UvA-DARE (Digital Academic Repository) …SIZE-STRUCTURED POPULATIONS A NDRE´ M. DE R OOS , 1,4 K JELL L EONARDSSON , 2 L ENNART P ERSSON , 2 AND G ARY G. M ITTELBACH 3 1 Institute

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Ontogenetic niche shifts and flexible behavior in size-structured populations

de Roos, A.M.; Leonardsson, K.; Persson, L.; Mittelbach, G.G.

Published in:Ecological Monographs

DOI:10.2307/3100028

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Citation for published version (APA):de Roos, A. M., Leonardsson, K., Persson, L., & Mittelbach, G. G. (2002). Ontogenetic niche shifts and flexiblebehavior in size-structured populations. Ecological Monographs, 72(2), 271-292.https://doi.org/10.2307/3100028

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271

Ecological Monographs, 72(2), 2002, pp. 271–292q 2002 by the Ecological Society of America

ONTOGENETIC NICHE SHIFTS AND FLEXIBLE BEHAVIOR INSIZE-STRUCTURED POPULATIONS

ANDRE M. DE ROOS,1,4 KJELL LEONARDSSON,2 LENNART PERSSON,2 AND GARY G. MITTELBACH3

1Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, P.O. Box 94084,1090 GB Amsterdam, The Netherlands

2Department of Ecology and Environmental Science, Umea University, S-901 87 Umea, Sweden3W. K. Kellogg Biological Station and Department of Zoology, Michigan State University,

Hickory Corners, Michigan 49060 USA

Abstract. Flexible behavior has been shown to have substantial effects on populationdynamics in unstructured models. We investigate the influence of flexible behavior on thedynamics of a size-structured population using a physiologically structured modeling ap-proach. Individuals of the size-structured population have a choice between living in a riskybut profitable habitat and living in a safer but less profitable habitat. Each of the two habitatshouses its own resource population on which the individuals feed. Two types of flexiblebehavior are considered: discrete habitat shifts, in which individuals instantaneously andnonreversibly shift from living in the safe habitat to the more risky/profitable habitat, andcontinuous habitat choice, in which individuals can continuously adapt their habitat choiceto current resource/mortality conditions. We study the dynamics of the model as a functionof the mortality risk in the risky/profitable habitat. The model formulation and parameteri-zation are derived using data on Eurasian perch (Perca fluviatilis) and describe reproductionas a yearly event at the beginning of summer, while all other processes are continuous intime. The presence of two habitats per se, with unique resources that are shared among allconsumers, does not change model dynamics, when compared to the one-resource situation.Flexible behavior increases the range of mortality levels for which the population can persist,because it allows individuals to hide from high mortality in the risky habitat. In contrast,flexible behavior does not significantly change the dynamics for mortality risks, where theconsumer population also persists without it. Discrete habitat shifts result in model dynamicsthat are largely similar to the dynamics observed with continuous habitat choice, as long asindividuals strongly respond to small differences in habitat profitability. In these cases, con-sumers spend an increasing part of their first year of life in the safe habitat, when mortalityrisks in the risky habitat increase. Ultimately, consumers are driven out into the risky habitatby intercohort competition from their successive year class. Therefore, major mortality andrapid growth occur among 1-yr-old individuals. Younger individuals exhibit retarded growthdue to intracohort competition in the safe habitat, which may also induce large-amplitudefluctuations when the mortality risk is high in the risky habitat. With continuous habitat choiceand a low responsiveness to habitat profitability, consumer persistence is increased as well,but large-amplitude fluctuations are absent. In this case, consumers always spend a significantpart of their first year of life in the risky habitat, even at high mortality risks. Major mortalityand rapid growth occur among individuals younger than 1 yr, while the shift to the riskyhabitat is mainly induced by intracohort competition for resources. The high mortality andrapid growth at younger ages lead to an increase in maximum size and fecundity of survivingindividuals, as well as to larger total population biomasses. We argue that the pattern ofindividual habitat use is mainly determined by population feedback on resource levels.

Key words: cohort competition; flexible behavior; habitat use; ontogenetic niche shifts; persis-tence; physiologically structured population models; population feedback; size-structured populations;stability.

INTRODUCTION

Flexible behavior in animals has been suggestedto have substantial effects on population dynamics(Werner 1992, Abrams 1996, Abrams et al. 1996).In one predator–one prey systems, flexible refuge useby prey has been shown to have a stabilizing effect

Manuscript received 17 November 2000; accepted 9 March2001; final version received 20 June 2001.

4 E-mail: [email protected]

on the dynamics (Ives and Dobson 1987, Mangel andRoitberg 1992, Krivan 1998). In one predator–sev-eral prey systems, optimal foraging by the predatormay increase the potential for prey coexistence andalso dampen oscillatory dynamics (Gleeson and Wil-son 1986, Fryxell and Lundberg 1993, 1994, Krivan1996, 1997, Van Baalen et al. 2001). Optimally for-aging predators may, however, also cause the loss ofstable fixed point dynamics and induce cycles.Hence, the overall effects of flexible behavior may

272 ANDRE M. DE ROOS ET AL. Ecological MonographsVol. 72, No. 2

be increased population persistence but decreasedstability (Krivan 1996).

Theoretical studies examining the implications offlexible behavior for population dynamics have usedtraditional, nonstructured predator–prey models to linkindividual-level processes such as foraging rate to pop-ulation processes (Ives and Dobson 1987, Mangel andRoitberg 1992, Werner 1992, Fryxell and Lundberg1993, Abrams 1996, Abrams et al. 1996, Krivan 1996,1997). In contrast, many experimental studies of flex-ible predator and prey behavior involve species whosepopulations are typically size/stage-structured, partic-ularly fish (Werner and Hall 1977, Mittelbach 1981,Werner et al. 1983a, b, Persson 1985, Gilliam and Fra-ser 1987, Godin and Sproul 1988, Gilliam 1990, Pers-son and Greenberg 1990, Godin and Clark 1997), butalso other taxa such as crustaceans and amphibians(Stein and Magnusson 1976, Jaeger and Barnard 1981,Jaeger et al. 1982, Werner and Anholt 1993). This con-trast has become even more striking because of thedevelopment of a modeling framework, known as phys-iologically structured population models, that (1) ex-plicitly handles the presence of size/stage-structure,and (2) in a formal and straightforward way links in-dividual and population processes. Physiologicallystructured population models are based on a two-levelstate concept: an i-state that represents the state of theindividual in terms of a collection of characteristicphysiological traits (size, age, sex, energy reserves,etc.), and a p-state that represents the population stateas a frequency distribution over the space of possiblei-states (Metz and Diekmann 1986, de Roos 1988,1997, Metz et al. 1988, Caswell and John 1992,DeAngelis and Gross 1992). Processes at the individuallevel include individual foraging, energy partitioningbetween growth and reproduction and mortality, all ofwhich relate to basic state variables in animal decisionmaking. Due to their explicit link between individualperformance and population dynamics, physiologicallystructured models provide a natural modeling frame-work to study flexible behavior.

In the absence of flexible behavior, size-structuredone consumer–one resource models have been shownto generate highly fluctuating population dynamics, in-cluding single-generation cycles (Persson et al. 1998).These cycles mainly result from intercohort competi-tion between consumers of different sizes where small-er individuals are superior competitors to larger con-specifics. A question that naturally arises from the ob-served fluctuating dynamics is whether flexible behav-ior may affect the outcome of the intercohortcompetition and facilitate the coexistence of differentsize-cohorts. Such an influence of flexible behaviorcould thus potentially dampen oscillatory fluctuations,as found in several studies using nonstructured pred-ator–prey models (Gleeson and Wilson 1986, Fryxelland Lundberg 1993, 1994, Krivan 1996, 1997).

To study the consequences of flexible individual be-

havior for the stability and persistence of size-struc-tured consumer-resource systems, we analyzed an ex-tended version of a consumer-resource model studiedby Persson et al. (1998). This model was adapted toaccount for two resource populations, each living in itsown habitat. Movements by individual consumers be-tween the habitats were based on a decision rule takingboth growth and mortality risk into account. We in-vestigated two different scenarios for shifts betweenhabitats. First, we considered the case when consumersshift habitat (resource) instantaneously and only onceduring their lifetime. Second, we considered the casewhen consumers continuously adapt the fraction oftime they spend in each of the two habitats. The firstscenario is reminiscent of a life history in which theindividual goes through an ontogenetic niche shift dur-ing juvenile development. Gilliam (1982; see also Wer-ner and Gilliam 1984, Ludwig and Rowe 1990, Hous-ton and McNamara 1999) has determined the optimaltiming for organisms to make such an ontogenetic nicheshift, when individual growth and mortality are sizedependent and consumer and resource populations arenot changing. Here, we analyze the ontogenetic nicheshift in a population dynamic context, taking into ac-count the fact that consumer shifting itself has a feed-back on resource levels in both habitats. The secondscenario is reminiscent of optimal foraging behavior,as investigated by Gleeson and Wilson (1986), Fryxelland Lundberg (1993, 1994), and Krivan (1996, 1997).An important difference between the two cases is thatin the second scenario the consumers can adjust theirhabitat use to changes in densities of the two resourcescaused by that same habitat shift. Thus, in this casethere are population feedbacks of resource dynamicson individual habitat use. The dynamics with these sce-narios of habitat use were compared with the dynamicswhen all consumers used only a single habitat or usedboth habitats in proportion to their volume.

MODEL FORMULATION

Physiologically structured models are based on astate concept at each of two levels of organization: ani-state, which represents the state of the individual (seeTable 1) and a p-state, which is the frequency distri-bution over the space of possible i-states (Metz andDiekmann 1986, Metz et al. 1988, Caswell and John1992, DeAngelis and Gross 1992). A mathematical de-scription of the behavior of a single individual (e.g.,its feeding, growth, development, reproduction, andmortality) as a function of its physiological character-istics and the current environmental conditions (e.g.,resource densities) constitutes the core element of anystructured-population model. The basic formulation ofthe model studied here, describing a size-structuredconsumer population and a nonstructured resource, isgiven by Persson et al. (1998, see also Claessen et al.2000, de Roos and Persson 2001). These papers analyzethe main dynamical properties of the model. We extend

May 2002 273BEHAVIOR IN SIZE-STRUCTURED POPULATIONS

TABLE 1. State variables of the standard, consumer-resource model with flexible behavior.

Symbol Units Interpretation

x, xi

y, yi

Ni

zl

zp

gg···g/m2

g/m3

irreversible mass (of an individual in the ith cohort)reversible mass (of an individual in the ith cohort)total number of individuals in the ith cohortbiomass density of littoral resourcebiomass density of pelagic resource

the model by introducing a second (unstructured) re-source population. The consumer population is consid-ered to live in a subdivided habitat, where in each ofthe two habitat parts it feeds on a unique resource pop-ulation. The model is parameterized for a populationof Eurasian perch (Perca fluviatilis) as the size-struc-tured consumer in a lake with a distinct open-water(pelagic) and vegetated (littoral) habitat. We assumethat the total lake volume equals 106 m3, of which 10%is taken up by the littoral habitat. The total bottomsurface area of the littoral habitat is set to 5.0 3 104

m2, implying an average water depth in the littoral hab-itat of 2 m. In the pelagic habitat, zooplankton (Daph-nia sp.) is considered the main resource, while ma-croinvertebrates (Sialis sp.) constitute the main re-source in the littoral habitat.

The individual level

Consumers are characterized by two physiologicalparameters, irreversible and reversible mass, and bythe fraction of time spent in the pelagic habitat. In theirreversible mass, x, compounds like bones and organsthat cannot be starved away by the consumer are in-cluded, whereas reversible mass, y, includes energyreserves such as fat, muscle tissue, and also gonads formature individuals. Reversible mass may be used tocover basic metabolism during starvation. Total bodymass of an individual consumer equals the sum of re-versible and irreversible mass, (x 1 y). The standard-ized body mass m(x) of a consumer is defined as

m(x) 5 (1 1 q )x (1)j

in which qj represents a characteristic ratio betweenreversible and irreversible mass for nonstarving indi-viduals, discounting energy reserves for reproduction(e.g., juveniles or adults right after spawning). The no-tion of standardized body mass is introduced, becausefunctional response experiments with size-structuredconsumers have shown a close relationship betweencapture rate and body length independent of body con-dition (Mittelbach 1981, Persson 1987). To estimateattack rate and handling time parameters from suchexperiments, we identify the measured individual massin the experiments with the standardized body massm(x).

Relations describing the foraging rate, metabolism,energy partitioning between growth and reproductivetissue, and starvation (including starvation mortality)as a function of irreversible and reversible mass, x and

y, respectively, were developed in Persson et al. (1998)and are summarized in Table 2. In contrast to the con-sumer-resource model by Persson et al. (1998), we treatthe resource populations in terms of biomass rather thannumber of individuals, which affects the formulationsof resource growth rate, consumer attack rates, andconsumer handling times. Below we briefly describethe part of the model that relates to consumer feedingand energy channeling.

The resource-specific attack rates [al(x) and ap(x) forthe littoral and pelagic resource, respectively] of anindividual consumer are functions of its standardizedbody mass m(x) only. Hence, we assume that a con-sumer’s condition, i.e., its reversible mass y, does notinfluence its foraging rate (see Persson et al. 1998 fora justification of this assumption). The attack rate onthe pelagic resource is a hump shaped function of stan-dardized body mass (see Fig. 1 and Table 2), reachinga maximum value determined by the parameter Amax ata body mass mopt. The attack rate on the littoral resourceis an allometric function of standardized body massm(x), following data on the feeding of perch on Sialis(Persson and Greenberg 1990). We assume that indi-viduals living in the littoral part of the lake feed entirelyon bottom-dwelling macroinvertebrates. Therefore, theattack rate on littoral resource is expressed as theamount of bottom surface area searched per day (cf.the volume searched per day for the pelagic attack rate).Fig. 1 shows the pelagic and littoral attack rate as afunction of standardized body mass.

Prey handling time is formulated in terms of diges-tion time per unit ingested biomass, and hence is iden-tical for both pelagic and littoral resources. The han-dling time is assumed to reflect digestive constraints,related to the gut capacity of an individual with a givensize (Claessen et al. 2000). It is described as an allo-metric function of standardized body mass m(x).

The foraging rate of individual consumers on thelittoral and pelagic resource is assumed to follow aHolling type II functional response, incorporating theresource-specific attack rate, handling time, and den-sities of the littoral (zl) or the pelagic (zp) resource,respectively (see Table 2 and Persson et al. 1998). Thetotal individual foraging rate equals the sum of theselittoral and pelagic foraging rates, weighted by a factor(1 2 F) and F, respectively, to account for the amountof time that consumers spend in either of the two hab-itats. Ingested food is assumed to be converted to en-ergy assimilate with a constant conversion efficiency.


TABLE 2. Functional relationships for the model of individual energy acquisition and use.

Function Equation

Habitat-independent functions

Standardized mass m(x) 5 (1 1 q )xj

Handling time j2H(x) 5 j m(x)1

Maintenance requirements r2E (x, y) 5 r (x 1 y)m 1

Starvation mortality s(q x /y 2 1) if y , q xs sd (x, y) 5s 50 otherwise

Pelagic feeding and mortality

Attack rate on pelagic resourcea

m(x) m(x)a (x) 5 A exp 1 2p max 1 2[ ]m mopt opt

Intake rate of pelagic resourcea (x)zp p

I (z , x) 5p p 1 1 H(x)a (x)zp p

Mortality in pelagic habitatx

d (x, y) 5 m 1 m exp 2 1 d (x, y)p b p s1 2xm

Profitability of pelagic habitatk I (z , x) 2 E (x, y)e p p m

Q (z , x, y) 5p p d (x, y)p

Littoral feeding and mortalityAttack rate on littoral resource n2a (x) 5 n m(x)l 1

Intake rate of littoral resourcea (x)zl lI (z , x) 5l l 1 1 H(x)a (x)zl l

Mortality in littoral habitat d (x, y) 5 m 1 d (x, y)l b s

Profitability of littoral habitatk I (z , x) 2 E (x, y)e l l mQ (z , x, y) 5l l d (x, y)l

Habitat choice, total intake, total mortality

Fraction of time in pelagic habitatVp

F(z , z , x, y) 5p l V 1 V exp{2s[Q (z , x, y) 2 Q (z , x, y)]}p l p p l l

Total intake rate I(z , z , x, y) 5 F(z , z , x, y)I (z , x) 1 [1 2 F(z , z , x, y)]I (z , x)p l p l p p p l l l

Total mortality d(z , z , x, y) 5 F(z , z , x, y)d (x, y) 1 [1 2 F(z , z , x, y)]d (x, y)p l p l p p l l

Net production, allocation, and reproduction

Net energy production E (z , z , x, y) 5 k I(z , z , x, y) 2 E (x, y)g p l e p l m

Fraction of net production used for growth inirreversible mass

1 yif x # x and E . 0f g(1 1 q )q xj j

k(x, y) 5 1 yif x . x and E . 0f g(1 1 q )q xa a

0 otherwise

Fecundity

k (y 2 q x)r j if x . x and y . q xf jf (x, y) 5 mb0 otherwise

Note: See Methods and Results, Persson et al. (1998), and Claessen et al. (2000) for discussion.

An individual’s current energy intake is first used tocover its metabolic requirements, which follow an al-lometric function of total consumer body mass (x 1y). The remaining part of the ingested energy (the netenergy intake or net production) is allocated to re-versible and irreversible mass such that a constant ratio(y/x) between the two is targeted for. This ratio forjuveniles (qj) differs from that for adults (qa) on thegrounds that reversible mass in mature individuals alsoincludes gonads (qa . qj) (Tables 2 and 3; Persson et

al. 1998). When energy intake does not suffice to covermetabolic requirements, growth in irreversible mass xstops and reversible mass y is used to cover the deficit.When net energy intake becomes positive again, energyis preferentially allocated to reversible mass in orderto restore the target ratio y/x.

We have assumed that all consumers experience thesame, constant background mortality in both the littoraland the pelagic habitat, which is determined by theparameter mb (Tables 2 and 3). The littoral habitat is


FIG. 1. Attack rate of an individual consumer on the pelagic(solid line) and littoral (dashed line) resource, as a function ofits standard body mass. Pelagic and littoral attack rate are ex-pressed in terms of the volume (in cubic meters) and area (insquare meters), respectively, searched per day. The vertical dot-ted line represents the standard body mass at maturation.

TABLE 3. Model parameters for Eurasian perch (Perca fluviatilis) feeding on pelagic zooplankton (Daphnia sp.) and littoralmacroinvertebrates (Sialis sp.).

Symbol Value Units Interpretation

qj

qa

aAmax

mopt

n1

n2

j1

0.741.370.62

30.08.24.00.405.0

·········m3/dg

2 21 2n2m ·d ·g···

(11j )2d/g

maximum juvenile condition y/xmaximum adult condition y/xallometric exponent in pelagic attack ratemaximum pelagic attack ratebody mass with maximum pelagic attack rateallometric constant in littoral attack rateallometric exponent in littoral attack rateallometric constant in handling time function

j2

r1

r2

ke

mb

xf

kr

qs

s

20.80.0330.770.610.00184.60.50.20.2

···/d(12r )2g

······gg······d21

allometric exponent in handling time functionallometric constant in maintenance rate functionallometric exponent in maintenance rate functioningestion–assimilation conversion efficiencytotal mass of an egg (newborn)irreversible mass at maturationgonad–offspring conversion efficiencycondition threshold y/x for starvation mortalityproportionality constant of starvation mortality rate

mb 0.01 d21 constant, background mortality rate in littoral and pelagic habitatmp varied d21 scaling constant in additional, size-dependent mortality rate in pelagic habitatxm 2.0 g characteristic size in additional, size-dependent mortality rate in pelagic

habitatsKp

Kl

rp

varied3.03.00.1

g21

g/m3

g/m2

d21

proportionality constant in habitat switching rateresource carrying capacity in pelagic habitatresource carrying capacity in littoral habitatresource regrowth rate in pelagic habitat

rl

Vp

Vl

Al

Y

0.19.0 3 105

1.0 3 105

5.0 3 104

90

d21

m3

m3

m2

d

resource regrowth rate in littoral habitattotal volume of pelagic habitattotal volume of littoral habitattotal bottom surface area of littoral habitatduration of growing season

assumed to provide a refuge from predation, especiallyfor small fish, whereas the pelagic habitat is more risky.Therefore, in the pelagic habitat consumers experiencean additional, size-dependent mortality rate equal to

xm exp 2 (2)p 1 2xm

where xm determines the size-scaling of this mortality

rate. We study the dynamics of the model as a functionof the scaling constant mp. Increasing values for mp

make the pelagic a more risky habitat. However, dueto its large size and hence high total productivity, thepelagic is potentially a more profitable habitat. Indi-viduals also experience starvation mortality whenevertheir reversible/irreversible mass ratio y/x drops belowthe starvation mortality limit qs (Tables 2 and 3). Thestarvation mortality is modeled in such a way that deathoccurs with certainty when an individual’s reversiblemass is depleted entirely.

Pulses of reproduction occur as discrete events intime, since it is assumed that individuals only spawnat the beginning of the growing season (summer).When they spawn, adults allocate all reversible massthat they accumulated in excess of their standardizedbody mass m(x) 5 (1 1 qj)x to the production of eggswith a constant conversion efficiency. Following a suc-cessful spawning event, an adult thus has the samereversible/irreversible mass ratio y/x as a nonstarvingjuvenile; after that, the buildup of gonadic mass to bereleased at the next reproduction event starts anew.Maturation of juvenile into adult consumers occurs onreaching a fixed threshold of irreversible mass xf.

Modeling flexible behavior

To incorporate flexible behavior into the structuredpopulation model, we formulate below a Markov modelfor the switching between habitats by a single individ-


FIG. 2. Fraction of time spent in the pelagic habitat (seeEq. 6) for different values of s as a function of the differencein habitat profitability between littoral and pelagic habitat,Ql(zl, x, y) 2 Qp(zp, x, y).

ual, assuming that the state of both habitats is notchanging. Subsequently, we assume that this switchingbehavior takes place at such a rapid time scale that thedistribution of individuals over both habitats is at anytime in pseudo-steady-state with the current habitatconditions. This yields an expression for the fractionof time that an individual consumer of a given sizespends in the pelagic zone, given the current densitiesof pelagic and littoral resource.

Gilliam (1982, see also Werner and Gilliam 1984)developed an individual-level theory for the timing ofhabitat shifts during ontogeny based on the minimi-zation of the ratio between mortality and growth rate(the ‘‘m/g-rule’’). We adopt the inverse of this ratio asa measure of habitat profitability to prevent problemsthat occur when growth rates reduce to 0. Hence, theprofitabilities Ql and Qp of the littoral and pelagic hab-itat are defined as:

k I (z , x) 2 E (x, y)e l l mQ (z , x, y) 5 (3)l l d (x, y)l

k I (z , x) 2 E (x, y)e p p mQ (z , x, y) 5 (4)p p d (x, y)p

respectively, where keIl(zl, x) 2 Em(x, y) and keIp(zp,x) 2 Em(x, y) are the net energy production (assimi-lation minus maintenance) rates in the littoral and pe-lagic habitat, respectively (see Table 2); ke is the con-version efficiency from food intake to energy assimi-lation rate; dl(x, y) and dp(x, y) are the total death ratesin the two habitats. We assume that an individual con-sumer leaves the littoral and pelagic habitat at a ratethat is proportional to

exp[2sQ (z , x, y)]l l

exp[2sQ (z , x, y)]p p

respectively. In addition, we will assume that the rateat which an individual leaves a habitat is inverselyproportional to the volume of that habitat.

If F denotes the fraction of time that an individualconsumer spends in the pelagic habitat, the dynamicsof F on a short time scale can be described by

exp[2sQ (z , x, y)]dF p p5 2 F

dt Vp

exp[2sQ (z , x, y)]l l1 (1 2 F). (5)Vl

In this equation Vp and Vl refer to the volume of thepelagic and the littoral zone of the lake, respectively.Assuming that at any time the individual-level switch-ing is in pseudo-steady-state with the current habitatconditions (i.e., dF/dt ø 0) yields the following ex-pression for F:

F(z , z , x, y)p l

Vp5 . (6)

V 1 V exp{2s[Q (z , x, y) 2 Q (z , x, y)]}p l p p l l

For s 5 0, F equals the ratio of the pelagic and totallake volume, which implies that individuals use boththe pelagic and the littoral habitat in proportion to theirvolume. We will refer to this case as ‘‘proportionalhabitat use.’’ Individuals will also use both habitatsproportionally if the habitats are equally profitable (Qp

5 Ql). For positive values of s, Eq. 6 represents asigmoid function, rising from 0 when the profitabilityof the littoral habitat is much higher than the pelagic(Qp K Ql), to 1 in the reverse situation (Qp k Ql). Theparameter s determines the steepness of the sigmoidcurve at equal habitat profitability (Qp 5 Ql). For s →`, individuals will always switch instantaneously tothe habitat with the highest profitability (see Fig. 2).

Given that an individual spends a fraction F of itstime in the pelagic habitat, its total intake rate of re-source biomass is

I(z , z , x, y) 5 F(z , z , x, y)I (z , x)p l p l p p

1 [1 2 F(z , z , x, y)]I (z , x) (7)p l l l

while its total mortality rate is an analogous, weightedaverage of the mortality rates experienced in either ofthe two habitats (see Table 2).

Newborn consumers always start life in the littoralhabitat. In addition to situations in which consumersare restricted to live only in the pelagic habitat, or inwhich they use both the littoral and pelagic habitatproportionally, we study two scenarios of responsivebehavior:

1. Discrete habitat shift.—With this term we referto a situation in which individual consumers onlyswitch habitat once during ontogeny. This switch isinstantaneous and irreversible and will occur as soonas the pelagic profitability is higher than the littoralprofitability (i.e., when Qp(zp, x, y) 2 Ql(zl, x, y) be-comes positive).

2. Continuous habitat choice.—In this case, individ-ual consumers continuously adapt their use of the pe-


lagic and littoral habitat to the current feeding and mor-tality conditions in both habitats. The fraction of timean individual of a given size and condition spends inthe pelagic habitat is given by Eq. 6.

The population level

The model only examines population dynamics dur-ing the growth season, corresponding to the summerin the temperate region. The changes in consumer andresource populations during the nongrowth season(winter) are assumed to be negligible.

Resources are assumed to reproduce continuouslythroughout the growth season. Consumers are assumedto feed, grow (or shrink in case of starvation), and diecontinuously during the summer season, but reproduceonly at the start of a growth season in a sharply pulsedevent. The model is thus a combination of a continuousdynamical system, describing growth and survival ofthe consumers and production and consumption of theresource during summer, and a discrete map describingthe pulsed reproduction of consumers in spring. When-ever only yearly model statistics are presented belowthey relate to the state of the system at the time ofreproduction.

The two resources are assumed to exist in separatehabitats. The population growth of both resource pop-ulations is assumed to follow semi-chemostat dynam-ics. Arguments for using this representation of resourcepopulation growth are given in Persson et al. (1998).Analytically the structured population model can beformulated as a system of integral equations (see Pers-son et al. 1998), which represents a way of bookkeepingthe dynamics of all individuals making up the popu-lation. Numerically, the model can be studied using theEBT (Escalator Boxcar Train) framework (de Roos1988, de Roos et al. 1992). The EBT method is spe-cifically designed to handle the numerical integrationof the equations that occur in physiologically structuredmodels. Below follows a short description of how theEBT method was applied to the model studied in thispaper. The Appendix presents the equations governingthe within- and between-seasons dynamics of the con-sumer and resource populations.

The pulsed reproduction process ensures that thereexists a natural subdivision of the population into co-horts of individuals. In one cohort all individuals havethe same age, reversible and irreversible mass, and allspend the same fraction of their time in the pelagichabitat. All individuals within a cohort are, moreover,assumed to grow at the same rate, i.e., individuals be-longing to a given cohort do not diverge in their al-location to reversible and irreversible masses. As a re-sult, each cohort consists of individuals that will remainidentical for the duration of their lives.

The dynamics of every cohort can be described bya system of ordinary differential equations, whichkeeps track of the number of individuals making upthe cohort, their age, reversible mass, and irreversible

mass. The dynamics of the entire consumer population,both in terms of its abundance and its composition, canbe followed throughout the summer season by numer-ically integrating the system of ordinary differentialequations for each cohort separately. In addition,changes in the resource population can be followed bynumerical integration of the ordinary differential equa-tion for the resource dynamics that incorporates thesemi-chemostat growth and the total resource con-sumption. The latter equals the summed foraging rateover all cohorts.

At the beginning of the growth season, new cohortsof individuals are added to the consumer populationdue to the reproductive process. This addition impliesthat the number of differential equations describing thepopulation dynamics is increased. At the same time,the current value of the reversible mass in the cohortsof reproducing individuals is reset, reflecting their in-vestment into offspring. Overall, the model simulationsthus involve the numerical integration of a (large) sys-tem of ordinary differential equations, which is ex-tended in dimension at the beginning of each seasonwith a concurrent reset of some of the variables. Thedimension of the system is reduced whenever the num-ber of individuals in a given cohort has become neg-ligible, at which time the differential equations for thisparticular cohort are removed.

Parameterization of the model

We parameterized the model based on the biologyof Eurasian perch (Perca fluviatilis) (Table 3). Perchis a suitable model system, as it has been demonstratedthat perch choose habitat in response to habitat prof-itability (Persson 1987, Persson and Greenberg 1990).Perch also take predation risk into account when choos-ing habitats, and it has been shown that the littoralhabitat serves as a refuge from predation for smallperch (primarily from large piscivorous perch; Persson1993, Persson and Eklov 1995). The size at which ju-venile perch switch between habitats has also beenshown to depend on cannibalistic perch density (P. Bys-trom, L. Persson, E. Wahlstrom, and E. Westman, un-published manuscript).

Parameterization of attack rates, handling times, andmetabolic demands was based on data given in Persson(1987), Persson and Greenberg (1990), and Bystromand Garcia-Berthou (1999: Table 2; see also Claessenet al. 2000). The open-water prey was represented byDaphnia, and littoral prey was represented by Sialis,which both are typical and dominant prey in the pelagicand littoral habitat, respectively (Persson 1987). Pa-rameter values were based on a temperature of 198C.

The size-independent, background mortality rate mb

in the littoral and pelagic habitat was set to 0.01 perday for all individuals. The model dynamics were stud-ied as a function of the proportionality constant in thesize-dependent, pelagic mortality rate mp (see Eq. 2).The length of the season was set to 90 d, corresponding


FIG. 3. Dynamics of the model when individuals use bothlittoral and pelagic habitat proportionally in the absence ofany additional, size-dependent mortality in the pelagic habitat(mp 5 0.0). Top: The solid line with circles represents indi-viduals younger than 1 yr (YOY); the dashed line with tri-angles represents juvenile individuals aged 1 yr and older;and the dotted-dashed line with squares represents adult in-dividuals. All consumer numbers are expressed as total num-ber in the entire lake. Bottom: The solid line with circlesrepresents the littoral resource (g biomass/m2); the dashedline with triangles represents the pelagic resource (g biomass/m3). Symbols in all curves mark the start of the season.

to the temperate region in middle Sweden. The twohabitats differed in size, with the littoral habitat makingup 10% and the pelagic habitat 90% of the total lakevolume. We assumed an average water depth of 2 mfor the littoral habitat to convert from habitat volumeto total bottom area, where the individuals were as-sumed to feed.

RESULTS

Single (pelagic) and proportional habitat use

For the default parameters (see Table 3) and withoutany differences in mortality between the littoral andpelagic habitat (mp 5 0.0), the model exhibits cycleswith a periodicity of 8 yr, where the population is al-most always made up by a single cohort of individuals.The details of these single-cohort cycles and the mech-anisms bringing them about have been discussed byPersson et al. (1998). Fig. 3 shows an example of thesedynamics for the case of proportional habitat use. Whenreproduction occurs, the large number of newborn in-dividuals (young-of-the-year, YOY) depresses the re-source levels in both the littoral and pelagic habitat tosuch low levels that all older individuals starve todeath. The competition for food among the YOY sub-sequently impedes their growth. When density declinesdue to background mortality, resource levels can in-crease and individual growth speeds up. The cohort

reaches maturation size when individuals are .7 yrold. The surviving individuals have accumulated suf-ficient reproductive mass by the end of their 8th yr toproduce a new dominant cohort. This new cohort out-competes the remaining adults.

The occurrence of single-cohort cycles can be ex-plained by the fact that individual competitiveness, asmeasured by the lowest resource levels that an indi-vidual can sustain without starving, increases withbody size (Persson et al. 1998). For a given irreversiblemass, x, individual growth stops at a critical resourcedensity where net energy production is 0 (i.e., when-ever keI 5 Em; see Table 2). Whether the individual isfeeding in the pelagic or littoral habitat, this criticalresource density is higher for larger individuals (resultsnot shown; see Persson et al. 1998). The dominantcohort completely controls the resource level and ex-ploits it to levels that are below the critical resourcelevel of older cohorts, inducing their starvation. In thecase of a single resource (habitat), Persson et al. (1998)showed that for increasing background mortality, thedensity of individuals in the dominant cohort declinesmore rapidly. As a result, resource levels increase fast-er, growth in size speeds up, and individuals matureearlier in the season. The dynamics change when in-dividuals mature 1 yr earlier at an age of 6 yr old. Inthis case, the period of the single-cohort cycle shortensto 7 yr. With increasing mortality, the single-cohortcycles thus shorten in a stepwise manner to lower andlower periodicities, ultimately reaching a fixed-pointdynamics, in which the state of the population andresource levels are identical at the beginning of eachseason.

Fig. 4 shows that this bifurcation pattern with step-wise transitions to single-cohort cycles of a one-year-shorter period occurs irrespective of whether consum-ers live only in the pelagic habitat, or use both pelagicand littoral habitat in proportion to their volume. Asthe size-dependent, pelagic mortality (mp) increases, thestepwise shortening of the single-cohort cycles occursover the range of values 0 # mp , 0.08. (Here andbelow all thresholds in mp, separating intervals withdifferent types of dynamics, should be interpreted asapproximate values.) For values of mp around the pointswhere the cycle periods shorten, some irregular dy-namics may be observed. At these transitions there mayalso be parameter intervals where alternative attractorscoexist. When consumers are restricted to live in thepelagic, the regular 2-yr cycle coexists over a signifi-cant range of mp values with a stable fixed point. Whenconsumers use both pelagic and littoral habitat pro-portionally, coexistence of alternative attractors occursbetween a regular 2-yr cycle and a stable fixed point(0.05 , mp , 0.08) and between a regular 3-yr cycleand either a stable fixed point or irregular, small-am-plitude cycles (0.02 , mp , 0.04).

From Fig. 4 we conclude that the introduction of asecond resource (habitat) that is shared among all con-


FIG. 4. Bifurcation diagram of the model when individ-uals are restricted to live only in the pelagic habitat (top) orto use both pelagic and littoral habitat proportionally (bot-tom). The total number of consumers of age 1 yr and olderin the entire lake is shown as a function of the pelagic mor-tality constant mp. Other parameters have their default value(see Table 3). For every value of the parameter, numericalsimulations were carried out over a period of 400 yr (seasons).Statistics of the system at the time of reproduction (i.e., thebeginning of the season) are shown for the last 50 yr of thesesimulations. Symbols in different shades of gray indicate theexistence of alternative attractors.

sumers, only leads to marginal differences in the bi-furcation pattern from the single-resource (habitat) sit-uation. The transitions in dynamics for both casesshown in Fig. 4 occur at the same average mortality.For example, when only the pelagic is used stable equi-libria occur for 0.045 , mp , 0.145, while they occurfor 0.05 , mp , 0.16 with proportional habitat use.Taking into account that in the latter situation an in-dividual spends 90% of its time in the pelagic and 10%in the littoral habitat with a lower mortality probability,these ranges of mp values are identical. The introductionof the second resource only leads to a higher propensityfor alternative attractors over a small range of mp-values(0.02 , mp , 0.04). The importance of these additional,alternative attractors (i.e., the stable fixed points andsmall-amplitude cycles), however, is not clear, as wedid not investigate which proportion of all possibleinitial states of the system would converge to the dif-ferent attractors.

Discrete habitat shifts (littoral to pelagic)

Dynamics for 0 # mp , 0.145.—When newborn in-dividuals start their life in the littoral habitat and thenswitch habitats once to continue living in the pelagic,the same bifurcation sequence of single-cohort cyclesis observed for low values of mp, as when individuals

are restricted to live only in the pelagic. Fig. 5 showsthat for 0 # mp , 0.145, habitat switching occurs atvery small sizes, i.e., only a few days after the indi-viduals are born. The littoral resource is only fed uponduring these first few days of an individual’s life andremains unexploited afterwards. Because we show re-source densities at the beginning of the growth season,the littoral resource density in Fig. 5 equals the carryingcapacity up to mp ø 0.175. For 0 # mp , 0.145, absolutedensities of consumers and resources are identical, ir-respective of whether individuals exhibit a discretehabitat shift (Fig. 5; top panel) or whether they arerestricted to live only in the pelagic (Fig. 4; top panel).Also, the values of mp at which stepwise decreases ofcycle length occur are the same between the two sit-uations. Comparing discrete habitat shifts with the sit-uation where individuals use both habitats proportion-ally, the total density of consumers is slightly lowerthan in the proportional-use scenario, because the pro-ductivity of the littoral habitat is not exploited, whilethe values of mp at which stepwise decreases of cyclelength occur are slightly smaller for the reasons ex-plained in the previous section. Over the entire rangeof mp values for which the consumer population persistswhen individuals only use the pelagic habitat, the in-corporation of a discrete habitat shift thus leads to re-sults that are graphically indistinguishable. The ro-bustness of the single-cohort cycles against the incor-poration of a discrete habitat shift may be related tothe differences in total productivity of the pelagic andlittoral habitat. For the default parameter set these pro-ductivities are 270 and 15 kg prey biomass/d, respec-tively. This difference is primarily due to the differentvolumes of the two habitats, i.e., 90% and 10% of thetotal lake volume, respectively. As a result, consumerpopulation feedback in the littoral habitat is alwaysintensified compared to the feedback in the pelagic. Itshould be noted that the immediate switching of new-born individuals to the pelagic is entirely a population-level effect, as the foraging capacity of newborn in-dividuals on the littoral resource is even slightly higherthan on the pelagic resource.

Another consequence of the high productivity of thepelagic habitat is that the number of newborn (YOY)individuals produced is invariably large (Fig. 5). As-suming that the newborn cohort is the only one ex-ploiting the littoral resource, the differential equationfor this resource density (see Eq. A.3b in the Appendix)can be used to derive the pseudo-steady-state valueimposed by the large YOY pulse. For this pseudo-steady-state resource density the right-hand side of Eq.A.3b vanishes. All our numerical studies so far haveindicated that this pseudo-steady-state value is a goodapproximation to the actual density of resource onwhich the YOY is foraging at a specific time. Fig. 6shows that the pseudo-steady-state resource density inthe littoral habitat rapidly decreases when the numberof YOY is .107 and drops below their starvation


FIG. 5. Bifurcation diagram of the model when individuals move once and instantaneously during their life from littoralto pelagic habitat: (A) the total number of YOY and (B) consumers aged 1 yr and older in the entire lake; (C) their irreversiblemass at habitat switching (in grams); and the resource biomass density (g biomass/m3) in the (D) pelagic and (E) littoralhabitat are shown as a function of the mortality constant mp in the open water. Other parameters have their default value (seeTable 3). Symbols in different shades of gray indicate the existence of alternative attractors.

threshold if the number of YOY is .4 3 107. At thetime of reproduction the pelagic resource density issufficiently large to allow adult individuals to grow andreproduce, and is at least larger than the critical re-source density for a maturing individual. Under theseconditions the littoral resource density at which thenewborn individuals switch to the pelagic is just abovetheir critical, starvation density (see Fig. 6). From Fig.6 it can therefore be concluded that YOY densities .43 107 will always lead to an almost immediate nicheshift to the pelagic, induced by its higher profitability.

Fig. 5 shows that this situation occurs for 0 # mp ,0.145.

The robustness of the single-cohort cycles hence re-sults from the marginal importance of the littoral hab-itat for the consumer population, even though at anindividual level the littoral resource is sometimes moreprofitable. The high total productivity of the pelagichabitat leads to the birth of large numbers of offspringin the littoral habitat. The population feedback forcesthese individuals out into the pelagic after a negligibletime span. The niche shift is triggered by the littoral


FIG. 6. (A) The pseudo-steady-state, littoral resource density (g biomass/m2) as a function of the total number of newbornindividuals (x 5 xb; solid line). This pseudo-steady-state density is computed by solving the resource density at which theright-hand side of ordinary differential equation A.3b (see the Appendix) vanishes, assuming that only the newborn cohortfeeds on the littoral resource. The dotted, horizontal line represents the resource density below which newborn individualsstarve. (B) Littoral resource densities (g biomass/m2) at which newborn individuals switch to the pelagic habitat, when thepelagic resource density equals carrying capacity (zp 5 Kp; thick solid line) and when it equals the critical pelagic resourcedensity for maturing individuals (thick dashed line).

resource dropping to such low levels that the currentconditions in the pelagic become more profitable forthe newborn individuals. Essentially, the dynamics arethus very close to the dynamics of the single resource–single consumer system studied by Persson et al.(1998).

Dynamics for 0.145 , mp , 0.24.—For mp . 0.145,the consumer population cannot persist if individualsonly live in the pelagic habitat, but can do so if theygo through a discrete habitat shift. For mp increasingfrom 0.145, the number of YOY and consumers aged1 yr and older sharply decreases and becomes so lowthat the pelagic resource density roughly equals car-rying capacity (Fig. 5). This implies that there is nofeedback any longer of the consumer population on thepelagic resource density and consequently also on thelife history of consumers after they have passedthrough the habitat shift. This leads to a constant fe-cundity for adult individuals. Therefore, for mp .0.145, the consumer population is regulated by the in-teractions taking place within the littoral habitat, morespecifically by the competition among YOY individ-uals and the size at which these individuals shift to thepelagic. The lower YOY density allows these individ-uals to start growing while living in the littoral habitat(Fig. 7A). For 0.145 , mp , 0.175 the pelagic becomesmore profitable after 10–20 d, because the attack rateon littoral resources increases less rapidly with increas-ing body size as compared with the attack rate on thepelagic resource (Fig. 1). The initial growth in the lit-toral habitat causes the increase in switch size observedover the interval 0.145 , mp , 0.175 (Fig. 5). Afterswitching to the pelagic, the high, size-dependent mor-tality decimates the density of YOY. As a consequence,they have hardly any impact on the pelagic resource

density and their growth is maximal. Within their firstyear of life the YOY reach the maturation size and startreproducing as 1-yr-old individuals. For mature indi-viduals the size-dependent mortality is negligible,which allows them to reach maximum ages of 8–9 yr.Reproduction peaks at the age of 2 and declines rapidlyafterwards. Individuals older than 5 yr do not reproduceany longer, as they use all assimilated food to covertheir maintenance requirements.

Partly coexisting with the attractor described abovein which YOY pass through the habitat shift duringtheir first year of life, another type of dynamics is ob-served for 0.15 , mp , 0.24, in which YOY switch tothe pelagic almost instantaneously after reaching theage of one year (Fig. 7B). This type of dynamics occurswhen the density of YOY is so low that during theirfirst year of life they do not deplete the littoral resourceto levels where the pelagic becomes more profitable.A substantial fraction of the YOY survive up to age 1,when the new YOY cohort causes a very rapid, smalldecrease in the littoral resource density (Fig. 7B). Thissmall decrease is sufficient to force the 1-yr-old indi-viduals into the pelagic. Because of the habitat shiftthe littoral resource density increases again consider-ably, which allows the new YOY cohort to stay in thelittoral habitat. The high survival of YOY up to age 1is reflected in the high densities of consumers aged 1yr and older, shown in Fig. 5. However, this densityrapidly drops after the habitat shift because of the size-dependent mortality experienced by 1-yr-old individ-uals (Fig. 7B). Due to the high pelagic resource densitythe growth of the surviving 1-yr-old individuals speedsup such that they mature in their 2nd year and repro-duce for the first time as 2-yr-old individuals. Repro-duction peaks at the age of 3 and declines to 0 for


FIG. 7. Dynamics of the discrete habitat shift model for (A) mp 5 0.17 and (B) mp 5 0.23. Top: The solid line with circlesrepresents individuals younger than 1 yr (YOY); the dashed line with triangles represents juvenile individuals aged 1 yr andolder; and the dotted-dashed line with squares represents adult individuals. Bottom: The solid line represents the littoralresource; the dashed line represents the pelagic resource. Symbols mark the start of the season. Units are as in Fig. 3.

individuals older than 6 yr. The maximum age is again8–9 yr.

The two types of dynamics differ in the timing ofthe habitat shift (as YOY vs. as 1 yr old), the mech-anism of its induction (within- vs. between-cohort com-petition) and the individual survival pattern (majormortality during 1st or 2nd yr of life). In addition, whenindividuals switch as YOY, the high mortality duringtheir 1st yr of life induces a rapid release from intra-cohort competition, and consequently a rapid growthrate up to age 1. When switching occurs as 1-yr-oldindividuals, a comparable mortality, release from com-petition, and rapid growth only occurs after reachingage 1. As a consequence, individuals reach larger bodysizes at the age of 1 yr when they switch as YOY. Thesurvival up to the time of first reproduction and thefecundity at that and later reproduction events are, how-ever, completely identical for the two attractors.

Dynamics for mp $ 0.24.—At mp ø 0.24, the fixedpoint at which individuals shift to the pelagic at theage of 1 yr due to the between-cohort competition withthe subsequent YOY, destabilizes and gives rise tolarge-amplitude cycles in the number of consumers andthe littoral resource density. The cycles in pelagic re-source density have much smaller amplitude and re-main close to the carrying capacity value at all times(see Fig. 5). For the body size at which the habitat shiftoccurs, values are observed that roughly separate intotwo ranges: individuals either switch at an irreversiblemass x between their mass at birth (x 5 0.001) and 10times that value (x 5 0.01), or they switch habitats ata size close to the maturation size.

Since the pelagic resource density is invariably closeto the carrying capacity, and any feedback on consumerlife history is absent after the age of 1 yr, the consumerpopulation is only regulated through the body size atwhich 1-yr-old individuals shift to the pelagic. The sizeis completely determined by the year class strength ofthe YOY, which controls the littoral resource densityat its pseudo-steady-state value. The fixed point desta-bilizes through overcompensation in this body size athabitat shift. With decreasing numbers of YOY overthe range 0.145 , mp , 0.24, the size at habitat shiftincreases and the subsequent mortality from age 1 tomaturation decreases. Cohorts of individuals thatswitch at larger body sizes produce larger pulses ofoffspring. If too large, these offspring cohorts them-selves switch at a smaller body size, experience higherpelagic mortality, and hence produce again smaller off-spring cohorts. Ultimately at mp ø 0.24 this overcom-pensation mechanism destabilizes the fixed point. Be-cause of the discontinuity inherent in the instantaneousand complete shift of an entire cohort of individuals,the cycle, which arises from the destabilization throughovercompensation and which is initially of small am-plitude (see, for example, Fig. 5, panel C), almost im-mediately gives way to fluctuations with very largeamplitudes. The conclusion about the route to insta-bility is corroborated by the fact that a variant of thecurrent model, in which individuals are forced to al-ways live in the littoral habitat up to age 1 and in thepelagic afterwards, exhibits the same destabilization atmp ø 0.24.

The model dynamics in this range of mp-values is


FIG. 8. Dynamics of the discrete habitat shift model formp 5 0.28. Top: The solid line with circles represents indi-viduals younger than 1 yr (YOY); the dashed line with tri-angles represents juvenile individuals aged 1 yr and older;and the dotted-dashed line with squares represents adult in-dividuals. Bottom: the solid line indicates the littoral re-source; the dashed line indicates the pelagic resource. Sym-bols mark the start of the season. Units are as in Fig. 3.

illustrated in Fig. 8 and can be described as follows:shortly before T 5 5 several cohorts of juvenile indi-viduals that have been living in the littoral habitat ma-ture almost simultaneously (T represents the time inyears). The individuals switch to live in the pelagic ata size close to the maturation size and hence have notexperienced the size-dependent pelagic mortality. Insubsequent years these adult individuals produce twolarge cohorts of offspring that migrate out into the pe-lagic early in life at small body sizes. The few survivorsof these offspring cohorts, amounting to only 1026%of their initial density, grow rapidly, mature within theirfirst year of life, and join the cohort of adults. Becausebackground mortality decreases the number of adultindividuals in the pelagic over the years, the cohortborn at T 5 7 is again sufficiently small to preventdepletion of the littoral resource. This and all subse-quent, smaller, cohorts born up to T 5 13 stay for alonger time in the littoral habitat and escape the pelagicmortality. Crowding effects obviously retard theirgrowth, since no maturation occurs until shortly beforeT 5 14 (Fig. 8), when the same simultaneous matu-ration pulse occurs as the one that started off the cycle.Detailed analysis of the changes in the population stateover the last 6 yr of the cycle show that the cohortsthat stay in the littoral habitat up to maturation stronglyconverge in body size. Cohorts that are born later catchup with older cohorts, whose growth is retarded due tocrowding. Fig. 8 also shows that the first cohort thatis small enough to prevent overexploitation of the lit-toral habitat is numerically dominating the entire cycle.

The dynamic pattern described above, including thepattern of ‘‘cohort stacking’’ in the littoral habitat andthe numerical dominance of the first cohort that man-ages to stay in the littoral habitat, is characteristic forthe entire range of dynamics between mp ø 0.24 and0.3. The dynamics may be a regular cycle with a periodof 8–9 yr or may be less regular fluctuations: the heightof the peaks may differ, and the period may jump be-tween two values. By and large, however, the dynamicsconsist of an alternation of periods with a dominationof juvenile cohorts in the littoral habitat without anysignificant reproduction and maturation, and periods inwhich the dominant cohort has matured and producesrather large pulses of YOY. These YOY individuals aredestined to die off rapidly as they quickly move outinto the pelagic. Only later on in life, the dominantcohort produces smaller YOY cohorts that are not tooabundant to deplete the littoral resource and can henceremain in the littoral habitat.

Continuous habitat choice

High responsiveness to profitability differences.—Tomodel continuous habitat choice, we used Eq. 6 to de-scribe the fraction of time that individuals spend in thepelagic habitat, as a function of the difference in habitatprofitability between pelagic and littoral. For high val-ues of s individuals strongly respond to small differ-ences in habitat profitability (see Fig. 2). High valuesof s can also be interpreted as modeling omniscientconsumers. As in the case of discrete habitat shifts,consumers react almost instantaneously to differencesin profitability. However, in contrast to the discretehabitat shift scenario, consumers may distribute theirtime over the two habitats and may also reverse theirhabitat choice. We first discuss the population dynam-ics for large s values.

Similar to the case of a discrete habitat shift, thenumber of YOY produced during a pulse of reproduc-tion with continuous habitat choice is high as long asmp , 0.14 (see Fig. 9). The high density of YOY in-duces an almost immediate depletion of littoral re-sources soon after reproduction. As a consequence, af-ter birth YOY quickly move out into the pelagic. Theirlarge number allows the YOY not only to control en-tirely the littoral, but also the pelagic resource density,depressing both to such levels that habitat profitabilitiesare roughly equal. Shortly after birth YOY thus tendto impose resource conditions in both littoral and pe-lagic that allow them to divide their time in such a waythat they approximate an ideal-free distribution.

Because of the difference in productivity betweenthe littoral and pelagic habitat, which is mainly due totheir difference in volume, YOY achieve this ideal freesituation only by spending most of their time (roughly90%) in the pelagic (Fig. 9). For low values of mp, theresulting dynamics are hence close to the dynamicswith proportional habitat use (cf. Fig. 4). For 0 # mp

, 0.08 the same bifurcation pattern of single-cohort


FIG. 9. Bifurcation diagram of the model when individuals continuously adapt their habitat choice to the current profitabilityof littoral and pelagic habitat: (A) the total number of YOY and (B) consumers of age 1 yr and older in the entire lake; (C) thefraction of time that YOY spend in the pelagic habitat; and the resource biomass density (g biomass/m3) in the (D) pelagic and(E) littoral habitat are shown as a function of the mortality constant mp in the open water. s 5 100, while other parameters havetheir default value (see Table 3). Symbols in different shades of gray indicate the existence of alternative attractors.

cycles is observed as before with a stepwise shorteningof the periodicity. These stepwise transitions occur atalmost the same values of mp as in the case of propor-tional habitat use. Obviously, single-cohort cycles arealso robust against the incorporation of continuous hab-itat choice, for reasons that are largely similar to thosediscussed in the previous section on discrete habitatshifts.

With increasing values of mp the density of YOYdeclines rapidly while they spend time in the pelagic.It is this decline in density, in conjunction with the

lower mortality in the littoral, that causes YOY to makegreater use of the littoral habitat as mp increases: Eventhough they almost immediately after birth move outinto the pelagic habitat, they will do so to a lesser extentand for a shorter time. As a whole the average fractionof their first year of life spent in the pelagic habitatdecreases and exhibits a sharp drop around mp ø 0.14–0.15 (Fig. 9). Fig. 10 shows that this sharp drop co-incides with rapid changes in the survival of differentyear classes: Up to mp ø 0.13 the average mortalityrate experienced by YOY increases with increasing val-


FIG. 10. Top: Average mortality rate per day over the first (solid line), second (dashed line), and third (dot-dashed line)year of life as a function of the size-dependent, pelagic mortality constant mp for three different values of s. Bottom: Average,irreversible body size x (g) at the age of 1 (solid line), and 2 yr (dashed line), and the average, maximum body size reached(dot-dashed line) as a function of the size-dependent, pelagic mortality constant mp for three different values of s. All otherparameters have their default value.

ues of mp, but sharply decreases over the range mp ø0.13–0.15. Simultaneously, the average mortality rateof 1-yr-old individuals strongly increases for mp ø0.13–0.15, while the mortality rate of older individ-uals equals the size-independent background mortal-ity mb (Fig. 10). Over this range of mp values, the sizethat individuals reach at the age of 2 and, especially,1 yr old (Fig. 10) also decreases strongly, as does thenumber of YOY produced in each reproduction pulse(Fig. 9).

At these mp values, relatively low densities of YOYdo not strongly deplete the littoral resource. Combinedwith the increasing riskiness of the pelagic, this keepsthe YOY restricted to the littoral habitat and leads totheir higher survival but retarded growth. The changesare similar to those that occur around this value of mp

when consumers exhibit a discrete habitat shift, al-though in the latter case the changes involved a discretejump of dynamics from one attractor to the other. Here,the changes are continuous, but again involve a shiftfrom (1) YOY spending the largest part of their timein the pelagic habitat, (2) experiencing the highest mor-tality, and (3) fastest growth in body size, to (19) YOYspending the largest part of their time in the littoral,(29) 1-yr-old individuals experiencing the highest mor-tality, and (39) fastest growth in body size. In the firstcase, intracohort competition is the main factor causingYOY to move into the pelagic; in the second case,

competition with the subsequent YOY forces 1 yr oldsinto the pelagic.

At mp ø 0.22 the fixed point dynamics change tosmall-amplitude cycles, which at mp ø 0.25, blow upinto large-amplitude fluctuations (Fig. 9). As in caseof discrete habitat shifts, this destabilization arises be-cause of overcompensation in the number of YOY pro-duced: A large pulse of YOY reaches only small bodysizes at age 1, resulting in lower subsequent survivaltill maturation and the production of small offspringcohorts. Vice versa, smaller pulses of YOY reach largerbody sizes at age 1 and exhibit higher subsequent sur-vival and reproduction. Fig. 11 illustrates this high(low) survival of small (large) YOY cohorts. For ex-ample, more individuals, even in absolute numbers,survive till maturation from the small cohorts born atT 5 3 and 8 than from the large cohorts born at T 50, 5, and 11 (Fig. 11). For intermediate-sized YOYcohorts, an additional factor plays a role (for example,those born at T 5 1, 6, and 7): their survival is inter-mediate, but their growth is retarded to such an extentthat they only manage to reproduce for the first timeas 3-yr-old individuals, as opposed to small and largeYOY cohorts that reproduce as 2-yr-old individuals forthe first time (Fig. 11).

Although different in details, the large-amplitudefluctuations that occur for mp $ 0.25 show a similarpattern as observed for the model incorporating a dis-


FIG. 11. Dynamics of the continuous habitat choice modelfor mp 5 0.23 and s 5 100. Top: The solid line with circlesrepresents individuals younger than 1 yr (YOY); the dashedline with triangles represents juvenile individuals aged 1 yrand older; and the dotted-dashed line with squares representsadult individuals. Bottom: The solid line indicates the littoralresource; the dashed line indicates the pelagic resource. Sym-bols mark the start of the season. Units are as in Fig. 3.

FIG. 12. Dynamics of the continuous habitat choice modelfor mp 5 0.28 and s 5 100. Top: The solid line with circlesrepresents individuals younger than 1 yr (YOY); the dashedline with triangles represents juvenile individuals aged 1 yrand older; and the dotted-dashed line with squares representsadult individuals. Bottom: The solid line indicates littoralresource; the dashed line indicates the pelagic resource. Sym-bols mark the start of the season. Units are as in Fig. 3.

crete habitat shift (cf. Fig. 12 and Fig. 8). Shortly beforeT 5 5 several cohorts of juvenile individuals that havenot suffered much from size-dependent, pelagic mor-tality mature almost simultaneously, forming a domi-nant group of adult individuals. In subsequent yearsthese adults produce two large cohorts of offspring ofwhich only few, if any at all, survive because of thesmall size these individuals reach at age 1. Their thirdoffspring cohort (born at T 5 7; Fig. 12) is still rela-tively large, but does not experience competition by alarge YOY cohort in the year after and can hence spendmost of its juvenile period in the littoral habitat. Allsubsequent offspring cohorts produced by the dominantadult group are smaller and hence also spend most oftheir juvenile period in the littoral habitat. All theseoffspring cohorts converge in size and mature virtuallysimultaneously at T 5 13, making up the new dominantgroup of adult individuals. As in the case of discretehabitat shifts, the dynamics are characterized by a pat-tern of ‘‘cohort stacking,’’ simultaneous maturation ofa number of different cohorts and numerical dominanceby the first one or two YOY cohorts that are not tooabundant to deplete the littoral resource, and hencemanage to evade the high pelagic mortality.

Low and intermediate responsiveness to profitabilitydifferences.—As shown in the previous section, highresponsiveness to differences in habitat profitability incase of continuous habitat choice leads to extendedpersistence of the consumer population and to desta-bilization at higher pelagic mortalities. Similar popu-lation-level consequences are observed when consum-

ers exhibit a unidirectional and instantaneous habitatshift. Both type of responses are characterized by al-most stepwise dependence on the difference in profit-ability between the pelagic and littoral habitat (Fig. 2).For lower values of s consumers respond less stronglyto differences in profitability; when s 5 0 they useboth habitats proportionally to habitat volume irre-spective of habitat profitability.

Fig. 13 shows that the unstable dynamics occurringat high values of mp are a consequence of the rapidresponse to differences in habitat profitability. For s5 10 the small-amplitude cycles are still present for0.24 , mp , 0.29, but the high-amplitude fluctuationsshown in Fig. 8 no longer occur. For s 5 1, stablefixed point dynamics occur over the entire range of mp

values .0.08 (Fig. 13). The consumer population does,however, persist for all values of mp investigated (mp

# 0.30), even though with s 5 1 the responsivenessof the consumers to profitability differences does notdiffer greatly from that characterizing proportionalhabitat use (i.e., s 5 0; see Fig. 2).

The cyclic dynamics at high mp values disappear,because with lower responsiveness to profitability dif-ferences YOY will never be completely constrained tousing only the littoral habitat and will therefore alwaysexperience the high pelagic mortality for part of theirtime. For example, with s 5 1, YOY spend at least25% of their first year in the pelagic (results notshown). As a consequence, the first year of life remainsthe year with the highest mortality and the fastestgrowth in body size (Fig. 10). For s 5 1, regulation


FIG. 13. Bifurcation diagram of the model when individ-uals continuously adapt their habitat choice to the currentprofitability of littoral and pelagic habitat for (top) s 5 1 and(bottom) s 5 10. The total number of consumers of age 1 yrand older in the entire lake is shown as a function of thepelagic mortality constant mp. Other parameters have theirdefault value (see Table 3). Symbols in different shades ofgray indicate the existence of alternative attractors.

FIG. 14. Average adult fecundity (top) and total consumerbiomass (bottom; g biomass in entire lake) as a function ofthe size-dependent, pelagic mortality constant mp for s 5 1(solid line), s 5 10 (dashed line), and s 5 100 (dot-dashedline), while other parameters have their default value. Indi-viduals continuously adapt their habitat choice to the currentprofitability of littoral and pelagic habitat.

of the consumer population is therefore mainly throughthe survival of YOY. Because of their substantial mor-tality and the concurrent rapid growth, the body sizethat consumers reach as 1-yr-old individuals is suffi-ciently large to escape high pelagic mortality after-wards. In contrast, for s 5 100 and intermediate tolarge values of mp, regulation of the consumer popu-lation is mainly through the body size reached by con-sumers at age 1, which strongly determines the sub-sequent survival of 1-yr-old individuals that live en-tirely in the pelagic habitat. Only this latter regulationincorporates an overcompensation mechanism thatgives rise to unstable dynamics.

Another consequence of the high YOY mortalitywhen s 5 1 is that resource densities in the littoralhabitat are roughly 5–10 times larger than is the casefor s 5 100 (data not shown, but see the difference inlittoral resource density for 0.08 , mp , 0.14 and 0.15, mp , 0.22 in Fig. 9). The surviving YOY thereforedo not any longer depress littoral resource levels belowthe subsistence density for adults. The littoral evenbecomes the preferred habitat for individuals of 3 yrand older, as the attack rate on littoral resources con-tinues to increase with body size, whereas pelagic at-tack rates decrease to 0 (Fig. 1). For example, for s 51 and mp 5 0.30, YOY spend roughly 27% of their firstyear of life in the pelagic, shift to live entirely in thepelagic at the age of 1 year when the next YOY cohortis born, and shift back again to spend all their time in

the littoral habitat at 2–3 yr old. The increasing attackrate on littoral resources subsequently allows them toattain significantly larger body sizes than those occur-ring for s 5 100 when individuals respond strongly toprofitability differences (compare the left-most andright-most lower panel in Fig. 10). Concomitant withthese larger body sizes, both mean individual fecundityand total population biomass of the consumers are larg-er for lower values of s (Fig. 14).

Summarizing, when consumers only respond mod-erately to habitat profitability differences, the majorityof newborn individuals die as YOY at small body sizes,which constitutes a smaller energy drain to the systemthan when major mortality occurs at the age of oneyear or older. Even though the survival probability tillmaturation is lower than in the case where individualsrespond strongly to profitability differences, the sur-vivors can reach higher body sizes and fecundities andestablish a population with a larger total biomass.

DISCUSSION

Intercohort competition, population persistence,and stability

In size-structured consumer-resource systems, com-petition among and within cohorts has been shown tobe an important factor shaping the population dynamics(Persson et al. 1998). In the absence of high YOYmortality, the competitive pressures exerted by smallbut abundant YOY may force larger and older indi-viduals to starve, which eventually leads to the occur-rence of single-cohort cycles. Persson et al. (1998) sug-gested that the presence of alternative resources could


FIG. 15. Schematic overview of the types of dynamics occurring in the size-structured model with continuous habitatchoice for different values of the behavioral responsiveness s and the pelagic mortality constant mp. The gray area indicatesparameter combinations for which stable fixed points occur. Different shades of gray distinguish between fixed points withdifferent characteristics. White areas indicate either cyclic dynamics or extinction. The designated parameter regions onlyqualitatively indicate where representative patterns of dynamics are to be expected, since the boundaries between them havenot been located in any quantitative detail. The text presents an extensive discussion of these types of dynamics and thecomplex shifts between them.

potentially stabilize these single-cohort cycles. The re-sults in this paper indicate that the addition of a secondresource that is shared among all consumers does notprevent the occurrence of single-cohort cycles. Fig. 3shows that YOY exert tight control over both pelagicand littoral resources, depleting them far below thelevel that adult individuals need to survive. The onlyway that adults can escape this intercohort competitionis when they have access to a resource that is not de-pleted by the YOY. Because of the large YOY densities,this will only occur if the YOY attack rate on one ofthe resources is virtually 0. Hence, only exclusive re-sources will alleviate the intercohort competition be-tween adult and newborn individuals. When the secondresource is not shared among all cohorts evenly, butits exploitation is affected by ontogenetic niche shiftsor flexible habitat choice, single-cohort cycles remainthe dominant outcome of population dynamics at lowlevels of mortality. More specifically, when habitats donot differ in mortality rate (i.e., in the case of pureoptimal foraging), one- and two-resource models yieldthe same single-cohort cycle with a period of 8 yr.

Fig. 15 summarizes the types of dynamics that we

have identified to occur in the size-structured modelwith continuous habitat choice for different values ofthe behavioral responsiveness s and the pelagic mor-tality constant mp. The characteristics of these dynamicpatterns have been discussed in detail in the Resultssection. We did not attempt to derive precise criteriato distinguish between different types of dynamics, nordid we accurately locate the boundaries between re-gions of parameters that lead to different dynamics.Therefore, Fig. 15 gives only a rough overview ofwhich type of dynamics can be expected to occur fora given combination of behavioral responsiveness andpelagic mortality, presented as a cartoon bifurcationdiagram. The dynamics predicted by the model withdiscrete habitat shifts resemble the patterns occurringfor the model with continuous habitat choice in casethe behavioral responsiveness is high (s 5 100).

The model results presented in this paper predict thatsubstantial population effects of flexible behavior areonly to be expected when pelagic mortality levelswould cause extinction of the consumer population ifthe individuals did not respond to difference in habitatconditions (Fig. 15). Under these conditions, YOY


spend longer periods of time in the littoral habitat, bothin case of discrete and continuous habitat shifts, beforemoving out into the pelagic habitat. This increased useof the littoral habitat allows the YOY to evade highpelagic mortality and ultimately leads to increased pop-ulation persistence. A comparable increase in persis-tence has been found in unstructured population modelsincorporating flexible behavior (Gleeson and Wilson1986, Fryxell and Lundberg 1993, 1994, Krivan 1996,1997, Van Baalen et al. 2001).

Our results indicate, however, that the increased per-sistence occurs independent of how strongly consumersrespond to differences in habitat profitability (Fig. 15).If consumer responsiveness to habitat profitability ishigh, as is the case when individuals exhibit a discretehabitat shift or for steep response functions with con-tinuous habitat choice (i.e., high s values; see Fig. 2),YOY spend most of their first year in the littoral habitatand are only driven out into the pelagic by competitionfrom the subsequent YOY cohort. This may causelarge-amplitude fluctuations at high levels of pelagicmortality, because the competition among the YOYthemselves in the littoral habitat incorporates a poten-tially overcompensating mechanism: large YOY co-horts may produce small offspring cohorts as their bodysize at the moment of habitat shift is low through re-tarded growth (Fig. 15). If consumer responsiveness tohabitat profitability is low (i.e., low s values in caseof continuous habitat choice; see Fig. 2) fixed-pointdynamics occur instead. In this case, YOY alwaysspend at least 25% of their time in the pelagic habitat.Although this causes substantial YOY mortality (thesurvival until maturation is significantly lower thanwhen consumers respond more rapidly to profitability),the surviving individuals fare better. They reach largerbody sizes and higher fecundities, such that the totalpopulation biomass is also significantly higher than forlarger s values. An increase in consumer responsive-ness to differences in habitat profitability thus decreas-es growth and fecundity of surviving consumers, de-creases total population biomass, and increases thelikelihood of population fluctuations.

Depending on responsiveness, the increased consum-er persistence, which results from flexible behavior,may thus be associated either with large-amplitudefluctuations or with fixed-point dynamics (Fig. 15).These results resemble the results of unstructured mod-els incorporating optimal foraging or switching behav-ior of consumers. When optimal foraging by consumersfollows a stepwise response to changes in resource lev-els, population cycles may be induced by the behavioralresponse per se (Gleeson and Wilson 1986, Krivan1996, Van Baalen et al. 2001). In contrast, switchingbehavior of predators, representing a more gradual re-sponse to changes in resources, tends to stabilize pop-ulation dynamics (Murdoch and Oaten 1975, Van Baa-len et al. 2001). For an unstructured model Van Baalenet al. (2001) show that a lower responsiveness leads to

a functional response with a more convex shape, whichfulfills the necessary condition for stability of the equi-librium. This route to equilibrium stability contrastswith the mechanism operating in the size-structuredconsumer-resource model, since in the latter the changein stability is clearly determined by the competitionwithin and among different-sized cohorts.

Patterns of habitat use and ideal free distributions

Littoral resource densities are depleted almost in-stantaneously after each reproduction pulse, because ofthe large number of YOY produced. If individuals arecapable of continuously adapting their use of both hab-itats, they will move out into the pelagic habitat forpart of their time and distribute themselves in a waythat is close to an ideal free distribution. A dynamic,ideal free distribution also occurs in unstructured mod-els of identical consumers (Krivan 1997). The idealfree distribution of YOY allows them to control bothresource densities, drive larger individuals to starva-tion, and hence induce single-cohort cycles. As in thesize-structured, consumer-resource model studied byPersson et al. (1998), abundant YOY cohorts thus playa crucial role in the dynamics and induce single-cohortcycles at low levels of pelagic mortality. This co-oc-currence of ideal free distributions and high-amplitudefluctuations contrasts with the often implicit assump-tion that ideal free distributions ensure population dy-namic stability (Fretwell and Lucas 1970, Sutherlandand Parker 1985, 1992).

While under most conditions YOY will initially dis-tribute their time over both habitats, they will residealmost exclusively in the littoral zone when individualsrespond strongly to differences in habitat profitabilityand pelagic mortality is high. Older and larger indi-viduals are most often restricted to the pelagic habitat,unless responsiveness to profitability differences islow, in which case the pelagic is used by 1- and 2-yr-old individuals, with still older individuals exclusivelyforaging on the littoral resource. These observationsresemble patterns of habitat use in, for example, blue-gill sunfish (Lepomis macrochirus; Dimond and Storck1985, Werner and Hall 1988) where the YOY often useboth the littoral and open-water habitat during someperiod, while during the rest of their life they seem tobe more bound to one of the habitats at a time. Theusual hypothesis concerning the use of the pelagic byYOY bluegill, is that very small individuals are in-vulnerable to predation by larger piscivores. Mortalityrisks in the pelagic are hence low for a short periodafter hatching, which allows the YOY to exploit it. Ourmodel predicts a similar pattern of habitat use, whichis, however, entirely driven by the feedback of the YOYon littoral and pelagic resources and does not dependon an invulnerable size window early on in life. Whichof these explanations is more relevant for the habitatuse of YOY bluegill, can only be determined by de-tailed measurements of the resource dynamics during


the period in which bluegill shifts to the pelagic andback. Existing data (Mittelbach 1981) do not have suf-ficient resolution, but do hint at a concomitant declinein both littoral and pelagic resource during the first partof the growing season. Thus, they seem to support theidea that the population feedback of YOY on both re-sources may be a crucial factor in their pattern of hab-itat use.

The effect of refuge size

The relative sizes and productivities of the littoraland pelagic habitat parts may play an important rolein determining the population dynamics. The large vol-ume of the pelagic constitutes the basis for a high totalproductivity, and therefore a high total population fe-cundity. The feedback of the large number of YOY issignificantly intensified in the littoral habitat, becauseit only occupies 10% of the total lake volume. Thisspecific size of the littoral refuge as fraction of the totallake volume was chosen to reflect a typical lake mor-phology in northern Scandinavian lakes. Larger littoralzones, such as occur at lower latitudes, and especiallylower pelagic productivities may allow for a more ex-tensive influence of flexible behavior on the populationdynamics.

To test for the influence of refuge size, we also stud-ied the bifurcation pattern of the model with both thediscrete habitat shifts and continuous habitat choicescenario, assuming that the littoral refuge part occupied40% of the total lake volume. These results (not shown)indicate that an increase in refuge size generally in-creases total population biomass for all values of thepelagic mortality constant mp and all values of con-sumer responsiveness s. However, an increased refugesize does not induce qualitatively different dynamics.For low values of mp single-cohort cycles occur, albeitover a slightly smaller range of parameters as comparedto a 10% littoral zone fraction. Flexible behavior, boththe discrete habitat shift and continuous habitat choicescenario, increases population persistence independentof the strength of consumer responsiveness to profit-ability differences. For high consumer responsiveness,population dynamics become unstable at a value of thepelagic mortality constant mp that turns out to be vir-tually the same for the case with a large (40%) and asmall (10%) littoral zone. Finally, also when the refugesize is larger, an increase in consumer responsivenessto habitat profitability differences decreases total pop-ulation biomass and mean annual fecundity of the sur-viving individuals.

Thus, in contrast to intuitive expectations, the sizeof the refuge does not play an important role in deter-mining the population dynamic outcome. Total popu-lation biomass, and thus the population feedback onindividual life history, adjusts itself to the larger refugesize, such that the changes at the individual level areminimal. As a consequence, the larger refuge size also

induces little change to the competitive relationshipswithin the consumer population.

Different rules of individual behavior

The results presented in this paper do not seem todepend strongly on the choice of the profitability mea-sures (Eqs. 3 and 4) for the littoral and pelagic habitat,respectively. We have obtained roughly the same dy-namic pattern when using the difference between themass-specific growth rate and the mortality rate as prof-itability measure. For example, for the littoral habitatthe profitability would in this case be described by

[k I (z , x) 2 E (x, y)]e l l m 2 d (x, y).lx 1 y

By maximizing the above quantity, a cohort of in-dividuals would at any moment in time maximize itsexpected rate of change in total cohort biomass (der-ivation not presented). In addition, we have also ob-tained roughly similar results when assuming that pe-lagic mortality is a stepwise function of body size,adopting a constant but high value for juvenile indi-viduals and a value equal to the background mortalityin the littoral habitat after maturation. All model var-iants that we have studied show that ontogenetic nicheshifts and flexible behavior increase consumer persis-tence, but that at low levels of pelagic mortality theycannot stabilize single-cohort cycles, while at high lev-els of pelagic mortality they may induce large-ampli-tude cycles in which the abundance of a YOY cohortstrongly determines its subsequent fate (cf. Figs. 8 and12). Stable fixed-point dynamics occur only for inter-mediate levels of pelagic mortality. This region offixed-point dynamics is significantly larger when con-sumers do not strongly respond to differences in habitatprofitability.

The results presented here (1) reveal that flexiblebehavior may play an important role in determining theoutcome of competition within and between cohorts,and (2) lead to predictions about patterns of habitat useduring their life history. In contrast to previous studies(Werner and Gilliam 1984) these predictions not onlyincorporate dependence on body size and habitat-spe-cific resource and mortality conditions, but also takeinto account the feedback of consumers on both re-sources, as it is modulated by their flexible behavioritself. Given that different measures of habitat profit-ability lead to qualitatively similar results, we concludethat the precise way in which individual consumersweigh mortality risks against feeding advantages intheir choice of habitat is of minor importance. Rather,it is the population feedback, especially from small,competitively superior individuals, in combinationwith the responsiveness to this feedback, that ulti-mately determines the pattern of habitat use of indi-vidual consumers during their life history.


ACKNOWLEDGMENTS

A.M. de Roos is financially supported by a grant from theNetherlands Organization for Scientific Research (NWO). L.Persson is financially supported by grants from the SwedishNatural Science Research Council and the Swedish Councilfor Forestry and Agriculture. This project was initiated whileG. G. Mittelbach visited the University of Umea on a visitingscholarship from the Swedish Council for Forestry and Ag-ricultural Science. This is contribution number 954 from theW. K. Kellogg Biological Station, Michigan State University.

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APPENDIX

POPULATION-LEVEL MODEL FORMULATION

The pulsed reproduction process ensures that there existsa natural subdivision of the consumer population into a var-iable number (c) of cohorts. Every cohort is characterizedby a set of variables (Ni, xi, yi), representing the total numberof individuals, Ni, making up the cohort, their irreversiblemass, xi, and reversible mass, yi, respectively. The index i 51, . . . , c refers to the cohort number, which is only used forbookkeeping purposes. The state of the resource populationsin the littoral and pelagic habitat are represented by the bio-mass densities zl and zp, respectively.

Within-season dynamics

The dynamics of the consumer population during the grow-ing season is described by a set of ordinary differential equa-tions (ODEs) for the change in number of individuals in thevarious cohorts, their reversible and their irreversible mass.Under growing conditions, i.e., Eg(zp, zl, xi, yi) . 0, the ODEsfor Ni, xi and yi are given by (see also Table 2):

dNi 5 2d(z , z , x , y )N (A.1a)p l i i idt

dxi 5 k(x , y )E (z , z , x , y ) (A.1b)i i g p l i idt

dyi 5 [1 2 k(x , y )]E (z , z , x , y ) i 5 1, . . . , c. (A.1c)i i g p l i idt

When food is scarce and individual maintenance require-ments are higher than the current energy assimilation rate,i.e., Eg(zp, zl, xi, yi) , 0, growth in irreversible mass stops andthe energy deficit is covered from reversible mass:

dNi 5 2d(z , z , x , y )N (A.2a)p l i i idt

dxi 5 0 (A.2b)dt

dyi 5 E (z , z , x , y ) i 5 1, . . . , c. (A.2c)g p l i idt

The dynamics of the resource populations in both the lit-toral and pelagic habitat are governed by ODEs, which arecoupled to the equations above by the foraging of the con-sumer cohorts:

cdz 1p5 r (K 2 z ) 2 F(z , z , x , y )I (z , x )N (A.3a)Op p p p l i i p p i idt V i51p

cdz 1l 5 r (K 2 z ) 2 [1 2 F(z , z , x , y )]I (z , x )N .Ol l l p l i i l l i idt A i51l

(A.3b)

Regrowth of the resource populations follows a semiche-mostat dynamics with parameters rl and Kl or rp and Kp forthe littoral and pelagic resource, respectively (see Table 3).The factors 1/Al and 1/Vp occur in these equations, becauseresource densities are expressed as biomass per unit bottomsurface and lake volume, respectively, while the consumerpopulation is expressed in terms of the number of individualsin the entire lake.

When individuals use both habitats proportionally, s equals0 in Eq. 6 and the value of the function, F(zp, zl, x, y), is equalto 0.9 for all cohorts. In case of the discrete habitat shift sce-nario, F(zp, zl, x, y) assumes the value of 0 for newborn indi-viduals and is set to 1, as soon as the profitability of the pelagichabitat becomes larger than the littoral profitability, Qp(zp, x,y) . Ql(zl, x, y). For the continuous habitat choice scenario thevalue of F(zp, zl, x, y) for each cohort is specified by Eq. 6.

Between-seasons dynamics

For every cohort of consumer individuals a set of ODEs(A.1) (or A.2 for starving cohorts) is solved numerically fromthe beginning of the growing season until the beginning ofthe next growing season. The ODEs (A.3) for the dynamicsof the resources are solved simultaneously. At the beginningof the growing season consumer reproduction takes place,which is described by a discrete map. A new consumer cohortarises with the following state:

c

N 5 f (x , y )N (A.4a)O1 i i ii51

1x 5 m (A.4b)1 b1 1 q j

q jy 5 m (A.4c)1 b1 1 q j

while the state of all reproducing cohorts is reset such thatyi 5 qjxi. Simultaneously, the indices of all existing cohortsare renumbered for bookkeeping reasons:

N 5 N (A.5a)i11 i

x 5 x (A.5b)i11 i

q x if f (x , y ) . 0j i i iy 5 i 5 1, . . . , c. (A.5c)i11 5 y otherwisei

These operations increase the number of consumer co-horts by 1. The number of cohorts is kept limited by re-moving cohorts from the population that have become ofnegligible size (i.e., Ni , 1) due to background or starvationmortality.

Documents

UvA-DARE (Digital Academic Repository) …SIZE-STRUCTURED POPULATIONS A NDRE´ M. DE R OOS , 1,4 K JELL L EONARDSSON , 2 L ENNART P ERSSON , 2 AND G ARY G. M ITTELBACH 3 1 Institute