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Effects of zooplankton diel vertical migration
on a phytoplankton community: a scenario
analysis of the underlying mechanisms
Thomas Petzoldt a,∗, Lars Rudolf a,b, Karsten Rinke a,c,
Jurgen Benndorf a
aTechnische Universitat Dresden, Institute of Hydrobiology, 01062 Dresden,
Germany.
bMax Planck Institute for the Physics of Complex Systems, 01187 Dresden,
Germany.
cUniversity of Konstanz, Limnological Institute, 78457 Konstanz, Germany.
Preprint submitted to Elsevier 31.01.2009
Abstract
A mechanistic model was applied to study the influence of diurnal vertical migra-tion (DVM) of planktonic crustaceans on the succession and composition of thephytoplankton community. While zooplankton was restricted to only one functionalgroup, the phytoplankton community was divided into two functional groups whichare distinguished by their maximum growth rates and vulnerability to zooplanktongrazing. DVM causes a pulsed grazing regime and may also entail a correspondingreduction of the cumulative daily rates of ingestion and losses of zooplankton. Tostudy the relative importance of these two mechanisms of DVM to phytoplanktonwe performed a scenario analysis consisting of 5 different scenarios. The resultsshow that DVM has a strong influence on the phytoplankton community. Well ed-ible algae benefit during the first three to four weeks of summer stratification byreduced daily grazing. The typical shift from small, well edible algae to larger,poorly or non-edible phytoplankton is distinctly delayed. Under the assumption ofunchanged daily grazing, however, a pulsed grazing regime has nearly no influenceon the resulting phytoplankton composition. As similar effects are also found forcompletely non-edible phytoplankton, indirect effects via phosphorus availabilitymust be assumed. Thus, the scenario analysis reveals that the observed effects ofDVM on phytoplankton can be explained by a combination of two mechanisms:(1) reduction of the daily zooplankton grazing, and (2) changed assimilation andremineralisation of phosphorus. Surprisingly and in contradiction to earlier reportsthere is almost no DVM effect on phytoplankton due to the sole action of a pulsedgrazing regime.
Key words: Daphnia, DVM, phytoplankton competition, top-down, zooplankton,differential equation model
1 Introduction
Diurnal vertical migration (DVM) of zooplankton has been observed in oceansand numerous lakes and has received considerable attention in limnologicaland oceanographic research for many decades (McLaren, 1963; Petipa, 1969;Geller, 1986; Lampert, 1993; Ringelberg, 1999). Zooplankton migrate intodeeper water layers at dawn and spend the whole day in this dark, cold habitat.Although for some habitats elevated food concentrations in these deep layershave been reported due to the occurrence of deep chlorophyll maxima (Winder
∗ Corresponding author: Thomas Petzoldt, Technische Universitat Dresden, Insti-tute of Hydrobiology, 01062 Dresden, Germany. Tel: +49-351-463-34954;
Email address: [email protected] (Thomas Petzoldt).
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et al., 2004), the animals are usually exposed to very low food concentrationsat their daytime depth (e.g. Stich, 1989).
During dusk they migrate upward into surface layers to feed on phytoplank-ton. This behaviour is proximately triggered by the relative change in lightintensity during gloaming and by the presence of fish infochemicals (Haney,1993; Ringelberg, 1999). Visual predation by fish was shown to be the ulti-mate factor of DVM (Zaret & Suffern, 1976; Stich & Lampert, 1981; Lampert,1993), making it adaptive to stay within deep water layers with lower preda-tion risk owing to low light intensities in such depths. Since top-down pressureby planktivorous fish is known to be a key factor for Daphnia population dy-namics (Mills & Forney, 1983) DVM is a very effective strategy in habitatswith high fish stocks (Rinke & Petzoldt, 2008). This is also mirrored by thefact that in extreme cases it may become adaptive for Daphnia to stay thewhole day in deep layers, as reported by empirical and theoretical studies (Pi-janowska & Dawidowicz, 1987; Vos et al., 2002). However, DVM is also costlybecause of lower temperatures and usually poor food conditions in deeper lay-ers (Stich & Lampert, 1981; Loose & Dawidowicz, 1994) whereby the costsmediated by the temperature-effect appeared to be much stronger than thefood related costs (Loose & Dawidowicz, 1994; Rinke & Petzoldt, 2008). Thesecosts are ultimately caused by a prolonged egg development time and a re-duced food intake, which both result in lower population birth rates (Stich &Lampert, 1984; Rinke, 2005).
In contrast to the prospering research on proximate and ultimate factors ofDVM, only very few studies focused on the influence of these diel migrationson the phytoplankton community. This is surprising because DVM leads toa pulsed grazing and already four decades ago, Petipa (1969) and McAllister(1969) derived from mathematical models that phytoplankton production isenhanced under a pulsed grazing regime as compared to continuous grazing.Later, Lampert (1987) pointed out that this pulsing could possibly effect phy-toplankton succession and community structure. He hypothesised that algalcommunities should respond to vertical migration of zooplankton by increas-ing the share of fast growing, well edible algae. Recently, this hypothesis wasscrutinised experimentally by Reichwaldt et al. (2004) and Reichwaldt & Sti-bor (2005). Reichwaldt et al. (2004) examined the influence of pulsed grazingby two Daphnia species on several well ingestible algal species in short-termexperiments of up to 15 days. They pointed out that DVM led to increasedalgal biomass under a pulsed grazing regime even when the daily grazingrate resembled the continuous daily grazing. They were unable to identify themechanisms that accounted for the phytoplankton advancement due to a noc-turnal grazing regime, but suggested that the periods of undisturbed growthduring the day might explain this phenomenon.
In their second study, an enclosure experiment lasting about three weeks,
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Reichwaldt & Stibor (2005) focused on the influence of DVM on the phyto-plankton community structure. The results supported the hypothesis of Lam-pert (1987) that well edible phytoplankton benefit from DVM and inediblealgae benefit from nonmigratory grazers.
Although all studies cited in the preceding paragraph contributed essentiallyto the understanding of DVM-effects on phytoplankton, they were constrainedby the short time scale. Moreover, in these studies constant forcing factors(zooplankton biomass, light, nutrients) did not allow any feedback effects ex-erted from phytoplankton on these forcing factors which, in fact, should beregarded as variables. It is conceivable that these constraints produce unreal-istic phytoplankton responses. Thus, to achieve deeper and potentially morerealistic insights into DVM-effects on phytoplankton, investigations are desir-able taking into account (1) a longer time scale, and (2) the dynamic natureof forcing functions. Regarding the first aspect (longer time scale), the typicalseasonal succession of phytoplankton in stratified temperate lakes takes placebetween late spring and early autumn during summer stratification (Sommeret al., 1986). In many central-European lakes the time window of this period isabout 100 days. Thus, any methodology investigating DVM effects on phyto-plankton should allow to cover at least a 100-day period. However, the lengthof enclosure experiments is strictly limited by periphyton growth on the cagewalls (Carpenter, 1996; Benndorf et al., 2002; Reichwaldt & Stibor, 2005).The effort to conduct field studies over such a long time would be very large,the choice of scenarios would be limited and a definite comparison betweensimilar systems, with and without DVM, would be hardly possible. However,since important parameters of typical zooplankton and phytoplankton speciesare well known it is possible to construct a presumably realistic model withcomparatively minor effort. Such a model is also most suitable to cover thesecond aspect mentioned above (dynamics of forcing) by treating them eitheras dynamic forcing functions or as state variables. The latter possibility al-lows also to include feedback effects. Different scenarios can be simulated byadjusting the corresponding parameters of the model and simulation resultscan be compared and evaluated. Even some theoretical scenarios that cannotbe realised in field studies or experiments can be explored using modelling.
The selection of a suitable model type with appropriate level of complexity isan important, but also a disputable decision. To address our question it wouldbe, in principle, possible to use complex bio-geo-chemical lake models, eitherwith generalized parametrization, like SALMO (Benndorf & Recknagel, 1982)or BELAMO (Mieleitner & Reichert, 2008) or to calibrate models like PRO-TECH (Elliott & Thackeray, 2004) to various real plankton successions. On theupper end it would be even possible to adapt coupled hydrophysical-ecologicalmodels like ELCOM-CAEDYM (Robson & Hamilton, 2004) or SALMO-HR(Rolinski et al., 2005). For many situations, such an approach would be quiteadequate, for example if sediment-mediated nutrient cycling is to be con-
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sidered (e.g. Petzoldt & Uhlmann, 2006) or if complex interactions betweenzooplankton, nutrients and hydrophysics are modelled for one specific lake(e.g. Bruce et al., 2006). The disadvantage of such models is, however, thattheir inherent complexity can make it difficult to attribute observed effects toparticular mechanisms. That’s why we focussed on the main elements (phy-toplankton, zooplankton, phosphorus) relevant for testing our hypothesis (seebelow) and constructed a population dynamic model, specifically designed forour question (cf. Ellner & Guckenheimer, 2006 or Jørgensen, 2008 for addi-tional arguments).
The first step of this study consists in constructing a mathematical model. Themodel should allow both, a long-term simulation and the consideration of thedynamic nature of forcing (including feedback effects). This model will then beused in a scenario analysis to test different assumptions regarding the differentgrazing regimes on phytoplankton. Thereby the study aims to contribute to adeeper understanding of the key mechanisms that are responsible for the ob-served DVM-effects on phytoplankton. We hypothesize that two different keymechanisms may cause the effects, either separately or jointly: (i) the tempo-ral grazing regime (pulsed vs. non-pulsed), and/or (ii) the differences in dailyingestion rates resulting from different residence times of zooplankton in theepilimnion. The effects of these two mechanisms are compared for three phy-toplankton types realising different strategies regarding the trade-off betweengrowth rate and non-edibility, and for two different situations regarding thebalance between ingestion rate and other zooplankton rates.
2 Methods
2.1 Model description
Since phytoplankton dynamics in eutrophic systems typically take place withinthe epilimnion we straightforwardly defined the model to be restricted to theepilimnion of a eutrophic lake. The epilimnion was assumed to have a con-stant mixing depth zmix of 10m and a fixed water temperature of 20 ◦C. Themodel describes the dynamics of the state variables phytoplankton (X1; X2),zooplankton (Z) and phosphorous (P ) in a eutrophic lake by means of or-dinary differential equations. A schematic structure of the model is given inFig. 1. Two functional groups of phytoplankton were included: small, well ed-ible algae (or cyanobacteria) with high maximal growth rate (X1) and large,poorly or not ingestible algae (or cyanobacteria) that achieve only moderatemaximal growth rates (X2; X2
′). The second functional group is parameterisedin two different ways. Case A describes X2 as a poorly edible form, whereasin case B X2
′ is assumed to be completely inedible by zooplankton. Case B
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was defined in order to test indirect DVM-effects on X2′ via changes in the
assimilation or regeneration of the limiting nutrient (P). This becomes possi-ble in case B by the complete exclusion of any direct grazing effects to X2
′.In summary, X1 is characterized as a r-strategist by high growth rates andhigh vulnerability to grazing, whereas slow growth and low (or completelyabsent) vulnerability to grazing can be found in the K-strategist X2 or X2
′.Phytoplankton assimilate carbon by photosynthesis depending on the amountof ambient phosphorous and light intensity and, therefore, vary over a diurnalcycle. Respiration and sedimentation represent further loss processes of phy-toplankton. To avoid unrealistic low values of algae we added a small influx.Further we assumed a continuous import of phosphorus, e.g. by an inflow, cor-responding to conditions typical for eutrophic lakes. Phosphorus assimilationthrough phytoplankton growth occurs with a constant ratio between phospho-rus and carbon within phytoplankton biomass. Phosphorus is remineralised byzooplankton and phytoplankton respiration, grazing and mortality.
Zooplankton feed on the phytoplankton groups X1 and X2 depending on theirconcentration. In order to model the spatio-temporal dynamics of vertical mi-gration we assumed migrating zooplankton to stay in the epilimnion fromsunset (22:00 h) to sunrise (06:00 h). During the day all zooplankton stayin hypolimnetic water layers and no zooplankton remain in the epilimnion.Ingestion and loss rates were assumed to be zero during 16 hours of dwellingthe hypolimnion. That is, the same biomass of zooplankton, which migrateddownward during dawn, returns into the epilimnion during dusk of the sameday. To link the model with higher trophic levels we assumed a constant abun-dance of planktivorous fish which feed on zooplankton with a Holling type IIfunctional response, that is a dynamic density depended zooplankton mortal-ity. A detailed model specification including all equations is given in Table1. For the sake of comparability all model parameters (with the exception ofmaximum growth and ingestion rates of phytoplankton) are kept constant.Parameters and boundary conditions can be found in Table 2.
2.2 Definition of the scenarios
To study and compare the effects of DVM on phytoplankton and the under-lying mechanisms we defined five scenarios (s1 to s5):
s1 WITHOUT DVM: Continuous grazing on phytoplankton takes place.The instantaneous ingestion rate varies only slightly in dependence on phy-toplankton biomass. We consider this scenario to study phytoplankton suc-cession under conditions of non-migrating zooplankton.
s2 WITH DVM: As a consequence of the diurnal vertical migration of zoo-plankton, grazing on phytoplankton is pulsed and occurs during night only.
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For a day-night rhythm of 16:8h, it follows that the cumulative grazing in-tensity per zooplankton biomass and day is reduced to 33 % in comparisonto s1. Because of the assumption that the zooplankton net growth is zero inthe hypolimnion it follows that this reduction also accounts to all other zoo-plankton related process rates on a cumulative daily base. We introducedthis scenario to study typical phytoplankton dynamics under the influenceof vertically migrating zooplankton.
s3 DVM WITH 3-FOLD ENHANCED GRAZING RATE: This scenariois similar to s2 but the maximum ingestion rate at night was assumed tobe increased by a factor of three. The resulting three-fold increase of theinstantaneous ingestion rate of zooplankton during night compensates thereduction in the daily grazing as simulated in s2, i.e. the daily grazingintensity per unit zooplankton biomass is the same as in s1.
s4 DVM WITH 3-FOLD ENHANCED GRAZING AND LOSS RATES OFZOOPLANKTON: This scenario is similar to s2 but all rates of zooplank-ton were increased by a factor of three. Hence, all cumulative daily rates aresimilar to the rates in s1, but the grazing regime is pulsed. With this sce-nario we studied the influence of pulsed grazing alone without the normallyassociated changes in cumulative daily rates.
s5 WITHOUT DVM AND 3-FOLD DECREASED GRAZING AND LOSSRATES OF ZOOPLANKTON: This scenario is similar to s1 but all ratesof zooplankton were reduced to one third. Hence, all cumulative daily ratesare similar to the rates in s2, but now without pulsed grazing. We con-sider this scenario to distinguish between the effects caused by the reducedzooplankton rates and those caused by the pulsed grazing mode.
The resulting model parameters and the effective rates of zooplankton over theday in relation to the scenario without migration (s1=100%) are summarizedin Table 3.
This set of scenarios was analysed with respect to the competition betweenthe two phytoplankton groups. In a first run of these scenarios, the phyto-plankton group X2 was assumed to be a K-strategist slightly ingestible byzooplankton (case A). In a second run (case B) this phytoplankton group(X2) was replaced by a completely inedible group (X2
′). We want to stressthat our standard scenarios for non-migrating and migrating zooplankton, re-spectively, are scenarios s1 and s2 of case A. The other scenarios and caseB are further analyses to get a deeper insight that account for the observedmodel response.
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2.3 Scenario evaluation
To quantify the effects of vertical migration and to differentiate between con-sequences of the temporal grazing regime and changed daily rates for the phy-toplankton community we defined eight cardinal characteristics to depict dif-ferences between scenarios. The characteristics were derived from daily meanvalues of the state variables, to remove short-term oscillations resulting fromexternal forcing (light cycle and pulsed grazing):
maxZoo: The maximum of zooplankton biomass.meanPhyto: Average biomass of total phytoplankton during the simulation
period.Xratio: Ratio of mean biomass between X1 and X2.shiftX: Date of shift in dominance from well edible (X1) towards poorly
edible phytoplankton (X2; X2′).
maxX1: Maximum biomass of the well edible phytoplankton group X1.firstbloom: Date of the beginning of blooming (more than 0.25 mg C l−1 and
positive derivative) of the poorly or not edible phytoplankton (X2; X2′).
meanPhos: Average concentration of dissolved phosphorus during the sim-ulation period.
minPhos: Minimum concentration of dissolved phosphorus.
Averages and maxima were calculated for a 100 day summer stratificationperiod in general. Values of firstbloom which may be greater than 100 indicatethat a bloom of the poorly edible resp. inedible phytoplankton was suppressedduring these 100 days.
2.4 Initial Conditions
An ecologically plausible set of initial conditions was selected for the four statevariables to characterise spring conditions of slightly eutrophic lakes, withX1,0, X2,0, Z0 = 0.05 mg C L−1 and P0 = 30 µgPL−1 dissolved phosphorus.
Although this study is focussed on analysing transient behaviour during thevegetation period, long-term simulations were carried out over an extendedtime period of 1000 resp. 2000 days with standard initial conditions to identifyequilibria of daily mean values for the state variables.
The domain of attraction for the equilibria was checked by performing k =50 repetitions with randomly selected initial values over a period of 48000time steps (2000 days) for all scenarios. The initial values were choosen bylatin hypercube sampling (McKay et al., 1979) with X1,0, X2,0, Z0 ∼ U(0, 0.2)and P0 ∼ U(0, 100) where biotic state variables are in mg C L−1, dissolved
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phosporus is in µg P L−1 and U(min, max) is a continuous uniform distributionover the specified range. The daily mean values of the final state were recordedfor all simulations. The coefficient of variation cvscenario,state,case was calculatedfor each state variable, each scenario and each series (case A and case B) andthe maximum cv-value of each case cvmax,case was used as an indicator ofstability.
With respect to our hypothesis, transient behaviour of the model is more im-portant than equilibrium state, so it was necessary to examine if and how ourconclusions would depend on particular values of the initial state. The analy-sis was done for our main scenarios s1 and s2 by plotting the most importantcardinal characteristics against systematically changed initial values of X1,0
and X2,0 resp. Z0 and P0.
2.5 Computational methods
The model was implemented in the R programming language for statisticalcomputing (R Core Team, 2009) with the object oriented package simecol (Pet-zoldt & Rinke, 2007); the source code is available on the internet. Numericalintegration was performed with the lsoda algorithm, originally developed byPetzold (1983) and Hindmarsh (1983) and the deSolve package of R (Soetaert& Herman, 2009). Output data were recorded with an external time step of1 h, while the internal time step was automatically adjusted by the lsoda al-gorithm to maintain the required tolerance conditions. Relative and absolutetolerances were set to < 10−6.
3 Results
3.1 Effects of DVM (standard scenarios)
The effects off DVM were first studied by comparing scenarios 1 (no DVM) and2 (with DVM) in case A (X2 = poorly edible phytoplankton; Fig. 2). Duringthe first month DVM is beneficial for well edible phytoplankton. As a result ofthe nocturnal feeding phytoplankton mortality by grazing is strongly reducedand the phytoplankton maximum is mostly limited by a lack of nutrients (P).However, DVM increases the biomass of zooplankton as well, but with a delayin time. The maximum value of well edible phytoplankton increases by 81%and the accompanying zooplankton maximum by 20% (Table 4). The averageratio between well and poorly edible phytoplankton biomass strongly increases(196%), and there is a distinct delay by 60 days (122%) of the date when
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poorly edible phytoplankton form dense blooms (exceeding 0.25 mg C L−1).However, towards the end of the succession the phytoplankton compositionand biomass of the two groups become more and more similar and the systembecomes dominated by X2 in both scenarios.
3.2 DVM with increased nocturnal grazing rate
The assumption that starvation in deep layers during daytime may increasethe grazing rate in upper layers during the night leads to scenario s3 (Fig.3). At the beginning of the succession the zooplankton strongly benefits fromthe advantageous conditions. The resulting high zooplankton biomass initiallyincreases grazing pressure on phytoplankton. The succession becomes domi-nated by this zooplankton maximum and the average biomass of both groupsof phytoplankton is strongly reduced (Table 4). Also the first maxima of thetwo phytoplankton groups are reduced and the phytoplankton collapse takesplace faster as compared with scenario 1 and 2 (Fig. 3). In addition, there isstill a delay of dense blooms of the poorly edible phytoplankton group X2,although it is somewhat shorter. The critical level of 0.25 mg C L−1 is ap-proached 23 days later in s3 as in s1, compared to a delay of 60 days betweens2 and s1 (Table 4). Overall, the assumption of increasing ingestion rate dueto hunger results in an increased fitness of the zooplankton since energy acqui-sition is threefold increased whereas loss processes (respiration, maintenance)are kept constant. This enhanced physiological efficiency leads to different dy-namics in comparison to the other four scenarios, which is expressed by overallincreased zooplankton biomass, reduced phytoplankton biomass, and elevatedconcentrations of dissolved phosphorus. Furthermore, this physiological adap-tation provides the zooplankton with the ability to control the poorly ediblephytoplankton (Fig. 3). This is not the case if X2 is assumed to be completelyinedible (case B, see Fig. 5) The high availability of phosphorous indicatesthat bottom-up processes play only an inferior role in scenario 3.
3.3 Influence of a pulsed grazing regime
A comparison of the scenarios simulating the same daily zooplankton rates butdifferent grazing regimes (s1 vs. s4 and s2 vs. s5, respectively) is shown in Fig.3. There are strong similarities between scenarios with the same daily rates.So simulations of s4 give the same succession pattern like s1 and the typicaldynamics of s2 resemble that of s5. A change in the grazing regime (pulsed orcontinuous) results in only small changes in the dynamics, when the daily ratesare equal. This is reflected in all quantitative characteristics like the averagebiomass of phytoplankton, date of shift of dominance and date of exceeding
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the threshold of 0.25 mg C L−1 of the poorly edible phytoplankton(Table 4).
3.4 Completely inedible phytoplankton
In case B a completely inedible group X2′ was used instead of poorly edible
phytoplankton X2 of case A. Similar to case A, DVM supports well ediblephytoplankton and zooplankton during the early succession (Fig. 4). Althoughin case B X2
′ cannot be directly grazed by zooplankton, the development of X2′
is delayed by DVM (scenario 2) due to intensification of phosphorus limitation(Fig. 4). With the exception of this enhanced P limitation, the case B-scenariosresemble the findings from case A (Fig. 3). There are quite similar results forscenarios simulating the same daily rates regardless of the two different grazingregimes (Fig. 5: s1 is similar to s4; s2 is similar to s5).
3.5 Influence of initial conditions
After an extended simulation period all scenarios tend to approach steadystate (Fig. 6). For case A, values of P and X2 are nearly the same for allscenarios with the exception of s3. With respect to X1 and Z s1 is similar tos4 and s2 to s5, which is in full agreement with the above mentioned successionpatterns. For case B these findings can be confirmed (Fig. 6, right), but thereis still some visible oscillation for s2 and s5, that diminishes if the simulationis extended once again for additional 1000 days.
For all simulations with randomly varied initial conditions (latin hypercubesampling) the same equilibria (focal points as shown in Fig. 6) are approached,regardless of a particular set of initial values. The maximum coefficients ofvariation cvmax for the 50 repetitions were below 1.6 · 10−4 for case A resp.3.6 · 10−4 for case B after a simulation period of 2000 days .
Not only equilibria, also transient patterns are independent from varying ini-tial values of X1 and X2. Though distinctions exist in the quantitative re-sponse surfaces (Fig. 7), cardinal characteristics are similar and, in particular,the sign of change (differences of respective characteristics between scenarios)is unchanged (see difference surfaces in Fig. 7 c, f, i). For example differencesbetween s2 and s1 are positive in all cases for maxX1, meanPhyto and max-Zoo, indicating an increase of edible phytoplankton, a relative suppression ofpoorly edible phytoplankton (meanPhyto is less affected than maxX1) andan increase of zooplankton under DVM conditions. Similar comparisons weremade for the other characteristics and scenarios (not shown here), which intheir essential properties are in full agreement with the results above. The dif-ference surfaces between scenarios s2 and s5 resp. s1 and s4 are relatively close
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to zero, so the above stated similarity is confirmed. Sometimes, the responsesurfaces exhibited nonlinear effects, however, a detailed discussion is beyondthe scope of this paper.
The influence of start values for P and X was examined by box-plots for therelative change of all characteristics (not shown) and by plotting diagrams ofabsolute differences between s2 and s1 for important characteristics (Fig. 8).Similarly to the analysis of phytoplankton start values, these differences didnot change their sign for non-zero initial values (e.g. Fig. 8, left and right),with the exception of meanPhyto, that showed an obvious change in sign forhigh initial values of zooplankton and an even stronger effect for high initialvalues of P .
4 Discussion
A comparison of the typical succession of plankton in eutrophic lakes (Som-mer et al., 1986) with simulations of the standard scenarios s1 and s2 of caseA (Fig. 2) shows the plausibility of the model. In wide accordance with thePEG model (Sommer et al., 1986) simulation results in Fig. 2 show a max-imum of well edible phytoplankton in spring and a subsequent maximum ofzooplankton. Then follows the collapse of the phytoplankton and a clear waterphase. Finally, the free resources can be used by the slow-growing but poorlyedible phytoplankton, which dominate the later succession. Thus, it seems tobe justified to apply the model to simulate realistically the consequences ofDVM on the phytoplankton succession during a whole summer stratificationperiod of 100 days.
The response of the plankton community to DVM is highly dependent onthe assumptions regarding the daily zooplankton rates. Only such scenariosassuming reduced grazing rate (s2) or loss rates (s2, s3) as a consequence ofDVM reveal strong response of the phytoplankton community to DVM. How-ever, even scenario 5, assuming a reduction of the daily zooplankton ratesunder a continuous grazing regime, shows the same strong response. On theother hand, scenario 4, assuming DVM without any cutback of the zooplank-ton rates, almost equals the results of scenario 1. The only sound explanationof this findings is that the reduction of the daily rates of zooplankton dueto reduced (nocturnal) grazing time rather than the pulsed grazing regime isthe crucial mechanism leading to the observed DVM effects. In this respect,our results contradict findings from experimental investigations by Reichwaldtet al. (2004) and Reichwaldt & Stibor (2005) as well as predictions derivedfrom earlier models (McAllister, 1969; Petipa, 1969; Lampert, 1987). All theseinvestigations revealed enhanced phytoplankton production under a noctur-nal grazing regime even at daily grazing rates equal to continuous grazing.
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However, the modelling studies cited above all apply a constant zooplanktongrazing and do not account for nutrient dynamics, whereas our approach ex-plicitly simulates zooplankton and nutrients as state variables. These essentialdifferences in the model structure make the results hard to compare. Underconsideration of this higher complexity of the pelagic system we were able toshow that reduced daily grazing of zooplankton, and not the pulsed grazingregime per se, is the trigger of DVM effects on phytoplankton. As this hasbeen found for both cases A and B (X2 represented by either poorly edibleor completely inedible phytoplankton) this conclusion is valid for a variety ofphytoplankton, independent on their edibility. But nevertheless, there still re-mains the contradiction between the outcome of our model and the empiricalfindings of Reichwaldt et al. (2004) and Reichwaldt & Stibor (2005).
According to our results nocturnal grazing does not lead to increased phyto-plankton production (compared to continuous grazing) as long as daily graz-ing rate is held constant whereas the experiments showed in fact an increasedproduction. There are two explanations of this difference: (1) Although, inprincipal, experimental shortcomings cannot be ruled out a priori (e.g. miss-ing feedback effects on zooplankton due to constant experimental zooplanktonabundance), the difference could be interpreted by further indirect effects in-cluded in the predator-prey interaction between algae and zooplankton thatlead to a benefit of the algae. Possible mechanisms could be predator-inducedlife-cycle adaptations or different resource usage of the algae. (2) The responseof the plankton community to DVM is highly dynamic and differs stronglyduring the whole simulated stratification period of 100 days. During an initialtime window of about 20 days, our results corroborate experimental studiesby Reichwaldt et al. (2004) and Reichwaldt & Stibor (2005) as well as modelpredictions by McAllister (1969) and Lampert (1987) that DVM leads to anincreased algal biomass and, in particular, benefits well edible, small phyto-plankton. In fact, for the first three weeks we observe similar effects of DVMon the phytoplankton community. Thus, the difference between our simulationresults and empirical findings of Reichwaldt et al. (2004) and Reichwaldt &Stibor (2005) could be only apparent and unravel when the same time scaleis applied. But anyway, our results clearly show that the sole effect of pulsedgrazing cannot be the causal factor of this observation. In line with thesefindings we consider in the further discussion only such DVM scenarios as “ef-fective” DVM scenarios which combine a reduction of the daily grazing ratesof zooplankton with a pulsed grazing mode (scenario 2 and 3).
The benefit of well edible phytoplankton by DVM disappears completely af-ter 30 days. At this point phytoplankton group X1 becomes overexploited bythe subsequent zooplankton maximum and collapses to almost zero. With theprogression of the succession the similarity between the scenarios with andwithout “effective” DVM is increasing. Finally, one can hardly find any differ-ence in phytoplankton biomass and composition by DVM after a succession
13
of 80 days (exception: scenario 3, see below). Nevertheless, the considerabledelay of the shift of dominance from well edible to poorly edible or inediblephytoplankton by effective DVM may play an important role in eutrophica-tion control issues, e.g. biomanipulation (Benndorf, 1990). A DVM-induceddelay of poorly edible algae or cyanobacteria may explain several discrepan-cies in biomanipulation succes (or failure) in deep lakes (compare for instanceBenndorf et al., 2002 and Lathrop, 2002). The apparent disadvantage of DVM-enhanced biomass of edible phytoplankton during the early succession (Figs.2, 4) vanishes under biomanipulation conditions due to high zooplankton startvalues in both modelling results (see Fig. 8) and field studies (Jeppesen et al.2004; Kasprzak et al. 2007). Thus, DVM is an important factor controllingthe competition between different phytoplankton groups, especially in tem-perate regions with a typical winter reset and a defined time window for thesuccession.
In contrast to earlier investigations regarding the impact of DVM on phyto-plankton production and composition (McAllister, 1969; Petipa, 1969; Lam-pert, 1987; Reichwaldt et al., 2004; Reichwaldt & Stibor, 2005) the scenarioanalysis presented in this work allows to identify two possible mechanismsaccounting for the DVM effects on phytoplankton. A first mechanism consistsin the reduced daily grazing rate (scenario 2, see above). This leads to aneffective delay of the zooplankton dynamics (scenario 2). During daytime thezooplankton decouple from the epilimnic phytoplankton and couple back intothe system during dusk with only a marginal change in biomass. Hence for aday-night rhythm of 16:8 hours the zooplankton time scale is extended by afactor of almost three, whereas the phytoplankton development is not. Thisresults in a time lag between phytoplankton and zooplankton biomass at thebeginning of the succession (Fig. 2). However, the delay of the zooplanktonmaximum can be beneficial for phytoplankton only for a distinct time as zoo-plankton grazing, even with DVM, will finally surpass a critical value for welledible phytoplankton. This provides an explanation for the observed dynamicson longer time scales.
A second mechanism consists in the indirect effects of DVM on the metabolism(assimilation, remineralisation) of dissolved phosphorus as the limiting nutri-ent. As can be seen in scenario 2 in case B (X2
′ = completely inedible phyto-plankton), the intensified P-assimilation by well edible phytoplankton causeslow P concentrations which, in turn, delay the development of the inediblephytoplankton. An even more pronounced delay can be observed in scenario 2of case A (X2 = a poorly edible phytoplankton) as intense grazing due to highzooplankton biomass may extend the delay. The important role of indirect ef-fects mediated by the limiting nutrient is also revealed in scenario 3 of case A.The early increase in zooplankton suppresses both groups of phytoplankton.Thus, reduced P assimilation together with enhanced P remineralisation byzooplankton provides very favourable conditions for phytoplankton growth.
14
These conditions can be used by both phytoplankton groups. However, due tohigh zooplankton biomass the poorly edible phytoplankton develop a secondmaximum, which in turn facilitates a second zooplankton peak. The ultimateconsequence of this chain of direct and indirect DVM effects is a very low totalphytoplankton biomass at the end of the simulated stratification period.
Another indicator that supports the key role of the daily rates is the, at a firstlook, surprising reversal of the DVM effect on mean phytoplankton biovolume(difference of meanPhyto for s2-s1) in the vegetation period (Fig. 8). Whilewith low initial values of Z DVM results in an increase of phytoplanktonbiomass (meanPhyto), high initial values of Z result in a decrease comparedto the non-migration scenario. This phenomenon shows clearly that migratingzooplankton can only control phytoplankton if the reduced grazing efficiencyis compensated by high biomass. The slower growth of migrating zooplanktonresults in enduring predation pressure over the vegetation period. Lastly theeffect is highest for high Z0 and high P0 because high Z at the beginning resultsin faster P-elimination and in consequence stronger P-limitation. Again, this isnot an effect of pulsed vs. non-pulsed grazing but an effect of the effective rates,expressed as the product of zooplankton biomass and zooplankton activity(grazing).
From the model simulations and the scenario analysis presented in this studythe conclusion can be drawn that zooplankton DVM is beneficial for small,well edible phytoplankton during a relatively short initial period of three tofour weeks, whereas at longer time scales this benefit disappears. However,DVM leads to a considerable time delay of the shift from well edible to poorlyedible/inedible algae or cyanobacteria when compared with continuous epil-imnic grazing. This time delay can exert a positive impact on eutrophicationcontrol issues aiming at enhanced grazing losses of phytoplankton (i.e. bioma-nipulation). These effects of DVM on the plankton dynamics base on twomechanisms: (1) the reduced daily grazing rates of zooplankton leading to andelay of the zooplankton dynamics and (2) the indirect effects mediated bythe limiting nutrient.
5 Acknowledgements
We thank Stephan Hulsmann for discussing Daphnia related questions duringmodel setup. We are grateful to two reviewers for their constructive com-ments, which greatly improved an earlier version of this manuscript. Our spe-cial thanks are attributed to Wolf Mooij for his suggestions how to sharpenour argumentation and in particular his recommendation to examine the im-pact of initial conditions. Kirsten Olrik reminded us that we should not drifttoo far away from biological realism.
15
The modelling activities are associated to topics supported by Deutsche Forschungs-gemeinschaft (DFG), grants Be 1671/8-2 (Biomanipulation), Be 1671/10-2and Pa 1202/1-2 (AQUASHIFT priority programme).
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Table 1System of model equations
Zooplankton:
dZ
dt=
(
ae
2∑
i=1
ingi(X1,X2) − resz(X1,X2) − mort(Z)
)
· Z
ingi(X1,X2) = ingmaxi·
Xi∑2
i=1 Xi + king
resz(X1,X2) = reszmax
(
0.5 + 0.5 ·
∑2i=1 ingi(X1,X2)
max(ingmax1, ingmax2
)
)
mort(Z) = mortmax ·Z
Z + kmort
Phytoplankton:
dXi
dt=(
photi(I, P,X1,X2) − sedi(Xi) − resxi(Xi))
· Xi
− ingi(X1,X2) · Z + impXi
photi(I, P,X1,X2) = photmaxi·
P
P + kP·
1
ε(X1,X2)· ln
I0 + kI
kI + I0 · e−ε(X1,X2)·zmix·
1
zmix
I0 = Iphot · (1 − Iref ) · Imax · 0.5
(
1 + sin(π
b· (t − a)
)
+
√
d + sin2(π
b(t − a)
)
−√
d + 1
)
ε(X1,X2) = εmin +
2∑
i=1
(εX · Xi)
sedi =vsed
zmix
resxi = 0.6 photmaxi·
Ikomp
kI + Ikomp
Phosphorus:
dP
dt= yieldCP
(
remin(X1,X2, Z)
−
2∑
i=1
(
photi(I, P,X1,X2) − resxi(Xi))
· Xi
)
+ impP
remin(X1,X2, Z) = yieldfae
(
mort(Z) + (1 − ae)
2∑
i=1
ingi(X1,X2) · Z)
Abbreviations: ε, spectral extinction coefficient; I0, photosynthetic active radiation below water surface;ingi, ingestion of phytoplankton species i by zooplankton; mort, zooplankton mortality; photi, growthof phytoplankton species i by photosynthesis; sed, phytoplankton sedimentation; resx, phytoplanktonrespiration (analytical solution for zero net growth at I = Ikomp and no phosphorus limitation); resz,zooplankton respiration; P , phosphorus; Xi, phytoplankton group i; Z, zooplankton
21
Table 2List of model parameters and boundary conditions for case A. For case B maximalingestion rate of phytoplankton group X ′
2 was set to ingmax2= 0.
Parameter Interpretation Value Unit Reference
a Time of sunrise 6 h -
ae Assimilation efficency 0.7 nondimensional a,b,c
b Length of day 16 h -
d Gloaming constant 10−3 nondimensional d
εmin Background turbidity 0.6 m−1 e,f,i,n,o
εX Light absorption by phytoplankton 0.5 m−1 mg C−1 L e,f,n,o
Ikomp Light threshold for algal growth 0.58 W m−2 e
Imax Maximum value of irradiation 560 W m−2 -
Iphot Photosynthetic active part of global radiation 0.5 nondimensional e,w
Iref Reflexion of light at water surface 0.1 nondimensional e,w
impP P import 0.5 g m−2a−1 -
impXiImport of algal biomass for group i 5 · 10−4; 5 · 10−4 mg C L−1 -
ingmaxiMaximal ingestion of phytoplankton group i 1.0 ; 0.2 d−1 g
mortmax Maximal mortality of Z 0.045 d−1 e
kI Half saturation coefficient light 3.36 W m−2 e
king Half saturation coefficient ingestion 0.164 mg C L−1 h
kmort Half saturation coefficient mortality of Z 0.016 mg C L−1 e
kP Half saturation coefficient phosphorus 5 µ g PL−1 e,f,i
photmaxiMaximum gross photosynthesis rate for Xi 2.5 ; 1.0 d−1 e,f,i - t
reszmax Maximum respiration for Z 0.21 d−1 u,v
vsed Sinking velocity of phytoplankton 0.1 m d−1 e
yieldCP ratio of carbon to phosphorus 24.4 mg C µg P−1 e
yieldfae ratio of resolved phosphorus from feaces 0.7 nondimensional e
zmix Depth of epilimnion 10 m -
References: (a) McCauley et al. (1996); (b) Urabe & Watanabe (1991); (c) Lynch et al. (1986); (d)Richards et al. (1996); (e) Benndorf (1979); (f) Scavia (1980); (g) Geller (1975); (h) Gurney et al. (1990);(i) Lehman et al. (1975); (k) Dauta et al. (1990); (l) Bouterfas et al. (2002) ; (m) Sorokin & Kauss (1958);(n) Hamilton & Schladow (1997); (o) Omlin et al. (2001); (p) Mechling & Kilham (1982); (q) Tilman &Kilham (1976); (r) Reynolds (1999); (s) Van Donk & Kilham (1990); (t) Fogg (1973); (u) Urabe & Wanabe(1990); (v) Lampert (1977); (w) Benndorf et al. (2001)
22
Table 3Summary of parameters for the five scenarios and cumulative daily rates referencedto the conditions in scenario s1.
Scenario
s1 s2 s3 s4 s5
Model paraneters
ingmax1(d−1) 1.0 1.0 3.0 3.0 1.0/3
ingmax2(d−1) 0.2 0.2 0.6 0.6 0.2/3
reszmax (d−1) 0.21 0.21 0.21 0.63 0.21/3
mortmax (d−1) 0.045 0.045 0.045 0.135 0.015
Grazing regime continuous pulsed pulsed pulsed continuous
Cumulative rates of zooplankton
Ingestion 100% 33.3% 100% 100% 33.3%
Respiration and mortality 100% 33.3% 33.3% 100% 33.3%
23
Table 4Comparison of the characteristics described in section2.3 for case A (X2 = poorlyedible phytoplankton).
scenario s1 s2(vs. s1) s3(vs. s1) s4(vs. s1) s5(vs. s1)
maxZoo(mg C L−1) 0.42 0.51(+20%) 0.45(+6%) 0.39(-8%) 0.51(+20%)
meanPhyto(mg C L−1) 0.26 0.30(+13%) 0.10(-62%) 0.28(+5%) 0.31(+19%)
Xratio 0.35 1.04(196%) 0.19(-46%) 0.28(-21%) 1.20(242%)
shiftX(d) 13.00 27.00(108%) 8.00(-38%) 12.00(-8%) 31.00(138%)
maxX1(mg CL−1) 0.47 0.85(+81%) 0.18(-61%) 0.40(-16%) 0.88(+86%)
firstbloom(d) 49.00 109.00(122%) 72.00(+47%) 43.00(-12%) 115.00(135%)
meanPhos(mg m−3) 6.04 5.41(-10%) 16.45(173%) 6.15(+2%) 5.32(-12%)
minPhos(mg m−3) 2.44 2.56(+5%) 7.17(194%) 2.44(+0%) 2.62(+8%)
24
Figure captions
Figure 1. Diagram of the model. The state variables Zooplankton (Z), SmallPhytoplankton (X1), Large Phytoplankton (X2) and Phosphorus (P ) are rep-resented in a solid box, the boundary conditions planktivore fish and lightin a dashed box. Direct connections are plotted as solid arrows. Dashed ar-rows indicate connections playing a role only in case A (X2 = poorly ediblephytoplankton).
Figure 2. Simulations of the two standard scenarios s1 (without DVM) and s2(with DVM) for case A (X2 = poorly edible phytoplankton). See Table 3 forexplanation of scenarios 1 and 2.
Figure 3. A comparison of the daily means of the four simulated state variablessmall phytoplankton (X1), large phytoplankton (X2), zooplankton (Z) andphosphorus (P ) for case A (X2 = poorly edible phytoplankton). See Table 3for explanation of scenarios 1 to 5.
Figure 4. Simulations of scenario s1 and s2 for case B (X2′ = completely
inedible phytoplankton). See Table 3 for explanation of scenarios 1 and 2.
Figure 5. Comparison of the daily means of the four simulated state variablessmall phytoplankton (X1), large phytoplankton (X2
′), zooplankton (Z) andphosphorus (P ) for case B (X2
′ = completely inedible phytoplankton). SeeTable 3 for explanation of scenarios 1 to 5.
Figure 6. System state after an extended simulation period (days 700 to 1000)for all scenarios. A horizontal line indicates equilibrium state. While all statesfor case A (X2 = poorly edible phytoplankton) reached steady state before day700, case B (X2 = inedible phytoplankton) still shows damped oscillations. SeeTable 3 for explanation of scenarios.
Figure 7. Response surfaces for the dependence of characteristic values (seesection 2.3) on variations of the initial values of phytoplankton groups X1,0 andX2,0 (all variables are given in mg C L−1). The + on the bottom plane indicatesthe initial values of the standard scenarios. Note that the right column (c, f,i) shows the difference between scenarios s2 and s1 (case A).
Figure 8. Diagrams for the dependence of characteristic values (see section2.3) on variations of the initial values of phosphorus (P0 in µ g L−1, x-axis)and zooplankton (Z0 in mg C L−1), individual curves). Note that the diagramsshow differences between scenarios s2 and s1 (case A).
25
Fig. 1.
26
Phytoplankton, small Phytoplankton, large
Zooplankton PhosphorusPhosphorus
0.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
s1
0.0
0.2
0.4
0.6
0.8
1.0
Time (d)
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
Pho
spho
rus
(mg
m−
3 )s2
Fig. 2.
27
Scenario S1 Scenario S2 Scenario S3
Scenario S4 Scenario S5
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
X1
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
X2
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
Z
0 20 40 60 80 1000
5
10
15
20
25
30
Time (d)
Pho
spho
rus
(mg
m−
3 )
0 10 20 30 40 50 60 70 80 90 100
P
Fig. 3.
28
Phytoplankton, small Phytoplankton, large
Zooplankton PhosphorusPhosphorus
0.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
s1
0.0
0.2
0.4
0.6
0.8
1.0
Time (d)
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
Pho
spho
rus
(mg
m−
3 )s2
Fig. 4.
29
Scenario S1 Scenario S2 Scenario S3
Scenario S4 Scenario S5
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
X1
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
X2
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
Time (d)
Bio
mas
s (m
g C
L−
l )
0 10 20 30 40 50 60 70 80 90 100
Z
0 20 40 60 80 1000
5
10
15
20
25
30
Time (d)
Pho
spho
rus
(mg
m−
3 )
0 10 20 30 40 50 60 70 80 90 100
P
Fig. 5.
30
Case A, poorly edible phytoplankton
0
1
2
3
4
5
Pho
spho
rus
(mg
m−
3 )
0.0
0.1
0.2
0.3
0.4
0.5
Bio
mas
s (m
g C
L−
1 )
s1 s2 s3 s4 s5 s1 s2 s3 s4 s5 s1 s2 s3 s4 s5 s1 s2 s3 s4 s5
X1 X2 Z P
Case B, inedible phytoplankton
0
1
2
3
4
5
Pho
spho
rus
(mg
m−
3 )
0.0
0.1
0.2
0.3
0.4
0.5
Bio
mas
s (m
g C
L−
1 )
s1 s2 s3 s4 s5 s1 s2 s3 s4 s5 s1 s2 s3 s4 s5 s1 s2 s3 s4 s5
X1 X2 Z P
Fig. 6.
31
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.2
0.4
0.6
0.8
1.0
X1,0
X2,0
max
X1
(s1)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.2
0.4
0.6
0.8
1.0
X1,0
X2,0
max
X1
(s2)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.2
0.4
0.6
0.8
1.0
X1,0
X2,0
max
X1
(s2
− s
1)
(a) (b) (c)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
X1,0
X2,0
mea
nPhy
to (
s1)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
X1,0
X2,0
mea
nPhy
to (
s2)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
X1,0
X2,0
mea
nPhy
to (
s2 −
s1)
(d) (e) (f)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
X1,0
X2,0
max
Zoo
(s1
)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
0.6
X1,0
X2,0
max
Zoo
(s2
)
0.020.04
0.060.08
0.10
0.02
0.04
0.06
0.080.100.0
0.1
0.2
0.3
0.4
0.5
X1,0
X2,0
max
Zoo
(s2
− s
1)
(g) (h) (i)
Fig. 7.
32
0 20 40 60 80 100
0.0
0.5
1.0
1.5
P0 (mg m−3)
Bio
mas
s (m
g C
L−
1 )
0.005
0.010
0.020
0.0300.0400.0500.0700.100
Z0 (mg m−3)
maxX1 (s2−s1)
0 20 40 60 80 100
−0.4
−0.2
0.0
0.2
0.4
0.6
P0 (mg m−3)
Bio
mas
s (m
g C
L−
1 )
0.010
0.0200.0300.0400.050
0.0600.065
0.070
0.075
0.080
0.085
0.090
0.100
Z0 (mg m−3)
meanPhyto (s2−s1)
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
1.2
P0 (mg m−3)
Bio
mas
s (m
g C
L−
1 )
0.01
0.02
0.03
0.04
0.05
0.06
0.070.08
0.090.10
Z0 (mg m−3)
maxZoo (s2−s1)
Fig. 8.
33
