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Limnologica 38 (2008) 269–285

www.elsevier.de/limno

Individual-based simulation of diel vertical migration of Daphnia:A synthesis of proximate and ultimate factors

Karsten Rinkea,b,�, Thomas Petzoldta

aInstitut fur Hydrobiologie, Technische Universitat Dresden, 01062 Dresden, GermanybLimnological Institute, University of Konstanz, Mainaustrasse 252, 78464 Konstanz, Germany

Received 28 April 2008; accepted 22 May 2008

Abstract

Diel vertical migration (DVM) of Daphnia is a well-studied inducible defence mechanism against predation by fish.Our study is anchored in constructing an individual-based model of DVM in order to bring established knowledgeabout essential key processes into a synthesis. For that purpose, we combined information about both proximate andultimate factors of DVM with the intention to unify published results from these historically separated lines ofresearch. The model consists of three submodels: (i) movements, (ii) growth and reproduction, and (iii) mortality. Thesubmodel ‘‘movements’’ includes algorithms for light-dependent migration behaviour of Daphnia that were able toreproduce spatiotemporal distribution patterns of DVM. By means of scenario analyses, we tested the predatoravoidance hypothesis by comparing population growth rates of migrating and non-migrating populations over a rangeof fish biomasses in the habitat. This enabled us to quantify the adaptive value of DVM under various environmentalsettings. Simulation results supported the predator avoidance hypothesis and showed a particularly high adaptivevalue of DVM if fish predation is intense. However, since DVM is associated with costs, a certain predation pressure inthe habitat has to be prevailing in order to turn DVM into an adaptive strategy. Otherwise, if fish predation is weak,migrating populations realize lower population growth rates than non-migrating populations. In a second scenario, wetested the influence of vertical gradients of temperature and food on the adaptive value of DVM. We found a greatpotential to maximize the adaptive value of DVM if daphnids are able to modify their migration amplitude independence of the vertical structure of their habitat. For example, a deep chlorophyll maximum (DCM) can be astrong attractor for Daphnia to modify the migration amplitude in such a way that the daytime depth corresponds tothe depth of the DCM. However, flexibility of the migration amplitude is only advantageous if the predation intensityis moderate – if predation is intense, only maximum migration amplitudes maximize fitness.r 2008 Elsevier GmbH. All rights reserved.

Keywords: Zooplankton; Adaptive value; Swimming behaviour; Vertical gradients; Food; Temperature; Fish predation; Deep

chlorophyll maximum

e front matter r 2008 Elsevier GmbH. All rights reserved.

no.2008.05.006

ing author at: Limnological Institute, University of

naustrasse 252, 78464 Konstanz, Germany.

882930; fax: +49 7531 883533.

ess: karsten.rinke@uni-konstanz.de (K. Rinke).

Introduction

Diel vertical migration (DVM) of plankton organismsis a fascinating phenomenon that attracted ecologists formore than 100 years. Already 130 years ago, Weismann

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285270

(1877) firstly described DVM of freshwater zooplanktonin Lake Constance and Carl Chun, the organizer of thefirst German deep sea expedition, was intrigued by thevertical migrations of marine zooplankton he observedduring his expedition (Chun, 1890). These pioneers werefascinated by the fact that such relatively smallorganisms like crustacean zooplankters migrate overdozens of metres every day for spending the night closeto the surface and diving down into deep waters duringthe day. A nowadays particularly well-studied exampleof DVM is provided by the genus Daphnia, whichperforms intensive DVM with amplitudes that canexceed 40m (Stich, 1989). This huge migration ampli-tude relative to the small size of Daphnia has motivatedlimnologists to disentangle which ultimate (making thebehaviour evolutionary adaptive) and proximate (elicit-ing and steering the behaviour) factors are associatedwith DVM behaviour.

Regarding the ultimate factors of DVM, there hasbeen a long debate with competing hypotheses. Someauthors claimed there is a metabolic advantage ofstaying in the cold, deep water layers (Enright, 1977;Geller, 1986; McLaren, 1963), e.g. by growing to largerbody sizes at cold temperature resulting in a higherfecundity or by an improved starvation resistance incold environments. Another theory on ultimate factorsof DVM is the predator avoidance hypothesis, whichwas already discussed for a long time but firstly clearlyformulated by Zaret and Suffern (1976). This long-lasting debate between both competing hypotheses wasresolved by the experiments of Stich and Lampert(1984), who undoubtedly refuted the metabolic advan-tage hypothesis and showed that DVM is associatedwith considerable costs due to low temperature andpoor food conditions in deep water layers. Finally, theexperimental proof that fish infochemicals trigger DVM(Dodson, 1988; Loose, 1993a, b) paved the way for abroad and enduring acceptance of the predator avoid-ance hypothesis (Lampert, 1993).

Since DVM is always tightly coupled to the daily lightcycle the role of light as a proximate factor was neverreally questioned. The ‘‘light preferendum hypothesis’’assumed individuals to migrate vertically in order tokeep ambient light intensity constant (staying in thesame isolume, Richards et al., 1996) and was used tosimulate the spatiotemporal progression of DVM.However, extensive field observations (Ringelbergand Flik, 1994; Ringelberg et al., 1991) and detailedlaboratory experiments (van Gool, 1997; van Gool andRingelberg, 1997) unambiguously elaborated that DVMof Daphnia is not directly controlled by light intensitybut by the relative change in light intensity (RLC, seeRingelberg, 1999). A second proximate factor is thepresence of infochemicals released by fish (kairomones),which are obligatory for inducing a DVM in Daphnia

(Dodson, 1988; Loose, 1993a, b) and manifest DVM as

an inducible defence mechanism against fish predation(Harvell, 1990). Finally, food availability in the habitatand developmental stage of individual Daphnia areknown factors that can modify migration behaviour ofDaphnia (Johnsen and Jakobsen, 1987; Pijanowska andDawidowicz, 1987). Surprisingly, no experimental evi-dence was found for an endogenous rhythmicity inDaphnia associated with the timing of DVM (Loose,1993a, b).

Besides the very obvious advantage of performingDVM in a risky habitat with strong fish predation in theepilimnion during the day, there are a couple of factorsthat potentially modify the adaptive value of DVMconsiderably. In fact, Lampert (1993) concluded thatDVM is not a fixed behaviour but a flexible strategy asthere is a trade-off between maximum protection andmaximum energy intake. Other factors may furtheraffect this trade-off, for example, toxic cyanobacteria inthe epilimnion may suppress ascent, or oxygen depletionin the hypolimnion may constrain descent during DVM(Forsyth et al., 1990).

The major cost of DVM primarily arises from the lowtemperatures in deep waters and, to a usually lesserextent, from poor food conditions (Dawidowicz, 1994;Loose and Dawidowicz, 1994). Whereas in stratifiedsystems temperature necessarily always decreases withdepth, food concentration can also increase with depth,e.g. if a deep chlorophyll maximum (DCM) exists (Molland Stoermer, 1982). In such a case, the actual verticalfood gradient in the habitat would have a strong effecton the adaptive value of the DVM; if the daytime depthof the daphnids match the depth of the DCM, costs ofDVM would be reduced relative to the opposite case.From experiments in Plankton Towers, we know thatDaphnia is able to sense vertical gradients and optimizetheir depth distribution in order to minimize costs (idealfree distribution with costs, Lampert et al., 2003).Generally speaking, without an external cue forcingindividuals to perform DVM Daphnia is capable tomaximize individual fitness by finding an optimal depthdistribution (Kessler and Lampert, 2004). Particularlyfor the dominating factors temperature and foodconcentration it was shown that Daphnia is able toactively find optimal patches (Calaban and Makarewicz,1982; Jensen et al., 2001). Unfortunately, we do not havea clear picture whether or to which extent, respectively,Daphnia is able to adapt its migration behaviour toactual vertical gradients of food in the habitat.

The aims of this study are threefold. Firstly, we intendto construct a model that is able to incorporatemajor outcomes of recent DVM research providing aplatform to bring this knowledge into synthesis. Sinceresearch on proximate factors of DVM and ultimatefactors of DVM was often well separated from eachother we felt particularly motivated to combine resultsfrom these two sides of the same coin. We chose an

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 271

individual-based modelling approach (IBM, DeAngelisand Mooij, 2005; Grimm, 1999) because the mostrelevant processes are associated with individual levelcharacteristics, empirical knowledge about DVM ismainly derived from experiments with individuals, andan extensive characterization of the individual is there-fore required. We also decided to use a spatially explicitmodel since we intended to apply continuous verticalgradients of temperature and food. Secondly, we aim onusing the model for quantifying the adaptive value ofDVM at different levels of fish predation. The third goalis to study whether flexible migration behaviour, e.g.reflected by flexible migration amplitudes, could poten-tially increase the adaptive value of DVM. For thisscenario, we are particularly interested in the potentialeffect of a DCM on the migration behaviour.

Model

The individual-based model is focused on Daphnia

galeata�Daphnia hyalina and consists of three sub-models: (i) submodel movements, (ii) submodel growthand reproduction, and (iii) submodel mortality, whichare described in detail as follows. A flow chart of themodel structure is provided in Fig. 1A. Calculations ofthe submodels (ii) and (iii) are performed with a

Fig. 1. (A) An overview over the model structure and the individ

(B) Flow chart of the submodel movements. The parameterization

simulation time step of one hour whereas calculationsof the submodel movements are done at finer time-steps(1 or 10min, respectively, see below). A complete listof parameters is provided in Table 1 and a brief outlineof the characteristics of the model following thestandard protocol for describing individual-basedmodels (Grimm et al., 2006) is provided in the appendix.The main aim of the model was to provide anestimate of the fitness (expressed as population growthrate r) under the respective environmental setting,defined by the factors temperature, food, and predationrisk. Note that these environmental factors are not onlydescribed by single values but are represented by one-dimensional gradients within the habitat and that ourapproach includes an explicit spatial representation. Inorder to keep the model simple and to allow thequantification of adaptive values in a fixed environ-mental setting, no feedback of Daphnia on its foodresource was included.

Submodel movements

We provided the individuals with four different modesof movement: food search, surface avoidance, negativegeotaxis, and vertical migration (Fig. 1B).

The standard movement is a food search basedon a random walk along the vertical axis of the lake.

ual life-loop executed by each individual during simulations.

of the model is based on empirical measurements.

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Table 1. Environmental factors and constant parameter

values of the individual-based model

Parameter Value Unit

Characterisation of the habitat of the population

F Vertical gradient mgCL�1

T Vertical gradient 1C

I Daily light cycle mEm�2 s�1

e 0.61 m�1

Bfish 0–2 gm�3

Parameters of the Daphnia model: submodel movements

v N(m ¼ 0, s ¼ 0.34) m

vdownward 0.29 m

Icrit 2� 10�7 mEm�2 s�1

Kfish 0.28 gm�3

fse �1.47 LmgC�1

fsc 0.2 –

rheobase 0.04 min�1

uintercept 0.027 mmin�1

uslope 0.307 m

accintercept �0.395 –

accslope 1.408 –

decintercept 0.437 –

decslope 0.541 –

MVmax,a 0.45 mmin�1

MVmax,j 0.225 mmin�1

Parameters of the Daphnia model: submodel growth and

reproduction

SAM 1.30 mm

SON 0.55 mm

a1 1.167 mm

a2 0.573 mgCL�1

a3 1.42 mm

a4 2.397 day

b1 1.09� 10�2 day�1

b2 0.122 (1C)�1

L0,Hall 0.35 mm

amax 23.83 Eggs

aks 0.65 mgCL�1

b 29.28 Eggs

bottrella 3.3956 –

bottrellb 0.2193 –

bottrellc 0.3414 –

Parameters of the Daphnia model: submodel mortality

amax N(m ¼ 50, s ¼ 10) days

mf 0.92� 10�3 m3 s g�1 mE�1

mi 10�4 mg�1 h�1

y �0.25 –

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285272

This movement is simulated with a time-step of 10min.The vertical displacement v of an individual within a10min interval was randomly chosen from a normaldistribution (based on assumptions made by Richardset al., 1996). This vertical swimming speed is multipliedby a food-dependent factor resulting in verticaldisplacement V to be inversely proportional to food

concentration F:

v ¼ Nðm ¼ 0;s ¼ 0:34Þ (1)

V ¼ vðexpðfseF Þ þ fscÞ (2)

Values for fse ( ¼ �1.47) and fsc ( ¼ 0.2) were derivedfrom experiments and the assumption of an ideal freedistribution of individuals with respect to food concen-tration (Larsson, 1997; Larsson and Kleiven, 1996). Dueto the negative exponent in Eq. (2) vertical swimmingvelocity declines with increasing food concentration.

If an individual approaches the lake surface (depth z

is below zcrit ¼ 2.5m, Fig. 1B) surface avoidance isactivated in order to avoid the occurrence of negativedepths. Ecological reasons for surface avoidance byDaphnia are given by possible damages from UVradiation or increased shear stress due to wave activity(Ringelberg, 1999). In addition to the vertical displace-ment calculated by the food search algorithm (Eqs. (1)and (2)), a downward movement (down) is added asgiven by Eq. (3). According to Ringelberg (1995), weassumed a value of 0.29m for the parameter vdownward:

if z ozcrit : down ¼ cospz

2zcrit

� �vdownward (3)

Similarly to the surface avoidance behaviour weintroduced a negative geotaxis for individuals residingat very low light intensities in order to avoid individualsdrifting in extremely large depths. Empirical evidencefor negative geotaxis is given by Ringelberg (1964), whofound that negative geotactical movements are ex-pressed if light intensities are very low, so we introduceda critical light intensity Icrit. If ambient light intensityI(z) of an individual at depth z falls below Icrit the valueof vdownward is subtracted from the vertical displacementcalculated by the food search algorithm (Eqs. (1) and(2)) inducing an upward movement. A value ofIcrit ¼ 2� 10�7 mEm�2 s�1 was assumed, based on thedaily light cycle and the fact that migrating individualsduring day should experience light intensities above Icritin their deep water refuge.

To account for light-mediated effects on the migra-tion behaviour we applied a daily light cycle that wasderived from average conditions in early spring incentral Europe (light model adopted from Richardset al., 1996). Since RLCs are particularly high duringtwilight where commonly used light sensors are notsensitive enough we used measured light intensitieskindly provided by J. Ringelberg (NIOO-Centre forLimnology, Nieuwersluis, The Netherlands, pers. com.,but see also Ringelberg et al., 1991) and inserted thesemeasurements into an average daily light cycle duringtwilight times (Fig. 2). During night a light intensity of10�5 mEm�2 s�1 was assumed.

According to van Gool and Ringelberg (1997), thevertical migration behaviour starts as soon as the

ARTICLE IN PRESS

Time (h)

0

100

200

300

0

Ligh

t int

ensi

ty (µ

E m

−2 s

−1)

6 12 18 24Time (h)

−0.10

−0.05

0.00

0.05

0.10

0.15

RLC

(min

−1)

0 6 12 18 24

Fig. 2. Daily cycle of light intensity (A) and relative change in light intensity (RLC, B) applied in the simulations. The thick line

segment during dawn and dusk is based on measurements conducted by J. Ringelberg (pers. com.).

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 273

absolute value of the RLC exceeds a critical value(rheobase ¼ 0.04min�1, Fig. 1B) with a migrationvelocity (MV, mmin�1) given by (refer to Table 1 forparameter values):

if jRLCj4rheobase : MV ¼ uintercept þ uslopeRLC

(4)

Further studies of this group showed that migrationvelocities are accelerated and decelerated if a sequenceof non-constant RLCs is applied – as it is the caseduring dusk or dawn (compare Fig. 2). If the absolutevalue of the RLC is increasing with time, i.e.jRLCt1 j4jRLCt0 j, van Gool and Ringelberg (1997)found an accelerating migration velocity (Eq. (5) andvice versa for decreasing RLCs (Eq. (6), van Gool,1997), respectively:

if jRLCt1 j4jRLCt0 j:

MVt1 ¼ accintercept þ accslopeRLCt1

RLCt0

� �MVt0 (5)

if jRLCt1 jojRLCt0 j:

MVt1 ¼ decintercept þ decslope �RLCt1

RLCt0

� �MVt0 (6)

For further details of Daphnia swimming velocityduring DVMs we refer to the original papers (van Gool,1997; van Gool and Ringelberg, 1997) and the overviewprovided by Ringelberg (1999). Note that duringsimulation positive RLCs have to be explicitly includedto result in downward movements and negative RLCs inupward movements, respectively. The time step ofmigration moves is 1min according to the measurementsprovided by the original papers (van Gool, 1997; vanGool and Ringelberg, 1997). Since migration behaviourwas shown to be also influenced by the light extinctioncoefficient of the water body (Dodson, 1990), weassumed a light extinction coefficient to be 0.61m�1,which was the respective value in the experiments onDaphnia swimming behaviour cited above (E. van Gool,NIOO-Centre for Limnology, Nieuwersluis, The Neth-erlands, pers. com.). Under specific conditions, the

outlined migration behaviour can lead to very highmigration velocities due to the acceleration effect. Inorder to keep migration velocities in a reasonable rangewe introduced a maximum migration velocity (MVmax,a)that cannot be exceeded and assigned a value of0.45mmin�1 to this parameter (Young and Watt, 1993).

Predator kairomones affect migration behaviour aswell and are taken into account by following the resultsobtained by Loose (1993a, b). Since kairomone concen-tration cannot be measured directly he used fish biomass(Bfish) as a proxy for kairomone concentration andfound a hyperbolic response of Daphnia migrationamplitude with increasing Bfish. We adopted this resultby introducing a relative scaling factor (fishscale) that ismultiplied with the calculated migration velocity inresponse to ambient light conditions:

fishscale ¼Bfish

K fish þ Bfish(7)

A value of 0.28 gm�3 was associated with theparameter Kfish (Loose, 1993a, b).

Finally, field observations and experiments showedthat migration behaviour of Daphnia is stage-dependent(Johnsen and Jakobsen, 1987; Ringelberg et al., 1991).Neonates were found to reside in the epilimnion for thewhole day without performing DVM. It was furtherfound that larger juvenile individuals realize smallermigration amplitudes than adults. Johnsen and Jakob-sen (1987) divided their experimental population intoadults (L(t)4SAM), juveniles (SAM4L(t)4Lj,min), andneonates (L(t)oLj,min), whereby they used the arith-metic mean between size at maturity (SAM) and SON(size of neonates) for Lj,min:

Lj;min ¼SAMþ SON

2(8)

We adopted this categorization and enabled DVMbehaviour only for juveniles and adults, not forneonates. Furthermore, maximal migration velocity ofjuveniles (MVmax,j) was reduced by 50% in comparisonto the respective value of adults (MVmax,a). Thiscomplies with published relationships between body

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285274

size and maximal swimming velocity (Baillieul andBlust, 1999; Dodson and Ramcharan, 1991). A reduc-tion of the maximum migration velocity by 50% resultsunder the environmental setting applied to a reductionof the migration amplitude by approximately 30%,which is in agreement with observations (Johnsen andJakobsen, 1987).

Submodel growth and reproduction

Growth and reproduction of Daphnia was modelledas a function of ambient food concentration andtemperature by following the approach of Rinke andPetzoldt (2003). Calculations were performed on a time-step of 60min. Since movements of the individuals tookplace on a smaller time-step we used mean temperatureand food conditions averaged over the past 60min asinputs into this submodel. Somatic growth and eggproduction are calculated according to the model ofRinke and Petzoldt (2003) with somatic growth:

L ¼ SON� L0;Hall þ Lmax � ðLmax � L0;HallÞe�kt (9)

with

Lmax ¼a1F

a2 þ Fþ a3 � a4k (10)

k ¼ b1eb2T (11)

and egg production:

E ¼amaxF

aks þ FL� bð1� e�1F Þ (12)

The individual level model of Rinke and Petzoldt(2003) was originally developed to predict growth andreproduction under conditions of constant temperatureand food concentration. However, this is not the case inour application since ambient temperature and foodconcentration change with time due to the individualswimming behaviour. In order to adopt this model tovarying temperature and food conditions we applied theconcept of physiological age (for somatic growth,Geller, 1989) and the rate summation method (for eggproduction, Keen, 1981).

At each time-step when the submodel growth andreproduction is invoked the average temperature and theaverage food concentration over the past 60min and thecurrent length of the individual was used to calculate asomatic growth curve by means of the formulas givenabove. Given the actual body length L of the individualand the calculated somatic growth curve a physiologicalage a can be computed. For the current time-step, thebody length is then just updated to the body lengthL(a+60min) 1 h later, which can be directly calculatedfrom the Eqs. (9)–(12). In case of L(a+60min)oL(a) itwas assumed that the body length remains constant andthe old value of L(a) is retained for this time-step. Such a

situation can easily arise if an individual has moved intoa food-depleted depth where somatic growth is ceased.Since Daphnia has a rigid exoskeleton they do not shrinkin size under food shortage.

Egg production in Daphnia is a discontinuous process;eggs are deposited directly after moulting in a broodpouch wherein they are carried until the next moult. Eggdevelopment time D (days) is a temperature-dependentfunction (T, temperature, 1C) and described by anempirical formula by Bottrell et al. (1976):

D ¼ expðbottrella þ bottrellb lnðTÞ � bottrellcðlnðTÞÞ2Þ

(13)

At each time-step the submodel growth and repro-duction is invoked; a hypothetical clutch size Ehypothetical

is calculated from the average temperature and foodconditions over the last 60min and the actual bodylength by Eqs. (9)–(12). The current number of eggsin the brood pouch of the individual E was thenincremented by this hypothetical clutch size multipliedby the reciprocal of egg development time:

Etþ60 min ¼ Et þ EhypotheticalD�1 (14)

In parallel, the stage within the current egg develop-ment cycle of the individual (eggstage) is traced bysumming D�1 over time:

eggstagetþ60 min ¼ eggstaget þD�1 (15)

As soon as eggstage reaches a value larger than onethe current egg development cycle is finalized and theeggs in the brood pouch are converted in neonates,which are added to the current population. At the sametime, eggstage is decremented by subtracting 1 from itscurrent value.

Note that in this model formulation body length isdescribed as a continuous process (it can potentiallyincrease at each time-step) whereas reproduction is adiscontinuous process, which arises from a continuousproduction of eggs that are discontinuously released asneonates at distinct times.

Submodel mortality

Three sources of mortality are included in the model:(i) senescence, (ii) predation by positively size-selectivelyfeeding fish, and (iii) predation by negatively size-selectively feeding invertebrate predators.

Life expectancy of Daphnia is highly variable. Whilethey can become relatively old in the laboratory (up to70 days, e.g. Lynch, 1980) the life span under fieldconditions is considerably shorter (Hulsmann andVoigt, 2002). For example, Vos et al. (2002) showedthat daphnids may face a 36% per day mortality duringthe peak of fish predation, even if they perform DVM.For that reason, we set individual maximum lifespan(days) as amax�N(m ¼ 50, s ¼ 10) as a random variable.

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 275

Predation effects were implemented by modifying anapproach chosen from Fiksen (1997). The predation ratedfish (h

�1) is assumed to be proportional (proportionalityfactor mf) to the ambient light intensity I(t,z) at time t

and depth z, fish biomass in the habitat (Bfish), andcross-sectional body area (squared body length L(t)) ofthe daphnid. The latter is given as a relative area scaledto a given reference body area. In our case, we chose theSAM as the reference for this cross-sectional area:

d fish ¼ mfIðt; zÞBfishð1=2LðtÞÞ2

ð1=2 SAMÞ2(16)

The ambient light intensity I(t,z) of an individual iscalculated from the surface radiation at time t and depthz ¼ 0, I(t,z ¼ 0) taken from the daily light cycledepicted in Fig. 2, and light extinction coefficient of thewater (e ¼ 0.61m�1) according to Lambert–Beer’s lawI(t,z) ¼ I(t,z ¼ 0) e(�e z). Since the time-step for individualmovements (1 or 10min) is shorter than the time-step inthe submodel mortality (60min), ambient light intensity ofeach individual was calculated as the geometric mean (dueto the exponential decrease of light intensity with depth)of the light intensities experienced within the last 60min.

Besides positive size-selective predation by fish, we alsoincluded a term representing predation by invertebratepredators, which usually select small prey (Lynch, 1979).Since we consider an environment where predationpressure on Daphnia is dominated by fish, only a weakpredation by invertebrates was included. Based on anapproach by McGurk (1986) and Fiksen (1997), inverte-brate predation rate dinvertebrate (h

�1) was calculated by

d invertebrate ¼ miðW � 10�6Þy (17)

Fig. 3. Contour plot of individual predation risk (h�1) over the

ranges of light intensity and body sizes relevant during the

simulations.

We adopted parameter values from Fiksen (1997) forthe proportionality factor mi ( ¼ 10�4) and the exponenty ( ¼ �0.25). The body dry weight W of individualDaphnia was calculated from body length L byW ¼ 11.705L2.52 according to Bottrell et al. (1976). Sincecommon invertebrate predators like Chaoborus are tactilepredators their predation rate is not related to ambientlight intensity.

Finally, predation rates were converted to an indivi-dual predation risk pi (h

�1) by

pi ¼ 1� expð�d fish � d invertebrateÞ (18)

Calculated individual predation risks over the rele-vant range of body sizes and light intensities aredepicted in Fig. 3. With the exception of extremelylow light intensities, where fish predation is negligibleand slight predation by invertebrates dominates, preda-tion risk increases with increasing light intensity andindividual body size, making the epilimnion duringdaytime a risky habitat for large Daphnia.

Simulations

The model was implemented in the programminglanguage JAVA. Each simulation was started with 80individuals (40 neonates, 16 juveniles, and 24 adults)with random individual properties (length, etc.). Thestarting depths of the initial individuals were distributedrandomly over the epilimnion between 1 and 9m depth.For each scenario, 20 replicate simulations wereperformed and evaluated by calculating the populationrate of increase r (day�1) for each simulation by

r ¼lnðNt1 Þ � lnðNt0 Þ

ðt1 � t0Þ(19)

with Nt1 as the Daphnia abundance at the end of thesimulation and Nt0 as the Daphnia abundance at thestart of the simulation. Finally, a mean value of r wascalculated over all 20 simulations per scenario. The endof a simulation (t ¼ t1) was reached as soon as one ofthe following three criteria was fulfilled: (i) if thepopulation went extinct, (ii) if total populationsize exceeded 300 individuals, or (iii) at day 60 of thesimulation. Maximum number of individuals in thesimulations was limited in order to keep computationtime in a sensible range (approximately 20min on adesktop PC).

Scenario definitions

Besides studying the model behaviour in a pattern-oriented analysis (Grimm et al., 2005) we studied twoscenarios in detail: (i) adaptive value of DVM, and(ii) variation of the migration amplitude. In the firstscenario, we conducted simulations with a migratingand a non-migrating (migration behaviour is switched

ARTICLE IN PRESS

1.4

1.6

)

neonatejuvenileadult

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285276

off and animals perform food search in the epilimnionover the whole day) Daphnia population and comparedtheir population growth rates over a range of reasonablefish biomasses (0–2 gm�3, Mehner et al., 1998). In thesecond scenario, we studied how a changing migrationamplitude (by modifying migration velocities by meansof a scaling factor) affects the population growth rate ofa migrating population.

0

0.6

0.8

1.0

1.2

Bod

y le

ngth

(mm

10 20 30 40 70Time (d)

50 60

15

10

5

0

Dep

th (m

)

Vertical gradients

For all model applications used in this study, weapplied vertical gradients of temperature and food asshown in Fig. 4. These vertical gradients are consideredto represent typical conditions during late spring/earlysummer in meso-eutrophic lakes (e.g. compare Flik andRingelberg, 1993). Temperature distribution is charac-terized by a shallow epilimnion and a strong thermoclineat ca. 10m depth with much weaker temperaturestratification beneath. We assumed a bimodal fooddistribution with high phytoplankton abundance in theupper epilimnion and a smaller peak (DCM, see Molland Stoermer, 1982) below the thermocline.

030

25

20

Time (d)

adult

2 4 6 8 10

neonate juvenile

Fig. 5. (A) Population development in a simplified simulation

with predation switched off starting with one neonate

individual at time ¼ 0 (body length ¼ SON). Population

development is shown as body length of all individuals plotted

against time. Colours are used to separate the three develop-

mental stages (neonate, juvenile, and adult). (B) Spatiotem-

poral distributions of individuals during a simulation with the

full model (migrating population, fish biomass ¼ 1 gm�3).

Colours are used to separate the three developmental stages

(neonate, juvenile, and adult).

Results

Pattern-oriented analysis of model behaviour

In order to visualize general patterns in the emergingindividual and population dynamics of migratingDaphnia and to explore the model behaviour, weperformed a simulation starting with one neonate andwithout predation by fish or invertebrates (Fig. 5A).Somatic growth of the initial neonate took place quicklyand continuously because neonates did not migrate andresided in the upper warm and food-rich water layers.After approximately 3 days the individual became

50

40

30

20

10

0

5

Dep

th (m

)

Temperature (°C)

0.0

Food (

(A)

201510 0.0.2

Fig. 4. Vertical distribution of temperature (A), food concentration

scenarios. The dotted line in (C) depicts the vertical gradient of pre

juvenile and started to migrate. Consequently, growthwas slowed down and occurred in a step-wise mannerbecause growth rates at night (while residing in the

mgC L−1)

(B)

10−9 10−6 10−3 1 100

Predation risk (h−1) andLight (µmol m−2s−1)

(C)

light intensitypredation risk

0.80.64

(B), and light intensity at noon (C) applied in the simulation

dation risk (h�1) arising from light-dependent fish predation.

ARTICLE IN PRESS

0No. of birth events

30

25

20

15

10

5

0

Dep

th (m

)

50 100 150

Fig. 7. Depth distribution of birth events during the course of

one simulation (migrating population, fish biomass ¼ 1 gm�3),

n ¼ 383. Note that adults show a markedly different depth

distribution since they spent roughly two-third of their life in

deep waters and only one-third in the epilimnion.

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 277

epilimnion) are much higher than during day (whileresiding in the hypolimnion). Roughly 10 days later theindividual matured into an adult and deposited its firstclutch of eggs into its brood pouch. After one moultingcycle, 7 days later, this first clutch was released asnewborns (day 19). All newborns, per definition, had astarting body length equal to the SON. However,immediately after birth these newborns start to swimindividually through their environment, i.e. they experi-ence different conditions of temperature and foodconcentration depending on their individual swimmingpaths. These different environmental conditions trans-lated into different somatic growth curves of theindividuals and caused substantial variation in theirlength development. As a consequence, the age atmaturity of all individuals varied over a range of 7 days(Fig. 6A) and, though exponentially growing in aconstant environment, a desynchronized populationwith a continuous size structure emerged at the end ofthe simulation (Fig. 6B).

Neonates did not migrate and remained in theepilimnion over the entire day, juvenile and adultindividuals performed DVM with the former displayingdistinctly lower migration amplitudes (Fig. 5B, standardmodel simulation with 1 gm�3 fish biomass). Theimplemented migration behaviour appeared to beasymmetrical; the upward migration during dusk tookplace with higher migration velocities than the corre-sponding downward migration at dawn. This asymme-trical migration arises from slight differences in thephenomenology of the RLC peaks during dusk anddawn, i.e. the RLC peaks are skewed. As a consequence,individuals migrated upward until approaching thesurface in the evening. Newborns that hatched duringday in the hypolimnion needed a certain time until theyinvaded the food-rich and warm epilimnetic layers(approximately 1 day by performing negative geotaxis,see days 1–2 in Fig. 5B). However, this event occurred

Age at maturity (d)

Freq

uenc

y

12

0

10

20

30

40

13 14 15 16 17 18 19

Freq

uenc

y

Fig. 6. (A) Variation in age at maturity derived from the simplified si

Daphnia population at the end of the simplified simulation depicted

rarely since most newborns were born in the epilimnion(Fig. 7). This is a consequence of faster developmentalrates in the epilimnion due to higher temperatures. Onaverage, approximately three-fourths of all neonatesborn during the simulation hatched in the epilimnionalthough their mothers spent only one third of the day inthis habitat.

Scenario analysis

Population growth rates of a migrating Daphnia

population remained positive over the entire range of

Body length (mm)0.6

0

20

40

60

80

100

1.0 1.4 1.8

mulation depicted in Fig. 5A (n ¼ 153). (B) Size structure of the

in Fig. 5A (n ¼ 582).

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0.0

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

Fish biomass (gm−3)

Pop

ulat

ion

grow

th ra

te (d

−1)

DVMno DVM

0.5 1.0 1.5 2.0

Fig. 8. Population growth rate r (day�1) of a migrating (open

cycles) and a non-migrating (closed cycles) Daphnia population

over a range of fish densities in the habitat. Note that a

population growth rate at zero fish biomass can only be

calculated for non-migrating populations because the intensity

of the migration behaviour scales with the fish biomass

(compare Eq. (7)) and therefore DVM behaviour is ceased if

no fish is present.

0

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

Migration amplitude (m)

Pop

ulat

ion

grow

th ra

te (d

−1)

Fish biomass = 0.25 gm−3

Fish biomass = 2.00 gm−3

5 10 15 20

Fig. 9. Resulting population growth rates r (day�1) for

varying migration amplitudes under high (2 gm�3) and

moderate fish biomass (0.25 gm�3). In the standard model

run the realized migration amplitudes are approximately 20m

at the high fish biomass and approximately 10m at the

moderate fish biomass. The depicted migration amplitudes

correspond to those performed by adult individuals.

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285278

fish biomass applied (Fig. 8). In contrast to this, a non-migrating population suffered severely from increasingfish biomass and the resulting population growth rateturned out to be negative if fish biomass surpassed0.4 gm�3. By comparing population growth rates ofmigrating and non-migrating populations, a highadaptive value of DVM emerged, particularly underconditions of intense fish predation. However, there arealso significant costs associated with DVM due to coldtemperature and poor food conditions during the day.Under conditions of relatively low fish predation, thesecosts caused population growth rates of migratingpopulations to be lower than the correspondingpopulation growth rates of non-migrating populations.As a result, a certain predation pressure in the habitathad to be surpassed in order to turn DVM into anadaptive behavioural defence mechanism against fishpredation. In our setting, this critical fish predationcorresponded to a fish biomass of about 0.22 gm�3.

If no fish is present in the habitat and no DVM takesplace, the realized population growth rate of Daphnia inthe simulated environment is 0.25 day�1. In order toquantify the costs of DVM due to lower ambient foodavailability and temperature, we performed scenarioswith migrating populations but without fish predation.In a first scenario, we simulated DVM with a migrationamplitude of 6m (corresponding to the DVM amplitudeat a fish biomass of ca. 0.10 gm�3), which resulted in apopulation growth rate of 0.147 day�1, i.e. costs ofDVM are reflected by a decrease in population growth

rate from 0.25 to 0.147day�1. At a migration amplitude of10m (corresponding to a fish biomass of ca. 0.25gm�3)the resulting population growth rate was further decreasedto 0.105day�1. A further increase in migration amplitude(e.g. 20m, corresponding to 2 gm�3 fish biomass) didchange the resulting population growth rate only slightly(0.099day�1) due to very shallow gradients of temperatureand food below the thermocline.

Changing the migration behaviour by increasing ordecreasing the effective migration amplitude resulted invariation of the population growth rate, too. If fishbiomass is moderate (0.25 gm�3, migration amplitudewith standard parameterization is about 10.5m) areduction of the migration amplitude resulted inincreased population growth rates (Fig. 9). Maximumpopulation growth rates were realized at migrationamplitudes around 7m. This effect is due to the fact thatat those migration amplitudes the daytime depth ofmigrating individuals approached the second food peakat a depth of 12m (DCM). Additionally, averagetemperature conditions improved with decreasing mi-gration amplitudes, which speeded up developmenttimes. However, further decreases in migration ampli-tudes appeared to be costly since predation ratesby fish increased exponentially with decreasing depthdue to the exponential light gradient. Therefore,population growth rates declined as soon as migrationamplitudes fall below 6m. Interestingly, if migrationbehaviour is ceased completely (migration amplitude of0m) population growth rate increased, again, which is

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 279

the consequence of an undisturbed food search that iscontinuously active without being suspended by migra-tion movements. A different picture is drawn at high fishbiomass (2 gm�3, migration amplitude with standardparameterization is about 20m). Although we detected asmall and relatively flat peak in population growth ratesat migration amplitudes around 10m, the potentialadaptive value of decreasing migration amplitudesremained relatively small (although such small differ-ences can be relevant over evolutionary time-scales). Incontrast, costs arising from intensified fish predationwith decreasing migration amplitudes were pronouncedand population growth rates decreased sharply as soonas migration amplitudes fall below 10m. Also, potentialbenefits of the food search, if the migration behaviourwas turned off (migration amplitude of 0m), were notrealizable due to the strong predation effects.

Discussion

For a comprehensive understanding of the phenom-enon of DVM several ecological, physiological, andbehavioural mechanisms come simultaneously into play.Individual daphnids display a repertoire of differentswimming modes allowing distinct responses in theirdepth distribution to various environmental factors. Atthe same time, individual growth and egg productionstrongly depend on this behaviour since their habitat isusually characterized by vertical gradients of tempera-ture and food that determine individual fecundity. Thethird component, predation risk from visually huntingpredators such as planktivorous fish, is associated withambient light intensity, creating spatial as well astemporal gradients of predation risk. The presentedmodel provides a framework spanning over all thesedifferent aspects of DVM.

Our results are in accordance with the predatoravoidance hypothesis (Lampert, 1993; Stich and Lam-pert, 1981) and clearly indicate the high adaptive valueof DVM if predation by fish is intense in the habitat.This outcome complies with the results of furthermodelling approaches focused on the adaptive value ofDVM (Fiksen, 1997; Gabriel and Thomas, 1988; Voset al., 2002). Furthermore, our model provides aquantitative estimation of the costs of DVM (Looseand Dawidowicz, 1994; Stich and Lampert, 1984)arising from reduced temperature and poor foodconditions during the day. (Note that there are nodirect costs arising from the vertical swimming duringDVM, Dawidowicz and Loose, 1992.) These costs makeDVM disadvantageous in an environment with low fishpredation because a migrating population can realizeonly a lower population growth rate than a non-migrating population. As a consequence, a certain

predation pressure has to be present in the habitatin order to turn DVM into an adaptive behaviour.The adaptive value of DVM changes gradually withincreasing predation risk and the shape of this gradualchange depends on the environmental setting (e.g.vertical gradients, migration behaviour). Such cost-mediated thresholds in predation intensities have beenalso shown for other inducible defences (e.g. life-historyshifts, Rinke et al., 2008), which provides a goodexplanation why these defence mechanisms have evolvedas an inducible strategy (see also Tollrian and Harvell,1999).

The presented model incorporates a comprehensivespectrum of relevant mechanisms ranging frombehavioural ecology, individual ontogenesis and pre-dator–prey interactions within a spatially explicit,individual-based approach. We see a great potential insuch approaches bringing knowledge from differentfields of research into synthesis. An indispensableprerequisite for this step is a sound body of mechanistic,process-oriented fundamental research. Another majorcontribution modelling approaches can add is to gobeyond a purely qualitative perspective, and to providea quantitative view. Models do not more and not lessthan showing to which extent a given phenomenon can(or cannot) be explained or predicted, given the knowl-edge (processes and parameters) put into the model.That is why discrepancies between predicted andobserved patterns can provide valuable informationabout missing links or promising further directions ofresearch. The other side of the coin of using models for asynthesis is that the resulting models are becomingrather complex which at first look seems to contradictthe principle of maximum parsimony (Popper, 1992).

Our approach is inevitably limited by the knowledgethat is provided by the literature (e.g. in the form ofparameter values). Besides the rather direct synthesizingidea that we followed here, an alternative is provided byartificial evolution approaches. By implementing beha-vioural traits in individual-based models through neuralnetworks and genetic algorithms one can study whichstrategies evolve in a defined environmental setting(Huse et al., 1999). This can be used as an independenttest for an observed phenomenon, e.g. food search byrandom walk or realized migration amplitude duringDVM in a given vertical environment. In a study abouthabitat choice and optimal life-history of a marineplanktivorous fish, Strand et al. (2002) demonstrated theabilities of this artificial evolution approach. In a similarstudy based on computational techniques to findoptimal migration strategies, Sekino and Yamamura(1999) suggested that the migration behaviour ofzooplankton depends on the amount of accumulatedenergy of the individuals. Those having high reservecontent were expected to perform DVM for saving thereserves (e.g. for reproduction) whereas individuals with

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285280

low reserve content should not migrate in order tomaximize food intake. Optimal vertical distributionpatterns in copepods, trading-off between predation riskand foraging were also investigated by Leising et al.(2005).

Spatial and temporal distribution patterns of indivi-duals in the simulations resembled those observed infield and mesocosm studies (Johnsen and Jakobsen,1987; Ringelberg et al., 1991; Stich and Lampert, 1981)indicating that the applied algorithm for DVM is able tocatch the fundamental aspects of daphnid migrationbehaviour. The applied submodel for growth andreproduction was validated extensively (Rinke andPetzoldt, 2003) and enabled us to explicitly accountfor food effects. In a comparable individual-basedmodelling approach by Vos et al. (2002), which alsoquantified the adaptive value of DVM, the underlyinggrowth model of Daphnia accounted for food concen-tration only indirectly. In that study measured dataabout individual fecundity were needed as a proxy forambient food conditions (standard egg-productionapproach, Mooij et al., 2003). Since the authors appliedthe same time-series of individual fecundity measured ona migrating Daphnia population in Lake Maarsseveen toall scenarios – irrespective of the modelled migrationstrategy – they excluded food-mediated effects in thealternative migration/life history scenarios. Comparedwith this, our study provided an estimate of DVM costsbased on both, food and temperature. Moreover, due tothe spatially explicit setting, our approach allowedstudying the effects of vertical gradients in the habitaton the costs and benefits of DVM. However, since Voset al. (2002) used measured predation rates, tempera-tures, and food concentrations as forcing factors in theirsimulation they explicitly accounted for the well-documented annual variability of these factors, aprocess that we cannot account for in our simulationswith a static environment. It would therefore beworthwhile to extend our model in such a way thatpredation risk, temperature, and concentration are givenas dynamic forcing factors to the simulation. This wouldenable a full comparison between model outputs andfield data.

Besides behavioural defences like DVM, Daphnia mayalso display life-history shifts as inducible defenceagainst fish predation (De Meester and Weider, 1999)– a fact that in our model is not yet accounted for.However, since the major life-history parameters in-volved in such a life-history shift (e.g. SON and SAM)are input parameters to our model, the inclusion of life-history shifts in the existing framework is easilyrealizable (compare Rinke and Petzoldt, 2003; Petzoldtand Rinke, 2007). In fact, life-history adaptations andbehavioural adaptations are often expressed togetherand other modelling studies have shown that thecombination of both strategies is a very effective anti-

predator strategy (Fiksen, 1997; Fiksen and Carlotti,1998; Vos et al., 2002). In particular, simulations in aseasonal environment with changing environmentalfactors (e.g. light intensity, food availability, andpredator abundance) generate further interactions(Varpe et al., 2007).

The applied predation model represents a conceptualframework and calculated predation risks were notcalibrated on measurements. In fact, such a calibrationwould be hard to achieve since fish predation onzooplankton also depends on physiological and beha-vioural properties of the fish (Lazzaro, 1987), which arenot included in the model. A more mechanisticrepresentation of the predation process that includesprey detection and selection (Werner and Hall, 1974;Eggers, 1982; Link and Keen, 1999) may be analternative approach for the submodel predationalthough an empirically based parameterization of theseprocesses remains hard to achieve. As a consequence,quantitative outcomes with respect to fish biomass haveto be interpreted carefully. For example, the existence ofa critical predation risk which has to be exceeded in thehabitat in order to make DVM to pay off is clearlyindicated by our results, but its value of 0.22 gm�3 maybe interpreted with caution. Also note that the effect offish predation on Daphnia may become more detri-mental if selective predation on egg-bearing females isassumed (Tucker and Woolpy, 1984; Sekino andYamamura, 1999).

Spatiotemporal distribution of predation risk onzooplankton is further modified by spatiotemporaldistribution patterns of planktivorous fish. In somelakes, the main planktivores were found to remainalways in the warm epilimnion (Flik et al., 1997).In such a case, the depth of the thermocline wouldbe – besides light – an important factor determiningdepth-dependent predation risks. However, behaviouralproperties of planktivores seem to be flexible and inother systems the same species intrude into thehypolimnion (Cech et al., 2005).

In field studies, it was often found that young fishprey upon migrating Daphnia during twilight times(Gliwicz and Jachner, 1992; Flik et al., 1997). Juvenilefish may avoid the open water zone during day in orderto escape predation by piscivores or water fowl (Gliwiczand Jachner, 1992) and feed during twilight whenpredation risk from piscivores is diminished but theirplanktonic prey still available in the upper water layers.However, our simulations show that light-dependentmigration behaviour of Daphnia schedule the migrationin such a way that they are never exposed to elevatedlight intensities, even not during dusk or dawn. As aconsequence, predation risk does not peak duringtwilight (Fig. 10). In contrast to this, we found predationrisk to be highest for neonates since they do not migrate,which indicate neonate survival to be the bottleneck for

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10−3

10−2

10−1Neonates

Pre

datio

n ris

k (h

−1)

10−3

10−2

10−1Juveniles

1

10−3

10−2

10−1Adults

Hour of the day4 8 12 16 20 24

Fig. 10. Daily distribution of predation risks (h�1) of different

stages of migrating Daphnia individuals. Note that predation

risks are available only on a temporal resolution of 1 h (see

model description). Predation risks were taken from a

simulation over 60 days with DVM and a fish biomass of

2 gm�3. Boxplots depict median, 25% and 75% percentile and

data range.

Table 2. Comparison of calculated individual and population

level characteristics of Daphnia computed by the empirical

model of Rinke and Petzoldt (2003) and the energy allocation

model of Rinke and Vijverberg (2005); Lmax ¼ maximal

body length, Fcrit ¼ critical minimal food concentration,

bmax ¼ maximal birth rate

Criteria Unit Rinke and

Petzoldt

(2003)

Rinke and

Vijverberg

(2005)

Lmax mm 2.40 2.50

Fcrit mgCL�1 0.05 0.04

bmax day�1 0.30 0.32

K. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 281

Daphnia population growth in the model. This findingremains to be studied in the field. The role of non-migrating Daphnia neonates should moreover be ofparticular importance during times when feeding by veryyoung fish is gape-limited (e.g. Mehner et al., 1998),resulting in selective predation on small individuals.Changing the category-based differentiation of themigration behaviour of neonates, juveniles and adultsinto a continuous length-dependent function woulddecrease predation risks for small individuals but wouldnot change the outcome that the very young individualssuffer the highest predation risks and constitute abottleneck for population growth.

The most complex part within the presented frame-work is the submodel movements consisting itself offour different modes of movements. Although weintentionally followed the literature as closely aspossible, the resulting algorithm for the migrationbehaviour turned out to be biased unless an additionalparameter for maximum migration velocity (MVmax)

was introduced. This observation was made also byothers (van Gool, 1997) and indicates the need forfurther research. Furthermore, the introduction of theparameter Icrit within the context of negative geotaxis,seems to be a vague concept since it remains question-able whether daphnids can really sense these low lightintensities. However, experiments showed that theupward migration of Daphnia at dusk is not governedby an internal clock (Loose, 1993a, b) but indeed bylight-dependent swimming behaviour (as reviewed byRingelberg, 1999). Consequently, we have to assumethat individuals residing at 40m depth, i.e. in almostdarkness, can indeed sense light intensities sufficientlysince they display an upward migration in the evening.This supports the assumption of an extremely sensitivelight perception in Daphnia.

Somatic growth of individuals in the model takesplace in a continuous manner whereas in reality growthof Daphnia occurs only during moulting, i.e. discontinu-ously. A more realistic, instar-based somatic growthcould be realized if the existing approach would besubstituted by an energy allocation model (Rinke andVijverberg, 2005). In order to analyse possible devia-tions between these alternative approaches, we usedboth models for calculating important individual andpopulation level characteristics of Daphnia. In thisanalysis, we found no major differences between thesemodels (Table 2). Therefore, we do not expect inac-curacies from using the simpler model of Rinke andPetzoldt (2003). However, using an energy allocationmodel would facilitate the inclusion of a feed-back ofDaphnia on its algal resource enabling the simulation ofconsumer-resource dynamics and its consequences oncompeting zooplankters in multi-species simulations(compare Hulsmann et al., 2005).

Our scenario analysis shows that the vertical gradientsin the habitat are important moderators of costs andbenefits of DVM. If the migration behaviour wouldallow an adaptation of the daytime depth distribution ofmigrating individuals in dependence of the verticalgradients prevailing, the adaptive value of DVM could

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285282

be further increased. This is particularly true if fishpredation in the epilimnion is not too strong. We knowthat Daphnia indeed optimize their depth distribution inthe absence of predators (Lampert et al., 2003) and itwould be worth to study whether this is also the case inthe presence of predators. Our results also showed thatother aspects of Daphnia swimming behaviour, particu-larly behavioural mechanisms to find food-rich waterlayers, are important for maximizing their fitness. Fromexperimental studies Daphnia’s ability to find food-richpatches and optimal temperature conditions is welldocumented (Calaban and Makarewicz, 1982; Jensenet al., 2001). In fact, the picture drawn by these resultsclearly shows that individual behavioural traits provideextremely potential mechanisms for the maximization offitness and in this respect DVM displays a prominentexample linking behavioural to evolutionary ecology.For that reason, we need an improved understanding ofdistribution patterns of Daphnia in the field and theunderlying behavioural mechanisms. The application ofhydroacoustic methods for a continuous observation ofspatiotemporal distribution pattern of zooplankton mayprovide a promising tool for this line of research (Lorkeet al., 2004; Rinke et al., 2007).

The application of energy allocation models forsimulating growth and reproduction and the possibilityto study consumer resource dynamics would open aninteresting field of research. The behavioural mechan-isms outlined above do not only act on the fitness ofDaphnia but, of course, also affect trophic interactions.For example, the energy flow between zooplankton andfish may be considerably lower if zooplankters areperforming DVM (Vos et al., 2002), which should haveconsequences for reproduction of fish. A pulsed grazingregime, as prevalent in environments with verticallymigrating zooplankton, was shown to influence phyto-plankton growth and the competition between differentphytoplankton species (Reichwaldt and Stibor, 2006;Reichwaldt et al., 2004). These potential effects on thecommunity level and the underlying mechanisms may bealso relevant for applied issues focused on food webmanipulation (Benndorf, 1990, 1995) and sustainablewater resource management.

Acknowledgements

We wish to kindly thank Jurgen Benndorf, who wasalways enthusiastic about this work and thus providedencouraging support. Very valuable comments on themigration behaviour of Daphnia and helpful additionaldata were provided by Joop Ringelberg and Eric vanGool (NIOO-Centre for Limnology, Nieuwersluis, TheNetherlands). We are grateful for this uncomplicatedway of exchanging ideas and data. Stephan Hulsmann,Torsten Schulze, Marco Matthes, and Dietmar Straile

provided valuable comments to the model. We wish tothank Matthijs Vos and one anonymous reviewer fortheir constructive criticism and helpful comments on anearlier version of the manuscript. KR was partlysupported by Deutsche Forschungsgemeinschaft underGrant Ro 1008/11-1.

Appendix A

Brief outline of the model by following the standard

protocol for describing individual-based models

(Grimm et al., 2006)

Overview

a.

Purpose: Calculating population growth rates of aDaphnia population under different environmentalconditions. Quantifying the adaptive value of DVM.

b.

State variables and scales: Individuals are character-ized by the following state variables: age, bodylength, number of eggs in the brood pouch, stage,state within the moulting cycle, location. The spatialscale of the habitat (lake) is given by one-dimensionalvertical gradients entirely defined by the model input(refer to 3b). The depth of the lake is theoreticallyunlimited but simulations effectively cover a depthrange of about 0–50m. The temporal scale of asimulation is at maximum 60 days and internal time-steps are at the scale of minutes.

c.

Process overview and scheduling: An overview isprovided in Fig. 1. There is no structure forscheduling of individuals included in the model, i.e.all individuals are handled at each time-step.

Design concepts

Emergence: population dynamics, vertical distribu-tion patterns.Sensing: ambient food concentration.Adaptation: no.Stochasticity: stochasticity in individual movements(random walk) and mortality.Spatial representation: explicit.Observation: all states at a time-step of 1 h.

Details

a.

Initialization: A start population consisting of 40neonates, 16 juveniles, and 24 adults is initialized withrandom values for state variables.

b.

Input: Model inputs are, besides model parameters asprovided in Table 1, a time-series of light intensity atthe surface of the lake and information about thehabitat. The latter includes vertical gradients of

ARTICLE IN PRESSK. Rinke, T. Petzoldt / Limnologica 38 (2008) 269–285 283

temperature and food concentration, the light extinc-tion coefficient of the water, as well as the abundanceof predators in the habitat (fish biomass).

c.

Submodels: Three submodels (movements, growthand reproduction, mortality) are described in detail inthe model-description above.

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