《生态学》课程教学资源（书籍教材）Theoretical Ecology Principles and Applications，Third Edition

15HOWPOPULATIONSCOHEREand Gintis, 2004; Traulsen et al.,2005).We onlyhaveto make sure that its basic requirementsare fulfilled in a particular situation (Levin andDKilmer,1974;MaynardSmith,1976).ExactlywhatDthese requirements are can be illustrated with aDsimple model (Traulsen and Nowak,2006).Imagine a population of individuals subdividedDinto groups.For simplicity,we assume that thenumber ofgroupsisconstantandgivenbym.EachCCgroup contains between 1 and n individuals.Thetotal population size can fluctuate between theDbounds m and nm.Again, there are two types ofindividual,cooperators and defectors.Individualsinteract with others in their group and therebyreceivea payoff.At eachtimestepa random indi-Figure 2.6A simple model of group selection.A populationvidual from the entire population is chosen propor-consists of m groups of maximum size n. Individuals interact withothers in their group in the context of an evolutionary game. Here wetional to payoff in order to reproduce.The offspringconsiderthe game between cooperators,C,and defectors,D.Foris added to the same group.If the group size is lessreproduction, individuals are chosen from the entire population withthan orequaltonthennothingelsehappens.Ifthea probability proportional to their payoff. The offspring is added togroup size, however, exceeds n then with probabilitythe same group. Iif a group reaches the maximum size, n, then itgthegroupsplits intotwo.Inthiscase,arandomeithersplitsintwoorarandomindividual fromthatqroupisgroup is eliminated (inordertomaintaina constanteliminated. If a group splits, then a random group dies, in order tokeepthetotal populationsizeconstantThismetapopulationnumber ofgroups).Withprobability1-q,thegroupstructureleadstotheemergenceoftwolevelsofselection,althouqhdoes not divide, but instead a random individualonly individuals reproduce.from that group is eliminated (Figure 2.6).This minimalist model of multilevel selectionwhat we would obtain for a neutral mutant.Anhas some interesting features. Note that the evo-analytic calculation is possible in the interestinglutionary dynamics areentirelydriven by indivi-limit q1, where individuals reproduce muchdual fitness.Only individuals are assigned payoffmore rapidly than groups divide. In this case, mostvalues.Only individuals reproduce.Groups canofthegroupsareattheirmaximumsizeandhencestay together or split (divide) when reaching athe total population size is almost constant andcertain size.Groups that contain fitter individualsgiven by N=nm.We find that selection favorsreach the critical size faster and therefore splitcooperators and opposesdefectors,Pc>1/N>Pp,ifmore often.This concept leads to selection among(2.5a)b/c>1 + n/(m - 2)groups, although only individuals reproduce. Thehigher level selection emerges from lower levelThis result holds for weak selection.Smaller groupreproduction. Remarkably,the two levels ofsizesandlarger numbersof competinggroupsselection can oppose each other.favor cooperation.Wealso notice that the numberAsbefore,we can compute the fixation prob-of groups, m, must exceed 2.There is an intuitiveabilities,Pcand Pp,ofcooperators anddefectors toreason for this threshold. Consider the case ofcheckwhetherselectionfavorsoneorthe other.Ifm=2 groups with n=2 individuals. In a mixedwe add a single cooperator to a population ofgroup,the cooperatorhas payoff-candthedefectors,thenthis cooperator mustfirsttake overdefector has payoff b;the defector/cooperatora group.Subsequently the group of cooperatorsdifferenceisb+c.Inahomogeneousgroup,twomust take over the entire population.Thefirst stepcooperators havepayoff b-c,while two defectorsisopposedbyselection,thesecondstepisfavored'This is mot the same q as in section 2.3; we have run out ofby selection.Hence,we need to find out if theconvenient letters.overall fixation probability isgreatertoor less than

and Gintis, 2004; Traulsen et al., 2005). We only have to make sure that its basic requirements are fulfilled in a particular situation (Levin and Kilmer, 1974; Maynard Smith, 1976). Exactly what these requirements are can be illustrated with a simple model (Traulsen and Nowak, 2006). Imagine a population of individuals subdivided into groups. For simplicity, we assume that the number of groups is constant and given by m. Each group contains between 1 and n individuals. The total population size can fluctuate between the bounds m and nm. Again, there are two types of individual, cooperators and defectors. Individuals interact with others in their group and thereby receive a payoff. At each time step a random indi￾vidual from the entire population is chosen propor￾tional to payoff in order to reproduce. The offspring is added to the same group. If the group size is less than or equal to n then nothing else happens. If the group size, however, exceeds n then with probability q the group splits into two. In this case, a random group is eliminated (in order to maintain a constant number of groups). With probability 1
q, the group does not divide, but instead a random individual from that group is eliminated (Figure 2.6)* . This minimalist model of multilevel selection has some interesting features. Note that the evo￾lutionary dynamics are entirely driven by indivi￾dual fitness. Only individuals are assigned payoff values. Only individuals reproduce. Groups can stay together or split (divide) when reaching a certain size. Groups that contain fitter individuals reach the critical size faster and therefore split more often. This concept leads to selection among groups, although only individuals reproduce. The higher level selection emerges from lower level reproduction. Remarkably, the two levels of selection can oppose each other. As before, we can compute the fixation prob￾abilities, rC and rD, of cooperators and defectors to check whether selection favors one or the other. If we add a single cooperator to a population of defectors, then this cooperator must first take over a group. Subsequently the group of cooperators must take over the entire population. The first step is opposed by selection, the second step is favored by selection. Hence, we need to find out if the overall fixation probability is greater to or less than what we would obtain for a neutral mutant. An analytic calculation is possible in the interesting limit q << 1, where individuals reproduce much more rapidly than groups divide. In this case, most of the groups are at their maximum size and hence the total population size is almost constant and given by N¼ nm. We find that selection favors cooperators and opposes defectors, rC> 1/N> rD, if b=c > 1 þ n=ðm
2Þ ð2:5aÞ This result holds for weak selection. Smaller group sizes and larger numbers of competing groups favor cooperation. We also notice that the number of groups, m, must exceed 2. There is an intuitive reason for this threshold. Consider the case of m ¼ 2 groups with n ¼ 2 individuals. In a mixed group, the cooperator has payoff
c and the defector has payoff b; the defector/cooperator difference is b þ c. In a homogeneous group, two cooperators have payoff b
c, while two defectors C C D C D D D D C D C C C Figure 2.6 A simple model of group selection. A population consists of m groups of maximum size n. Individuals interact with others in their group in the context of an evolutionary game. Here we consider the game between cooperators, C, and defectors, D. For reproduction, individuals are chosen from the entire population with a probability proportional to their payoff. The offspring is added to the same group. If a group reaches the maximum size, n, then it either splits in two or a random individual from that group is eliminated. If a group splits, then a random group dies, in order to keep the total population size constant. This metapopulation structure leads to the emergence of two levels of selection, although only individuals reproduce. * This is not the same q as in section 2.3; we have run out of convenient letters. HOW POPULATIONS COHERE 15

16THEORETICALECOLOGYhave a payoff of O.Thus the disadvantage forgroups and migration between groups is not toocooperators in mixed groups cannot be compen-frequentsated for by the advantage they have in homo-geneous groups.Interestingly,however,forlarger2.6 Conclusionsplitting probabilities, q, we find that cooperatorscan be favored even for m=2 groups.The reasonWe end by listing the five rules that we mentionedis the following: for very small q, the initialin the beginning.These rules represent lawscooperator must reach fixation ina mixed group;of nature governing the natural selection ofbut for largerq,a homogeneous cooperatorgroupcooperation.can also emerge if a mixed group splits, giving rise1.Kin selection leads to cooperation ifb/c>1/r,to a daughter group that has only cooperators.where r is the coefficient of genetic relatednessThus, larger splitting probabilities make it easierbetween donor and recipient.forcooperationtoemerge.2.Direct reciprocity leads to cooperation ifLet us also consider the effect of migrationb/c>1/wo,where w is theprobability of playingbetweengroups.Theaveragenumberofmigrantsanother round in the repeated Prisoner's Dilemma.accepted by agroup during its lifetime is denoted3.Indirect reciprocity leads to cooperation ifby z.We find that selection favors cooperationb/c>1/q,whereqistheprobabilityofknowingprovided thatthe reputation of a recipient.b/c>1+z+n/m(2.5b)4.Graph selection (ornetwork reciprocity)leads tocooperation if b/c>k, wherek is the degree of theIn order to derive this condition we have assumedgraph; that is, the average number of neighbors.weak selection and q<1, asbefore, but also that5.Groupselectionleadstocooperationifboth the numbersofgroups, m,and themaximumb/c>1+z+n/m,wherez isthenumber ofmig-group size, n, are much larger than 1.For morerants accepted by a group during its lifetime, n isinformation, see Traulsen and Nowak,2006.the group size, and m is thenumber ofgroups.Group selection (or multilevel selection) is aIn all five theories, b is the benefit for the recipientpowerful mechanism for the evolution of coop-eration if there is a large number of relatively smalland c the cost for the donor of an altruistic act

have a payoff of 0. Thus the disadvantage for cooperators in mixed groups cannot be compen￾sated for by the advantage they have in homo￾geneous groups. Interestingly, however, for larger splitting probabilities, q, we find that cooperators can be favored even for m ¼ 2 groups. The reason is the following: for very small q, the initial cooperator must reach fixation in a mixed group; but for larger q, a homogeneous cooperator group can also emerge if a mixed group splits, giving rise to a daughter group that has only cooperators. Thus, larger splitting probabilities make it easier for cooperation to emerge. Let us also consider the effect of migration between groups. The average number of migrants accepted by a group during its lifetime is denoted by z. We find that selection favors cooperation provided that b=c > 1 þ z þ n=m ð2:5bÞ In order to derive this condition we have assumed weak selection and q << 1, as before, but also that both the numbers of groups, m, and the maximum group size, n, are much larger than 1. For more information, see Traulsen and Nowak, 2006. Group selection (or multilevel selection) is a powerful mechanism for the evolution of coop￾eration if there is a large number of relatively small groups and migration between groups is not too frequent. 2.6 Conclusion We end by listing the five rules that we mentioned in the beginning. These rules represent laws of nature governing the natural selection of cooperation. 1. Kin selection leads to cooperation if b/c > 1/r, where r is the coefficient of genetic relatedness between donor and recipient. 2. Direct reciprocity leads to cooperation if b/c > 1/w, where w is the probability of playing another round in the repeated Prisoner’s Dilemma. 3. Indirect reciprocity leads to cooperation if b/c > 1/q, where q is the probability of knowing the reputation of a recipient. 4. Graph selection (or network reciprocity) leads to cooperation if b/c > k, where k is the degree of the graph; that is, the average number of neighbors. 5. Group selection leads to cooperation if b/c > 1 þ z þ n/m, where z is the number of mig￾rants accepted by a group during its lifetime, n is the group size, and m is the number of groups. In all five theories, b is the benefit for the recipient and c the cost for the donor of an altruistic act. 16 THEORETICAL ECOLOGY

CHAPTER3Single-species dynamicsTim Coulson and H.Charles J.GodfrayWhat determines the densities of the differentgrowth rates fordifferent types of population, andspecies of plants,animals, and micro-organismsexplorehow suchcalculations,even thoughtheywith which we share the planet, why do theirarebased onthesimplisticassumptionofconstantnumbers fluctuateand extinctions occur,and howdemographic rates, can be very useful for a varietydo different species interact to determine eachof problems in applied populationbiology.The fact that populations persist over appreci-other's abundance?These are some of the ques-tions addressed by the science of ecologicalable periods of time inescapably means thatpopulation dynamics, the subject that underpinsdemographic rates-births and deaths, immigra-all the chapters in this book. In this chapter wetion and emigration-do not remain constant.introduce some of the basic principles of theIn fact,population persistence in the long termsubject by concentrating on the dynamics of single-requires that as populations increase in density thespecies systems.These are species whose popula-death rate must rise relative to the birth rate andtion biology can be studied without also explicitlyeventually exceed it. In real populations, suchdemographic rates are what ecologists call density-includingthe dynamics of other species in thecommunity.The chief justification for this brutaldependentandmathematiciansnonlinear,abstraction is that it allows many of the underlyingwhereas an engineer might talk about negativefeedback. It is this nonlinearity that can give rise toprocessestobedescribedsimplyandmoreclearlyMoreover, arguments based on the analysis ofa stable equilibrium, the population density atsingle-species population dynamics are often surwhich birth rates precisely equal death rates. But aprisingly useful in understanding real populations,farmorediversemenagerieof dynamicbehavioursis possible; the population may not settle on aespeciallythoseinrelativelysimpleenvironmentssuch as agro-ecosystemsstable equilibrium but show persistent cycles.At the core of population dynamics is a simpleStranger still, these cycles may not be regular buttruism: the density or numbers of individuals in acomplex and unpredictable in detailthey mayshow mathematical chaos. The possibility ofclosed population is increased by births,anddecreased by deaths.If the population is not closedchaotic dynamics in simplepopulations was firstthen we need also to include immigration andappreciated inthe1970s,and ecologicalproblemsemigration in our calculation. A population inwere very significant in the development of thisnew field of mathematics.The second section ofwhich births exceed deaths will tend to increaseand one where the reverse is truewill tend tothis chapter explores the consequences of deter-decrease. But more significant is the mode ofministic density-dependent demographic rates,change.If birth and death rates remain constantand explores chaos in ecology.then the consequent increase or decrease inInthelastsentence,bydeterministicwemeanpopulation numbers occurs exponentiallypopu-that demographic rates are constant or simplylation dynamics occurs on a geometric rather thandetermined by density, and do not also vary byan arithmetic scale. In the first section of thischance. Of course, all real populations are subjectchapter we describe the calculation of exponentialto random effects.When the average birth rate is17

CHAPTER 3 Single-species dynamics Tim Coulson and H. Charles J. Godfray What determines the densities of the different species of plants, animals, and micro-organisms with which we share the planet, why do their numbers fluctuate and extinctions occur, and how do different species interact to determine each other’s abundance? These are some of the ques￾tions addressed by the science of ecological population dynamics, the subject that underpins all the chapters in this book. In this chapter we introduce some of the basic principles of the subject by concentrating on the dynamics of single￾species systems. These are species whose popula￾tion biology can be studied without also explicitly including the dynamics of other species in the community. The chief justification for this brutal abstraction is that it allows many of the underlying processes to be described simply and more clearly. Moreover, arguments based on the analysis of single-species population dynamics are often sur￾prisingly useful in understanding real populations, especially those in relatively simple environments such as agro-ecosystems. At the core of population dynamics is a simple truism: the density or numbers of individuals in a closed population is increased by births, and decreased by deaths. If the population is not closed then we need also to include immigration and emigration in our calculation. A population in which births exceed deaths will tend to increase and one where the reverse is true will tend to decrease. But more significant is the mode of change. If birth and death rates remain constant then the consequent increase or decrease in population numbers occurs exponentially—popu￾lation dynamics occurs on a geometric rather than an arithmetic scale. In the first section of this chapter we describe the calculation of exponential growth rates for different types of population, and explore how such calculations, even though they are based on the simplistic assumption of constant demographic rates, can be very useful for a variety of problems in applied population biology. The fact that populations persist over appreci￾able periods of time inescapably means that demographic rates—births and deaths, immigra￾tion and emigration—do not remain constant. In fact, population persistence in the long term requires that as populations increase in density the death rate must rise relative to the birth rate and eventually exceed it. In real populations, such demographic rates are what ecologists call density￾dependent and mathematicians nonlinear, whereas an engineer might talk about negative feedback. It is this nonlinearity that can give rise to a stable equilibrium, the population density at which birth rates precisely equal death rates. But a far more diverse menagerie of dynamic behaviours is possible; the population may not settle on a stable equilibrium but show persistent cycles. Stranger still, these cycles may not be regular but complex and unpredictable in detail—they may show mathematical chaos. The possibility of chaotic dynamics in simple populations was first appreciated in the 1970s, and ecological problems were very significant in the development of this new field of mathematics. The second section of this chapter explores the consequences of deter￾ministic density-dependent demographic rates, and explores chaos in ecology. In the last sentence, by deterministic we mean that demographic rates are constant or simply determined by density, and do not also vary by chance. Of course, all real populations are subject to random effects. When the average birth rate is 17

18THEORETICAL ECOLOGYtwo offspring per year, some individuals will havereproduce immediately). Let the rate at whichfewer or more offspring; and in some years, or inindividuals produce female offspring be b (wesome sites within the species' range, the averageassume for now that males have no effect onmaybe slightlymoreor slightlyless.More radi-population growth rate,something that is true forcally, the birth rate may be two, year on year,most but not all organisms)and the rate at whichexcept for the time the meteorite hit and no-onethey die be d. Define the difference between thesereproduced. The last 10 years has seen notabletwo ratesas r=b-d.Thepopulation increases ifadvances in the studyof populationsthattakeintor>0and decreases ifr<0.Moreover,ifthe currentaccount stochasticeffects,and thesearethesubjectpopulation size is No then the population sizeftime units into the future is N,=No explrt]. Theof our third section.population increases or decreases at a rate deter-mined bythepower of r.Notethat inthis simple3.1Therateofpopulationgrowthmodel,topredictpopulationgrowthrateswedoAll subjects need their founding myths, withnot need toknow birth and death rates separately,appropriate heroes, and while physics has thejust their net difference.giants Newton and Einstein, and evolution theNot all species reproduce continuously. Con-peerless Darwin, students of population dynamicssider a population of an animal or plant withare stuck with the far less appetising Thomasdiscrete generations that produces female off-Malthus.Malthus was not the first person tospring before dying. It is straightforward to seeappreciate the geometric nature of populationthat if population size is now Nothen t generationsgrowth but he was thefirst clearly to work throughin the future it will be N,=Noa,which can beits consequences.In his famous pamphlet An Essaywritten N,=No exp[ln(>) tl, the latter expressionon the Principle of Population of 1798, he vividlyemphasizing the similarity with the continuousillustrated the powerof geometric growth in termscase, with In(2) replacing r.that mirror the modern cliches that if populationThe parameter r (or In(2) is the intrinsic growthgrowth is unchecked it would take only a fewrate of the population; it allows us to projectyearsforthetotal numbersofaphid/cod/elephantpopulation numbers into the future. Of course weor your favourite animal to weigh more than thedo not believe r will stay constant forever-aprojection should not be confused with a fore-Earth (Malthus, 1798). It was the power of thisargument that so influenced Darwinsuch greatcastbut it tells us something about what willpotential fecundity must be balanced by greathappen intheshortterm,giventhecurrentbirthmortality,and any heritable trait that favoured oneand death rates.This can be a veryimportantindividual over another would increase in fre-management tool.Supposefor example oneisquency ineluctably. In contrast, the message thattrying to assess the potential vulnerability of aMalthus, an upper-class vicar, drew from his ownseries of populations of an endangered species.insight was theneed to do somethingabout theCalculating their different population growthirresponsibly fecund lower classes (as well asrates will not give you a complete answer to thisabout other problems suchas women and thequestion,but it will provide an importantFrench).TypeMalthus into Google and youfindclue to their different vulnerabilities. Estimationshima herotoanunpleasant consortium ofmod-of population growth rates for more complic-ern-day social engineers.ated population structures(see below)lie at theBut despite its shady origin, the rate of expo-heart of population viability analysis,a frequentlynential population growth basedon currentused tool in conservation biology.Epidemiologydemographic rates is an immensely useful quan-provides a rather different example of the impor-tity.Consider first a simple, unstructured popula-tanceof populationgrowthrate.Consider ation; by unstructured we mean that birth andpopulation of susceptiblehosts exposed to a smalldeath rates are identical across individuals (clearlynumberof infectious individuals.Fromthepointofview of the disease,births consist of newinfectionsan approximation,asanewborn individualcannot

two offspring per year, some individuals will have fewer or more offspring; and in some years, or in some sites within the species’ range, the average may be slightly more or slightly less. More radi￾cally, the birth rate may be two, year on year, except for the time the meteorite hit and no-one reproduced. The last 10 years has seen notable advances in the study of populations that take into account stochastic effects, and these are the subject of our third section. 3.1 The rate of population growth All subjects need their founding myths, with appropriate heroes, and while physics has the giants Newton and Einstein, and evolution the peerless Darwin, students of population dynamics are stuck with the far less appetising Thomas Malthus. Malthus was not the first person to appreciate the geometric nature of population growth but he was the first clearly to work through its consequences. In his famous pamphlet An Essay on the Principle of Population of 1798, he vividly illustrated the power of geometric growth in terms that mirror the modern cliche´s that if population growth is unchecked it would take only a few years for the total numbers of aphid/cod/elephant or your favourite animal to weigh more than the Earth (Malthus, 1798). It was the power of this argument that so influenced Darwin—such great potential fecundity must be balanced by great mortality, and any heritable trait that favoured one individual over another would increase in fre￾quency ineluctably. In contrast, the message that Malthus, an upper-class vicar, drew from his own insight was the need to do something about the irresponsibly fecund lower classes (as well as about other problems such as women and the French). Type Malthus into Google and you find him a hero to an unpleasant consortium of mod￾ern-day social engineers. But despite its shady origin, the rate of expo￾nential population growth based on current demographic rates is an immensely useful quan￾tity. Consider first a simple, unstructured popula￾tion; by unstructured we mean that birth and death rates are identical across individuals (clearly an approximation, as a newborn individual cannot reproduce immediately). Let the rate at which individuals produce female offspring be b (we assume for now that males have no effect on population growth rate, something that is true for most but not all organisms) and the rate at which they die be d. Define the difference between these two rates as r ¼ b
d. The population increases if r > 0 and decreases if r < 0. Moreover, if the current population size is N0 then the population size t time units into the future is Nt ¼ N0 exp[rt]. The population increases or decreases at a rate deter￾mined by the power of r. Note that in this simple model, to predict population growth rates we do not need to know birth and death rates separately, just their net difference. Not all species reproduce continuously. Con￾sider a population of an animal or plant with discrete generations that produces l female off￾spring before dying. It is straightforward to see that if population size is now N0 then t generations in the future it will be Nt ¼ N0 lt , which can be written Nt ¼ N0 exp[ln(l) t], the latter expression emphasizing the similarity with the continuous case, with ln(l) replacing r. The parameter r (or ln(l)) is the intrinsic growth rate of the population; it allows us to project population numbers into the future. Of course we do not believe r will stay constant forever—a projection should not be confused with a fore￾cast—but it tells us something about what will happen in the short term, given the current birth and death rates. This can be a very important management tool. Suppose for example one is trying to assess the potential vulnerability of a series of populations of an endangered species. Calculating their different population growth rates will not give you a complete answer to this question, but it will provide an important clue to their different vulnerabilities. Estimations of population growth rates for more complic￾ated population structures (see below) lie at the heart of population viability analysis, a frequently used tool in conservation biology. Epidemiology provides a rather different example of the impor￾tance of population growth rate. Consider a population of susceptible hosts exposed to a small number of infectious individuals. From the point of view of the disease, births consist of new infections 18 THEORETICAL ECOLOGY

19SINGLE-SPECIES DYNAMICSand deaths occurwhenthehosteither recovers orwhile the probabilities of surviving until the newactually dies. The disease will only spread ifseason form the lower subdiagonal. Thematrix Ar=b-d>O,where b and d are the rates of diseaseis an example of a population-projection matrix,'births'and'deaths'.In the epidemiological litera-and this particular form, where the population isture this condition is normally stated as exp(r)=structured by age, is called a Leslie matrix (Leslie,Ro>1, which has the simple interpretation that for1945).spread to occur every initial infection must leave atThesimplestwaytoexplorethegrowthrateofaleast one secondary infection.As Grenfell andpopulation described by eqn 3.1 is to iterate it on aKeeling(Chapter10inthisvolume)discussinmorecomputer,an option not availabletothe origina-detail, calculation of Ro, usually called the basictors of these techniques in the 1940s. But there arereproductiveratio byecologists or,morecorrectlysome importantmathematical resultsthat allowthe basic reproductivenumber (it is dimensionless)much greater insight into the population growthbyepidemiologists,liesattheheartofmuchhumanprocess.We do not have the space to derive theseand animal health population analysis.results or explain them in detail, but attempttogive some flavour of their elegance and import-ance3.1.1Structured populationsThe matrix A includes all the information weneed toknow about thepopulation's demographicThe assumption that all populations aremade upparameters(Caswell,1989,2001).From thismatrix,of identical individuals with the same demo-a polynomial equation in an arbitrary variable (saygraphic rates is clearly a gross oversimplification.n) can be derived. The order of the polynomial isHow can population growth rates be calculated indetermined by the number of age classes. If theremore complex structured populations?are five age classes than the equation will be ofWe introduce this topic by considering a popu-order five (terms up to ) and if there are 20 agelation that has discrete breeding seasons so that itclasses then there will be terms up to 20. Just asmakes sense to census it once a year. We alsothe familiar quadratic equation (order two) hassuppose that the population is age-structured:two roots (values of for which the equationdemographic rates vary with age but are constantequals O) then these larger polynomials of order xwithin an age class. To describe population num-have exactly x roots, though unlike the quadraticbers attimet we nowneed to write down a vector,they can only be calculated numerically (except forn(t)=fni.n2,nl(t),wheren,isthenumbersome special cases).A collection of mathematicalor density of individuals in their ithyear at timetresultscalled thePerron-Frobeniustheoremtells(and x is the oldest age class).To explore howus that for Leslie matrices (with some minorpopulationnumberschangeover timeweneedtoexceptions that we will return to)there will alwaysknow the probability that an individual of age ibe one root that is larger than all the others.will survive to the next year (p)and the number ofMoreover,thisroot,which is a complicated func-offspring itproduces eachyear (fi).Thention of the different elements of the matrix A,f2f3represents the long-term growth rate of thefi(m).fx(00o13P1population.In matrix theory the roots are called00...n13n3p2eigenvalues and calculation of the largest root, the(t+1)(t)........dominanteigenvalue,provides the asymptotic.+.00population projection that we require.(nx)0nxPx-1This powerful result tells us that whateverthe(3.1)initial distribution of individuals across age classeswhich can be written more succinctly n(t+1)=the population will eventually grow or decline at aA n(t). Note that the numbers in the youngest agerate set bythe dominant eigenvalue (this indeclass are given by the numbers in each age classpendence of starting conditions is called ergodi-the season before multiplied by their fecundity,city). Another ergodic property of these population

and deaths occur when the host either recovers or actually dies. The disease will only spread if r ¼ b
d > 0, where b and d are the rates of disease ‘births’ and ‘deaths’. In the epidemiological litera￾ture this condition is normally stated as exp(r)¼ R0> 1, which has the simple interpretation that for spread to occur every initial infection must leave at least one secondary infection. As Grenfell and Keeling (Chapter 10 in this volume) discuss in more detail, calculation of R0, usually called the basic reproductive ratio by ecologists or, more correctly, the basic reproductive number (it is dimensionless) by epidemiologists, lies at the heart of much human and animal health population analysis. 3.1.1 Structured populations The assumption that all populations are made up of identical individuals with the same demo￾graphic rates is clearly a gross oversimplification. How can population growth rates be calculated in more complex structured populations? We introduce this topic by considering a popu￾lation that has discrete breeding seasons so that it makes sense to census it once a year. We also suppose that the population is age-structured: demographic rates vary with age but are constant within an age class. To describe population num￾bers at time t we now need to write down a vector, nðtÞ¼fn1; n2; . ; nxgðtÞ, where ni is the number or density of individuals in their ith year at time t (and x is the oldest age class). To explore how population numbers change over time we need to know the probability that an individual of age i will survive to the next year (pi) and the number of offspring it produces each year (fi). Then n1 n2 n3 . . . nx 0 BBBBB@ 1 CCCCCA ðtþ1Þ ¼ f1 f2 f3 fx p1 0 0 0 0 p2 0 0 . . . . . . . . . . . 0 0 px
1 0 0 BBBBB@ 1 CCCCCA n1 n2 n3 . . . nx 0 BBBBB@ 1 CCCCCA ðtÞ ð3:1Þ which can be written more succinctly n(t þ 1) ¼ A n(t). Note that the numbers in the youngest age class are given by the numbers in each age class the season before multiplied by their fecundity, while the probabilities of surviving until the new season form the lower subdiagonal. The matrix A is an example of a population-projection matrix, and this particular form, where the population is structured by age, is called a Leslie matrix (Leslie, 1945). The simplest way to explore the growth rate of a population described by eqn 3.1 is to iterate it on a computer, an option not available to the origina￾tors of these techniques in the 1940s. But there are some important mathematical results that allow much greater insight into the population growth process. We do not have the space to derive these results or explain them in detail, but attempt to give some flavour of their elegance and import￾ance. The matrix A includes all the information we need to know about the population’s demographic parameters (Caswell, 1989, 2001). From this matrix, a polynomial equation in an arbitrary variable (say Z) can be derived. The order of the polynomial is determined by the number of age classes. If there are five age classes than the equation will be of order five (terms up to Z5 ) and if there are 20 age classes then there will be terms up to Z20. Just as the familiar quadratic equation (order two) has two roots (values of Z for which the equation equals 0) then these larger polynomials of order x have exactly x roots, though unlike the quadratic they can only be calculated numerically (except for some special cases). A collection of mathematical results called the Perron–Frobenius theorem tells us that for Leslie matrices (with some minor exceptions that we will return to) there will always be one root that is larger than all the others. Moreover, this root, which is a complicated func￾tion of the different elements of the matrix A, represents the long-term growth rate of the population. In matrix theory the roots are called eigenvalues and calculation of the largest root, the dominant eigenvalue, provides the asymptotic population projection that we require. This powerful result tells us that whatever the initial distribution of individuals across age classes the population will eventually grow or decline at a rate set by the dominant eigenvalue (this inde￾pendence of starting conditions is called ergodi￾city). Another ergodic property of these population SINGLE-SPECIES DYNAMICS 19