Theoretical Population Biology 54, 270�293 (1998)

Ontogenetic Scaling of Foraging Ratesand the Dynamics of a Size-StructuredConsumer-Resource Model

Lennart Persson and Kjell LeonardssonDepartment of Animal Ecology, University of Umea# , Umea# S-90187, SwedenE-mail: Lennart.Persson�animecol.umu.se

Andre� M. de RoosDepartment of Pure and Applied Ecology, University of Amsterdam, Kruislaan 320,Amsterdam NL-1098 SM, The NetherlandsE-mail: aroos�bio.uva.nl

Mats GyllenbergDepartment of Applied Mathematics, University of Turku, Turku FIN-20500, FinlandE-mail: matsgyl�utu.fi

and

Bent ChristensenDepartment of Animal Ecology, University of Umea# , Umea# S-90187, Sweden

Received July 4, 1997

The ontogenetic scaling of foraging capacity strongly influences the competitive abilityof differently sized individuals within a species. We develop a physiologically structuredmodel to investigate the effect of different ontogenetic size scalings of the attack rate on thepopulation dynamics of a consumer-resource system. The resource is assumed to reproducecontinuously whereas the consumer only reproduces at discrete time instants. Depending onthe ontogenetic size scaling, the model exhibited recruit-driven cycles, stable fixed pointdynamics, non-recruit juvenile-driven cycles, quasiperiodic orbits, or chaotic dynamics. Thekind of dynamics observed was related to the maintenance resource levels required ofdifferently sized individuals. Stable fixed point dynamics was, besides at the persistenceboundary, only observed when the minimum resource levels were similar for newborns andmature individuals. The tendency for large population fluctuations over a wide range of theparameter space was due to the consumer's pulsed reproduction. Background mortality andlength of season were major determinants of cycle length. Model dynamics strongly resembledempirically observed dynamics from fish and Daphnia populations with respect to both pat-terns and mechanisms. The non-recruit juvenile-driven dynamics is suggested to occurin populations with size-dependent interference or preemptive competition like cicadapopulations. ] 1998 Academic Press

Article No. TP981380

2700040-5809�98 K25.00

Copyright ] 1998 by Academic PressAll rights of reproduction in any form reserved.

1. INTRODUCTION

Organisms of most animal taxa undergo growth andmorphological change during a substantial part of thelife cycle. Due to the taxonomic dominance of insectsand marine phyla, Werner (1988) estimated thatapproximately 800 of all animal species undergometamorphosis. Even if only vertebrates are considered,individuals of 750 of all taxa grow during most of theirlife cycle (Ebenman and Persson, 1988). The rate atwhich the organism grows often depends on resourcelevels and observed variations in size structure betweenpopulations can in many cases be related to resourcedependent individual growth (Norberg, 1988; Persson,1988; Wilbur, 1988).

The phenomenon of individual growth entails that twofundamental components of its competitive ability, itsforaging rate and its metabolic demands, change overthe organism's life cycle where the competitive abilityis defined by (1) the ability of the individual to con-sume the resource and (2) its ability to withstandresource depression (depending on metabolic demands).(Persson, 1985; Werner, 1988; Lundberg and Persson,1993). Changes in these components are expected to havemajor impacts on ecological interactions as size-depend-ent foraging capacity has been shown to have majoreffects on habitat use and performance of differentlysized individuals (Mittelbach, 1981; Werner et al., 1983;Persson, 1987). An analysis of how size-dependentchanges in the two components of competitive abilityover ontogeny affect population dynamics is therefore ofmajor importance in order to increase our understandingof ecological interactions in natural systems.

Of the two components, metabolic demands have beenfound to scale to body weight according to a powerfunction with an exponent generally varying between 0.7and 0.8 between taxa (Peters, 1983; Calder, 1984). Theforaging rate as a function of body weight is alsocommonly described by a power function (Schoener,1969; Wilson, 1975; Peters, 1983; Calder, 1984; Werner,1988). However, this intraspecific scaling of foraging rateto body size (hereafter the ontogenetic scaling of foragingrate) varies much more between different taxa than thecorresponding exponent for metabolic demands (Werner1988). Reasons for this are that the foraging rate of anorganism is the integrated result of a number of traits(visual acuity, locomotor ability, gape size and digestivecapacity) each of which exhibits substantially differentontogenetic scaling relations in different taxa (Schoener,1969; Wilson, 1975; Peters, 1983; Calder, 1984; Hyatt,1979; Sebens, 1982; Hoyle and Keast, 1987; Persson,1987).

The effects of the ontogenetic scaling relation, andchanges therein, on the competitive abilities of differentlysized individuals have been discussed in several papers(cf. Werner, 1988; 1994; Lundberg and Persson, 1993).Empirical evidence that size-dependent competitiveabilities may generate population cycles has also beenprovided (Hamrin and Persson, 1986; McCauley andMurdoch, 1987). In addition, a number of theoreticalpapers have considered the implications of stage-basedcompetitive relationships for population dynamics(Ebenman, 1988; Cushing and Li, 1991, 1992; Nisbetand Onyiah, 1994). However, an explicit and completetreatment of how the ontogenetic scaling of foraging rateaffects population dynamics is lacking.

In this paper, we analyse the effects of differentontogenetic scalings of foraging rate as the most variablecomponent of competitive ability on the populationdynamics in a size-structured one consumer��one resourcemodel. We use a physiologically structured model, anapproach that explicitly links the dynamics that takesplace at the level of population to the behavior ofindividual organisms (Metz and Diekmann, 1986; Metzet al., 1988; DeAngelis and Rose, 1992; de Roos 1997).This modelling approach allows us to mechanisticallylink individual (size) dependent foraging rate andmetabolic demands to population dynamics. We showfurther that the results of our modelling provide anunderstanding of the mechanisms behind the populationoscillations in populations where intercohort interac-tions have been suggested to drive the dynamics ofconsumer-resource interactions. These examples includeplanktivorous fish, zooplankton and cikadas (Bulmer1977; Hamrin and Persson 1986; McCauley andMurdoch 1987; Townsend et al., 1990).

2. THE PHYSIOLOGICALLYSTRUCTURED MODEL

2.1. Consumer-Resource Dynamics

Our physiologically structured model is built along thelines advanced by Metz and Diekmann (1986; see alsoMetz et al., 1988; de Roos 1988, 1997; de Roos et al.,1990, 1992, 1997). These models are based on a stateconcept at each of two levels of organisation: an i-statewhich represents the state of the individual in terms of acollection of characteristic physiological traits (size,age, sex, energy reserves etc.), and a p-state which isthe frequency distribution over the space of possiblei-states (Metz and Diekmann, 1986; Metz et al., 1988;Caswell and John, 1992; DeAngelis and Gross, 1992).

271Ontogenetic Scaling and Population Dynamics

We have formulated our model in ``cumulative'' terms(see Diekmann et al., 1993, 1995), as opposed to thetraditional approach based on instantaneous rates.

In the model, the consumer population is size-struc-tured whereas the resource population is not. We assumethat the resource is continuously reproducing through-out the season. The consumer is assumed to grow (orshrink in case of starvation) continuously during theseason, but reproduces at discrete time instants only. Thegrowth season of the consumer corresponds to thesummer in the temperate region and reproduction takesplace at the start of the growing season (i.e. spring). Inthe present paper, we ignore the winter season which isequivalent to the situation when winter adds nothing buta scaling down of all rates. Our model is a combinationof a continuous dynamical system, describing growthand survival of the consumer and production and con-sumption of the resource during summer, and a discretemap describing the pulsewise reproduction of consumersin spring. This combination takes the size distribution ofthe consumer population and the resource density at thebeginning of a season to their corresponding values at thebeginning of the next season. We thus have all the usualconcepts of the theory of dynamical systems, e.g. ``fixedpoint,'' ``periodic orbit'' and ``attractor'' to our disposal.For instance, a fixed point, or equilibrium, implies thatthe consumer size distribution and the resource densityalways has the same value at a fixed time within eachseason, even though during the season these will ofcourse change. Similarly, an orbit of period p is a solutionthat repeats itself at a fixed time every pth season. In ouranalyses, we chose the end of the day prior to thereproductive event as reference time and consequently allresults were extracted at this instant. At this particulartime, it was possible to study reproductive investment ofeach consumer cohort in the population as well as thetotal number of offsprings produced the following day.

As is described above, the core part of the structuredpopulation model is a description of the individualbehavior of the consumers, that is, individual feeding,growth, development, reproduction and mortality, as afunction of the current state of both the environment(resource) and the individual itself. In the followingsections we will discuss this model of individualenergetics in more detail.

2.2. Individual State and Individual State Space

We assume that the consumer's foraging, metabolism,growth, survival, and reproduction for each givenresource density can be described as functions of twophysiological variables, irreversible and reversible mass.

In irreversible mass x we include compounds like bonesand organs which cannot be starved away by the con-sumer. In reversible mass y we include energy reservessuch as fat, muscle tissue and gonads. These reserves maybe used to cover basic metabolism during starvation. Thetotal mass of the individual hence equals x+ y. For asimilar approach, see Broekhuizen et al. (1994).

We assume that there exists a fixed irreversible mass xf

( f for fertile) that distinguishes juveniles from adults.Individuals with x�xf are juveniles and individuals withx>xf are adults that can produce young. It is furtherassumed that there is a maximal ratio of reversible massto irreversible mass and that this ratio is different forjuveniles and adults as the latter also allocate mass togonads. Denoting these ratios by qj and qa for juveniles andadults, respectively, the i-state space is contained in the set

0=[x0�x��, 0� y� y*(x)], (1)

where x0 is the (given) minimum irreversible mass anindividual can have (i.e. at birth) and

y*(x)={qjxqa x

if x�xf ,if xf<x.

(2)

The law of mass allocation to be discussed in Section 2.4imposes further restrictions on the collection of possibleindividual states which turns out to be substantiallysmaller than the set 0. See Fig. 1.

2.3. Consumption Rate, Attack Rate,and Handling Time

We assume that a consumer captures the resourcefollowing a Holling type II functional response. In general,the attack rate and the handling time depend on bothi-state variables in which case y functions as a measure ofthe condition of the individual. In this paper, however,we assume that the attack rate and the handling timeonly depend on x through the quantity

w=x+qjx. (3)

We do this as functional response experiments with size-structured consumers have shown a close relationshipbetween capture rate and body length independent ofbody condition (Mittelbach 1981; Persson 1987). Wecall the quantity w effective body mass. The functionalresponse is hence described by:

#(w, R)=a(w) R

1+a(w) h(w) R, (4)

272 Persson et al.

where a(w) is the attack rate, h(w) is the handling time,and R is the resource density.

For a given prey size, attack rate a(w) has been foundto be a hump-shaped function of predator size (Werner,1988; Tripet, and Perrin 1994). The initial increase of theforaging capacity with predator size can be explained byan increase in visual acuity and locomotor ability, bothof which will affect the encounter with prey (Schoener,1969; Wilson, 1975; Peters, 1983; Noakes and Godin,1988; Persson, 1987; Werner, 1988). The decreasing partof the function can be related to a decrease of rod densityand hence the capacity to discern small prey (Breck andGitter, 1983). Furthermore, attack efficiency may alsodecrease above a given body size due to decreased abilityto make fine tuned manoeuvres (Hyatt, 1979; Breck andGitter, 1983; Persson, 1987; Noakes and Godin, 1988).Adopting this general shape, we assume that the attackrate depends on effective body mass according to thefollowing formula:

a(w)=A \ wwo

exp \1&wwo++

:

, (5)

which relationship is governed by three (positive) para-meters: (1) the maximum rate, A, (2) the body size atwhich the maximum rate is achieved, wo , and (3) a sizescaling exponent, :, which affects the rate by which theattack rate increases below and decreases above wo . Forlarge values of : the attack rate will increase more rapidlywith w and have a sharp peak, whereas the attack ratewill increase slowly with w and have a broad peak forsmall values of : (Fig. 2a).

Commonly, handling time h(w) per prey first decreaseswith body size due to an increase in gape size anddigestive capacity (Mittelbach, 1981; Hoyle and Keast,1987; Persson, 1987) (Fig. 2b). Thereafter handling timeincreases; possibly due to difficulties in handling smallprey (Persson, 1987). The specific function we use todescribe this pattern is:

h(w)=/1+/2 w&/3 exp(/4w), (6)

where /1 , /2 , /3 and /4 are positive constants. Thehandling time has a minimum when w=/3 �/4 .

2.4. Mass Allocation, Growth, Reproduction,and Survival

Based on Peters (1983), Calder (1984) and Lundbergand Persson (1993), we assume that the metabolic

demands per unit of time Em as a function of body massx+ y can be described by a power function:

Em(x, y)=m1(x+ y)m2, (7)

where m1 and m2 are positive constants. The net massintake per unit of time Eg(x, y, R) equals the mass intakeper unit of time E minus metabolic demands per unit oftime, hence

Eg(x, y, R)=E(w, R)&Em(x, y), (8)

where E(w, R) equals the consumption rate #(w, R)multiplied by a conversion factor kl which takes intoaccount prey weight, assimilation efficiency and otherconversion costs (Table I). If mass intake exceeds costsfor metabolism, the surplus mass is invested in growth.We assume that a fraction, }(x, y), of the surplus massEg is allocated to growth in irreversible mass (and therest into growth of reversible mass) according to thefunctions:

}(x, y)={1

(1+qj) qj

yx

1(1+qa) qa

yx

if x�xf

if x>xf

(9)

Individuals are assumed to be born with i-state(x0 , y0) where y0=qjx0 , that is, they are born with themaximum ratio of reversible mass to their irreversiblemass (Fig. 1). As long as the individual does not starve,that is as long as Eg�0, a juvenile individual will alwayshave this maximum ratio of reversible versus irreversiblemass. When reaching the size-dependent maturity x=xf ,it is assumed that the allocation function changes andthat the individual in addition to fat reserves alsoallocates mass to gonads. The maximum amount ofreversible mass for an adult individual is thereforeproportionally larger and equals qax (Table I, Fig. 1). Ineffect, the allocation rule (9) ensures that the individualswill approach this boundary but seldom reach it. Theindividual allocates mass surplus to reversible mass at arate which is proportional to ( y*& y)�y* (see Eq. 9 andFig. 1).

If the net mass intake, Eg is negative, the individualstarves and reversible mass (but not irreversible mass)decreases (Fig. 1). We assume that the individual canstand a certain amount of starvation, down to the pointwhen y=qsx before its death rate increases due to star-vation. The rate of starving to death, +s(x, y), when the

273Ontogenetic Scaling and Population Dynamics

FIG. 1. The set of reachable individual states, as determined by the allocation of consumed energy to irreversible (x) and reversible ( y) mass.The individual is born with an irreversible mass of x0 and a reversible mass of qj x0 and will grow in mass along the line y=qjx as long as it doesnot starve. When reaching maturity (at irreversible size xf), it increases its maximum amount of reversible mass to qax. When the individual spawns,reversible mass drops to qj x and after spawning the individual allocates mass according to the overall rule for partitioning between reversible andirreversible masses (Eq. 9). When starving, the individual only decreases in reversible mass, while during recovery after a starvation period, it allocatesmass preferentially to reversible mass, as expressed by Eq. 9. Below the line y=qsx, the individual starts to die of starvation. Observe that theindividual may be positioned anywhere in the (x, y)-space (grey area) within the limits set by y=qj x, y=qax, the x axis and to the right by themaintenance limit set by the solution for K from the equation Eg(x, y, K )=0 (cf. Eq. 8).

FIG. 2. Attack rate (a) and handling time (b) as functions of consumer effective body mass (w). For the attack rate function, curves for the threedifferent : values used in the simulations are shown. Parameter values: wo=17.42 g wet weight and A=10&4 Lake day&1. Other values as given inTable I.

274 Persson et al.

reversible mass is below qsx is assumed to increase withthe proportion x�y according to the following equation:

+s(x, y)={s \qs

xy

&1+0

ifyx

�qs

ifyx

>qs

(10)

where s is a (positive) proportionality constant. Thisequation ensures that death is certain before y=0.

In addition to starvation mortality, we assume that theindividual may die of other causes than starvation at arate +0 which is independent of the i-state. The total percapita death rate is thus the sum of the starvation andbackground death rates.

As described above, adult individuals allocate mass togonads and hence have a relatively higher maximumreversible mass value than juveniles. The amount y&qjxof reversible mass exceeding qj x at the end of the growingseason is used for reproduction. This gonad mass istransformed into newborns on the first day of the follow-ing growing season according to the birth functionspecified in the appendix and Table 1. At the same time,the parent's reversible mass is reduced to y=qjx (Fig. 1).If at the end of the growing season a mature individual'sreversible mass is below y=qjx, it will not reproduce.

2.5. The Model at the Population Level

At the population level the system under considerationconstitutes a strongly coupled feedback loop between anunstructured resource and a size-structured consumerpopulation. The resource population decreases in densitydue to the size-dependent foraging of the consumers andincreases by means of an intrinsic growth process. Wehave assumed that this resource growth can be describedby semi-chemostat dynamics. Semi-chemostat dynamicsmay be more realistic than the commonly used logisticgrowth dynamics when (1) the resource has a physicalrefuge or (2) the resource includes invulnerable (smaller),albeit mature size classes which grow into a vulnerablesize range. The latter is, for example, the case forzooplankton fed upon by planktivorous fish. In turn, theresource density determines the current food intake bythe consumers and hence indirectly governs consumergrowth, reproduction and thus foraging capacity. Itshould be noted that semi-chemostat dynamics in theresource has a stabelising effect on the populationdynamics compared to logistic growth dynamics (deRoos et al., 1990).

Analytically, the model at the population level can becast into the so-called cumulative formulation of physio-logically structured population models (Diekmann et al.,1993, 1995). This cumulative formulation uses measures(see appendix) to describe the state of the consumerpopulation at a specific time, a variable R to refer to theresource density and integral equations to describe theirdynamics. Essentially, the integral equations constitute away of bookkeeping the dynamics of all individualsmaking up the population. In the appendix we give a fullstatement and discussion of the model equations. Herewe restrict ourselves to sketching the representation ofthe model within the EBT (Escalator Boxcar Train)framework (de Roos, 1988 version 2.0; de Roos, 1993;1997; de Roos et al., 1992; see appendix for the formula-tion of the model in the EBT) that was used to carry outthe numerical simulations. The EBT numerical method isspecifically designed to handle the numerical integrationof the equations that occur in physiologically structuredmodels. The simulation results were verified by using dif-ferent precisions in the numerical integrations re-runningmany of the parameter combinations used.

The pulsed reproduction process ensures that thereexists a natural subdivision of the population intocohorts of individuals that all have the same age, revers-ible and irreversible mass. All individuals within a cohortare moreover assumed to grow at the same rate, i.e.individuals belonging to a given cohort do not diverge intheir allocation to reversible and irreversible masses. Thedynamics of every cohort can therefore be described by asystem of ordinary differential equations, which keepstrack of the number of individuals making up thecohort, their reversible mass and irreversible mass (seeappendix). The dynamics of the entire consumer popula-tion, both in terms of its abundance and its state, cannow be followed throughout the summer season bynumerically integrating the system of ordinary differen-tial equations for each cohort separately. In addition,changes in the resource population can be followedby numerical integration of the ordinary differentialequation for the resource dynamics that incorporates thesemi-chemostat growth and the total resource consump-tion. The latter equals the summed foraging rate over allcohorts. At the beginning of the season, new cohorts ofindividuals are added to the consumer population due tothe reproductive process. This addition implies that thenumber of differential equations describing the popula-tion dynamics is increased. At the same time, the currentvalue of the reversible mass in the cohorts of reproducingindividuals is reset, reflecting their investment intooffspring. Overall, the model simulations thus involve thenumerical integration of a (large) system of ordinary

275Ontogenetic Scaling and Population Dynamics

TABLE I

Definitions of Constants and Variables, and Their Numberical Values used in the Model

Symbol Value Unit Description References�comments

EnviromentT 90 day Length of season, summer in temperature regions.Lake 109 L Size of the lake.

ResourceR ind 10&9 L&1 Resource density.\ 0.5* day&1 Resource growth, inflow rate. *\-value of 0.05

gave qualitatively similar results as 0.5.K 5 1010 ind 10&9 L&1 Carrying capacity.

Consumerc ind 10&9 L&1g&2 Consumer i-state distribution.A varied 109 L day&1 Maximum attack rate, i.e., at size wo .: varied Size scaling exponent of the attack function.wo 8.71, 17.42, 34.84 g Effective body masses wo=x+qj x at which max-

imum attack rate is atained./1 4.0 10&6 day Constant used in the handling time function./2 8.19 10&5 day gx3 Constant used in the handling time function. Lessmark (1983)./3 0.68 Slope of decline in handling time at small con-

sumer sizes./4 1.15 10&3 g&1 Slope of increase handling time at large consumer

sizes.m1 0.033 g(1&m2) day&1 Metabolic constant.m2 0.77 Metabolic exponent. Schiemer et al. (1989), Kaufmann

(1990), Keckeis and Schiemer(1990), Kock et al. (1992).

k1 6.71 10&6 g Conversion factor; (assimilation efficiency(=0.75)&Specific Dynamic Action (=0.14))*prey mass (=1.1 10&5 g).

k2 0.5 Conversion efficiency; converting reversible massto offspring mass. Loss due to males not con-tributing to egg mass (30�350), and costs fortransforming somatic mass to gonad mass andgonad connective tissue (15�200).

Assimilation efficiency Elliott(1976), SDA Beamish (1974).

� New born size distribution, see also x0 . Threecohorts were used constituting 350, 540, and110, respectively.

x0 0.798, 0.804, 0.809 mg Irreversible masses of the three size cohorts ofnewborn individuals. Total newborn massbecomes w=x0(1+qj).

Tong (1986), Wieser et al. (1998)

xf 5.0 g Maturation threshold in terms of irreversiblemass.

x g Irreversible mass.y g Reversible mass.qj 0.742 Proportionality constant determining maximum

juvenile reversible mass.qa 1.0 Proportionality constant determining maximum

adult reversible mass (including gonads). Yields amaximum gonadosomatic index (GSI) of 150that compensates for the absence of males in themodel.

GSI slightly lower that that foundfor female roach Goldspink (1979),Papageorgiou (1979).

qs 0.2 Proportionality constant determining at whichlevel of reversible mass starvation mortality starts.

s 0.2 day&1 Constant for starvation mortality. This value of simplies that, without food, the number ofindividuals in a starving newborn cohort does notapproach zero before the reversible massapproaches zero.

+0 0.01, varied day&1 Constant background mortality rate.

276 Persson et al.

differential equations, which is extended in dimension atthe beginning of each season with a concurrent reset ofsome of the variables (see de Roos et al. 1992). Thedimension of the system is reduced whenever the numberof individuals in a given cohort has become negligible, atwhich time the differential equations for this particularcohort are removed.

3. MODEL PARAMETERIZATION

Our main interest was to investigate the effects ofthe size scaling of the attack function, a(w), on the pop-ulation dynamics of the consumer-resource system.Specifically we focused on, and hence varied, the param-eters affecting the size scaling exponent : and the maxi-mum attack rate A in the expression for a(w) (Eq. 5). To alesser extent, we also cover variation in the body size wo atwhich maximum attack rate occurs (Table I). Further-more, because background mortality turned out to havesubstantial effects on stability, we varied the mortality rate,+0 , over a wide range of parameter values.

Although metabolic demands and handling time areparameterised for the interaction between a plankti-vorous fish population of roach Rutilus rutilus and arelatively small-bodied zooplankton population asresource (Table I), the model should generally be inter-preted as describing the interaction between a small preyand its specialist size-structured predator. All parametervalues are given in Table I. All rates representingdynamic processes were scaled to a daily basis. Allparameter values related to energetics which were notvaried are based on a reference temperature of 19%C.With this temperature, a year (season) in the model isassumed to last 90 days. As mentioned above, we ignoreprocesses taking place during the winter season in thepresent paper. It should, however, be noted that a scalingdown of all rate parameters to the same extent to accountfor the lower temperatures in winter is analogous tochanging the length of the summer season. We havefound no qualitatively new types of dynamics whenvarying the season length in our model.

In the model, all individuals are assumed to be ahybrid between males and females. We therefore adjustedthe reproductive investment qa to be intermediatebetween males and females (Table I). All consumer off-spring are released and hatch simultaneously at the startof the biological season and the offspring start to feedimmediately after birth. The size distribution 9 of thenewborn individuals is assumed to be rather clumpedand for computational reasons we chose to represent 9by three discrete size cohorts.

4. DYNAMICS OF THE CONSUMER-RESOURCE MODEL

To delineate the parameter combinations of interestwithin the (:, A)-plane, we derived two necessary condi-tions for consumer persistence. First, under optimalfeeding conditions (i.e. when the resource density equalsK ) newborn consumers should not starve to death, whichrequires Eg(x0 , qsx0 , K )>0. Second, maturing con-sumers (individuals whose irreversible mass x passesthrough the threshold xf) must be able to build up atleast a minimum of gonads, requiring Eg(xf , qjxf , K)>0. On a semi-logarithmic scale these boundaries con-sist of two straight lines with different slopes (see Fig. 3).The first condition yields the boundary to the right inFig. 3 and the second the boundary at the bottom ofFig. 3. Numerical simulations verified that the actual per-sistence boundaries were close to the ones derived fromthe necessary conditions above. As wo is increased, theintercept of the persistence boundary of newborns isshifted upwards whereas the slope of the persistenceboundary of maturing individuals increases. The regionof consumer persistence in the (:, A)-plane thus becomessmaller as wo is increased.

Effects of :. The time series of consumer andresource density at the pre-reproductive census time inspring showed a wealth of dynamical patterns, rangingfrom stable fixed points, regular cycles, quasi-periodicdynamics on invariant closed curves to chaos. Thetransitions between stable fixed points and quasi-periodic dynamics indicated the occurrence of Hopf(Neimarck�Sacker) bifurcations, while transitionsbetween regular cycles and chaotic dynamics exhibiteda cascade of period-doubling bifurcations. To pointout the effects of the ontogenetic scaling of foraging rateon populations dynamics and its consequences at theindividual level, we will concentrate on the biologicalsignificance of three major types of dynamics that weremost frequently found: recruit-driven cycles, equilibriumand non-recruit juvenile-driven cycles, exemplified by thepopulation time series at three different : values (0.5,0.93, 1.10). For the default background mortality(+0=0.01) the recruit-driven and non-recruit juvenile-driven cycles had a cycle period of 7 years, although thecharacteristics of the dynamics do not depend on thecycle length (see the discussion on the effects +0 later inthis section).

The type of dynamics found at a specific combinationof : and A turns out to be strongly determined by thescaling of competitive ability with consumer size. As

277Ontogenetic Scaling and Population Dynamics

FIG. 3. Region in the (:, A)-plane bounded by the necessary conditions for consumer persistence. The lower and right-hand limits for persistenceof the consumer constitute the solutions for : and A, for which Eg=0 for a newborn individual of weight x0+qsx0 , and for a maturing individualof weight xf+qjxf , respectively, with the resource equal to its carrying capacity. Within the persistence region a broad classification of dynamicsobserved with semi-chemostat resource dynamics is given indicating the parameter combinations for which the 3 major types of dynamics (stable fixedpoint dynamics, recruit-driven and non-recruit juvenile-driven cycles) occurred and the parameter combinations for which other types of dynamicswere found, such as quasi-periodic dynamics, cycles with higher periodicity (due to a period-doubling bifurcation) and chaotic dynamics. Both therecruit-driven and the non-recruit juvenile-driven cycles are indicated as stable cycles. All stable cycles for : values below 0.93 are recruit-driven cycles,while the stable cycles for higher a values are non-recruit juvenile-driven. Parameter values: wo=17.42 g wet weight, K=5 1010 Lake&1 and +0=0.01day&1.

outlined in the introduction, competitive ability is afunction of the ability of an individual to consume theresource and its ability to withstand resource depression(depending on metabolic demands). Specifically, thecompetitive ability of an individual of a given size can becharacterised by the resource density at which it can justmeet maintenance demands, that is, the solution for Rfrom the equations Eg(x, qjx, R)=0. We will refer to thisspecific resource density as the maintenance resourcedensity. We did not study variation in the functions formetabolic demands and handling, hence only changes inthe parameters of the attack rate function (Eq. 5) willlead to changes in the competitive abilities of differentlysized individuals. For relatively low : values (:<0.9) themaintenance resource density is monotonically increas-ing with body mass, for intermediate : values (i.e. : closeto the intersection point, see Fig. 3) newborns and

maturing individuals have similar maintenance resourcedensities and for high : values (:>1) the maintenanceresource density first decreases rapidly with body massand thereafter increases (Fig. 4). The : value at the inter-section point of the existence boundaries of newbornsand maturing individuals is the value of : at which thenewborns and adults have similar competitive abilities(cf. Fig. 3). Hence, they cannot outcompete each other.At lower : values, the young-of-the-year have a highercompetitive ability than the adults, and vice versa athigher : values.

Regular, Recruit-Driven Cycles. The regular cyclesfound at low : values (:=0.5) commenced with a strongreproduction of a newly matured age class (Fig. 5). Thenumerical dominance of recruits over adults caused astrong reduction of the resource level immediately after

278 Persson et al.

FIG. 4. Maintenance resource density (the solution for R from the equations Eg(x, qjx, R)=0) as a function of effective body mass for three dif-ferent : values (0.50, 0.93, 1.10). Parameter values: w0=17.42 g wet weight and A=10&4 Lake day&1.

recruitment resulting in the adults starving to death(Fig. 5, 6). It was the combination of the newborn'srelatively high attack rate (due to a low :, see Fig. 2a)and the swamping effect of offspring due to the pulsedreproduction in combination with high fecundity thatmade the resource level decrease below the level theadults need to grow (Fig. 4, 5). The newborns were ableto grow at this low resource level due to their low main-tenance resource density (Fig. 4, 5). The resource densityremained below the adults maintenance resource densitysufficiently long to cause the adults to starve to death. Asa result, only cohorts belonging to one age class werepresent in the consumer population (Fig. 6). The growthof the individuals in this age class increased as theresource recovered from the initial decline. Theindividual growth rate and rate by which the resourcepopulation recovered was related to the disappearanceof consumers due to background mortality. Hence, thecycles tend to become shorter with increasing back-ground mortality (see below). When the juvenile cohortmatured and reproduced, their offspring caused a dramaticdecrease in the resource again leading starvation death of

adults. As a result, the maximum size reached by adultswas below wo (17.42 g) and recruitment only occurredevery 7th year (Fig. 5, 6). That is, the consumers were func-tionally semelparous despite being defined as iteroparous.

With increasing :, the competitive ability of the adultsrelative to the recruits increased as the recruits' ability toreduce the resource density declined. As a consequence,the adults managed to survive in the population forseveral years, although reproduction still only occurredevery 7 years. Finally, at still higher : values, the adultssucceeded in reproducing twice and the stable cyclescollapsed into low-amplitude, quasi-periodic dynamicsor a stable equilibrium depending on A and wo (Fig. 3).

Stable Equilibria. At intermediate :-values (:=0.93)close to the : where the persistence boundaries intersectedstable equilibria were observed (Fig. 5, 6). Annualrecruitment was considerably lower than recruitmentoccurring in the recruitment year in the cycles (Fig. 7).The equilibrium resource density was just above themaintenance resource density for the newborns allowingcoexistence between juvenile and adult cohorts (age

279Ontogenetic Scaling and Population Dynamics

280 Persson et al.

FIG

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FIG. 6. Weight distributions of the consumer population in each year for the population dynamics shown in Fig. 5. The upper row representsthe year in the cycle (if present) when recruitment occurs. Black histobars represent juveniles, grey histobars reproducing individuals and openhistobars mature, but non-reproducing individuals. Note that individuals may shrink between years for :=1.10. Cohorts with less than one individualhave been excluded.

281Ontogenetic Scaling and Population Dynamics

FIG. 7. The number of individuals recruited (whole histobars) and still alive 90 days after recruitment (black part of histobars) for the populationdynamics shown in Fig. 5 and 6. The values at the x-axis represent the different years in the 7 year cycles present for :=0.50 and :=1.10, where year1 refers to a year in which recruitment is successful.

FIG. 8. Bifurcation plot of the resource density (R) versus the slope of the attack rate (:). The graph is compiled from the resource densities atthe pre-reproductive census time in spring that were observed in the numerical simulations after transient dynamics had disappeared. The resourcedensities from these time series are represented with a dot plotted at the particular :-value. Other parameter values wo=17.42 g wet weight, A=10&4

Lake day&1, K=5 1010 Lake&1, and +0=0.01 day&1.

282 Persson et al.

classes) (Fig. 4, 5, 6). An important factor contributing tothe observed equilibrium conditions was that the main-tenance resource density was relatively independent ofbody size over a wide range of body sizes preventing anysize cohort from outcompeting the others (Fig. 4).Despite a resource density close to the maintenance limit,the consumers reached maturity after 6 years. The adultsinitially increased their reproductive output with size.After a few reproduction occasions, the gonads declinedin size and the adults reached a stage where no furtherreproductive investment was possible. The maximumadult size was also in this region below wo(17.42 g), but adultreproduction ceased because of the retardation in the in-crease in attack rate when body size approaches wo (Fig. 2a).No starvation mortality was present for any cohorts withinthe stable equilibrium region, hence the background mor-tality totally determined the life span of the consumers.

Regular, Non-recruit Juvenile-Driven Cycles. Thecycles at higher : values (:=1.1) to the right of the

FIG. 9. Bifurcation plot of the resource density (R) versus the maximum attack rate (A). The graph is compiled from the resource densities atthe pre-reproductive census time in spring that were observed in the numerical simulations after transient dynamics had disappeared. The resourcedensities from these time series are represented with a dot plotted at the particular A-value. Other parameter values: wo=17.42 g wet weight, :=0.7,K=5 1010 Lake&1, and +0=0.01 day&1.

parameter region where stable equilibria occurred weresimilar to the cycles to the left of that region in that astrong juvenile year class matured and gave rise to a newstrong recruit year class of similar size (Fig. 6, 7).However, due to the low attack rate of recruits (Fig. 2a),this strong year class could not depress the resource levelto the same extent as could a strong recruit year class inthe recruit-driven region where the attack rate of recruitswas higher (Fig. 5). As a result, the adults could obtainenough energy to reproduce in the following next twoyears (Fig. 7). As the maintenance resource density ofnewborn individuals is high, offspring born in the yearsfollowing a strong recruitment starved to death (Fig. 4,7). The reason for the reappearance of the cycle is thusthat at a certain :, previously recruited juveniles depressthe resource to a level at which the newborns cannot sur-vive despite the fact that that resource levels may allowthe adults to produce gonads. The total feeding rate ofthe successfully entered strong year class increases duringits first four years of life, because the individuals grow at

283Ontogenetic Scaling and Population Dynamics

a rate fast enough to more than compensate for thedecrease in their numbers due to background mortality.As a result, population consumption rate increased andthe resource level continued to decrease for four yearsbefore it started to recover (Fig. 5). This situation con-trasts to the one with recruit-driven cycles (Fig. 4, 5, 6).While the population dynamics at low : values wasdriven by recruits, the cycles at higher : values were thusdriven by juveniles already recruited preventing newrecruits to enter the system. This difference in dynamicsbetween low and high : values can be related to thedifferent forms of the relationship describing competitiveability as a function of body size (Fig. 4).

The adults reached larger sizes than under stableequilibrium conditions and the size often exceeded wo .Consequently, the largest adults suffered from starvationduring the years after a successful recruitment eventwhich is revealed by the change in mass distribution forthe largest cohorts during years 1�5 (Fig. 6). At stillhigher : values the stable cycles collapsed into chaoticdynamics, due to the reduced attack rate and hencecapacity of a previously recruited juvenile year class todepress the resource density allowing the following year'soffspring to be successfully recruited into the population(Fig. 3).

Figure 8 summarises these changes in the observedpopulation dynamics with increasing : values. This bifur-cation plot shows at every : value considered theresource densities at the pre-reproductive census time inspring that were observed in the numerical simulationsafter transient dynamics had disappeared. Stable, recruit-driven cycles with a period of 7 years persisted from low:-values (0.1) to : values slightly above 0.9 where thecycles collapsed into stable fixed point dynamics (Fig. 8).An alternative pattern of quasi-periodic dynamics waspresent from :r0.8, and bifurcated from the stable fixedpoint through a Hopf bifurcation at :r0.9. Increasing :above the stable parameter region more or less mirroredthe previously described pattern, but the quasi-periodicfluctuations increased faster in amplitude than for thelower : values. The stable 7-year cycles with recruitmentonce every 7 year reappeared and gave rise to quasi-periodic dynamics for : values above 1.14. Finally, theconsumer went extinct when reaching the boundary setby Eg(x0 , qsx0 , K)=0 (see Fig. 3).

Effects of A. Along a transect from low to high maxi-mum attack rate (A) at :=0.70, the observed dynamicschanged from a stable fixed point close to the lowerpersistence boundary set by Eg(xf , qjxf , K)=0 (seeFig. 3) to quasi-periodic dynamics, due to the occurrenceof a Hopf bifurcation (Fig. 9). With increasing A, these

quasi-periodic dynamics collapsed into stable 6 yearcycles, which underwent a cascade of period-doublingbifurcations and finally gave rise to chaotic dynamicsaround A=10&5 Lake days. The consumer populationconsisted of several coexisting age classes for all thesetypes of dynamics, although in the 6 years cycles theadult cohorts remaining in the population after a success-ful recruitment year were not capable of furtherreproduction due to their low reversible mass. Whenincreasing A even further the chaotic dynamics collapsedinto 7-year, recruit-driven cycles consisting of only oneage class. The transition of stable, recruit driven cyclesto chaotic dynamics around this value of A can be under-stood on the basis of the detailed description of therecruit-driven cycles discussed above. At this combina-tion of : and A values, the newborn individuals are notcapable of driving down the resource to a large extent. Asa consequence, juveniles earlier enter the phase wherethey are more limited by their handling time than by theirattack rate leading to an increased growth rate. Thestable cycles collapse when the consumers are able tomature at an age of 6 years as the per capita fecundity ofthese 6-year old adults is low. As a consequence, resourcedepression by newborns is low as well and allows foranother successful reproduction event of the adult cohortthe following year.

We found that the same pattern of bifurcations occurswhen varying the carrying capacity K of the resource (notshown here). Indeed, the population level equationspresented in the appendix can all be rescaled by dividing

FIG. 10. Regions in the (:, wo)-plane where different types ofdynamics (7-year cycles, quasi-periodic dynamics, stable fixed points)occurred. The boundaries are drawn on the basis of simulations with0.7�:�1.1, wo=8.71 (equal to the maturation size), 17.42 and34.84 g, A=10&4 Lake day&1, K=5 1010 Lake&1 and +0=0.01 day&1.

284 Persson et al.

both the resource and the consumer densities by K. Theparameters A and K in these scaled population equationsoccur always together as the product AK, implying thatequivalent changes in either A or K should induce thesame changes in the observed population dynamics.

Figure 3 summarises the regions in the (:, A)-plane wherethe different types of population dynamics occurred. Stablefixed point dynamics appeared along the lower boundary setby Eg(xf , qfxf , K)=0, which is a necessary condition foradult allocation to gonads, and along a minor part of theright boundary set by Eg(x0, qsx0, K)=0, which is anecessary condition for newborn growth. Stable fixed pointdynamics was moreover found in a region expanding abovethe intersection point of these two boundaries. Most promi-nent is the extensive set of parameter combinationsfor whichstable, recruit-driven cycles occur.

Effects of wo . The ontogenetic scaling of the foragingrate we are concerned with in this paper also includes theeffects of wo , the size at which the maximum attack rateis achieved. Changes in wo are negatively correlated withthe extent of the region in the (:, A)-parameter space, forwhich consumers can persist. Also the regions with thedifferent types of dynamics are affected by varying wo .

FIG. 11. Changes in the length of the recruit-driven cycles in relation to the background mortality rate +0 . The graph is compiled from the con-sumer densities at the pre-reproductive census time in spring that were observed in the numerical simulations after transient dynamics had disap-peared. The consumer densities from these time series are represented with a dot plotted at the particular +0-value. With decreasing mortality rateregular cycles go through a cascade of period doubling bifurcations to chaotic dynamics prior to a prolongation of the cycle period with an extra year.Parameter values: :=0.67, A=10&4 Lake day&1, K=5 1010 Lake&1 and wo=17.42 g.

With decreasing wo the region of consumer persistenceexpands mainly because the persistence boundary ofnewborns is shifted (the right-hand boundary in Fig. 3).wo has, moreover, pronounced effects on the regionswhere stable equilibrium and quasi-periodic dynamicsoccur (Fig. 10). As wo decreases towards the effectivebody size at maturation, xf+qjxf the parameter regionwhere stable equilibria occur becomes narrower andmoves towards larger : values. When wo equals theeffective body size at maturation, no stable equilibriumwith consumer-resource coexistence are found. As wo

decreases, quasi-periodic dynamics occur over a narrowerrange of : values, and the : value at which the quasi-periodic orbits collapse into recruit-driven cyclesincreases (Fig. 10). In contrast, the : value at which thequasi-periodic dynamics collapse into non-recruitjuvenile-driven cycles is less affected by a change in wo .When wo approaches the effective body size at matura-tion, the region with stable cycles persists in the entirerange of : values analysed (Fig. 10). This pattern resultsfrom the fact that a decrease in wo will decrease the scopefor adults to reproduce several times. The postulatedconnection between several reproductive events (func-tional iteroparity) and the presence of stable fixed point

285Ontogenetic Scaling and Population Dynamics

dynamics is confirmed by an analysis of the behaviourof a modified semelparous version of the model. In thislatter model, the cycles never collapse into stable fixedpoint dynamics independent of wo . In contrast, chaoticdynamics are still present at high : values suggestingthat the latter dynamics depend on the offspring sizedistribution (several cohorts present simultaneously)and not iteroparity per se.

Effects of +0 . The background mortality (+0) has amajor impact on the stability of the system and theperiodicity of the recruit or non-recruit juvenile-drivencycles. In general, increasing +0 increases stability. At asufficiently high +0 (0.033 for :=0.67), only stable fixedpoint dynamics occur (Fig. 11). Reducing +0 from thispoint leads to 2-year cycles, which, in turn undergo aseries of period doubling bifurcations resulting in chaoticdynamics. At still lower +0 , the chaotic dynamics changeinto 3-year cycles. This pattern repeats itself with furtherreduction in +0 . After each phase of regular cycles, acascade of period doubling bifurcations and chaoticdynamics, the newly appearing cycle is one year longerthan the previous one (Fig. 11). The dominating periodin the chaotic dynamics is the same as for the regularcycle.

Juveniles and adults can coexist whenever the popula-tion exhibits stable fixed point dynamics, 2-year cyclesor 3-year cycles with +0 larger than 0.022. With 2-yearcycles, reproduction occurs every year. With 3-yearcycles, reproduction is absent for one of the three years athigh +0 , whereas reproduction only occurs once duringthe 3-year cycle at low +0 . As a consequence, non-recruitjuvenile-driven cycles do not occur with a period of 2 and3 years. For regular cycles longer than 3 years, onereproductive occasion during a cycle is the rule. Allcycles with a period longer than 3 years have similarcharacteristics as the 7-year cycles discussed in detailabove.

5. DISCUSSION

5.1. Ontogenetic Scaling and PopulationDynamics

Changes in individual size and morphology occur inalmost all animal taxa and have profound consequencesfor the foraging capacity and metabolic requirements oforganisms in different stages of their life. The effectsof such ontogenetic scalings of foraging rate andmetabolism on the outcome of competitive interactionsbetween different organisms have been studied at the

individual level, in, for example, anurans (Werner, 1994).To our knowledge, our paper is the first to address theconsequences of these size-specific, individual-leveldifferences at the population level in the case of pulsed, asopposed to continuous reproduction.

Our analysis shows major effects of the ontogeneticscaling of attack rate on the population dynamics, oftenleading to cohort-driven population fluctuations. Theinvestigated model is based on size-dependent energygathering (gain) in terms of size-dependent attack rateand handling time and size-dependent spending (costs)in terms of a size-dependent metabolism. It is the sizedependency of the attack rate and metabolic functions(see Section 4) that leads to differences in competitiveabilities between different life stages of the consumers.The value of the exponent for the size scaling of meta-bolic demands is relatively conservative between taxa(Peters, 1983; Calder, 1984; Werner, 1988). Therefore, itis mainly differences in the size-related components of theattack rate investigated in this paper that will leadto a different ontogenetic scaling of the individual'scompetitive ability.

The differences in competitive ability between differentsize classes lead to dynamic patterns which, over a largeregion of the parameter space, are characterised bydominance and suppression by a single cohort. Stableequilibria are, apart from close to the persistence boun-daries or when background mortality is high, only foundwhen the competitive ability is similar throughout theentire life cycle of an individual, otherwise populationcycles are likely to occur. These cycles are of two formsthat are nonetheless based on the same principle: onelifestage of individuals completely outcompeting theothers. Recruit-driven cycles occur when the minimumresource density to just meet maintenance increasesmonotonically with body mass. In contrast, non-recruitjuvenile-driven dynamics occur when this criticalresource density is lowest for an intermediate body mass(Fig. 4, 5).

A number of empirical cases can be found in theliterature that suggest the occurrence of recruit-drivendynamics. Regular or possibly damped populationoscillations have, for example, been found in several fishspecies (Cryer et al., 1986; Townsend 1989; Townsendet al., 1990). These species include roach, for which weparameterized the functions for metabolic demands,handling time, fecundity and offspring size. Simulation ofan age-based model and empirical studies of the popula-tion dynamics of roach suggest that the presence of cyclesin this species depends heavily on a low adult survivalrate due to high background and starvation mortality,because a high survival rate will allow individuals to

286 Persson et al.

reach a size where they shift to other diets. This, in turn,will decrease the possibility for density dependent effectsof recruits on adult fecundity to occur (Townsend et al.,1990). Our results suggest that the coupling between ahigh survival rate of adults and the absence of regularcycles may arise because adults then have more than asingle successful reproduction event during their life.Accordingly, the absence of stable fixed point dynamicsand the enlargening of the part of parameter space withcycles following a decrease in wo are related to the factthat the scope for adult mass allocation to gonads isdiminished.

A classical example of population oscillations in fish isfound in zooplanktivorous cisco (vendace) (Coregonusalbula) (Ja� rvi,1930; Sva� rdson, 1956; Aass, 1972; Ustyogov,1972; Viljanen, 1986; Hamrin and Persson, 1986;Saloja� rvi, 1987; Helminen et al., 1993; Auvinen, 1994).The cycle length in these populations are often 2 yearsbut an oscillation period of 3 years or more are alsopresent (Svardson 1956, Aass 1972). Empirical evidencethat these cycles were recruit-driven comes from (i) thestrong impact that a strong recruiting age class had onthe zooplankton resource, (ii) the depression of thezooplankton resource which caused starvation in oldercisco whereas recruits (young-of-the-year) cisco alwayshad a positive growth, and (iii) the positive correlationbetween mortality rate and body size during periods withhigh resource limitation (Hamrin and Persson, 1986).There was also a positive correlation between theabundance of cisco, one year old and older, andzooplankton density which is not expected if the cycleswere non-recruit juvenile-driven (cg. Fig. 5).

Single generation cycles caused by dominance of onecohort and suppression of adult fecundity as is the casein the recruit-driven cycles have also been found inDaphnia. Models of individual behaviour of Daphniashow that Daphnia foraging capacity scales to body masswith a slope which lies within the values (0.56�0.67)where recruit-driven dynamics is predicted by our model(Kooijman and Metz, 1984; McCauley et al., 1990;Gurney et al., 1990; de Roos et al., 1992; Kooijman,1993). Additionally, analyses of laboratory cultures,stock tanks and lake populations of Daphnia corroboratethe existence of single generation cycles driven bydominance and suppression of adult fecundity (Murdochand McCauley, 1985; McCauley and Murdoch, 1987,1990; McCauley et al., 1988; McCauley, 1993). McCauleyand Murdoch (1987) also mention other systems whichmay exhibit single generation cycles driven by domi-nance and suppression.

The non-recruit juvenile-driven cycles as well as recruit-driven cycles in our analysis were specific in containing

only one age class of juveniles. The available data citedabove and a priori expectations advanced by Wilson(1975) suggest that the scaling exponent for searchcapacity to body mass in organisms is not likely toexceed 1 if only exploitative competition is considered(see also reviews of empirical data in Peters, 1983;Calder, 1984). Adding size-dependent interferencecompetition is expected to increase the competitiveability (foraging rate) of larger individuals versus smallerones, which is likely to increase the possibilities for non-recruit juvenile-driven cycles. Periodic cicadas in NorthAmerica form one potential example of non-recruitjuvenile-driven cycles. Although other mechanisms likepredation (Hoppensteadt and Keller, 1976) have beenadvanced to explain the cycles in cicadas, Bulmer (1977)showed theoretically that despite the importance ofpredator satiation for synchronising the cicada cycles,predation can not cause them. Instead, he suggested thatthe cycles were driven by cohort competition wherelarger (older) individuals were competitively superior tothe smaller ones as a size effect (interference) and becausethe larger individuals are most likely to be initially inposition in the roots (pre-emptive competition). Thisargumentation is supported by the results of our con-sumer-resource model in that the only situation whenrecruits could be prevented to enter a system with regularcycles is the situation with non-recruit juvenile-drivencycles (Fig. 7, 11).

The ontogenetic scaling of foraging rate to body sizevaries considerably between taxa, but part of thisvariation has been suggested to consist of systematicdifferences between functional groups (Wilson, 1975;Sebens, 1982; Werner, 1988; Lundberg and Persson,1993). As a growing body of literature is becomingavailable on the size scaling of foraging rate andmetabolic demands within taxa (Werner and Gilliam,1984; Sebens, 1982; Werner, 1988, 1994; Mittelbach,1981; Kooijman and Metz 1984; Gurney et al., 1990,McCauley et al., 1990; Jones, Richards and Southern 1992:Kooijman, 1993), it will be possible to analyse whethersuch systematic differences in the ontogenetic scaling offoraging rate can actually be corroborated. This would, inturn, allow predictions to be advanced regarding func-tional group dependent population dynamics.

5.2. Population Oscillations versus Stability

According to our analysis, one would expect manyspecies to exhibit large population fluctuations which isa situation not commonly seen in nature. In this sectionwe will consider some factors which will lead to largepopulation fluctuations and discuss some other factors

287Ontogenetic Scaling and Population Dynamics

that have the potential to stabilise the dynamics. Theoccurrence of the cycles depends heavily on a sufficientlyhigh reproduction pulse to allow either the recruiters toreduce the resource more than the older cohorts(recruiter driven- dynamics) can, or non-recruit juvenilesto reduce the resource more than recruiters (non-recruitjuvenile-driven dynamics) are capable of. The seasonalreproduction per se is not a necessary requirement for thecohort-driven cycles to be present (see de Roos et al.,1990 for a continuous case), but these are more easilyintroduced and far more prominent in systems withdiscrete reproduction than in systems with continuousreproduction. When regular single generation cyclesoccur, their periodicity, TC , must fulfil TC=Ln(F)�(+0Y),where F is the number of female offspring per adultfemale, Y is the season length in days, and +0 is thebackground daily mortality rate. This equation followsfrom the fact that on average only one of a parent'soffspring must survive to reproduction to give rise tostable cycles. The regular cycles occur when the adultscan build up sufficient fecundity, otherwise irregular orfixed point dynamics prevail (Fig. 11). Thus, for regularcycles to occur, the energy allocation must providenecessary space for gonad mass in combination withsmall offspring as in our model.

The periodicity of the cycles depends mainly onbackground mortality (see Fig. 11) while fecundity hasless impact. On the basis of our results some effects of sizedependent mortality can be outlined. For example, highmortality during the first few weeks after birth is expectedto exert a stabilising impact on the dynamics, sincethe few surviving offspring will have a low impact onthe resource and allow other cohorts to coexist. Sizedependent mortality is commonly observed in naturalpopulations (Litvak and Leggett, 1992; Fuijman, 1994;Fagan and Odell, 1996) and may be one important mecha-nism explaining the rare occurrence of the oscillatingdynamics that are expected with a constant, size inde-pendent mortality. In such situations, the dynamicalconsequences of differences in competitive ability will notoccur due to the absence of the necessary numericaleffects. This will also be the case when the adults have alow fecundity.

Another potentially stabilising mechanism may be thepresence of ontogenetic niche shifts (e.g. the presence ofa second resource in the same or in another habitat).Changes in habitat use are likely to also affect the scalingof foraging rate to body size. For example, a predatorsearching for prey in a three dimensional environment(e.g. a pelagic habitat) is expected to have a higher sizescaling of the foraging capacity to body size than thesame predator searching for prey in a two dimensional

environment (e.g. a benthic habitat). Correspondingly,the allometric relationship between encounter rate andbody weight for bluegill sunfish (Lepomis macrochirus)decreased from 0.67 for pelagic prey, to 0.23 for vegeta-tion prey to 0.09 for sediment prey (Mittelbach, 1981). Inthe context of ontogenetic niche shifts, it has also beensuggested that there exists a connection between theparameters of the attack rate function and the extent towhich organisms undergo ontogenetic niche shifts dueto ontogenetic trade-off costs (Werner and Gilliam,1984; Werner, 1988; Persson and Greenberg, 1990). Toinvestigate the population dynamical consequences ofthese aspects in a one consumer��two resources system isan interesting topic for future research.

Acknowledgments The research has been sponsoredby grants from the Swedish Council for Forestry andAgricultural Research to Mats Gyllenberg, KjellLeonardsson and Lennart Persson, The Bank of SwedenTercentenary Foundation to Mats Gyllenberg and theSwedish Natural Science Research Council to LennartPersson. Valuable comments on the manuscript weregiven by Bill Murdoch and an anonymous reviewer.

APPENDIX

Cumulative Formulation of the PopulationModel

Here we give a description of the equations, definingthe dynamical system at the population level. Theirreversible mass of an individual will be indicated by XR .The function XR(t, t0 , (x, y)) specifies the amount ofirreversible mass at time t of an individual which at timet0 had an amount of irreversible and reversible massequal to x and y, respectively. Therefore, the increase ofirreversible mass during the time period between t0 andt of such an individual equals

XR(t, t0 , (x, y))&x

For consistency reasons it is clear that

XR(t0 , t0 , (x, y))=x.

Analogously, the reversible mass of an individual will beindicated by YR . The function YR(t, t0 , (x, y)) specifiesthe amount of reversible mass at time t of an individualwhich at time t0 had an amount of irreversible and

288 Persson et al.

reversible mass equal to x and y, respectively. Again, it isclear that for consistency reasons:

YR(t0 , t0 , (x, y))= y

On the basis of the model components at the individuallevel discussed in the main text, the equations describingthe changes in the individual state can now be derived asfunctions of irreversible mass XR , reversible mass YR , thetime t, and resource density R. The growth in consumerirreversible mass is described by the following ordinarydifferential equation and initial condition:

}(XR , YR) Eg(XR , YR , R(t)),ddt

XR={ if Eg(XR , YR , R(t))>0

0, if Eg(XR , YR , R(t))�0,

XR(t0 , t0 , (x, y))=x,

where the }-function represents the mass allocation rule(Eq. 9) and the Eg -function (see Eq. 8) represents thenet resource uptake. The above equation reflects ourassumption that a fraction } of the net energy productionis channelled to the increase in irreversible mass, unlessnet production is negative, in which case the amount ofirreversible mass does not change. The correspondingequations for the growth in reversible mass are:

ddt

YR={(1&}(XR , YR)) Eg(XR , YR , R(t)),

if Eg(XR , YR , R(t))>0Eg(XR , YR , R(t)),

if Eg(XR , YR , R(t))�0,

YR(t0 , t0 , (x, y))= y.

which reflect our assumption that the remaining part(1&}) of net energy production is channelled to anincrease in reversible mass. All deficits in net energyproduction are deducted from the reversible mass.

The probability that an individual with an amountof irreversible and reversible mass equal to x and y,respectively, at time t0 will survive at least until time t,will be indicated with the function FR(t, t0 , (x, y)).This probability is given by the following equation as afunction of time, the background mortality rate +0 , andthe state dependent starvation mortality function+s(x, y):

FR(t, t0 , (x, y))

=exp \&+0(t&t0)&|t

t0

+s(XR({, t0 , (x, y)),

YR({, t0 , (x, y))) d{+ .

The first term in the exponent relates to the backgroundmortality over the time interval from t0 to t, while thesecond term is the integrated result of starvation mor-tality during this period.

The dynamics of the consumer population within aseason is given by the equation:

c(t)(w)=|[(x, y) # 0 : (XR(t, t0 , (x, y)), YR(t, t0 , (x, y))) # w]

FR(t,

t0 , (x, y)) c(t0)(dxdy).

Here, c(t)(w) represents the number of consumers at timet with an individual state (e.g. a combination of reversibleand irreversible mass) within the set w. Similarly, c(t0)(dxdy)can be interpreted as the number of consumersthat at time t0 have a combination of irreversible andreversible mass within a small subset dxdy. The integralexpression thus reflects the simple bookkeeping fact thatall individuals ending up at time t with an individualstate within w are those that started out at time t0 with anindividual state somewhere in the individual state space0, in addition survived during the time period from t0 tot and grew to end up with an individual state in w.

The resource balance within a season is a function ofthe initial resource density R(t0), the resource inflow rate\, the carrying capacity K, the consumer feeding ratefunction #(x, y), its survival probability FR(t, t0 , (x, y)),and the consumer density. Its dynamics is given by:

R(t)=R(t0)+\K(t&t0)&\ |t

t0

R({) d{

&|0|

t

t0

#(XR({, t0 , (x, y)), YR({, t0 , (x, y)),

R({)) FR({, t0 , (x, y)) d{c(t0)(dxdy).

This equation specifies that the change in resourcedensity over the period from t0 to t equals the inflow termof the assumed chemostat dynamics \K(t&t0), minus itsoutflow over this period, which equals \ � t

t0R({) d{,

minus the total amount of consumption by the consumerpopulation over the period. This total amount ofresource consumed is given by the last integral in the

289Ontogenetic Scaling and Population Dynamics

equation above. The integrand in this term represents theproduct of the number of consumers at time t0 with anindividual state in a small subset dx dy, their survivalprobability FR({, t0 , (x, y)) up to time { and their feedingrate #(x, y) at this last point in time. The integral henceaccumulates the feeding contributions over the periodfrom t0 to t of all consumer individuals that at time t0 hada combination of irreversible and reversible massanywhere within the individual state space 0.

The consumer population after each reproductive event,which is assumed to take place at the discrete times nT,is related to its state just before the event by the functionb(x, y). This function describes not only the productionof offspring but also the instantaneous change inthe reversible mass of the reproducing adults. Hence, theconsumer population before and after the reproductiveevent fulfils the equation:

c(nT+)=|0

b(x, y) c(nT&)(dxdy), n=1, 2, ...,

where b(x, y) is given by:

b(x, y)=k2( y&qjx)+ 9+$(x, y&( y&qj x)+) .

The first term in this expression reflects the assumptionthat adults invest all reversible mass that they accumu-lated on top of the amount qj x into the production ofoffspring. The conversion efficiency of this investmentequals k2 and 9 indicates the size distribution of theneonates. The second term in this equation contains thedelta function $(x, y) which specifies that all adults whichdo invest in reproduction have an immediate decreaseof reversible mass from their current amount y to thevalue qjx.

dN i

dt=&(+0++s(X i

R , Y iR)) N i

{dX iR

dt={}(X i

R , Y iR) Eg(X i

R , Y iR , R)

0if Eg(X i

R , Y iR , R)>0

if Eg(X iR , Y i

R , R)�0 i=1 } } } MdY i

R

dt={(1&}(X i

R , Y iR)) Eg(X i

R , Y iR , R)

Eg(X iR , Y i

R , R)if Eg(X i

R , Y iR , R)>0

if Eg(X iR , Y i

R , R)�0

dRdt

=\(K&R)& :M

i=1

#((1+qj) X iR , R) N i (t)

Starting at t=0, which is assumed to be the beginning of a growing season, this system of ODEs is integrated numeri-cally until t=T, corresponding to the end of the growing season. At t=T reproduction takes place, described by thefollowing mapping (T& and T+ in the next equations refer to the value of a variable directly before and after thereproduction event, respectively):

Escalator Boxcar Train Formulation of thePopulation Model

To study the dynamics of the population modelnumerically, its equations were recast into the EBT(Escalator Boxcar Train) formulation (de Roos et al.,1992). Because of the pulsed reproduction the consumerpopulation is naturally divided into distinct cohorts ofindividuals. All individuals within a single cohort areborn with the same irreversible and reversible mass. Theyalso remain identical to each other throughout theirlifetime because growth is deterministic. The consumerpopulation hence consists of a finite number of cohorts ofidentical individuals.

A growing season is assumed to last T time units fromt=(n&1) T to t=nT (n=1, 2, 3...). Within eachgrowing season the resource density, the number ofindividuals in each consumer cohort, their irreversibleand reversible mass all change continuously. Reproduc-tion takes place at the beginning of the growing season atthe discrete times t=nT. A discrete map describes thestate of the consumer population after the reproductionevent as a function of its prereproduction state. The EBTformulation of the population model is hence a combina-tion of a continuous-time process, specified in terms ofordinary differential equations, and a discrete-timeprocess, specified as a mapping (cf. model C, ``the food-dependent, size-structured model,'' in de Roos et al.,1992).

Let N i (t) denote the number of individuals in cohorti at time t, and X i

R(t) and Y iR(t) the irreversible and

reversible mass, respectively, of these individuals. Letthere be M cohorts of individuals present. The within-season dynamics of all consumer cohorts and theresource density R(t) is then governed by the followingsystem of (3M+1) ODEs:

290 Persson et al.

N1(T +)=0.35B(T&) N 2(T+)=0.54B(T&) N3(T +)=0.11B(T&)

X1R(T +)=0.798 X 2

R(T+)=0.804 X 3R(T+)=0.809

Y1R(T +)=0.592 Y 2

R(T+)=0.597 Y 3R(T+)=0.6

N i(T+=N i&3(T&)

X iR(T +)=X i&3

R (T&){ qj X i&3R (T&) if X i&3

R (T&)>xf i=3 } } } M+3

Y iR(T +)={ and Y i&3

R (T&)>q j X i&3R (T&)

Y i&3R (T&) otherwise

The first 3 cohorts represent newborn consumers, of which 350, 540 and 110 have an initial irreversible massx0=0.798, 0.804, and 0.809, respectively. This distribution represents the newborn size distribution 9 (see Table 1). Thereversible mass of newborn individuals equals qj times their irreversible mass. The function B(T&) represents the totalpopulation reproduction rate, given by:

B(T&)=k2

(1+qj) x� 0

:M

i=1{Y i

R(T &)&qjX iR(T &)

0if X i

R(T&)>xf and Y iR(T&)>qjX i

R(T&)otherwise

Here x� 0 refers to the average irreversible mass of anewborn individual, calculated from the newborn sizedistribution 9 (see Table I) as a weighted average overthe three initial masses x0=0.798, 0.804, and 0.809,respectively. The last equations of the reproductionmapping constitute a renumbering of all other cohortsand a reset of the reversible mass of adult, reproducingindividuals to qj times their irreversible mass, reflectingtheir contribution to the population birth rate B(T&).

The same mapping is applied at each of the reproduc-tion events following t=T+ (at t=nT, n=2, 3...) whilethe dynamics from one reproduction event to the next(from t=(n&1) T+ to t=nT&) is always obtained bythe numerical integration of the ODEs governing thewithin-season dynamics. In principle each reproductionevent may thus lead to the addition of 3 new cohorts tothe consumer population. The number of cohorts makingup the consumer population is decreased if the number ofindividuals in a particular cohort has become negligiblysmall. Such cohorts are removed from the population.

REFERENCES

Aass, P. 1972. Age determination and years class fluctuations of ciscoe(Coregonus albula) in the Mjo� sa hydroelectric reservoire, Rep. Inst.Freshw. Res. Drottningholm 52, 4�21.

Auvinen, H. 1994. ``Intra- and Interspecific Factors in the Dynamicsof Vendace (Coregonus albula (L.)) Populations,'' Ph.D. thesis,University of Helsinki, Finland.

Beamish, F. W. H. 1974. Apparent specific dynamic of largemouth bass,Micropterus salmoides, J. Fish. Res. Board Can. 31, 1763�1769.

Beck, J. E., and Gitter, M. J. 1983. Effect of fish size on the reactivedistance of bluegill (Lepomis macrohirus) sunfish, Can. J. Fish.Aquat. Sci. 40, 162�167.

Broekhuizen, N., Gurney, W. S. C., Jones, A., and Bryant, A. D.1994. Modelling compensatory growth, Funct. Ecol. 8, 770�782.

Bulmer, M. G. 1977. Periodical insects, Am. Nat. 11, 1099�1117.Calder, W. A., III 1984. ``Size, Function and Life History,'' Harvard

Univ. Press, Cambridge.Caswell, H., and John, A. M. 1992. From the individual to the popula-

tion in demographic models, in ``Individual-Based Models andApproaches in Ecology��populations, Communities and Eco-systems'' (D. L. DeAngelis and L. J. Gross, Eds.), pp. 36�61,Chapman 6 Hall, New York.

Cryer, M., Peirson, G., and Townsend, C. R. 1986. Reciprocal interac-tions between roach, Rutilus rutilus, and zooplankton in a smalllake: Prey dynamics and fish growth and recruitment, Limnol.Oceanogr. 31, 1022�1038.

Cushing, J. M., and Li, J. 1991. Juvenile versus adult competition,J. Math. Biol. 29, 457�473.

Cushing, J. M., and Li, J. 1992. Intra-specific competition and densitydependent juvenile growth, Bull. Math. Biol. 54, 503�519.

DeAngelis, D. L., and Gross, L. J., Eds. 1992. ``Individual-BasedModels and Approaches in Ecology��Populations, Communitiesand Ecosystems,'' Chapman 6 Hall, New York.

DeAngelis, D. L., and Rose, K. A. 1992. Which individual-based approach is most appropriate for a given problem, in``Individual-Based Models and Approaches in Ecology��populations, Communities and Ecosystems'' (D. L. DeAngelis andL. J. Gross, Eds.), pp. 67�87, Chapman 6 Hall, New York.

De Roos, A. M. 1988. Numerical methods for structured populationmodels: The escalator boxcar train, Num. Meth. Part. Diff.Equations 4, 173�195.

De Roos, A. M. 1993. The escalator boxcar train��User's manualversion 2.0, Department of Pure and Applied Ecology, University ofAmsterdam, Amsterdam, Netherlands.

De Roos, A. M. 1997. A gentle introduction to physiologicallystructured population models, in ``Structured Population Modelsin Marine, Terrestrial and Freshwater Systems'' (S. Tuljapurkar

291Ontogenetic Scaling and Population Dynamics

and H. Caswell, Eds.), pp. 119�204, Chapman 6 Hall, NewYork.

De Roos, A. M., Metz, J. A. J., Evers, E., and Leipoldt, A. 1990. A sizedependent predator-prey interaction: Who pursues whom?,J. Math. Biol. 28, 609�643.

De Roos, A. M., Metz, J. A. J., and Diekmann, O. 1992. Studying thedynamics of structured populations models: A versatile techniqueand its application to Daphnia, Am. Nat. 139, 123�147.

Diekmann, O., Gyllenberg, M. M., and Thieme, R. H. 1993. Perturbingsemigroups by solving Stieltjes renewal equation, Diff. IntegralEquations 6, 155�181.

Diekmann, O., Gyllenberg, M. M., and Thieme, R. H. 1993. Perturbingevolutionary systems by cumulative outputs an step responses, Diff.Integral Equations 8, 1205�1244.

Ebenman, B. 1988. Dynamics of age- and size-structured populations:Intraspecific competition, in ``Size-Structured Populations: Ecologyand Evolution'' (B. Ebenman and L. Persson, Eds.), pp. 127�139,Springer-Verlag, Heidelberg.

Ebenman, B., and Persson, L. 1988. Dynamics of age- and size-struc-tured populations��an overview, in ``Size-Structured Populations:Ecology and Evolution'' (B. Ebenman and L. Persson, Eds.),pp. 3�9, Springer-Verlag, Heidelberg.

Elliott, J. M. 1976. The growth rate of brown trout (Salmo trutta) fedon reduced rations, J. Anim. Ecol. 44, 823�842.

Fagan, W. F., and Odell, G. M. 1996. Size-dependent cannibalism inpraying mantids: using biomass flux to model size-structuredpopulations, Am. Nat. 147, 230�268.

Fuijman, L. A. 1994. The interplay of ontogeny and scaling in theinteractions of fish larvae and their predators, J. Fish Biol. 45, 55�79.

Goldspink, C. R. 1979. The population density, growth rate andreproduction of roach (Rutilus rutilus L.) in Tjeukemeer, theNetherlands, J. Fish Biol. 15, 473�498.

Gurney, W. S. C., McGauley, E., Nisbet, R. M., and Murdoch, W. W.1990. The physiological ecology of Daphnia: A dynamic model ofgrowth and reproduction, Ecology 71, 716�732.

Hamrin, S. F., and Persson, L. 1986. Asymmetrical competitionbetween age classes as a factor causing population oscillations in anobligate plaktivorous fish, Oikos 47, 223�232.

Helminen, H., Auvinen, H., Hirvonen, A., Sarvala, J., and Toivonen, J.1993. Year-class fluctuations of vendace (Coregonus albula) in LakePyha� ja� rvi, Southwest Finland, during 1971�1990, Can. J. Fish.Aquat. Sci. 50, 925�931.

Hoyle, J. A., and Keast, A. 1987. The effect of prey morphology and sizeon handling time in a piscivore, the largemouth bass (Micropterussalmoides), Can. J. Zool. 65, 1972�1977.

Hyatt, K. B. 1979. Feeding strategy, in ``Fish Physiology'' (W. S. Hoar,D. J. Randall, and J. R. Brett, Eds.), Vol. VIII, pp. 71�119, AcademicPress, New York.

Hoppensteadt, F. C., and Keller, J. B. 1976. Synchronization ofperiodical cicada emergences, Science 194, 335�337.

Jones, H. D., Richards, O. G., and Southern, T. A. 1992. Gill dimen-sions, water pumping rate and body size in the mussel Mytilus edulis,J. Exp. Mar. Biol. Ecol. 155, 213�237.

Ja� rvi, H. 1930. On the fluctuations in abundance exhibited by the stocksof lake herring (Coregonus albula) in Lake Keitle, Finland, J. Cons.Explor. Perm. Internat. Mer. 65, 182�188.

Kaufmann, R. 1990. Respiration cost of swimming in larval andjuvenile cyprinids, J. Exp. Biol. 150, 343�366.

Keckeis, H., and Schiemer, F. 1990. Consumption, growth and respira-tion of bleak Alburnus alburnus (L.) and roach, Rutilus rutilus (L.),during early ontogeny, J. Fish Biol. 36, 841�851.

Koch, F., Wieser, W., and Niedersta� tter, N. 1992. Interactive effects ofseason and temperature on enzyme activities, tissue and wholeanimal respiration in roach, Rutilus rutilus, Envir. Biol. Fish 33,73�85.

Kooijman, S. A. L. M. 1973. ``Dynamic Energy Budgets in BiologicalSystems,'' Cambridge Univ. Press, Cambridge.

Kooijman, S. A. L. M., and Metz, J. A. J. 1984. On the dynamics ofchemically stressed populations: The deduction of populationconsequences from effects on individuals, Ecotox. Environ. Saf. 8,254�274.

Lessmark, O. 1983. ``Competition between Perch (Perca fluviatilis) andRoach (Rutilus rutilus) in South Swedish Lakes,'' Ph.D. thesis, Univ.of Lund, Sweden.

Litvak, M. K., and Leggett, W. C. 1992. Age and size-selectivepredation on larval fishes: The bigger-is-better hypothesis revisited,Mar. Ecol. Progr. Ser. 81, 13�24.

Lundberg, S., and Persson, L. 1993. Optimal body size and resourcedensity, J. Theoret. Biol. 164, 163�180.

McCauley, E. 1993. Internal versus external causes of dynamics in afreshwater plant herbivore system, Am. Nat. 141, 428�439.

McCauley, E., and Murdoch, W. W., 1987. Cyclic and stable popula-tions: Plankton as a paradigm, Am. Nat. 129, 97�121.

McCauley, E., Murdoch, W. W., and Watson, S. 1988. Simple modelsand variation in plankton densities among lakes, Am. Nat. 132,383�403.

McCauley, E., Murdoch, W. W., Nisbet, R. M., and Gurney, W. S. C.1990. The physiological ecology of Daphnia: Development of amodel of growth and reproduction, Ecology 71, 703�715.

Metz, A. J., and Diekmann, O. 1986. ``The dynamics of physiologicallystructured populations,'' Lecture Notes in Biomathematics, Vol. 38,Springer-Verlag, Heidelberg.

Metz, A. J., De Roos, A. M., and Van Den Bosch, F. 1988. Populationmodels incorporating physiological structure: A quick survey of thebasic concepts and an application to size-structured populationdynamics in waterfleas, in ``Size-Structured Populations: Ecologyand Evolution'' (B. Ebenman and L. Persson, Eds.), pp. 106�126,Springer-Verlag, Heidelberg.

Mittelbach, G. G. 1981. Foraging efficiency and body size: Astudy of optimal diet and habitat use by bluegills, Ecology 62,1370�1386.

Murdoch, W. W., and McCauley, E. 1985. Three distinct types ofdynamics behaviour shown by a single planktonic system, Nature316, 628�630.

Murdoch, W. W., and McCauley, E. 1990. Predator-prey dynamics inenvironments rich and poor in nutrients, Nature 343, 455�457.

Nisbet, R. M., and Onyiah, L. C. 1994. Population dynamics conse-quences of competition within and between age classes, J. Math.Biol. 32, 329�344.

Noakes, D. L. G., and Godin, J. -G. J. 1988. Ontogeny of behavior andconcurrent developmental changes in sensory systems in teleostfishes, in ``Fish Physiology'' (W. S. Hoar and D. J. Randall, Eds.),Vol. XIB, pp. 345�395, Academic Press, New York.

Nornberg, R. A1 . 1988. Self-thinning of plant populations dictated bypacking density and individual growth geometry and relationshipsbetween animal population density and body mass governed bymetabolic rate, in ``Size-Structured Populations: Ecology andEvolution'' (B. Ebenman and L. Persson, Eds.), pp. 259�279,Springer-Verlag, Heidelberg.

Persson, L. 1985. Assymmetrical competition: Are larger animalscompetitively superior?, Am. Nat. 126, 261�266.

Persson, L. 1987. The effects of resource availability and distribution onsize class interactions in perch Perca fluviatilis, Oikos 48, 148�160.

292 Persson et al.

Persson, L. 1987. Assymmetries in competitive and predatory interac-tions in fish populations, in ``Size-Structured Populations: Ecologyand Evolution'' (B. Ebenman and L. Persson, Eds.), pp. 203�218,Springer-Verlag, Heidelberg.

Persson, L., and Greenberg, L. A. 1990. Juvenile competitive bottlenecks:The perch (Perca fluviatilis)-roach (Rutilus rutilus) interaction,Ecology 71, 44�56.

Peters, R. H. 1983. ``The Ecological Implications of Body Size,''Cambridge Univ. Press, Cambridge.

Polis, L. A. 1991. Complex trophic interactions in deserts: An empiricalcritique of food web theory, Am. Nat. 138, 123�155.

Saloja� rvi, K. 1987. Why do vendace (Coregonus albula L.) populationsfluctuate?, Aqua Fenn. 17, 17�26.

Schiemer, F., Keckeis, H., and Wanzenbo� ck, J. 1989. Foraging incyprinids during early development, Pol. Arch. Hydrobiol 36,467�474.

Schoener, T. W. 1969. Models of optimal size for solitary predators,Am. Nat. 103, 227�313.

Sebens, K. P. 1982. The limits to indeterminate growth: An optimalmodel applied to passive suspension feeders, Ecology 63, 209�222.

Sva� rdson, G. 1956. Lambafja� rdens siklo� ja, Svensk Fiskeritidskrift 65,73�80 [in Swedish].

Tong, H. Y. 1986. A qualitative and quantitative description of the earlygrowth of roach Rutilus rutilus, in the laboratory, Envir. Biol. Fish.15, 293�300.

Townsend, C. R. 1989. Population cycles in freshwater fish, J. Fish Biol.35, 125�131.

Townsend, C. R., Sutherland, W. J., and Perrow, M. R. 1990. A modell-ing investigation of population cycles in the fish Rutilus rutilus,J. Anim. Ecol. 59, 469�485.

Tripet, F., and Perrin, N. 1994. Size-dependent predation by Dugesialugubris (Turbellaria) on Physa acuta (Gastropoda): Experimentsand model, Funct. Ecol. 8, 458�463.

Ustyogov, A. F. 1972. The ecological and morphological characteristicsof the Siberian ciscoe (Coregonus albula sardinella) from the Yeniseybasin, J. Ichtyol. 12, 745�758.

Viljanen, M. 1986. Biology, propagation, exploitation and managementof vendace (Coregonus albula L.) in Finland, Arch. Hydrobiol. Beih.Ergebn. Limnol. 22, 73�97.

Werner, E. E. 1988. Size, scaling and the evolution of life, in ``Size-Structured Populations: Ecology and Evolution'' (B. Ebenman andL. Persson, Eds.), pp. 60�81, Springer-Verlag, Heidelberg.

Werner, E. E. 1994. Ontogenetic scaling of competitive relations: size-dependent effects and responses in two Anuran larvae, Ecology 75,197�231.

Werner, E. E., and Gilliam, J. F. 1984. The ontogenetic niche andspecies interactions in size-structured populations, Ann. Rev. Ecol.Syst. 15, 393�425.

Werner, E. E., Mittelbach, G. G., Hall, D. J., and Gilliam, J. F. 1983.Experimental tests of optimal habitat use in fish: The role of relativehabitat profitability, Ecology 64, 1525�1539.

Wieser, W., Forstner, H., Schiemer, F., and Mark, W. 1988. Growthrates and growth efficiencies in larvae and juveniles of Rutilus rutilusand other cyprinid species: Effects of temperature and food in thelaboratory and in the field, Can. J. Fish. Aquat. Sci. 45, 943�950.

Wilbur, H. M. 1988. Interactions between growing predators and grow-ing prey, in ``Size-Structured Populations: Ecology and Evolution''(B. Ebenman and L. Persson, Eds.), pp. 157�172, Springer-Verlag,Heidelberg.

Wilson, D. S. 1975. The adequacy of body size as a niche difference,Am. Nat. 109, 769�784.

293Ontogenetic Scaling and Population Dynamics