30THEORETICAL ECOLOGYthat you have to have sufficient data, which inexponent.This method has since found widepractice is usually a very demanding requirement.application beyond biology in econometricsThe first attempt to fit models to data did notModel-basedapproacheshavealso enjoyedusetime series but life-historydata on fecundityrenewed attention.One strand has sought toand density-dependent mortality. Hassell et al.develop more accurate mechanistic population(1976) fitted a two-parameter model to data frommodels,capitalizing on both the more powerful24 species of insects with reasonably discretecomputing tools now available and statisticalgenerations and concluded that the vast majorityadvances in extractingparameter values fromhad stable dynamics, indeed not even showing andata. A different strand, with similarities to Sugi-oscillatory return to equilibrium. Although thehara and May's approach, fits very flexible non-authors were at pains to stress the provisionalmechanisticpopulationmodelstotime-seriesdatanature of their conclusions, this paper had a verytypically using response surfaces that are opti-mized either by traditional least-squares methodsmajor impact, and to a certain extent inadvertentlylicensed ecologists to treat chaos as a theoreticalor more exotic techniques such as thin-platecuriosity for the next decade.splines or neural nets (Ellner and Turchin,1995).Thenextmajorattempttosearchfor chaos usedThemagnitudeofthedominantLyapunovexpomodel-free approaches and was spurred by thenent is calculated directlyfrom thefitted model.Itis still too earlyto judge the long-term value ofgrowth of empirical chaos studies in the physicalsciences(Schaffer,1985;Schafferand Kot,1985a,these methods, although they have revealed a1985b;Olsenand Schaffer,1990).Thebasicideanumber of systems with apparent chaotic dynam-wastoreconstructtheattractorbyembeddingtheics, in particularly involving human-disease andtime series in time-lagged coordinates and thenpredator-prey interactions.For single-species interactions, the best exampleseither to take a Poincare section and look for a one-of possible chaos involve laboratory systems,dimensional chaotic map, or to estimate theattractor dimension.In our daily lives we do notincluding Nicholson's famous long-term blow flynormally need tests totell us whether an object isexperiment.A very niceexperimental exampleisone-, two-, or three-dimensional but mathemati-thework of Costantinoetal.(1997)ontheflourcians who often work in much higher dimensionalbeetle, Tribolium castaneum.Recall we mentionedspace have derived algorithms to estimate arbi-above that strong interactions between differenttrary dimensionality.When these are applied tolife-history stages can give rise to complexfractal objects theyreturn a non-integer dimension.dynamics.In Tribolium, adults and larvae canni-A non-integer dimension implies a fractal and abalize eggs while adults also eat pupae. A popu-fractal implies chaos.Though clearly worth trying,lation model showed that by varying a singleultimately this research programme was defeatedparameter (pupal mortality)the dynamics of theby the quality of the data available.To quotesystemmoved from stabilityto chaos and then toaSchaffer (2000),'Only in theinstance of recurrentthree-point cycle. Figure 3.10 shows that experi-outbreaks of measles in human populations, wasmentally manipulating pupal mortality leads tothere sufficient data to justify our initial enthu-dynamics that look very like those predicted. It issiasm' and, he added, even here the argumenttrue that this is a highly artificial system, yet it isan impressivedemonstration that the dynamics ofchiefly rested on the comparison of time-seriesdata with the output of epidemiological models.these insects havebeen understood.In the last 15 years, interest has grown again inthe challenge of detecting chaos from time series.3.3RandomnessSugihara and May (1990)developed a techniquecalled nonlinearforecastingwhichmeasures the3.3.1Typesof random effectextent towhichpredictability decays with time.InReal animals,plants,and micro-organisms arechaotic systems this occurs in a characteristic waydetermined by the magnitude of the Lyapunovcontinuallybuffeted bytheeffects of random
that you have to have sufficient data, which in practice is usually a very demanding requirement. The first attempt to fit models to data did not use time series but life-history data on fecundity and density-dependent mortality. Hassell et al. (1976) fitted a two-parameter model to data from 24 species of insects with reasonably discrete generations and concluded that the vast majority had stable dynamics, indeed not even showing an oscillatory return to equilibrium. Although the authors were at pains to stress the provisional nature of their conclusions, this paper had a very major impact, and to a certain extent inadvertently licensed ecologists to treat chaos as a theoretical curiosity for the next decade. The next major attempt to search for chaos used model-free approaches and was spurred by the growth of empirical chaos studies in the physical sciences (Schaffer, 1985; Schaffer and Kot, 1985a, 1985b; Olsen and Schaffer, 1990). The basic idea was to reconstruct the attractor by embedding the time series in time-lagged coordinates and then either to take a Poincare´ section and look for a onedimensional chaotic map, or to estimate the attractor dimension. In our daily lives we do not normally need tests to tell us whether an object is one-, two-, or three-dimensional but mathematicians who often work in much higher dimensional space have derived algorithms to estimate arbitrary dimensionality. When these are applied to fractal objects they return a non-integer dimension. A non-integer dimension implies a fractal and a fractal implies chaos. Though clearly worth trying, ultimately this research programme was defeated by the quality of the data available. To quote Schaffer (2000), ‘Only in the instance of recurrent outbreaks of measles in human populations, was there sufficient data to justify our initial enthusiasm’ and, he added, even here the argument chiefly rested on the comparison of time-series data with the output of epidemiological models. In the last 15 years, interest has grown again in the challenge of detecting chaos from time series. Sugihara and May (1990) developed a technique called nonlinear forecasting which measures the extent to which predictability decays with time. In chaotic systems this occurs in a characteristic way determined by the magnitude of the Lyapunov exponent. This method has since found wide application beyond biology in econometrics. Model-based approaches have also enjoyed renewed attention. One strand has sought to develop more accurate mechanistic population models, capitalizing on both the more powerful computing tools now available and statistical advances in extracting parameter values from data. A different strand, with similarities to Sugihara and May’s approach, fits very flexible nonmechanistic population models to time-series data typically using response surfaces that are optimized either by traditional least-squares methods or more exotic techniques such as thin-plate splines or neural nets (Ellner and Turchin, 1995). The magnitude of the dominant Lyapunov exponent is calculated directly from the fitted model. It is still too early to judge the long-term value of these methods, although they have revealed a number of systems with apparent chaotic dynamics, in particularly involving human–disease and predator–prey interactions. For single-species interactions, the best examples of possible chaos involve laboratory systems, including Nicholson’s famous long-term blow fly experiment. A very nice experimental example is the work of Costantino et al. (1997) on the flour beetle, Tribolium castaneum. Recall we mentioned above that strong interactions between different life-history stages can give rise to complex dynamics. In Tribolium, adults and larvae cannibalize eggs while adults also eat pupae. A population model showed that by varying a single parameter (pupal mortality) the dynamics of the system moved from stability to chaos and then to a three-point cycle. Figure 3.10 shows that experimentally manipulating pupal mortality leads to dynamics that look very like those predicted. It is true that this is a highly artificial system, yet it is an impressive demonstration that the dynamics of these insects have been understood. 3.3 Randomness 3.3.1 Types of random effect Real animals, plants, and micro-organisms are continually buffeted by the effects of random 30 THEORETICAL ECOLOGY
SINGLE-SPECIES DYNAMICS31300(a)150ao300(b)150W0300Fiqure3.10 Time series of the number of larval(c)beetles in laboratory populations for different rates ofpupal mortality which were artificially manipulated.150Theoretical models predictthatthepopulationinpanel a should have a stable equilibrium, panel bshould be chaotic, and panel c should have a three-0point cycle. The experimental data show good020608040agreement with the predictions (after Costantinoet al., 1997).Timeinweeksprocesses and a critical question in populationLet us return to the discrete-time model of abiology is the extent to which insights gained frompopulation with non-overlapping generations,the analysis of deterministic models survive theN,+1=N, 2, and for the sake of argument assumeinsultsthrown at them by stochastic nature.that the value of is actually constant over timeThere are a variety of different ways in whichBut this does not mean that every single individualrandom or stochastic effects can influence popu-in the population will produce exactly femalelation dynamics (May,1973a).Perhapsthemostoffspring. In the real world there will always bestraightforward isenvironmental stochasticity,some between-individual variation or demographicwhere the value of a demographic parameterstochasticity.For example,consideraparasite thatchanges over time. Recall the density-independent,searches randomly for hosts into which it lays adiscrete-time model N+1=N,入 where 入 is thesingle egg; if the average parasite lays femaleannual population growth rate.This model impli-eggs then some will by chancediscover morehostscitly assumes that thevalue of is constant, but inand some by chance fewer. This is a Poisson pro-fact it will almost certainly vary from generation tocess where the variance is the same as the mean.generation; we might better write the equationOne can imagine other natural histories whereN+1=N,,to emphasize thisfact.Notethatthevariance is muchlessthan aPoisson processenvironmental stochasticityaffectsthedemo-(vertebrates that normally produce one offspringgraphic rates of all individuals in a population ina year)and others where thevariance is muchthe same way, and that this effect is independentgreater (organisms living in a highly hetero-of population size (Lande et al.,2003).Muchgeneous environment).Nowsupposethepopula-research in identifying factors generating envir-tion is small: by chance all individuals in oneonmental stochasticity has focused on climategenerationmayexperiencelowreproductionand(Stenseth et al.,2002),although inprinciple anyso thefollowingyearthepopulation sizewouldother factorwith unpredictable effects on popula-besignificantlyless thantheexpected N,.Ofcourse,the probability of simultaneous episodestionparameterscan contributetothisprocess
processes and a critical question in population biology is the extent to which insights gained from the analysis of deterministic models survive the insults thrown at them by stochastic nature. There are a variety of different ways in which random or stochastic effects can influence population dynamics (May, 1973a). Perhaps the most straightforward is environmental stochasticity, where the value of a demographic parameter changes over time. Recall the density-independent, discrete-time model Ntþ1 ¼ Ntl where l is the annual population growth rate. This model implicitly assumes that the value of l is constant, but in fact it will almost certainly vary from generation to generation; we might better write the equation Ntþ 1¼ Nt lt to emphasize this fact. Note that environmental stochasticity affects the demographic rates of all individuals in a population in the same way, and that this effect is independent of population size (Lande et al., 2003). Much research in identifying factors generating environmental stochasticity has focused on climate (Stenseth et al., 2002), although in principle any other factor with unpredictable effects on population parameters can contribute to this process. Let us return to the discrete-time model of a population with non-overlapping generations, Nt þ 1 ¼ Nt l, and for the sake of argument assume that the value of l is actually constant over time. But this does not mean that every single individual in the population will produce exactly l female offspring. In the real world there will always be some between-individual variation or demographic stochasticity. For example, consider a parasite that searches randomly for hosts into which it lays a single egg; if the average parasite lays l female eggs then some will by chance discover more hosts and some by chance fewer. This is a Poisson process where the variance is the same as the mean. One can imagine other natural histories where the variance is much less than a Poisson process (vertebrates that normally produce one offspring a year) and others where the variance is much greater (organisms living in a highly heterogeneous environment). Now suppose the population is small: by chance all individuals in one generation may experience low reproduction and so the following year the population size would be significantly less than the expected Nt l. Of course, the probability of simultaneous episodes 20 40 60 Number of insects (a) (b) (c) 0 150 300 0 150 300 0 150 300 0 80 Time in weeks Figure 3.10 Time series of the number of larval beetles in laboratory populations for different rates of pupal mortality which were artificially manipulated. Theoretical models predict that the population in panel a should have a stable equilibrium, panel b should be chaotic, and panel c should have a threepoint cycle. The experimental data show good agreement with the predictions (after Costantino et al., 1997). SINGLE-SPECIES DYNAMICS 31
32THEORETICALECOLOGYof good or bad luck become progressively morepopulation size is dominated by very rare, hugeunlikely in larger populations and hence demo-population sizes in the upper tail of the distribu-graphic stochasticity is most important in smalltion.In fact the modal population size, the popu-populations.In many ways, its action is similar tolation sizethat will actuallybe observed in thefield, grows not at a rate determined by the simpledrift in population genetics.A further random process that is sometimesarithmetic mean,,butthegeometricmean(-1-2----1)/distinguished is catastrophic stochasticity: randomevents that destroy the whole population irre-Several biologically interesting results followspective of its size or current demographicfrom this.First, as long as there is some variance inparameters. We shall not discuss this type of ran-thegeometric mean will always be lowerthandomness further here,although it is particularlythe arithmetic mean: poor years have a greaterrelevanttostudiesofmetapopulations(seenegative effect on population growth than theChapter4inthisvolume)andalsoinconservationpositive effect of good years.Second, a singleyearbiology where populations may be wiped out bywith zero net reproduction (α=0)renders thehuman action that can at least beapproximated aslong-term growth rate0.This makes intuitivearandomprocesssense as the population goes extinct,but note thatthis is not what a calculation based on the arith-metic mean would suggest.Finally,recall that in3.3.2Density-independent populationsthedeterministiccasepersistencewas verystraightforward:a population would increase ifLet us now see how stochasticity affects popula-入>1 and decrease if ^<1. The situation is nowtion growth rate and population projection.Formore complicated:populations with geometricease of explanation we shall stick to discrete-timemean growth rates less than one will always ulti-models although the same principles apply tomately go extinct,but some may persist for a longpopulations that reproduce in continuous timeperiod of time if byluck they experience a chain ofReturn once again to the model N,+1=N, 2, wherepropitious years.Similarly,althoughpopulationsthesubscript to thepopulation growthratewith geometric growth rates greater than 1 willemphasizes that it varies between generations,tend to persist, some will bybad luck go extinct. Inspecifically with mean and variance ;.This isfact populations which will, on average, grow tothe way that randomness is most frequently dealtinfinity also have a probability of extinction of1forwithinpopulationmodels,andhasbeenreferredverylongperiodsoftime.This canbe seenveryto as the equilibriumtreatment of noise (Coulson etsimply:ifE(N,)-T2,wheretrepresentstimeandTal.,2004).Ifwe takelogarithms then we can writeis the length of time since the simulation began,1the probability of extinction can be written as(3.9)Log[N] = log[No] + log[2x]-1-1/T.When T gets very large, the expectedx=0population size tends to infinity and the prob-If the values of vary independently over time,ability of extinction tends to unity as 1/T approa-then the right-hand term is the sum of indepen-ches 0. It is possible to calculate the distribution ofdent random variables, which the Central Limitpersistence times of populations governed by dif-Theorem tells us is asymptotically normally dis-ferent distributions of growth rates, and this maytributed. This implies that population size itself isbehelpful inpopulationmanagement.lognormally distributed. There are some com-In many ways the population effects of demo-plexitiesincalculatinglong-termpopulationgraphic stochasticity are similar to its environ-growth rates in this case (Lewontin and Cohen,mental counterpart. It will increase thevariance in1969).An intuitive procedure might be to seeand sotend toreducelong-termgrowthrates,how expected population size grows with time.Aand increasetheprobabilityofextinctionbybadsimple calculation reveals it increases exponen-luck.Themajordifferenceisthatitseffectsbecometially at a rate determined by 2. But the expectedvery weak as population size increases. Indeed, the
of good or bad luck become progressively more unlikely in larger populations and hence demographic stochasticity is most important in small populations. In many ways, its action is similar to drift in population genetics. A further random process that is sometimes distinguished is catastrophic stochasticity: random events that destroy the whole population irrespective of its size or current demographic parameters. We shall not discuss this type of randomness further here, although it is particularly relevant to studies of metapopulations (see Chapter 4 in this volume) and also in conservation biology where populations may be wiped out by human action that can at least be approximated as a random process. 3.3.2 Density-independent populations Let us now see how stochasticity affects population growth rate and population projection. For ease of explanation we shall stick to discrete-time models although the same principles apply to populations that reproduce in continuous time. Return once again to the model Nt þ 1 ¼ Nt lt where the subscript to the population growth rate emphasizes that it varies between generations, specifically with mean l and variance sl. This is the way that randomness is most frequently dealt with in population models, and has been referred to as the equilibrium treatment of noise (Coulson et al., 2004). If we take logarithms then we can write Log½Nt ¼ log½N0 þXt1 x¼0 log½lx: ð3:9Þ If the values of l vary independently over time, then the right-hand term is the sum of independent random variables, which the Central Limit Theorem tells us is asymptotically normally distributed. This implies that population size itself is lognormally distributed. There are some complexities in calculating long-term population growth rates in this case (Lewontin and Cohen, 1969). An intuitive procedure might be to see how expected population size grows with time. A simple calculation reveals it increases exponentially at a rate determined by l. But the expected population size is dominated by very rare, huge population sizes in the upper tail of the distribution. In fact the modal population size, the population size that will actually be observed in the field, grows not at a rate determined by the simple arithmetic mean, l, but the geometric mean ð Þ l0 l1 l2lt1 1=t . Several biologically interesting results follow from this. First, as long as there is some variance in l the geometric mean will always be lower than the arithmetic mean: poor years have a greater negative effect on population growth than the positive effect of good years. Second, a single year with zero net reproduction (l ¼ 0) renders the long-term growth rate 0. This makes intuitive sense as the population goes extinct, but note that this is not what a calculation based on the arithmetic mean would suggest. Finally, recall that in the deterministic case persistence was very straightforward: a population would increase if l > 1 and decrease if l < 1. The situation is now more complicated: populations with geometric mean growth rates less than one will always ultimately go extinct, but some may persist for a long period of time if by luck they experience a chain of propitious years. Similarly, although populations with geometric growth rates greater than 1 will tend to persist, some will by bad luck go extinct. In fact populations which will, on average, grow to infinity also have a probability of extinction of 1 for very long periods of time. This can be seen very simply: if E(Nt) ¼ T2 , where t represents time and T is the length of time since the simulation began, the probability of extinction can be written as 1 1/T. When T gets very large, the expected population size tends to infinity and the probability of extinction tends to unity as 1/T approaches 0. It is possible to calculate the distribution of persistence times of populations governed by different distributions of growth rates, and this may be helpful in population management. In many ways the population effects of demographic stochasticity are similar to its environmental counterpart. It will increase the variance in l and so tend to reduce long-term growth rates, and increase the probability of extinction by bad luck. The major difference is that its effects become very weak as population size increases. Indeed, the 32 THEORETICAL ECOLOGY
SINGLE-SPECIES DYNAMICS33total variance in reproductiverates can bethoughtdynamics via two routes.First,stochasticity has aofasthesumoftwocomponents,Ve(environ-directeffect on the size and structure of the currentmental stochasticity)and Vp/N (demographicpopulation. Second, these changes influence thestochasticity divided by population size). A rea-future trajectory of the population.This interactionSonableruleof thumb is that demographic sto-between stochasticity and the deterministic skele-chasticity can be ignored for populations withton is sometimes referred toas theactive treatmentmore than 50 or so female breeders, though noteof noise,and is currently an area of considerablethat the population sizeof large carnivores,even ininterest in population biology research. Suchextensivenaturereserves,canoftenbebelowthiseffectsalwaysreducethetendencyof thepopula-threshold.tion to reach a stable age distribution and, inWe stated above that we were assuming thatanticipation of the next section,can also havestochastic effects were uncorrelated over time.important consequences onpopulation regulationOften this will not be the case, especially for short-if the strength and action of density dependence islived organisms that might, for example, havealso influenced by population structure.several generations in a single summer. Quitefrequently there will be a positive correlation3.3.3Density-dependent populationsbetween the random component of populationIn a real stochastic environment a population isgrowthrates in successive seasons(theterm rednoise is sometimes used for these positively corre-highly unlikely to remain at theexact same equi-lated random effects).The most important effect oflibrium valuefrom onegeneration tothenext.Butcorrelated stochasticity is to increase the severity ofitisstill reasonabletotalkaboutan equilibriumifpoor breeding seasons that now tend to follow onepopulations above a certain value tend to declineanother.We note in passing that correlated redinnumbers,andthosebelowthesamevaluetendto increase. Conceptually we can think of annoisemayleadtopatternsinpopulationdynamicsthat maybe very hard to distinguish from anequilibrium not as afixed population density,butunderlying deterministic cause, especially inas a probability distribution that remains the samestructured populations. There can also be correla-overtimeandwhichdeterminesthelikelihoodoftions between environmental and demographicobserving the population at anyparticular level ofstochasticity, in particular the effect of demo-abundance (Turchin,2003).Of course, we shouldgraphic stochasticity on population growthmay bealso consider the possibility that a population,higher in years when the consequences of envir-even onethattendsto increasewhen rare,goesonmental stochasticity are most severe, a clearextinct through a run of bad breeding seasons.concern in conservation biology.More generally,stochastic effects can causeaThe arguments above apply also to structuredpopulation to shift from one type of dynamicpopulations,thoughwith some complications.behaviour to another. Figure 3.5 depicted thedynamics of a species with two locally stable equi-First, there is no longer a simple relationshipbetween arithmetic and geometric populationlibria; it is possible that a sufficiently large randomperturbation can movethepopulation from thegrowth rates, but a stochastic equivalent to thedeterministic growth rate can be calculateddomain of attraction of oneequilibrium to that of(Tuljapurkar,1982).As with the unstructured pop-the other. Similarly,where there is an Allee effect aspecies is unable to increase in density when rare soulation,adding stochastic effects always reduceslong-term growth rates. Second, certain age orzeropopulation density islocally stable;randomeffects can push a species density below the criticalstage classesmaybe muchmoresusceptible tostochasticperturbation thanothers..Randomthreshold that leads to extinction. It is also possibleeffectsmaythusleadtoperturbationsthatdisruptthata speciesthatfor somereasonhasfallenbelowtheage-structure of thepopulation (structuralthethreshold can berescued byarandom set ofvariance; Coulson et al., 2001; Lande ef al., 2002)good breeding seasons. Of course, even when aHere, stochasticity influences the populationspecies can increase when rare, stochasticextinction
total variance in reproductive rates can be thought of as the sum of two components, VE (environmental stochasticity) and VD/N (demographic stochasticity divided by population size). A reasonable rule of thumb is that demographic stochasticity can be ignored for populations with more than 50 or so female breeders, though note that the population size of large carnivores, even in extensive nature reserves, can often be below this threshold. We stated above that we were assuming that stochastic effects were uncorrelated over time. Often this will not be the case, especially for shortlived organisms that might, for example, have several generations in a single summer. Quite frequently there will be a positive correlation between the random component of population growth rates in successive seasons (the term red noise is sometimes used for these positively correlated random effects). The most important effect of correlated stochasticity is to increase the severity of poor breeding seasons that now tend to follow one another. We note in passing that correlated red noise may lead to patterns in population dynamics that may be very hard to distinguish from an underlying deterministic cause, especially in structured populations. There can also be correlations between environmental and demographic stochasticity, in particular the effect of demographic stochasticity on population growth may be higher in years when the consequences of environmental stochasticity are most severe, a clear concern in conservation biology. The arguments above apply also to structured populations, though with some complications. First, there is no longer a simple relationship between arithmetic and geometric population growth rates, but a stochastic equivalent to the deterministic growth rate can be calculated (Tuljapurkar, 1982). As with the unstructured population, adding stochastic effects always reduces long-term growth rates. Second, certain age or stage classes may be much more susceptible to stochastic perturbation than others. Random effects may thus lead to perturbations that disrupt the age-structure of the population (structural variance; Coulson et al., 2001; Lande et al., 2002). Here, stochasticity influences the population dynamics via two routes. First, stochasticity has a direct effect on the size and structure of the current population. Second, these changes influence the future trajectory of the population. This interaction between stochasticity and the deterministic skeleton is sometimes referred to as the active treatment of noise, and is currently an area of considerable interest in population biology research. Such effects always reduce the tendency of the population to reach a stable age distribution and, in anticipation of the next section, can also have important consequences on population regulation if the strength and action of density dependence is also influenced by population structure. 3.3.3 Density-dependent populations In a real stochastic environment a population is highly unlikely to remain at the exact same equilibrium value from one generation to the next. But it is still reasonable to talk about an equilibrium if populations above a certain value tend to decline in numbers, and those below the same value tend to increase. Conceptually we can think of an equilibrium not as a fixed population density, but as a probability distribution that remains the same over time and which determines the likelihood of observing the population at any particular level of abundance (Turchin, 2003). Of course, we should also consider the possibility that a population, even one that tends to increase when rare, goes extinct through a run of bad breeding seasons. More generally, stochastic effects can cause a population to shift from one type of dynamic behaviour to another. Figure 3.5 depicted the dynamics of a species with two locally stable equilibria; it is possible that a sufficiently large random perturbation can move the population from the domain of attraction of one equilibrium to that of the other. Similarly, where there is an Allee effect a species is unable to increase in density when rare so zero population density is locally stable; random effects can push a species density below the critical threshold that leads to extinction. It is also possible that a species that for some reason has fallen below the threshold can be rescued by a random set of good breeding seasons. Of course, even when a species can increase when rare, stochastic extinction SINGLE-SPECIES DYNAMICS 33
34THEORETICAL ECOLOGYis permanent if there are no sources of migrants tofor a significant period of time. Indeed, this beha-rescue the population.This treatment of stochasti-viour may go on for ever if stochastic perturba-city in population models has been called thepas-tions are large enough to prevent the system eversive treatment of noise.from settling on the stable cycles.The time seriesThe shape of the equilibrium probability dis-produced by such a process can be indistinguish-tribution of abundances will obviously be deter-able from chaos:it can show exactlythe samemined by the magnitude and direction of theextreme sensitivity to initialconditions,andstochastic perturbations to the demographic para-attemptstoreconstructtheattractorwould suggestmeters,butalsobythedynamicconsequencesofthat it had a non-integer number of dimensions.the perturbations; that is, the interaction of theIn discussing the bifurcation diagram in Figure3.8 we already noted how random effects wouldnoise with the deterministic dynamics.Considerunstructured populations with deterministicallyinteractwith thedeterministic component of thestable equilibria which are approached eitherdynamics to give chaotic population behavioursmoothly(Figure 3.3)or by damped oscillationsthroughout the region beyond the'point of accu-(Figure 3.7a). It is very likely that the first popu-mulation',even though here there are narrowlationwilltend to returntowardstheequilibriumwindows of cyclic behaviour.As withchaoticfaster than the population with damped oscilla-repellers this is another exampleof theimpossi-tions, and for the sameamount of environmentalbilityofseparatingthedeterministicand stochasticstochasticity will have a lower variance equili-aspects of population dynamics in general andbrium population density:Apopulation with anchaos in particular.Although it may seem unarguable that weoscillatory approach to a stable equilibrium canmore easily be prevented from reaching thatshould seek to developmodels with both sto-chastic and deterministic components, exactly howequilibriumand thus appear totheobserver tobepersistently cyclic. This type of dynamic behaviourto do this is not always obvious.For example,has been termed quasicyclic (Nisbet and Gurney)adding onetype of noise to a model witha1976) and has been seen in several experimentaldeterministicallystableequilibrium anda differentsystems, including the flour beetle study describedtype of noise to a model governed by a chaoticattractor can produce dynamics that equallywellaboveas an exampleof chaos (Costantino et al.,1997).match thetype of data that ecological field studiesConsider an unstructured dynamic system thatproduce. Also it is often not clear how stochasticityshould be introduced into the model, onto whichis at the edge of chaos,perhaps showing persistentdemographicparameters,and with whatcorrela-cycles.If one or more parameters were changedslightly,it would move from persistent cycles intotion structure. Nevertheless, we are optimisticthe region of chaos where its dynamics would beabout thefuture.For theanalysis of time seriesandgoverned by a strange attractor. Near this thresh-other observational data there are a variety of newold, the transient behaviour of the populationstatistical methods and techniques that will helpbefore it settles into persistent cycles can be veryidentify the major stochastic drivers, and revealcomplex. Although in this region there is not ahow they interact with the underlying biology ofstrange attractor, dynamics may be influenced bythe species (Coulson et al., 2001; Lande et al., 2003;an object called a strange repeller (Rand and Wilson,Turchin,2003; Stenseth et al.,2004).There is also an1995),which like a strange attractor is a fractal, butincreasingwillingness of ecologiststo experiment,repels rather attracts dynamic trajectories.One canboth in the laboratory and the field, and to inte-think of the system like the ball in a pinballgrate modelling with experimental design andmachine,careering from buffer to buffer,perhapsanalysis
is permanent if there are no sources of migrants to rescue the population. This treatment of stochasticity in population models has been called the passive treatment of noise. The shape of the equilibrium probability distribution of abundances will obviously be determined by the magnitude and direction of the stochastic perturbations to the demographic parameters, but also by the dynamic consequences of the perturbations; that is, the interaction of the noise with the deterministic dynamics. Consider unstructured populations with deterministically stable equilibria which are approached either smoothly (Figure 3.3) or by damped oscillations (Figure 3.7a). It is very likely that the first population will tend to return towards the equilibrium faster than the population with damped oscillations, and for the same amount of environmental stochasticity will have a lower variance equilibrium population density. A population with an oscillatory approach to a stable equilibrium can more easily be prevented from reaching that equilibrium and thus appear to the observer to be persistently cyclic. This type of dynamic behaviour has been termed quasicyclic (Nisbet and Gurney, 1976) and has been seen in several experimental systems, including the flour beetle study described above as an example of chaos (Costantino et al., 1997). Consider an unstructured dynamic system that is at the edge of chaos, perhaps showing persistent cycles. If one or more parameters were changed slightly, it would move from persistent cycles into the region of chaos where its dynamics would be governed by a strange attractor. Near this threshold, the transient behaviour of the population before it settles into persistent cycles can be very complex. Although in this region there is not a strange attractor, dynamics may be influenced by an object called a strange repeller (Rand and Wilson, 1995), which like a strange attractor is a fractal, but repels rather attracts dynamic trajectories. One can think of the system like the ball in a pinball machine, careering from buffer to buffer, perhaps for a significant period of time. Indeed, this behaviour may go on for ever if stochastic perturbations are large enough to prevent the system ever from settling on the stable cycles. The time series produced by such a process can be indistinguishable from chaos: it can show exactly the same extreme sensitivity to initial conditions, and attempts to reconstruct the attractor would suggest that it had a non-integer number of dimensions. In discussing the bifurcation diagram in Figure 3.8 we already noted how random effects would interact with the deterministic component of the dynamics to give chaotic population behaviour throughout the region beyond the ‘point of accumulation’, even though here there are narrow windows of cyclic behaviour. As with chaotic repellers this is another example of the impossibility of separating the deterministic and stochastic aspects of population dynamics in general and chaos in particular. Although it may seem unarguable that we should seek to develop models with both stochastic and deterministic components, exactly how to do this is not always obvious. For example, adding one type of noise to a model with a deterministically stable equilibrium and a different type of noise to a model governed by a chaotic attractor can produce dynamics that equally well match the type of data that ecological field studies produce. Also it is often not clear how stochasticity should be introduced into the model, onto which demographic parameters, and with what correlation structure. Nevertheless, we are optimistic about the future. For the analysis of time series and other observational data there are a variety of new statistical methods and techniques that will help identify the major stochastic drivers, and reveal how they interact with the underlying biology of the species (Coulson et al., 2001; Lande et al., 2003; Turchin, 2003; Stenseth et al., 2004). There is also an increasing willingness of ecologists to experiment, both in the laboratory and the field, and to integrate modelling with experimental design and analysis. 34 THEORETICAL ECOLOGY