ecological modelling 220 (2009) 343–350

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Data-directed modelling of Daphnia dynamics in a long-term micro- experiment

Johan Grasman ∗, Egbert H. van Nes 1, Kees Kersting 2

Wageningen University and Research Centre, Wageningen, The Netherlands article info abstract

Article history: The micro-ecosystem under consideration consists of three compartments forming a closed Received 12 March 2008 chain in which circulates. Three trophic levels are represented in different com- Received in revised form partments: (algae, mainly Chlorella vulgaris), (Daphnia magna) and 7 October 2008 microbial . From a 20 years experiment with this system, data has been selected Accepted 20 October 2008 for this study. The dynamics of algae and Daphnia magna in only one of the compartments Published on line 27 November 2008 were modeled by different systems of differential and difference equations. We describe the successive steps in the process of model development, and the fitting of parameters using a Keywords: Nelder-Mead simplex calibration method. Identification problems were overcome by taking Logistic dynamics values for physiological parameters in agreement with the literature. It turned out that a Structured population logistic type of model gives the best result for the structured Daphnia population because of Parameter estimation the set up of the experiment: algae grow and reproduce in the upstream compartment. For this reason well-known plant– models did not comply with the data. The results of the parameter estimation procedure are discussed. The estimated grazing rate by Daphnia was smaller than expected. Possibly the Daphnia fed also on and decomposing algae which were not measured. © 2008 Elsevier B.V. All rights reserved.

1. Introduction fied micro- were set up. The general idea of this research was to perform experiments on a long timescale. Fur- In experimental studies of ecosystems one is confronted with thermore, a tool was created to analyse essential processes the problem of deciding which biological entities are con- taking place within an ecosystem and to obtain more insight tained in the system and which should be considered external in the throphic levels that can be discerned. It provided new (Beyers and Odum, 1993; Taylor, 2005). When dealing with results on the response of an ecosystem to external influences, concepts as stability and resilience, one feels the necessity such as pesticides (Kersting, 1991a), toxic substances (Kooi et to consider these topics in relation with complexity of the al., 2008) and other stress sources (Hallam et al., 1997). Another ecosystem structure. Such an approach easily leads to sys- societal application of micro-ecosystems is found in waste tems with many components and interactions, see Peterson water treatment, see, e.g. Daims et al. (2006) and Grasman et et al. (1998), Carpenter et al. (2001) and Cropp and Gabric al. (2005). (2002). Long-term experiments are hard to carry out for such The micro-ecosystem, we analysed, resides in a structure systems and cause–effect problems tend to be very hard to consisting of three compartments forming a closed chain in unravel. For these reasons experiments with strongly simpli- which water is circulated. The compartments represent the

∗ Corresponding author at: Wageningen University and Research Centre, Biometris, P.O. Box 100, 6700 AC Wageningen, The Netherlands. Tel.: +31 317 484085; fax: +31 317 483554. E-mail addresses: [email protected] (J. Grasman), [email protected] (E.H. van Nes), [email protected] (K. Kersting). 1 Wageningen University, Aquatic and Water Quality Management Group, P.O. Box 47, 6700 AA Wageningen, The Netherlands. 2 Wageningen Imares, The Netherlands. 0304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2008.10.010 344 ecological modelling 220 (2009) 343–350

Fig. 1 – Schematic representation of the micro-ecosystem with a total volume of 9.6 L. (A) autotrophe compartment (6 L), (B) herbivore compartment (1.8 L), (C) compartment (1.8 L), (M) magnetic stirrer, (P) peristaltic pump and (S) rubber stoppers. The flow rate of the circulating water was 0.03 L/h. The whole micro-ecosystem was placed in a climatized room at 18 ◦C. A circular fluorescent light surrounded the autotrophic subsystem (light/dark cycle of 14 h/10 h). The herbivore subsystem was shielded from direct illumination, while the autotrophic subsystem was wrapped in aluminum foil to create complete darkness.

three trophic levels: autotrophs (algae, mainly of the green designed to be stable. Regarding this aspect of the experi- algal species Chlorella vulgaris), herbivores (Daphnia magna) and ment, in which zooplankton only grazes a part of the system, microbial decomposers, see Fig. 1. Only the compartment with it is noted that it could be a rather realistic representation of autotrophs was lighted. For more details we refer to Ringelberg a heterogeneous lake, in which this mechanism also has a and Kersting (1978) and Kersting (1997). stabilizing effect (Scheffer and De Boer, 1995). From data, covering a period of 20 years, it is concluded that The above considerations brought us to the following ques- in principle this micro-ecosystem has a stable equilibrium, tion: what type of interaction determines the dynamics of as over extended periods fluctuations tend to be low. Exter- this micro-ecosystem? Since a wealth of data on our micro- nal perturbations are followed by a damped fluctuation of the ecosystem is available, we decided to take these data as system. Many ecosystems in tend to exhibit persistent starting point and formulate the mathematical model that fits fluctuations. This may be due to external temporal changes the best. Although we speak of a “micro”ecosystem, we real- with the ecosystem tracking such an input (Hsieh et al., 2005) ized, after an inventory study that such a system is already or to interactions within the system. Oscillation of population too large to consider in a complete model that will survive densities in a prey–predator model is the classical example the confrontation with the data (Hilborn and Mangel, 1997). In of such a behavior (Edelstein-Keshet, 1988). The possibility of this study we only consider the dynamics of algae and Daph- chaotic dynamics has been investigated in a number of studies nia in the herbivore compartment with an inflow from the (Grasman and Van Straten, 1994). Several mathematical popu- autotrophe compartment. Our goal is to model the dynam- lation models with a strange attractor as stable limit solution ics in the herbivore compartment by a system of differential have been formulated (Cushing et al., 2003). Examples of real or difference equations and to fit such a model to data on world chaotic processes are scarce, see Becks et al. (2005) and algal concentrations and on the size of the Daphnia popula- Benincà et al. (2008). The latter study considers a multi-species tion. There are also studies (De Roos et al., 1997; Nisbet et of bacteria, phytoplankton and zooplankton species al., 2000; Rinke and Vijverberg, 2005; Vanoverbeke, 2008) that from the Baltic Sea. It was cultured in a one-compartment start at the level of the individual. Using methods such as the laboratory mesocosm. Under constant external conditions, dynamic energy budget theory (Kooijman, 2000) the physio- irregular, large amplitude fluctuations of chaotic type were logical state of individuals (age, size) is determined from the observed. Apparently our microcosm is not above the required available food and other conditions (temperature). The sizes level of complexity to allow such a dynamics: the set up of of groups with individuals having well-defined physiological the experiment prevents strong trophic interactions. It was states are described in dynamical equations. From the solution ecological modelling 220 (2009) 343–350 345

of these equations, e.g. the box car train model (De Roos et dZ =−mZ + pBZ, (1b) al., 1992), one may extract the size of the total population dt and its fluctuation in time. It is noted that we started from where r is the rate of the flow of the medium through the this end by finding an appropriate model that fits data of the compartment and m the death rate of Daphnia. The graz- population as a whole. Half a way we meet, as from both ing by zooplankton is represented as a simple Holling type approaches we may conclude about longevity of Daphnia in an I functional response. experiment as we have here. It is remarked that for individual In the second model we separate zooplankton in number of based modeling precise information about the food compo- juveniles (X) and adults (Y) to obtain a structured population sition and other conditions is required. Algal concentrations system: are estimated from particle concentrations. These particles are assumed to consist of a fixed fraction of algae of the type dB = r{A(t) − B}−(h + fX + gY)B, (2a) Chlorella vulgaris, the presence of an other species of algae may dt disturb the pattern. Therefore, it is not easy to make the step to the population level (De Roos et al., 1997). Starting from the dX =−(q + s)X + pBY, (2b) data, as we do, has as risk that large unobserved systematic dt perturbations may bring about incorrect model choices. dY In first instance we considered four models and fitted their =−mY + qX, (2c) dt parameters using data from a small time interval. Two of them, based on plant–herbivore interaction, failed to give a satisfac- where s and m are, respectively, the death rates of the juvenile tory result. It turned out that the leading mechanism is that of and adult Daphnia and q is the rate at which juveniles become nonlinear food limited population growth as we find in mod- adults. els in which a term is incorporated. Next we We next introduced the mechanism of limited growth used data from a larger time interval to fit the parameters of represented by a carrying capacity model element. Ignoring the one model from the four that had population structure as the algal concentration B in the herbivore compartment, we well as the logistic dynamics. analysed the dependence of the unstructured Daphnia pop- ulation upon the incoming algae A(t) by a logistic difference equation: 2. Material and methods   Zt Zt+1 = Zt + p − n Zt (3) An extensive literature on modeling the dynamics of biological At populations is at hand, see Edelstein-Keshet (1988), DeAngelis (1992) and, for Daphnia, see Lampert (2006). Fitting of data with 1 + p the intrinsic growth rate and (1 + p)At/n the time can be straight forward using least squares and by moving in dependent carrying capacity. It is noted that over 1 day parameter space along the gradient of a penalty function to a changes in the state variables are relatively small, so that this minimum of this function (Nelder-Mead simplex direct search difference model can be seen as a forward Euler approxima- method, as implemented in MATLAB). Comparable methods tion of the solution of the corresponding logistic differential are found in Ellner et al. (2002) and Emlen et al. (2003). Fur- equation. thermore, in the literature, for a similar type of model, a neural Moreover, we considered the following extended logistic network approach has been employed (Wu et al., 2005). Finally, system of difference equations: we remark that, as in other studies (Hengl et al., 2007), we have = + − − + + to deal with identification problems. Bt+1 Bt r(At Bt) (h fXt gYt)Bt, (4a) In order to stay as close as possible to the data we decided = − + + to use particle concentrations for the algal densities with at Xt+1 Xt (q s)Xt pYt, (4b) time t, respectively, a concentration A(t) in the flow from the   m + nYt autotrophe compartment to the herbivore compartment (used Y + = Y − Y + qX . (4c) t 1 t B t t as external variable) and a concentration B(t) within the her- t bivore compartment. Moreover, we consider numbers X for The last term of Eq. (4a) represents the loss rate of algae from Daphnia in the juvenile state and Y for the adult state. As a natural death and from grazing by Daphnia within the com- first step we consider two elementary models of autotrophe- partment. It is taken proportional to the algal density with herbivore interaction (or prey-predator relation) as presented a coefficient that is linear in the densities of juvenile and by DeAngelis (1992) and fitted these differential equations to adult Daphnia. The second term at the right hand side of Eq. data of an episode in which the system recovers from a large (4b) shows a loss rate s of juveniles and a transition rate q perturbation and returns to a (noisy) equilibrium state. It is from juvenile to adult; they are taken proportional with the expected that from such a damped oscillation more informa- size of the juvenile population. The birth rate p of Daphnia is tion about the parameter values can be extracted than in an taken proportional to the size of the adult population. Finally, equilibrium state. In the first model we consider an unstruc- the first two terms at the right hand side of Eq. (4c) are of tured Daphnia population of size Z(t)=X(t)+Y(t) at time t: logistic type with an intrinsic growth rate smaller than 1 (the growth comes from the last term). For the system Eq. (4) we dB also selected a larger time interval from the data set. It starts = r{A(t) − B}−gBZ, (1a) dt with a population of 7 adult Daphnia. After a growth of the 346 ecological modelling 220 (2009) 343–350

duration of the adult stage at time t:   −1 m + nYt kt = . (6) Bt

3. Results

For the smaller data set (Fig. 3a and b) in which the Daph- nia population exhibits a damped oscillation after recovering from a perturbation, a parameter estimation procedure has been applied to the models (Eqs. (1–4)). In Table 1 the results are given. For the best fitting parameter values and initial values the simulated value of the total Daphnia population Z of the four models are presented in Fig. 3b. A comparison between the models is made by computing the residual sum of squares RSS(Z) between the simulated values of the total Daphnia population and the observed values at the observa- tion times. For the structured models (Eqs. (2–4)) the difference between the simulated and observed values of the sizes of the juvenile and adult population are given as well by RSS(X, Y). If for the plant–herbivore models (Eqs. (1 and 2)) we ignore the

Fig. 2 – Observed numbers [N] of juvenile and adult Daphnia population at time t and particle concentrations [␮3/mL] measuring the algal densities A(t) in the autotrophe compartment and B(t) within the herbivore compartment. The concentration of algae in both compartments were measured weekly with a Coulter Counter©. In the herbivore compartment the numbers of juvenile and adult Daphnia were counted at the same occasion: (a) algal concentrations and (b) Daphnia magna population.

population to a level of hundreds of adults and juveniles the population decreases and dies out after 798 days. In Fig. 2 we present this data for which we fitted the extended logis- tic system of difference equations Eq. (4). The parameters are estimated from the data using the software package GRIND for MATLAB (freely available at http://www.aew.wur.nl/uk/grind/). The penalty function consists of the squared relative differ- ences between observations and computed values of the state variables for given values of the parameters and initial values. It is noted that the flow rate and the volume of the herbi- vore compartment are known and set to, respectively, 0.03 L/h and 1.8 L. We assume that the average duration of the juvenile stage is 10 days and 33 days for the adult stage giving a back- ground mortality of 1/43 = 0.023 (at 18 ◦C) which is in the order of values found in the literature, e.g. 0.03 at 20 ◦C(Vanoverbeke, 2008). The above considerations lead to the following fixed parameter values [d−1]inEq. (4): Fig. 3 – Simulation results of the total Daphnia magna population Z = X + Y in the herbivore compartment as q = 0.1,m= 0.03 and r = 0.4. (5) described by the dynamical systems (Eqs. (1–4))overatime interval in which the system recovers from a perturbation: The other parameters and the initial values are varied. In Eqs. (a) observed algal concentrations A and B in [␮3/mL] and (b) (1) and (2) we also left the parameter m free. The best fit is observed total Daphnia magna population Z compared with found for the parameter values that correspond with a mini- the values of the simulated models (Eqs. (1–4)) with mum of the penalty function. From Eq. (4c) we derive the mean parameter values and initial states given in Table 1. ecological modelling 220 (2009) 343–350 347

Table 1 – Estimates of the parameters and initial states of the systems (Eqs. (1–4)) using the data presented in Fig. 3. Furthermore the observed total number of Daphnia is compared with the values found in the simulation by means of the residual sum of squares RSS(Z). For the two systems with structured populations this sum is also computed from the size of the juvenile and adult subpopulations. Model Eq. (1) Eq. (2) Eq. (3) Eq. (4)

Data A, B, ZA, B, X, YA, ZA, B, X, Y

Fixed parameters r = 0.4 r = 0.4 r = 0.4 q = 0.1 q = 0.1 m = 0.03

Optimal parameters and initial values f = 0.00303 p = 0.09 f =0 g = 0.0232 g = 0.000153 n = 0.00545 g = 0.0143

p = 0.128 h = 0.0750 Z0 = 230 h = 0.608 m = 0.134 s = 0.260 s = 0.437

B0 = 0.529 p = 0.657 p = 0. 802 Z0 = 224 m = 0.158 n = 0.00497

B0 = 0.426 B0 = 0.716 X0 = 152 X0 = 162

Y0 = 60.4 Y0 = 66.6

RSS(Z) 80,902 78,934 71,603 62,099

RSS(X, Y) 54,039 43,401 data on the algal concentration B(t) we find a RSS(Z) close to to judge the result of the parameter estimation process. It the model logistic model (Eq. (3)). However, then in the pro- is noted that apart from the parameter n, also the parame- cess of parameter estimation for these two systems we were ter s, the loss rate of the juvenile Daphnia, fluctuates strongly confronted with identification problems. The parameters, that although it stays small at all times. The remaining parame- were left free to be changed, tended to drift away while the ters h, f, g and p reproduce very well over the different time penalty function decreased extremely slowly. For this reason intervals. we did not include results on these simulations. Comparing In Fig. 5 we give the varying mean duration of the adult the two structured population models (Eqs. (2) and (4))we stage as given by formula Eq. (6). It is noted that the mean note that with an equal number of parameters (ignoring the life time of the Daphnia in this experiment is considerably less parameter f in Eq. (4)) the model (Eq. (4)) is doing better. We than under optimal feeding conditions. The average duration next concentrated on the structured logistic type of difference of the adult stage is 6.75 days (average mortality rate 0.148) equation system and fitted model (Eq. (4)) to the larger data meaning that the net reproduction ratio is 1.14. This num- set given in Fig. 2. Parameters and initial values were esti- ber is based on Table 2 last line and on the assumption that mated separately for the entire time interval (798 days) and the reproduction is asexual (parthenogenisis) with a popu- for two subintervals, see Table 2.InFig. 4 we give the simula- lation of just females, which is indeed the case. During the tion results for the system based on the parameter estimation period for which we fitted model (Eq. (4)) no ephippia were for the entire interval. It is noted that only over the time inter- observed. val (450–600) the values obtained from the model simulation are systematically above the observed values. A good agree- 4. Discussion ment is found for the intervals (300–450) and (625–775). In the interval (0–300) the output follows more or less the mean of From a micro-ecosystem consisting of three compartments the strongly fluctuating observations. Because of the fluctu- forming a closed chain we studied the dynamics in the com- ating availability of food the choice of the value of m did not partment with a Daphnia population living on algae produced have a large effect: the term nY/B governs the fluctuations in in an upstream compartment. Starting with the concept the losses. In line with this observation we find large differ- of a plant–herbivore relation we tried to fit such a type of ences in the estimated value of n in the different intervals. By model. However, no set of parameter values could be found splitting the data over different intervals, in which we esti- for which the dynamics of a damped oscillation agreed with mated independently the parameters, we got an idea how the data: the peaks in the data did not match in a sufficiently

Table 2 – Estimates of the parameters of the system (Eq. (4)) over different time intervals. Use is made of the data presented in Fig. 2. Interval h (d−1) f (N−1 d−1) g (N−1 d−1) p (d−1) s (d−1) n (␮3/(mL N d))

0–399 1.100 0 0.00357 0.185 0.00970 0.00034 399–798 0.716 0 0.00662 0.160 0 0.00573

Total: 0–798 1.072 0 0.00393 0.171 0.00175 0.00335 348 ecological modelling 220 (2009) 343–350

Fig. 5 – The mean duration k(t) [d] at time t of the adult stage of Daphnia magna in the herbivore compartment derived from Eq. (4).

The fact that a model for the herbivore compartment based on plant–herbivore interaction did not produce an accept- able fit to the data can be understood from the set up of the experiment. The grazing Daphnia could not affect the algae as they grew separated under better conditions in the autotrophe compartment. In addition the algae that entered the herbi- vore compartment were exposed to dark conditions leading to a high death rate and a low reproduction rate. These facts made us change our model concept: a nonlinear model based on logistic dynamics turned out to give a much better result. In the process of parameter estimation we ran into an iden- tification problem which we handled by using physiological parameter values from the literature. The model (Eq. (4))was fitted to the data given in Fig. 2. The fitted parameters are pre- sented in Table 2. The estimated value of the death rate of juvenile Daphnia is quite uncertain as its value changed con- siderable both under small changes in the model and from the choice of the time interval. Nevertheless it remained small in all cases. The parameters for the Daphnia birth and death rate showed a restricted variation over the different time intervals. Moreover, the death rate of algae within the her- bivore compartment turned out to depend only weakly upon Fig. 4 – Simulation results for the state variables of the the Daphnia density. Dependence of the death rate and birth dynamical system (Eq. (4)) describing the processes taking rate of juveniles upon the algal density was considered by place in the herbivore compartment over the time interval adding a linear term in B(t) to these constants. In the process (0, 798): (a) algal concentration B [␮3/mL] in the herbivore of parameter estimation the coefficients of these additional compartment and algal concentration A [␮3/mL] in the terms turned out to be negligible. It is remarked that the value autotrophe compartment being input to the system (Eq. (4)); of h is larger than we expected at forehand while the values (b) adult Daphnia magna population [N] in the herbivore of f and g are smaller than foreseen. A plausible explanation compartment; (c) juvenile Daphnia magna population [N] in is that the algae in the herbivore compartment have a high the herbivore compartment. mortality rate because they are permanently in a respiratory state. Moreover, decomposing algae may not be counted if they die within a week before the first time the algal den- sity is measured after their entry. However, they still can be accurate manner the rhythm of the damped oscillation of the part of the diet of the Daphnia. Another factor that influ- model system. Thus, it is likely that the fluctuation in the algal ences the monitoring of intake of nutrients is the recycling density was governed by processes outside the herbivore com- of food by Daphnia through re-uptaking of defecated particles partment. The algal density in the autotrophe compartment (Kersting, 1991b). A better founded explanation may be strongly fluctuated and indeed the Daphnia density was highly obtained by employing a based on mass correlated with this input to the herbivore compartment. conservation (Kooijman et al., 2007). ecological modelling 220 (2009) 343–350 349

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