ecological modelling 220 (2009) 343–350 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/ecolmodel Data-directed modelling of Daphnia dynamics in a long-term micro-ecosystem 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 water circulates. Three trophic levels are represented in different com- Received in revised form partments: autotrophs (algae, mainly Chlorella vulgaris), herbivores (Daphnia magna) and 7 October 2008 microbial decomposers. 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–herbivore 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 detritus and decomposing algae which were not measured. © 2008 Elsevier B.V. All rights reserved. 1. Introduction fied micro-ecosystems 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 Ecology 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) decomposer 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 nature 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 food web 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.