of chaos: namely sensitivity to initial conditions. Start two trajectories very close together and sooner or later they will diverge. This is not due to a lack of computing power: no matter how close the two initial values they will come to diverge. More accurate estimation of initial values, so that the 1 1.5 2 Fecundity parameter r Population density 2.5 3 0 0.5 1 1.5 2 2.5 Figure 3.8 The bifurcation diagram for the Ricker population model (see the legend of Figure 3.7). SINGLE-SPECIES DYNAMICS 27
28THEORETICALECOLOGYmeasured value is close to the “true'value, canwill see a complex pattern of bifurcations, aper-delay the divergence, but not prevent it, and thisiodic and period trajectories; chose part of thismeans that there is an absolute limit to our abilitypicture and enlarge yet again and the same pat-topredict into the futurethebehaviourof chaoticterns appear inminiature,and soon ad infinitum.The beauty of bifurcation diagrams is fragile: addsystems.The dynamics of a system can be described witha little stochastic noise-inescapable in real biolo-a quantity known as the Lyapunov exponentgical systems-—and their more rococo patterns dis-(named aftera Russian mathematician whoseappear.However, the extreme sensitivity to initialname is also transliterated Liapunov or Ljapunov).conditions, the signature of chaos, remains.SoThe Lyapunov exponentdescribes the rate ofwhile it is not mathematically true that the Rickerseparation of infinitesimally close trajectories;amodel predicts chaos for all r>2.69 itmight aspositive value means that the trajectories divergewell be for any biological purposes. Another bio-exponentially,and thisextreme sensitivity to initiallogicallyrelevantpropertyofchaosisalsoshowninconditions is the hallmark of chaos. AlgorithmsFigure 3.8. Although precise prediction is nothave been derived to estimate Lyapunov expo-possible the different population trajectories arenents directlyfrom timeseries (Wolf et al.,1985)bounded,that is they cannot becomearbitrarilyand have proved very valuable, especially in thelargeorsmall.Apurerandomwalkwouldnotbephysical sciences where relatively long time seriesbounded (except of course by n=O). Indeed, it isaretypicallyeasiertoobtain.sometimes possible to calculate the probabilityBifurcation diagrams are beautiful objects thatdistribution of different population states. Depend-containa wealth of mathematical detail.Theyhaveingonthesystemthismaybevaluableinformationquite literally been the subject of tens or possiblyforecologistsand populationmanagers.hundreds of mathematics PhDtheses.Our focusChaos is not just a property of discrete-timehere is on theirrelevanceto biology and wehavesystems and chaos incontinuoustimesystemshasspace to mention only a very few more technicalalso been extensively studied. Consider the non-results.First, May and others showed that thetransient behaviour of a continuous system.If thepatterns in Figure 3.8do not just apply to thesystemis at equilibrium this will be a simplepointRickerequationbuttoaverybroad classofmodelsbutif thereare persistent cycles orchaos then itthat all show the same transition through period-will be a continuous line.For single-speciesdoublingfromordertochaos(May,1973a,1974cpopulations this line can beplotted in a spaceLiand Yorke,1975).Thereisa limited number ofwhere the coordinates are population densitiesroutes to chaos and one can derive general resultsnow and at times in the past. For example, on athree-dimensional graph the coordinates might bethatapplytoverymanysystems.Forexample,theratio of the interval of r values in which two-pointdensities now,1 month ago, and 2 months ago.Incycles are found and in which four-point cycles arethis space a cycle will be a closed loop while afound is 4.6692.In factthesameratio is found forchaotic trajectory will bean object such asthat onevery adjacent interval (four-point/eight-point,the left of Figure 3.9. This object looks like aetc.)not only for the Ricker equation but for euerytwisted diaphanous sheet and is a fractal: succes-map that shows this type of transition from ordersive magnifications of parts of the sheet show thetochaos (Feigenbaum,1978).Second,ifyou looksame self-similarpattern.Onepointtonote is thatclosely at thebifurcation diagram tothe right ofchaos occursin simple(ordinary)differentialthe accumulation point you see that the region ofequations onlyfor systems of three or morevari-ables: the dynamics of a two variable-system canchaoscontains intervalsof simplerdynamics,including period-three cycles that undergo theirbe described in a two-dimensional space whichown transition back into chaos.In factthereisandoes notallowforthetwistingand mixing oftra-infinite number of narrow,periodic windows.jectories that arethe hallmarks of chaosFinally,thebifurcationdiagram hasfractal strucThereisacloselinkbetween chaosand fractalsObjects that represent the non-transient behaviourture: enlarge part of the region of chaos and you
measured value is close to the ‘true’ value, can delay the divergence, but not prevent it, and this means that there is an absolute limit to our ability to predict into the future the behaviour of chaotic systems. The dynamics of a system can be described with a quantity known as the Lyapunov exponent (named after a Russian mathematician whose name is also transliterated Liapunov or Ljapunov). The Lyapunov exponent describes the rate of separation of infinitesimally close trajectories; a positive value means that the trajectories diverge exponentially, and this extreme sensitivity to initial conditions is the hallmark of chaos. Algorithms have been derived to estimate Lyapunov exponents directly from time series (Wolf et al., 1985) and have proved very valuable, especially in the physical sciences where relatively long time series are typically easier to obtain. Bifurcation diagrams are beautiful objects that contain a wealth of mathematical detail. They have quite literally been the subject of tens or possibly hundreds of mathematics PhD theses. Our focus here is on their relevance to biology and we have space to mention only a very few more technical results. First, May and others showed that the patterns in Figure 3.8 do not just apply to the Ricker equation but to a very broad class of models that all show the same transition through perioddoubling from order to chaos (May, 1973a, 1974c; Li and Yorke, 1975). There is a limited number of routes to chaos and one can derive general results that apply to very many systems. For example, the ratio of the interval of r values in which two-point cycles are found and in which four-point cycles are found is 4.6692. In fact the same ratio is found for every adjacent interval (four-point/eight-point, etc.) not only for the Ricker equation but for every map that shows this type of transition from order to chaos (Feigenbaum, 1978). Second, if you look closely at the bifurcation diagram to the right of the accumulation point you see that the region of chaos contains intervals of simpler dynamics, including period-three cycles that undergo their own transition back into chaos. In fact there is an infinite number of narrow, periodic windows. Finally, the bifurcation diagram has fractal structure: enlarge part of the region of chaos and you will see a complex pattern of bifurcations, aperiodic and period trajectories; chose part of this picture and enlarge yet again and the same patterns appear in miniature, and so on ad infinitum. The beauty of bifurcation diagrams is fragile: add a little stochastic noise—inescapable in real biological systems—and their more rococo patterns disappear. However, the extreme sensitivity to initial conditions, the signature of chaos, remains. So while it is not mathematically true that the Ricker model predicts chaos for all r > 2.69 it might as well be for any biological purposes. Another biologically relevant property of chaos is also shown in Figure 3.8. Although precise prediction is not possible the different population trajectories are bounded, that is they cannot become arbitrarily large or small. A pure random walk would not be bounded (except of course by n ¼ 0). Indeed, it is sometimes possible to calculate the probability distribution of different population states. Depending on the system this may be valuable information for ecologists and population managers. Chaos is not just a property of discrete-time systems and chaos in continuous time systems has also been extensively studied. Consider the nontransient behaviour of a continuous system. If the system is at equilibrium this will be a simple point but if there are persistent cycles or chaos then it will be a continuous line. For single-species populations this line can be plotted in a space where the coordinates are population densities now and at times in the past. For example, on a three-dimensional graph the coordinates might be densities now, 1 month ago, and 2 months ago. In this space a cycle will be a closed loop while a chaotic trajectory will be an object such as that on the left of Figure 3.9. This object looks like a twisted diaphanous sheet and is a fractal: successive magnifications of parts of the sheet show the same self-similar pattern. One point to note is that chaos occurs in simple (ordinary) differential equations only for systems of three or more variables: the dynamics of a two variable-system can be described in a two-dimensional space which does not allow for the twisting and mixing of trajectories that are the hallmarks of chaos. There is a close link between chaos and fractals. Objects that represent the non-transient behaviour 28 THEORETICAL ECOLOGY
SINGLE-SPECIES DYNAMICS29ofadynamicsystemarecalledattractors(becausein Figure 3.9. If the position along the section istrajectories originating elsewhere in state space aretreated as a variable, and if the position in theattracted to them).Incontinuoustime,pointscurrent traverse is plotted against that in the pre-(stable eguilibria)and closedloops (cycles)arevious, one arrives at a map exactly equivalent toexamples of normal attractorswhereas fractalthe chaotic Ricker map discussed above. Now,objects such as that in Figure 3.9are termed strangehowever, the r parameter is not simply a measureattractors. All chaotic systems are governed byof single-species fecundity,but a more complexstrange attractors and, as we shall return toamalgam of the life histories of all species orshortly,determining that a system's attractor isdevelopment stages that influence the dynamics.fractal is one way of identifying chaos in nature.How might one seek to decide whether naturalTheattractor inFigure3.9also provides an insightpopulations are chaotic? Typically this has to beintowhychaos isalwaysassociated with extremedone from time-series data, which at least insensitivity to initial conditions. The right-handcomparison with data fromthephysical sciencespanel in Figure 3.9 is a cartoon to illustrate theare inevitably of relatively short duration. Thereevolution of a set of initially veryare two broad approaches.The first is to try to fit asimilartrajecflexiblepopulation model to the time-series datatories:the bundlemarked 1whichshould beimaginedaslyingflatonthehorizontalsurfaceofand then to determine by iterating the modelwhetherthe dynamics are chaotic.The second istothe attractor in front of theline X.Flow on theattractoroccursinthecounter-clockwisedirectiontrydirectlytoreconstructtheattractorgoverningand sets of points are first stretched (2,3)and thenthe system and determine whether it is fractal.folded (4,5).If you imaginethis occurringBoth approaches are helped by a very importantnumerous times it is easy to see how trajectoriestheorem (Takens,1981)that states that the attractorthat start off near each other quickly becomeof a multi-species or complex single-species inter-separated. The degree of stretching in a system isaction can always be reconstructed from single-quantified by the Lyapunov exponent.variable time-series data in a space made up ofChaos in continuous-and discrete-time systemsa sufficient number of time-lagged dimensionsis intimately related. Consider the section X(i.e. the coordinates are densities at time t, t-t,(called a Poincare section)throughtheattractort-2t...wheretisalag).Themajorproviso isFigure3.9 Chaos in continuous time.Theobject on the left is a strange attractor describing theflowof trajectories of a continuous-timesystem in three-dimensional space (the Rossler attractor). X is a Poincare section discussed in the text. The cartoon on the right describeshow bunches of nearby trajectories become stretched and folded as they move around the attractor. See text for further details
of a dynamic system are called attractors (because trajectories originating elsewhere in state space are attracted to them). In continuous time, points (stable equilibria) and closed loops (cycles) are examples of normal attractors whereas fractal objects such as that in Figure 3.9 are termed strange attractors. All chaotic systems are governed by strange attractors and, as we shall return to shortly, determining that a system’s attractor is fractal is one way of identifying chaos in nature. The attractor in Figure 3.9 also provides an insight into why chaos is always associated with extreme sensitivity to initial conditions. The right-hand panel in Figure 3.9 is a cartoon to illustrate the evolution of a set of initially very similar trajectories: the bundle marked 1, which should be imagined as lying flat on the horizontal surface of the attractor in front of the line X. Flow on the attractor occurs in the counter-clockwise direction and sets of points are first stretched (2, 3) and then folded (4, 5). If you imagine this occurring numerous times it is easy to see how trajectories that start off near each other quickly become separated. The degree of stretching in a system is quantified by the Lyapunov exponent. Chaos in continuous- and discrete-time systems is intimately related. Consider the section X (called a Poincare´ section) through the attractor in Figure 3.9. If the position along the section is treated as a variable, and if the position in the current traverse is plotted against that in the previous, one arrives at a map exactly equivalent to the chaotic Ricker map discussed above. Now, however, the r parameter is not simply a measure of single-species fecundity, but a more complex amalgam of the life histories of all species or development stages that influence the dynamics. How might one seek to decide whether natural populations are chaotic? Typically this has to be done from time-series data, which at least in comparison with data from the physical sciences are inevitably of relatively short duration. There are two broad approaches. The first is to try to fit a flexible population model to the time-series data and then to determine by iterating the model whether the dynamics are chaotic. The second is to try directly to reconstruct the attractor governing the system and determine whether it is fractal. Both approaches are helped by a very important theorem (Takens, 1981) that states that the attractor of a multi-species or complex single-species interaction can always be reconstructed from singlevariable time-series data in a space made up of a sufficient number of time-lagged dimensions (i.e. the coordinates are densities at time t, t t, t 2t . where t is a lag). The major proviso is 1 5 4 3 2 X Figure 3.9 Chaos in continuous time. The object on the left is a strange attractor describing the flow of trajectories of a continuous-time system in three-dimensional space (the Ro¨ssler attractor). X is a Poincare´ section discussed in the text. The cartoon on the right describes how bunches of nearby trajectories become stretched and folded as they move around the attractor. See text for further details. SINGLE-SPECIES DYNAMICS 29