20THEORETICAL ECOLOGYmodels is that the proportions of individuals inplants, which we shall index as stages 1, 2, and 3different age classes assume constant values (therespectively. The projection matrix for such astable age distribution) irrespective of startingspecies might look like this:values.Thesevalues can be calculated directlyfrom the projection matrix:associated with the0b2b3(3.2)dominant eigenvalue is a pair of vectors of length xa12a22a32(413a23a33with each element corresponding to an age class.These are the dominant eigenvectors and the rela-The top row reflects stage-specific fecundity.tive magnitudeof the elements of onegives us thestableagedistribution (we note inpassing that theBetweenonetimeperiodandthenextseedlingsdootherdominanteigenvectorprovidesa measureofnot reproduce but smallplantsproduceb2andFisher's reproductive value for each age class).large plants bs offspring that survive to the seed-Whereas the population growth rateis anling stage.Thesubdiagonal a12and a23 tell us theimportantmanagement tool,applied ecologistsareprobability that seedlings become small plants,often also interested in howbirths and deaths atandsmallplantsbecomelargeplants,justasinthedifferent age classes contribute to the overall pro-Lesliematrix.But wenow have further transitions(or lack of transitions): a22 and a33 are the probjection.Aconservation biologist may need toknowabilities that small and largeplants remain thewhether to prioritize efforts on old or youngindividuals, while a game manager might need tosame size (we assume this option is not open toknow the consequences of allowing animals ofseedlings),whilea13allows someplants,perhapsdifferent ages to be shot. The marginal effects onin extremely favourable microhabitats,to transitthe population growth rate of changing thebirth orfrom seedlings to large plants in one go, and a32allows those unfortunate individuals that encoun-death rate at each age can be calculated, and suchter a rabbit actually to decrease in size.relationships,which can be defined in differentAll the results that apply to the Leslie matrixways depending onprecisely for what they aretransfer tothese more complicated structuredrequired, are called sensitivities or elasticities, termspopulations:we can calculate projected populationborrowedfromequivalentproblemsineconomics.The projection matrix can also be used to pro-growth rates, and what we should now call thevide information on the speed at which thestable stage distribution. Analysis of matrix modelsasymptotic growth rate and stable age distribu-of stage-structured populations has proven to beextremelyvaluable in many fields of ecology,buttions are attained.The key quantity here is calledthe damping ratio, which is defined as the ratioperhaps especially so in plant ecology.However,there can sometimes be difficulties in placingof the largesteigenvalue of the projection matrix toindividuals that varyin a continuousvariable suchthe second largest eigenvalue (Caswell, 2001).Thisas size into the discrete classes that are required ofroot may be a complex number, and thisprovidesthe matrix formulation, and also in choosing theinformation about whether there is a smooth oroscillatory approach to thelong-termgrowth rateappropriate time step for analysis (Easterling et al.,(Fox and Gurevitch, 2000).2000). The decision need not be entirely arbitrary,Classifying individuals in populations by theiras there are some theoretical results that suggestage is perhaps the most common wayto relax theoptimalchoices.We mentioned that there were a few exceptionsassumptionthateveryone sharesthesamedemo-to the simple application of the ergodic resultsgraphic parameters.But the matrix formulation ismuch more powerful than this and populationsof matrix theory,although they are easily dealtwithbystraightforward extensions.These includecan be classified by size, life-history stage, sex,geographical location,or essentially any otherpopulationswithpost-reproductiveageclasses orvariable.To illustratethisconsider a plant whosewithsinglereproductiveageclasses.For example,individuals can be classified in one of three lifeconsiderthecicadapopulations inNorthAmericahistory stages: seedlings, small plants, and large(Magicicada sp.) that take precisely 17 years to reach
models is that the proportions of individuals in different age classes assume constant values (the stable age distribution) irrespective of starting values. These values can be calculated directly from the projection matrix: associated with the dominant eigenvalue is a pair of vectors of length x with each element corresponding to an age class. These are the dominant eigenvectors and the relative magnitude of the elements of one gives us the stable age distribution (we note in passing that the other dominant eigenvector provides a measure of Fisher’s reproductive value for each age class). Whereas the population growth rate is an important management tool, applied ecologists are often also interested in how births and deaths at different age classes contribute to the overall projection. A conservation biologist may need to know whether to prioritize efforts on old or young individuals, while a game manager might need to know the consequences of allowing animals of different ages to be shot. The marginal effects on the population growth rate of changing the birth or death rate at each age can be calculated, and such relationships, which can be defined in different ways depending on precisely for what they are required, are called sensitivities or elasticities, terms borrowed from equivalent problems in economics. The projection matrix can also be used to provide information on the speed at which the asymptotic growth rate and stable age distributions are attained. The key quantity here is called the damping ratio, which is defined as the ratio of the largest eigenvalue of the projection matrix to the second largest eigenvalue (Caswell, 2001). This root may be a complex number, and this provides information about whether there is a smooth or oscillatory approach to the long-term growth rate (Fox and Gurevitch, 2000). Classifying individuals in populations by their age is perhaps the most common way to relax the assumption that everyone shares the same demographic parameters. But the matrix formulation is much more powerful than this and populations can be classified by size, life-history stage, sex, geographical location, or essentially any other variable. To illustrate this consider a plant whose individuals can be classified in one of three lifehistory stages: seedlings, small plants, and large plants, which we shall index as stages 1, 2, and 3 respectively. The projection matrix for such a species might look like this: 0 b2 b3 a12 a22 a32 a13 a23 a33 0 @ 1 A ð3:2Þ The top row reflects stage-specific fecundity. Between one time period and the next seedlings do not reproduce but small plants produce b2 and large plants b3 offspring that survive to the seedling stage. The subdiagonal a12 and a23 tell us the probability that seedlings become small plants, and small plants become large plants, just as in the Leslie matrix. But we now have further transitions (or lack of transitions): a22 and a33 are the probabilities that small and large plants remain the same size (we assume this option is not open to seedlings), while a13 allows some plants, perhaps in extremely favourable microhabitats, to transit from seedlings to large plants in one go, and a32 allows those unfortunate individuals that encounter a rabbit actually to decrease in size. All the results that apply to the Leslie matrix transfer to these more complicated structured populations: we can calculate projected population growth rates, and what we should now call the stable stage distribution. Analysis of matrix models of stage-structured populations has proven to be extremely valuable in many fields of ecology, but perhaps especially so in plant ecology. However, there can sometimes be difficulties in placing individuals that vary in a continuous variable such as size into the discrete classes that are required of the matrix formulation, and also in choosing the appropriate time step for analysis (Easterling et al., 2000). The decision need not be entirely arbitrary, as there are some theoretical results that suggest optimal choices. We mentioned that there were a few exceptions to the simple application of the ergodic results of matrix theory, although they are easily dealt with by straightforward extensions. These include populations with post-reproductive age classes or with single reproductive age classes. For example, consider the cicada populations in North America (Magicicada sp.) that take precisely 17 years to reach 20 THEORETICAL ECOLOGY
21SINGLE-SPECIES DYNAMICSmaturity.As long as there is absolutely no mixingwe need to know what size classes in the last yearof cohorts then eachyear classwill increase ormight give rise to y-sized individuals this year.This contribution can occur in two ways: first,decrease in densityas determined by the long-runpopulation growth rate.But the ratio of initialindividuals of size x may give birth to individualsfrequencies in year 1, 2,...,17 remains the same;that are size y at the next census point (call thisthey do not converge on a stable age distribution.b(y, x); second, individuals of size x may avoidIn America adult Magicicada are abundant onlydeath and grow to become size y (call this p(y, x).once in 17years,and thispattern could beThe integral on the right hand side of eqn 3.3explained by initial conditions (for example allsimply sums these contributions to the current yyear classesarewiped out except one)plus lackofclass over all possible size classes last year (indexedcohort mixing.In fact this is highly unlikely,thebyx).Analysesof equationssuchaseqn3.3pro-power of even minor cohort mixing to destroy theduce very similar resultsto thematrixformulationimprint of starting values is so strong that ecolomostbiologically realistic populations increase orgists have universally rejected this hypothesis anddecrease at a growth rate and with an age dis-sought active processesto maintain the synchrotribution that is independent of initial startingnized cohorts.Because the length of the life cycle isvalues (Easterling et al., 2000)a prime number (and 11-and 13-year cicadaFinally,we can study populations structured bypopulations are also found) the classical explana-a continuous variable incontinuoustimeusing thetion is that it is a means of escaping predation,as itfamousMcKendrick-vonForsterequation:is hard for predator populations with life cyclesOn(y,t)On(y,t)_that are not exact divisors of the cicada life cycle to(3.4)-μ(y)n(y,t)atoyincrease in density (Hoppensteadt and Keller,1976).However,a recent study has suggested thatAgain n(y,t) represents the density of individualsalthough predator satiation and/or competitionof sizey at time t.This expression simply statesamongnymphscanexplainwhythedynamicsofthat the numbers in a cohort of individuals bornMagicicada areperiodic, they do not explain whyat the same time decline with age and time astheperiod is a primenumberofyears (Lehmann-mortality (μ(y), which is likely to be age-specific)Ziebarth et al., 2005). The authors speculate that ainexorably whittles them away.To complete thephysiological or genetic mechanism or constraintmodel we need a birth process which is introducedmight be responsible.as a boundary condition:Wehave dwelt at length on the matrix for-mulation partly because it is relatively straight-n(0,t) =b(y)n(y,t) dt(3.5)forward to explain, but we finish this section bybriefly describing two alternative approaches.This states that the numbers of individual of age OSuppose firstthat we are contentto census our(i.e.newborns)are simply the numbers of currentpopulation at discrete time intervals (perhapsindividuals in the population multiplied by theiryearly) but that we are unhappy to shoe-hornage-specific birth rates (b(y).These equations canindividuals into discrete classes of size (or otherbe generalized to populations structured by sizeclassifying variable).Instead we want to work withand by other variables (Wood, 1994).the more natural continuous size distribution.ThisAgain, asyou would expectfrom achange ofleads naturally to an integral projection model offormalism rather than a change in biology,thetheformbehaviourofpopulationsdescribedbythismodelissimilar to those we have discussed above. Mostn(y,t+1) =[b(y, x) + p(y,x)]n(r,t)dx(3.3)reasonable assumptions give ergodic populationgrowth and age distributions.Although theHere n(y,t+1) represents the density of indivi-McKendrick-von Forsterequation hasbeenduals of size y at time t+1. To calculate this valueused extensively in many branches of population
maturity. As long as there is absolutely no mixing of cohorts then each year class will increase or decrease in density as determined by the long-run population growth rate. But the ratio of initial frequencies in year 1, 2, . , 17 remains the same; they do not converge on a stable age distribution. In America adult Magicicada are abundant only once in 17 years, and this pattern could be explained by initial conditions (for example all year classes are wiped out except one) plus lack of cohort mixing. In fact this is highly unlikely, the power of even minor cohort mixing to destroy the imprint of starting values is so strong that ecologists have universally rejected this hypothesis and sought active processes to maintain the synchronized cohorts. Because the length of the life cycle is a prime number (and 11- and 13-year cicada populations are also found) the classical explanation is that it is a means of escaping predation, as it is hard for predator populations with life cycles that are not exact divisors of the cicada life cycle to increase in density (Hoppensteadt and Keller, 1976). However, a recent study has suggested that although predator satiation and/or competition among nymphs can explain why the dynamics of Magicicada are periodic, they do not explain why the period is a prime number of years (LehmannZiebarth et al., 2005). The authors speculate that a physiological or genetic mechanism or constraint might be responsible. We have dwelt at length on the matrix formulation partly because it is relatively straightforward to explain, but we finish this section by briefly describing two alternative approaches. Suppose first that we are content to census our population at discrete time intervals (perhaps yearly) but that we are unhappy to shoe-horn individuals into discrete classes of size (or other classifying variable). Instead we want to work with the more natural continuous size distribution. This leads naturally to an integral projection model of the form nðy;tþ1Þ ¼ Z 1 x¼0 ½ bðy; xÞ þ pðy; xÞ nðx;tÞdx ð3:3Þ Here n(y, t þ 1) represents the density of individuals of size y at time t þ 1. To calculate this value we need to know what size classes in the last year might give rise to y-sized individuals this year. This contribution can occur in two ways: first, individuals of size x may give birth to individuals that are size y at the next census point (call this b(y, x)); second, individuals of size x may avoid death and grow to become size y (call this p(y, x)). The integral on the right hand side of eqn 3.3 simply sums these contributions to the current y class over all possible size classes last year (indexed by x). Analyses of equations such as eqn 3.3 produce very similar results to the matrix formulation; most biologically realistic populations increase or decrease at a growth rate and with an age distribution that is independent of initial starting values (Easterling et al., 2000). Finally, we can study populations structured by a continuous variable in continuous time using the famous McKendrick–von Fo¨rster equation: qnðy;tÞ qt þ qnðy;tÞ qy ¼ mðyÞnðy;tÞ ð3:4Þ Again n(y,t) represents the density of individuals of size y at time t. This expression simply states that the numbers in a cohort of individuals born at the same time decline with age and time as mortality (m(y), which is likely to be age-specific) inexorably whittles them away. To complete the model we need a birth process which is introduced as a boundary condition: nð0;tÞ ¼ Z 1 0 bðyÞnðy;tÞ dt ð3:5Þ This states that the numbers of individual of age 0 (i.e. newborns) are simply the numbers of current individuals in the population multiplied by their age-specific birth rates (b(y)). These equations can be generalized to populations structured by size and by other variables (Wood, 1994). Again, as you would expect from a change of formalism rather than a change in biology, the behaviour of populations described by this model is similar to those we have discussed above. Most reasonable assumptions give ergodic population growth and age distributions. Although the McKendrick–von Fo¨rster equation has been used extensively in many branches of population SINGLE-SPECIES DYNAMICS 21
22THEORETICALECOLOGYbiology, it is notoriously difficult to work with,either a decline in birth rates or an increase inboth analytically and numerically, and it is gen-death rates. An illustration of this is given inFigure 3.2.At low densities populations increaseerally a less popular approach than the othertwo described here.from generation to generation but as densityincreases the rate of increase slows and thenreverses. At one particular density each individual3.2Densitydependencefemale precisely replaces herself and the popula-The value of projections is that they tell us some-tion is at equilibrium.In these simple diagrams thething about current populations; to move fromequilibrium, marked bya star, is easily found asprojections to forecasts we need to know not onlythe density wherethe population growth curvecrosses the 45° line.the current values of demographic parameters,but how they may change in the future. In thisSimple though they are, the diagrams can besection we concentrate on howbirth and mortalityused to look at dynamic trajectories as well asrates may be affected by changes in populationequilibria through a geometrical trick calleddensity.cobrwebbing.Supposethatthe current populationWe shall begin to explore this topic using adensityisrepresented bythepoint0 inFigure3.3.simple, unstructured population model in discreteThe density in the next generation is read off thetime which we sample every generation. We willpopulation growth curve at point 1.To find theplot the population density in the next generationdensity in the following generation the trick is toas a function of that in the current generation.draw a line that forms a right angle with theFigure 3.1 shows the simplest case where popula-(dashed) 45°line and then intercepts the popula-tion growth rate is independent of density.Thetion growth curve (the line 1-2 in Figure 3.3).dashed line at 45°represents the situation whereThis process can be repeated indefinitely and bypopulation density remains the same from gen-noting the population densities given by the serieseration to generation. The solid line illustrates(1,2, 3, 4,...)we obtain the population trajectory.density-independentgrowth,the slope of the lineIn the case of a population whose biology isbeing , the discrete-time rate of populationsummarized byFigures3.2and3.3,thedynamicsgrowth.areadamped approachtoastableequilibrium.Inevitably,as populations grow they will comeIn Figure 3.3 there is a single equilibrium pointto exceed their resource base and this will result inThis is also called a globally stable equilibrium/11/1Population density time tPopulation density timefFigure 3.2 Mild density-dependent population growth. The starFigure 3.1 Density-independent population growth.indicates an equilibrium
biology, it is notoriously difficult to work with, both analytically and numerically, and it is generally a less popular approach than the other two described here. 3.2 Density dependence The value of projections is that they tell us something about current populations; to move from projections to forecasts we need to know not only the current values of demographic parameters, but how they may change in the future. In this section we concentrate on how birth and mortality rates may be affected by changes in population density. We shall begin to explore this topic using a simple, unstructured population model in discrete time which we sample every generation. We will plot the population density in the next generation as a function of that in the current generation. Figure 3.1 shows the simplest case where population growth rate is independent of density. The dashed line at 45 represents the situation where population density remains the same from generation to generation. The solid line illustrates density-independent growth, the slope of the line being l, the discrete-time rate of population growth. Inevitably, as populations grow they will come to exceed their resource base and this will result in either a decline in birth rates or an increase in death rates. An illustration of this is given in Figure 3.2. At low densities populations increase from generation to generation but as density increases the rate of increase slows and then reverses. At one particular density each individual female precisely replaces herself and the population is at equilibrium. In these simple diagrams the equilibrium, marked by a star, is easily found as the density where the population growth curve crosses the 45 line. Simple though they are, the diagrams can be used to look at dynamic trajectories as well as equilibria through a geometrical trick called cobwebbing. Suppose that the current population density is represented by the point 0 in Figure 3.3. The density in the next generation is read off the population growth curve at point 1. To find the density in the following generation the trick is to draw a line that forms a right angle with the (dashed) 45 line and then intercepts the population growth curve (the line 1 !2 in Figure 3.3). This process can be repeated indefinitely and by noting the population densities given by the series (1, 2, 3, 4, .) we obtain the population trajectory. In the case of a population whose biology is summarized by Figures 3.2 and 3.3, the dynamics are a damped approach to a stable equilibrium. In Figure 3.3 there is a single equilibrium point. This is also called a globally stable equilibrium Population density time t+1 Population density time t Figure 3.1 Density-independent population growth. Population density time t Population density time t+1 Figure 3.2 Mild density-dependent population growth. The star indicates an equilibrium. 22 THEORETICAL ECOLOGY
23SINGLE-SPECIES DYNAMICSooeno//1-Population density timetPopulation density timetFigure 3.4 The Allee effect: populations that start off above theFigure 3.3 Cobwebbing with mild density-dependent populationthreshold T move towards the upper equilibrium; those that startgrowth.off or fall below the threshold become extinct (a population size of ois a stable equilibrium).becausewhateverthe initial populationdensitythe/Bpopulation trajectory will inevitably home in onthesameequilibriumpoint(trycobwebbingfromansuaoenodifferent starting points). But depending on thenatural history of the population in question morethan oneequilibrium canoccur.Considerapopulation whose growth curve can be repre-sented byFigure 3.4.Below the threshold T thepopulation in the nextgeneration is smaller than inthe current generation. This might occur if indivi-duals in low-density populations find it difficult tolocate mates (or get pollinated),or perhaps ingroup-hunting or colonial species if cooperationbreaksdown when thereonlya few individualspresent. Any population that drops below T willPopulation densitytimetcontinueto fall in densityuntilitbecomesextinctFigure 3.5 Alternative stable states: there are two non-zeroMoreover,the population will not be able toequilibria (marked by stars); which equilibrium the population movesincrease from low densities.There are thus twoto depends on whether it starts at above or below the threshold T.locally stable equilibria,and which one isattaineddepends on whether the initial population densityis above or below the threshold T.The presence ofpopulation starts above T it moves to the locala minimumpopulationdensitybelowwhich thestable equilibrium B and if below to A.But thepopulation goes extinct is called an Allee effect andlowerequilibriumisnownotat O;whenveryrarefor obviousreasons is somethingthat conservationthe population is still able to increase in numbers.biologists arevery concerned about.Note that the population in Figure3.5actually hasA species'population dynamics may includefour equilibria: at densities of O, A, T, and B. Butmore than one non-zero stable equilibria.Considerwhile Aand B are stable, O and Tare unstable:aFigure 3.5; again we have a threshold (T)whichpopulation whose density is precisely O or T willdetermines the final state of the system. If theremain at that density forever, but the slightest
because whatever the initial population density the population trajectory will inevitably home in on the same equilibrium point (try cobwebbing from different starting points). But depending on the natural history of the population in question more than one equilibrium can occur. Consider a population whose growth curve can be represented by Figure 3.4. Below the threshold T the population in the next generation is smaller than in the current generation. This might occur if individuals in low-density populations find it difficult to locate mates (or get pollinated), or perhaps in group-hunting or colonial species if cooperation breaks down when there only a few individuals present. Any population that drops below T will continue to fall in density until it becomes extinct. Moreover, the population will not be able to increase from low densities. There are thus two locally stable equilibria, and which one is attained depends on whether the initial population density is above or below the threshold T. The presence of a minimum population density below which the population goes extinct is called an Allee effect and for obvious reasons is something that conservation biologists are very concerned about. A species’ population dynamics may include more than one non-zero stable equilibria. Consider Figure 3.5; again we have a threshold (T) which determines the final state of the system. If the population starts above T it moves to the local stable equilibrium B and if below to A. But the lower equilibrium is now not at 0; when very rare the population is still able to increase in numbers. Note that the population in Figure 3.5 actually has four equilibria: at densities of 0, A, T, and B. But while A and B are stable, 0 and T are unstable: a population whose density is precisely 0 or T will remain at that density forever, but the slightest Population density time t 1 2 3 4 0 Population density time t+1 Figure 3.3 Cobwebbing with mild density-dependent population growth. Population density time t T Population density time t+1 Figure 3.4 The Allee effect: populations that start off above the threshold T move towards the upper equilibrium; those that start off or fall below the threshold become extinct (a population size of 0 is a stable equilibrium). Population density time t Population density time t+1 T B A Figure 3.5 Alternative stable states: there are two non-zero equilibria (marked by stars); which equilibrium the population moves to depends on whether it starts at above or below the threshold T. SINGLE-SPECIES DYNAMICS 23
24THEORETICAL ECOLOGYbetween scramble competition where resources aredivided equally among all the individuals in thepopulation, and contest competition where someasaoindividuals get the resources they require, andothersgetnone.(Ofcourse,manypopulationswillshowintermediate patterns.)At the populationlevel, pure contest competition will give growthcurvesthatresembleC inFigure3.6and scramblecompetition curves more like s. With C, as den-sities get high, a fixed number of individuals sur-vive or reproduce to form the next generation;with S,at high densities everyone suffers andpopulationnumbersplummet.Wecan explorethedynamicconsequences ofPopulation density time tmovingfromcontesttoscramblecompetitionFigure 3.6 Possible population-level consequences of contestusing cobwebbing.In Figure 3.3 we had a fairly(C)andscramble($)competitioncontestformof competition and thisresulted in asmoothapproachtoa stable equilibrium.Inthethree panels of Figure 3.7 we progressivelyperturbation will cause the population to moveincrease the scramble component. In thefirst panelwe still have a stable equilibrium but now theawaytoeitherAand B.Arethereexamplesof multiplenon-zeroequili-approachto the equilibrium is not smoothbutbria? There are, but to be strictly accurate mostthrough damped oscillations. In the second panel,involve multispecies interactions that require awith more scramble, we no longer get a stablemorecomplexapproachthanthesimple,equilibrium but instead a stable two-point cycle.Finally, in the third panel we get curious dynam-unstructured single-species models we are dis-cussing here.A classic example is the spruceics:oscillations that never exactly repeat them-budworm moth,Choristoneura fumiferana,which isselves. This is dynamical chaos, which we shallhypothesized normallyto becontrolled atarela-return to again in the following subsection.tively low population density by a suite of spe-We have derived this series of population-dynamic behaviours by discussing the spectrum ofcialist invertebrate predators (Peterman et al.,1979).If, however, the population is perturbedtypesofcompetitionfromcontesttoscramble,butsuch that control breaks down,themoth increaseswhat is important is not the underlying mechan-in densityto a second equilibrium which is set byism but the shape of the population growth curve,bird predation (these predators ignore sprucehowever this comes about. In particular, the angleof the growth curve at the equilibrium where itbudwormunless itbecomes very common).Afurther perturbation isneededtoswitch itbacktointersects the 45° line is informative. If the angle isthe low-density equilibrium. The density marked Tmore than 45°(i.e.the growth curve is below theis thus a threshold or tipping point separating two45linetotheleftof theequilibrium)wehaveanlocally stableequilibria.unstable equilibrium, as at T in Figure 3.5.If theCompetition for food and other resources isangleisbetween45and0°asinFigure3.3wehavewithout doubt the most important process deter-a smooth approach to a stable equilibriummining the dynamics of animals and plants whosebetween0and-45asinFigure3.7awehaveadynamics can be said to be single-species. Thedamped oscillatory approach to a stable equili-curves in Figures 3.2-3.5 are population-levelbrium;andif lessthan -45°thentheequilibriumemergentphenomena based on complex interac-is unstable but persistent cycles or chaos maytions occurring at the level of the individual.occur.In analysing population models a commonFor example,ecologistshave long distinguishedpractice is to solve for the equilibrium values and
perturbation will cause the population to move away to either A and B. Are there examples of multiple non-zero equilibria? There are, but to be strictly accurate most involve multispecies interactions that require a more complex approach than the simple, unstructured single-species models we are discussing here. A classic example is the spruce budworm moth, Choristoneura fumiferana, which is hypothesized normally to be controlled at a relatively low population density by a suite of specialist invertebrate predators (Peterman et al., 1979). If, however, the population is perturbed such that control breaks down, the moth increases in density to a second equilibrium which is set by bird predation (these predators ignore spruce budworm unless it becomes very common). A further perturbation is needed to switch it back to the low-density equilibrium. The density marked T is thus a threshold or tipping point separating two locally stable equilibria. Competition for food and other resources is without doubt the most important process determining the dynamics of animals and plants whose dynamics can be said to be single-species. The curves in Figures 3.2–3.5 are population-level emergent phenomena based on complex interactions occurring at the level of the individual. For example, ecologists have long distinguished between scramble competition where resources are divided equally among all the individuals in the population, and contest competition where some individuals get the resources they require, and others get none. (Of course, many populations will show intermediate patterns.) At the population level, pure contest competition will give growth curves that resemble C in Figure 3.6 and scramble competition curves more like S. With C, as densities get high, a fixed number of individuals survive or reproduce to form the next generation; with S, at high densities everyone suffers and population numbers plummet. We can explore the dynamic consequences of moving from contest to scramble competition using cobwebbing. In Figure 3.3 we had a fairly contest form of competition and this resulted in a smooth approach to a stable equilibrium. In the three panels of Figure 3.7 we progressively increase the scramble component. In the first panel we still have a stable equilibrium but now the approach to the equilibrium is not smooth but through damped oscillations. In the second panel, with more scramble, we no longer get a stable equilibrium but instead a stable two-point cycle. Finally, in the third panel we get curious dynamics: oscillations that never exactly repeat themselves. This is dynamical chaos, which we shall return to again in the following subsection. We have derived this series of populationdynamic behaviours by discussing the spectrum of types of competition from contest to scramble, but what is important is not the underlying mechanism but the shape of the population growth curve, however this comes about. In particular, the angle of the growth curve at the equilibrium where it intersects the 45 line is informative. If the angle is more than 45 (i.e. the growth curve is below the 45 line to the left of the equilibrium) we have an unstable equilibrium, as at T in Figure 3.5. If the angle is between 45 and 0 as in Figure 3.3 we have a smooth approach to a stable equilibrium; between 0 and 45 as in Figure 3.7a we have a damped oscillatory approach to a stable equilibrium; and if less than 45 then the equilibrium is unstable but persistent cycles or chaos may occur. In analysing population models a common practice is to solve for the equilibrium values and C S Population density time t Population density time t+1 Figure 3.6 Possible population-level consequences of contest (C ) and scramble (S) competition. 24 THEORETICAL ECOLOGY