Ecological Modelling 162 (2003) 15–31
Individual-based model of the reproduction cycle of Moina macrocopa (Crustacea: Cladocera) Egor S. Zadereev∗, Igor G. Prokopkin, Vladimir G. Gubanov, Michail V. Gubanov Institute of Biophysics, Akademgorodok, 660036 Krasnoyarsk, Russia Received 9 October 2001; received in revised form 22 July 2002; accepted 30 September 2002
Abstract An individual-based model of cyclic development of Cladocera populations was developed on the basis of experimental data. The model takes into account the following processes describing the development of an individual animal: maturation, transition into other reproductive classes, selection of the reproduction mode (parthenogenetic or gamogenetic), release of parthenogenetic progeny and death. The model assumes that switching from asexual to sexual reproduction is controlled by the concentration of food and metabolic by-products of the animal population. Verification of the model by independent experiments demonstrated that (1) during population growth, metabolic by-products build up in the medium, and (2) the effect of metabolic by-products on gamogenesis induction depends on concentration. The hypothesis that the effect of regulating reproductive switching factors should synchronise the development of population with the change of environmental conditions in order to ensure production of the maximum number of diapausing eggs was tested. It is shown that combination of regulating reproductive switching factors maximises the production of diapausing eggs. © 2002 Elsevier Science B.V. All rights reserved.
Keywords: Individual-based model; Cladocera; Change of reproduction mode; Metabolic by-products
1. Introduction ability of this kind of models to describe populations with pronounced time periodicity of life cycles. Most Individual-based modelling is a rapidly develop- efficiently discrete models are used to describe fish ing method of mathematical analysis of population populations (Grimm, 1999). Cladocera are very inter- dynamics. A single animal with its individual charac- esting for the investigation of population dynamics and teristics acts as a modelling unit in individual-based application of discrete modelling. These organisms models. This approach describes properties of the are the key trophic elements of aquatic ecosystems. population through the sum and interaction of its parts. Cladocera populations (in particular those inhabiting Generally, development of such models employs the ephemeral water bodies) develop in cycles deter- theory of discrete models (Bolker et al., 1997). Wide mined by their ability to alternate parthenogenetic application of discrete models to describe the popula- and gamogenetic modes of reproduction. Gamoge- tion dynamics of aquatic animals is determined by the netic progeny enter the diapause in the embryonic stage of development and in this state are capable ∗ Corresponding author. Tel.: +7-3912-495839; of surviving drying and freezing (Makrushin, 1996). fax: +7-3912-433400. Thus, depending on conditions, Cladocera females E-mail address: [email protected] (E.S. Zadereev). form either parthenogenetic eggs (which develop into
0304-3800/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0304-3800(02)00348-4 16 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 females or into males without fertilisation) or gamo- females of M. macrocopa. These results allowed a genetic latent eggs (which require fertilisation and detailed parameterisation of gamogenesis induction develop into females after the stage of rest). by these factors. It is safe to conclude that an almost The ecological significance of these different modes complete data set pertaining to the M. macrocopa of reproduction is as follows: parthenogenesis allows life cycle is available. It makes possible the use of the population to increase rapidly to use favourable discrete modelling to analyse cyclic development of environmental conditions with maximum efficiency. this species. The aim of the presented work was to Gamogenesis, does not increase the growth rate of a develop an individual-based model of cyclic develop- population (as parthenogenesis does), but expands the ment of Cladocera population and on the basis of this genetic base of populations to form multiple geno- model: (1) based on comparative analysis to select the types; the ability of gamogenetic progeny to enter mechanism of control of gamogenesis induction by diapause allows a population to survive under adverse density-dependent factors, and (2) to analyse the role environmental conditions. Also, if the number of of reproductive switching in population dynamics. produced latent eggs is important for the short-term success of a population in the next cycle of its devel- opment, then the bank of latent eggs in the sediments 2. Description of the mathematical model of a water body “passes” the genetic material through the time (Marcus et al., 1994) and can ensure the 2.1. The general structure of the model long-term success of the population. The major factors that directly influence induction As a rule, discrete modelling divides a model pop- of gamogenesis are photoperiod, temperature, trophic ulation into classes, by age, weight, or sex criteria. In conditions and effect of population density (Alekseev, our case, the population of M. macrocopa is divided 1990). For inhabitants of temporary water bodies, into four classes of living animals (juvenile, gamo- the effect of density-dependent factors (food concen- genetic and parthenogenetic females and males) and tration and metabolic by-products) is of high eco- ephippial eggs (Fig. 1). logical significance (Zadereev and Gubanov, 1996). Each class is described by a single characteristic— NX Even though such complex nature of development of the number of animals j in the class X at time j. Cladocera species has been known for quite a time, Each i-th animal at time j of development of a popula- W the existing mathematical models of Cladocera popu- tion is described by its weight i,j (mg of dry weight), t g lations usually either do not incorporate switching to age i,j (days), index i,j and by a set of discrete vari- g = sexual reproduction (e.g. De Roos et al., 1997; Koh ables. According to the index i,j 1, 2, 3, 4, the i-th et al., 1997) or use population level approaches to animal belongs at time j to juvenile, gamogenetic or control reproductive switching (Acevedo et al., 1995; parthenogenetic females or males. Acevedo and Waller, 2000). As individual-based mod- The model takes into account the following pro- els are able to describe populations with pronounced cesses describing the development of an i-th animal maturity time periodicity of life cycle and as there are a number (Fig. 2): maturation (Pi,j ), transition into other change G→P of evolution and ecological questions connected with reproductive classes (Pi,j , Pi,j ), selection of the presence of the reproductive switching and dia- the reproduction mode (parthenogenetic or gamoge- P G pause in the life cycle of a population, we consider the netic, Pi,j or Pi,j ), release of parthenogenetic progeny development of an individual-based model of cyclic (Pbirth) and death (Pdeath), at time j of population reproduction of Cladocera to be absolutely necessary. i,j i,j development. Previously, we experimentally investigated Discrete variables PP , PG , Pdeath describing the (Zadereev, 1997; Zadereev and Gubanov, 1996, 1999; i,j i,j i,j development of an animal are determined through Zadereev et al., 1998) the effect of density-dependent comparison of the value of the function describing factors (food concentration and concentration of the given process with random number PR ∈ [0, 1] metabolic by-products) on the induction of gamogene- Pmaturity sis, the survival, fecundity, sex ratio in parthenogenetic (Eqs. (8), (9) and (13)). Discrete variables i,j , change birth G→P progeny, the period between clutches in individual Pi,j , Pi,j , Pi,j are connected to the steady pe- E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 17
Fig. 1. Schematic diagram of M. macrocopa population model structure. (Rectangles): population classes; (ovals): elements of the culture medium; (solid arrows): interactions between population classes; (dotted arrows): hatching of progeny.
Fig. 2. Model life cycle of a single M. macrocopa. (A) Birth; (B) maturation; (C) transition into the class of gamogenetic females; (D) transition into the class of parthenogenetic females; (E) transition from the class of gamogenetic female to the class of parthenogenetic females. 18 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 riodicity of the animal’s life cycle, i.e. the values of parthenogenetic female, Di is the proportion of fe- these characteristics depend on the animal’s age ti,j males in progeny, released by the i-th parthenogenetic and are not stochastically determined (Eqs. (6), (10), female. (18) and (19)). Each of the discrete variables can take The model allows the number of ephippial eggs to only two values: 0 or 1. When the value of a variable be tracked. However, at this stage of development of is 1, an animal enters the process described by this the model they are supposed to be in diapause indef- variable, with value 0 this process is not realised. For initely; that is why we excluded the equation of the death males all variables, except for Pi,j , are always zero. dynamics of ephippial eggs from the description of the Neonates have initial zero values for all variables. model. Thus, a model life cycle of an animal is varia- The proposed model applies to a laboratory culture tion of values ti,j , Wi,j (growth of an animal) and with artificially controlled environmental conditions. ∗∗∗ Pi,j , gi,j (development of an animal) through dis- The population dynamics was simulated for the fol- crete time j. Dynamics of different classes is stochastic lowing experimental conditions: temperature, 26 ◦C; and is based on individual characteristics of animals, photoperiod, 16 h light, 8 h dark; volume of experi- death birth change (Eqs. (1)–(4)). When variables Pi,j , Pi,j , Pi,j , mental flow-through system with non-growing food, G→P Pi,j take value 1, the appropriate population classes 400 ml; flow-through rate, 1200 ml per day. change their size at time j + 1 of population develop- ment. 2.2. Growth and development of a model animal The following mathematical model describes dy- namics of different classes in a population: Table 1 gives a brief description of all parameters used in the equations, their values and sources of val- NP NJ j j ues. NJ = NJ + D E Pbirth − Pchange j+1 j i i i,j i,j (1) i=1 i=1 2.2.1. Maturation As the developed model was to be verified by pop- NJ j ulation experiments, the model was initially adjusted NG = NG + PG Pchange to the data obtained in laboratory conditions. Matu- j+1 j i,j i,j i=1 ration can be determined by visual observation when NG NG the length of an animal is 0.7 mm. Therefore, the age j j of an animal at which it becomes mature (Ti) is deter- − PG→P − Pdeath i,j i,j (2) mined as: i=1 i=1 Ti = ti,j , ⊃ (Wi,j−1 Table 1 Summary of parameters Parameter Description Value (units) Reference Ks Half-saturation food concentration for growth rate 0.00035 (mg/ml) Gladyshev et al. (1993) Y Efficiency of food consumption 0.3 (adimensional) Jorgensen et al. (1978) µmax Maximum specific growth rate of an animal 0.019 (1/h) Gladyshev et al. (1993) sh Model count step 0.2 (h) W Weight difference between animals with body sizes 0.01096 (mg) 0.34 and 1 mm T Time interval required for animal to reach body 48 (h) Zadereev (1998) length of 1mm with optimal food requirement S1 Half-saturation total food quantity for life span 0.06 (mg) Experimentally determined constant S2 Half-saturation total food quantity for juvenile growth 0.15 (mg) Experimentally determined constant td Difference between maximal and minimal animal age 11 (days) Zadereev et al. (1998) tmin Minimal age when an animal can die 2 (days) Zadereev et al. (1998) Dmax Coefficient determining the maximal age of animals 1 (female), 0.76 (male) Korpelainen (1989) of opposite sex τ Age when females start to hatch males 4 (days) Zadereev (1998) DEX Moulting period 28 (h) Zadereev et al. (1998) V Experimental chamber volume 400 (ml) N Sequence of natural numbers 1, 2, 3, ..., N reproduction mode. Previous experiments with indi- The i-th juvenile female starts a gamogenetic repro- G vidual females established the relationship between duction (Pi,X = 1) when conditions described in these two factors and the proportion of females Eq. (8) are realised, or alternatively, the i-th juvenile switching from parthenogenesis to gamogenesis (PSj) female starts a parthenogenetic reproduction (Eq. (9)). at time j (Zadereev and Gubanov, 1996): 2.2.3. Transition of an animal into the different PSj = 0.72 + 0.0013 × CHj reproductive class 0.007 × Xreal − 0.000015 × CHj × Xreal − j j The reproduction mode can be determined by vi- V sual observation methods when the length of a female (7) body is not less than 0.9 mm. A marker of gamo- genetic reproduction is albescent bars in the brood where Xreal is total quantity of algae in the culture j pouch (Makrushin, 1971). For parthenogenetic repro- volume (mg dry weight), CHj is the concentration of duction, the marker is the absence of bars in the brood metabolic by-products (mg/ml) and V is the volume pouch and formation of parthenogenetic embryos. of the experimental chamber (ml). This reproduction mode is visually determined when This relationship is used in the model as the func- the length of a female body is about 1 mm. Therefore, tion determining reproduction mode of the i-th mature the variables that determine the given process are the female. Reproduction mode is determined at the mo- G weight of an animal Wi,j and discrete variables Pi,j i ment of maturation T . P and Pi,j . The i-th female transits into the different G maturity Pi,X = 1, ⊃ PSX >PR,Pi,X = 1,gi,X = 1 reproductive class when: (8) change G Pi,X = 1, ⊃ Wi,X ≥ W0.9,Pi,X = 1,gi,X = 1 P maturity P Pi,X = 1, ⊃ PSX ≤ PR,Pi,X = 1,gi,X = 1 ⊃ Wi,X ≥ W1,Pi,X = 1,gi,X = 1 (9) (10) 20 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 where W0.9 and W1 are the weights of animals with where τ is the age at which a female begins to release body length 0.9 and 1 mm, respectively. males. After achieving a body length of 1 mm, a female 2.2.4. Death is assumed to gain all weight Wi,j to form a clutch. Previous experiments have shown that maximal life A similar approach to calculate fecundity is not new span (tmax)ofM. macrocopa in favourable conditions in discrete models (Uchmanski, 1999). The average is equal to 11–13 days (Zadereev et al., 1998). The body length of a neonate is 0.34 mm. The number of model assumes that tmax depends on food availability neonates (Ei) in a clutch is determined as: as: Wi,j − W1 real Ei = , ⊃ gi,j = 3 (15) td × Xj W t = t + (11) 0.34 max min real S1 + Xj where W0.34 is the weight of an animal with body length 0.34 mm. where tmin is the minimal age at which an animal can die, td is the difference between an animal’s maximal W − W i,j 1 and minimal ages, S1 is the experimentally selected 8, ⊃ ≥ 8,gi,j = 2 W0.34 constant—value of the total food quantity at which Ei = (16) W − W the life span is half of its maximum. i,j 1 0, ⊃ < 8,gi,j = 2 Taking into account that according to Korpelainen W0.34 (1989) the male life span is 76% of the female life span, the proportion of animals of the same age which Eq. (15) determines the number of neonates released D will die till the age j (Fj , Fig. 2) is defined by: by a parthenogenetic female. A gamogenetic female releases two ephippial eggs. 0, ⊃ ti,j 1 to 1.5 days (Zadereev et al., 1998). Therefore, the It is common to use the Monod function (Eq. (22)) i-th female hatches neonates when: to describe the specific growth rate of a population. t However, this function can be used to describe the spe- Pbirth = , ⊃ i,j ∈ N, g = , i,X 1 i,X 2 3 (19) cific growth rate of a separate animal with the follow- DEX ing assumptions: compared to other animals neither where N is the set of natural numbers. external conditions nor individual (genetically caused) Depending on values of Di and Ei all hatched features of animals give them any advantages in feed- progeny go into classes of either males or juvenile ing and growth. In this case, all animals with body females. length more than 1 mm are similar in terms of food consumption and growth and this application of the 2.2.6. Growth Monod function is possible. An equation describing dynamics of an animal’s weight can be written as: 2.3. The description of culture medium birth Wi,j+1 = Wi,j + dWi,j − Pi,j × Ei × W0.34 (20) The model simulates artificially controlled food Eq. (20) takes into account the gain of an animal’s supply and environmental conditions. The culture weight at each calculation step dWi,j and the reduction medium where the population develops is described real of an animal’s weight due to hatching of progeny (for by the quantity of food Xj (mg dry weight) (in mature females only). our case, the source of food is unicellular green al- Our experimental observations have shown that gae C. vulgaris) and the concentration of metabolic under favourable conditions it takes 2 days (48 h) by-products CHj of a population (mg/ml) at time j. for a female of M. macrocopa to grow from size The food supply is modelled as a manipulated and real 0.34 mm (size of hatched parthenogenetic progeny) controlled variable. The quantity of food Xj con- to size 1 mm (the female begins to hatch progeny). tained in the system at time j is defined as: The animal with body length more than 1 mm grows X , ⊃ X > exponentially. real j j 0 Xj = (23) W × Xreal 0, ⊃ Xj ≤ 0 j , ⊃ Wi,j Table 2 Effect of different methods of describing metabolic by-product concentrations and proportionality coefficient on emergence of gamogenetic females Food concentration Method of describing the concentration Proportionality Emergence of gamogenetic (103 cells/ml) of metabolic by-products coefficient (PC) females (days) 100 Eq. (26a) 4.5 ± 1.0 Eq. (26b) 1 3.5 ± 0.7 0.6 4.0 ± 0.9 0.5 3.8 ± 1.1 0.1 4.6 ± 1.2 0 5.5 ± 1.4 Experimental 6 200 Eq. (26a) 6.8 ± 0.2 Eq. (26b) 1 4.7 ± 0.6 0.6 5.3 ± 0.1 0.5 5.4 ± 0.4 0.1 6.8 ± 0.2 0 6.8 ± 0.2 Experimental 6 400 Eq. (26a) 6.5 ± 0.1 Eq. (26b) 1 5.1 ± 0.1 0.6 5.2 ± 0.3 0.5 5.5 ± 0.4 0.1 6.5 ± 0.1 0 7.0 ± 0.8 Experimentala 5 800 Eq. (26a) 8.0 ± 0.3 Eq. (26b) 1 5.1 ± 0.1 0.6 5.4 ± 0.5 0.5 5.7 ± 0.4 0.1 7.1 ± 0.7 0 8.0 ± 0.3 Experimentalb 5 1600 Eq. (26a) 7.7 ± 0.2 Eq. (26b) 1 5.1 ± 0.1 0.6 5.4 ± 0.3 0.5 5.8 ± 0.4 0.1 7.5 ± 0.5 0 8.1 ± 0.6 Experimentala 6 a Experimental data till day 6. b Experimental data till day 5. difference in the emergence of gamogenetic females limit the development of the population (at least before between the model and the natural population does not gamogenesis induction) and diapause is controlled by exceed 12 h (Table 2). This two-factorial control of the infochemicals. the reproduction mode can be easily explained on the Out of the five analysed variants of food supplies, basis of the theory of limiting factors. When the food only one limits the development of population. Thus, concentration is low, it limits the development of the all subsequent calculations were performed taking into population and controls gamogenesis induction. With account the accumulation of metabolic by-products the increase in food concentration, this factor does not with proportionality coefficient PC = 0.5(Eq. (26b)). 24 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 Fig. 3. M. macrocopa population dynamics in batch culture (model = average of 30 runs (thick line) ±S.D. (thin lines); experiment = dots ± S.D.). Food concentration: (a) 100 × 103 cells/ml; (b) 200 × 103 cells/ml; (c) 400 × 103 cells/ml; (d) 800 × 103 cells/ml; (e) 1600 × 103 cells/ml. The population dynamics of M. macrocopa under concentration 200 × 103 cells/ml the population sta- five regimes of food supply is represented in Fig. 3. bilises on day 9. Until stabilisation juvenile animals The results of modelling are in good agreement with constitute about 60–80% of the population. During the laboratory data. With food concentration 100×103 the next several days, these juvenile cohorts get ma- cells/ml the maximum population numbers (∼400 an- ture and a majority of them reproduce by gamoge- imals) and the time of peak population numbers (days nesis. After this, the proportion of juvenile females 9–10) in the model and experiment coincide; with food in the population drops down to 50% and the pro- concentrations 400, 800 and 1600 × 103 cells/ml, the portion of gamogenetic females increases. At the end model is close to the experimental population numbers of experiment, the proportion of males in the popu- till day 7 of the development (Fig. 3). lation stabilises at about 10%. These key aspects of A more detailed comparison of the modelling and population development are reflected in the results of experimental results is given in Fig. 4. With food modelling. E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 25 Fig. 4. The development of M. macrocopa population in batch culture with food concentration 200 × 103 cells/ml. (A) Experiment; (B) model (average of 30 runs (thick line) ±S.D. (thin lines). (1) Total population numbers; (2) juvenile females; (3) males; (4) ephippial eggs). 4. Verification experiment perimental data. However, the model was verified on the set of data received in standardised experimental The examples presented in the previous section set-up. Simulation by the model verified on a limited show that the individual-based model of cyclical set of parameters may not adequately predict the de- development of M. macrocopa developed on the ba- velopment of a population with different regulating sis of experiments with individual females can be parameters. A verification experiment was performed used to obtain population simulations close to ex- to estimate the reliability of the developed model. 26 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 A brief introduction to the methodology of ver- The experimental set-up satisfying the aforemen- ification experiment is needed. Change of the re- tioned requirements was selected by the model. Model production mode is controlled by the co-operative simulations demonstrated that in cultivators with vol- effect of food concentration and the concentration of umes 200 and 800 ml (total food quantity 32 × 107 metabolic by-products. These are density-dependent cells per day is the same in both cases, food con- factors. Consequently, it is very difficult to sepa- centrations in cultivators are 1600 × 103 cells/ml and rately measure the effect of each factor in popula- 400 × 103 cells/ml, respectively), the reproduction tion experiments. The following experimental set-up mode in the 200 ml cultivator (day 4.7) changed 1.5 could separate the effect of food concentration and days earlier than in the 800 ml cultivator (day 6.2). metabolic by-products on the induction of gamoge- After this step, we conducted the laboratory experi- nesis. The population is maintained in a cultivator ment with the same conditions. Experimental popula- supplied with two flows of medium. The first supplies tions started with five juvenile females (body length the cultivator with food, the second dilutes the cul- 0.3–0.5 mm). The experiment was performed in three ture medium in order to decrease the concentration replicates. The experiment demonstrated that in the of metabolic by-products. The outflow goes through vessel with volume 200 ml gamogenetic females were a filter that keeps algae in the cultivator. The rate found at day 4 which is 2 days earlier than in the of the second flow is varied to estimate the effect vessel with volume 800 ml (day 6). It should be men- of different concentrations of metabolic by-products tioned that both the time differences in the induction on the induction of gamogenesis with the same food of gamogenesis between different volumes and the concentration. time of gamogenesis induction in both vessels ob- However, such an experiment is difficult to re- tained in the experiment are very close to the model alise. We can propose the following simplification of prognosis. this scheme. The batch culture of a model organism Results of these experiments suggest two impor- develops in cultivators of different volumes. The cul- tant conclusions. First, the time difference found in tivators are supplied with the same amount of food the induction of gamogenesis in different volumes daily. Thus, in cultivators with different concentra- with the same total food quantity lends support to the tions of metabolic by-products in the medium the following hypotheses: (1) during population growth, same total quantity of food is achieved. It is clear metabolic by-products build up in the medium, and that this experiment distinctly differs from the previ- (2) the effect of metabolic by-products on gamoge- ously described discriminatory experiment. The key nesis induction depends on concentration. Second, a difference is that the experiment with batch culture good agreement between the experimental data and will be performed under the effect of different food the model prognosis indicates that the mathematical concentrations, as the food density will vary with model was constructed with adequate conceptions cultivator volume. However, as it was discussed in about cyclical development of M. macrocopa.Itis Section 3 and demonstrated in Table 2, the effect of worth mentioning that the experimental design suit- food concentration on gamogenesis induction pre- able for the detection of time difference in gamogen- vails over the effect of metabolic by-products only esis induction was selected by model calculations. when the food availability is low (under the condi- Comparison of model simulations with experimen- tions of food limitation). As Cladocera species are tal data and successful verification experiment provide filter feeders, they can satisfy their food requirements grounds for the statement that the mathematical model by varying the filtering rate at different food con- developed reflects the main features of M. macro- centrations. Thus, if the food concentration does not copa cyclic development. However, as mentioned in limit the development of the population, animals will the review on individual-based modelling: “Blind faith be able to satisfy their food requirements within a in what a model predicts is not the purpose of the relatively wide range of food concentrations. Taking modelling”(Grimm, 1999). The mathematical model into account this argument, the experiment with batch of M. macrocopa cyclic development can serve to anal- cultures should be performed with non-limiting food yse theoretically several aspects of population dynam- concentrations. ics of this species. E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 27 5. The role of the change of reproduction mode in among all three tested variants. The proportion of ju- population dynamics of M. macrocopa: theoretical venile females after a population has stabilised is less analysis than in previous variants and fluctuates periodically within the range of 20–40%. The proportion of males First, we analyse the effect of reproductive switch- is stable and equals 30% of the population. The pro- ing on the development of population. Model simula- portion of parthenogenetic females fluctuates from 0 tion of the population development of M. macrocopa to 20%. The first gamogenetic females emerge on day with food concentration in the medium 400 × 103 5. The proportion of gamogenetic females reaches the cells/ml is shown in Fig. 5. Three versions of pop- maximum on day 11 (77%), decreases and performs ulation development are shown: (A) the sole way of regular fluctuations from 20 to 60% in antiphase with reproduction is parthenogenesis, (B) the change of the the proportion of juvenile females. reproduction mode is determined by food availability, Thus, because of the introduction of reproductive and (C) the change of reproduction mode is deter- switching and subsequent complication of the mech- mined by the co-operative effect of food availability anism of regulation of this processes, the population and metabolic by-products. declines, changes its size and age structures (decline During the first days of the development of the in the proportion of juvenile females, increase in the population “without gamogenesis” the proportion of proportions of males and gamogenetic females). juvenile females (Fig. 5A) is more than 90%. After the Diapause allows the population to survive in the first peak (1465 animals), the population declines and adverse environment. The first step of cyclic repro- stabilises at 1092 animals on days 13–14 of the devel- duction is the induction of diapause that is determined opment. Size and age structures of the population also by the environmental factors. When favourable envi- stabilise after the population reaches its plateau. The ronmental conditions are established, the diapausing proportion of juvenile females in the population is organisms are reactivated and the population cycle about 70%. The proportion of males fluctuates within repeats again. At the reactivation stage, a population 20–28%. The proportion of parthenogenetic females with the maximum number of diapausing organisms periodically fluctuates from 1 to 11%. has an advantage in competition. It means that the When the change of the reproduction mode is regu- effect of regulating reproductive switching factors lated by food availability (Fig. 5B), the first population should synchronise the development of a population peak is the same as in the previous case both in num- with the change of environmental conditions in order bers (1465 animals) and time (days 8–9). However, af- to ensure the production of the maximum number ter day 14, an average population, when it reaches the of diapausing eggs. In this case, combinations of plateau, is less and equals 1004 animals. The size and regulating factors that maximise the production of age structures are also different. The proportion of ju- diapausing eggs are selected and stabilised in the pro- venile females declines and fluctuates near 50%. The cess of evolution. Similar considerations have been proportion of males in the population is stable—about developed for some social insects using Pontryagin’s 20%. The proportion of parthenogenetic females fluc- maximum principle (Oster and Rocklin, 1979). tuates periodically from 5 to 10%. The gamogenetic The developed model of cyclic reproduction can be females emerge in the population on day 7. The pro- used to examine this hypothesis in Cladocera at least portion of gamogenetic females reaches the maximum for two factors regulating gamogenesis—food con- in day 13 (50% of total population), then declines and centration and metabolic by-products. The dynamics stabilises at 30%. of accumulation of diapausing eggs during the devel- When the change of the reproduction mode is reg- opment of M. macrocopa populations with different ulated by the combined effect of food availability and regulation mechanisms of gamogenesis induction is metabolic by-products (Fig. 5C), the population dy- shown in Fig. 6. When the change of reproduction namics and the size–age structures (after the popula- mode is controlled by the sole effect of food concen- tion is stabilised) differ from the previous runs. The tration, the number of diapausing eggs is minimal for maximum population (1357 animals) and the average all tested food concentrations. The number of diapaus- population after day 14 (943 animals) are the least ing eggs increases with the coefficient responsible for 28 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 Fig. 5. Model runs of M. macrocopa batch population. Cultivation volume = 400 ml, food concentration = 400 × 103 cells/ml, flow rate = 1200 ml per day. (1) Total population numbers; (2) juvenile females; (3) males; (4) parthenogenetic females; (5) ephippial eggs. (A) Population reproduces by parthenogenesis; (B) change of the reproduction mode is controlled by the concentration of food; (C) change of the reproduction mode is controlled by the concentration of food and metabolic by-products. E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 29 Fig. 6. The effect of PC coefficient on the production of ephippial eggs in the batch culture of M. macrocopa. Culture volume = 400 ml; food concentrations: (1) 100 × 103 cells/ml, (2) 400 × 103 cells/ml, (3) 800 × 103 cells/ml; flow rate = 1200 ml per day; y-axis = the total number of ephippial eggs (±S.D., 30 model runs) accumulated during 40 days of M. macrocopa population development; x-axis = PC coefficient. Bars denoted with different letters significantly (P<0.01) differ according to t-test for independent samples. the accumulation of metabolic by-products. The num- by-products is more than 0.5 and close to 1. The ber of diapausing eggs stabilises when the coefficient same range was selected for this coefficient on the reaches 1. Mathematically, the coefficient might have basis of different principles during the verification any value. However, the increase of coefficient (PC = of the model. Thus, now it is possible, within the 2) that follows does not increase the production of the above-mentioned limits, to give a biological justifica- diapausing eggs with any tested food concentration tion for the regression with the selected coefficient: (Fig. 6). With the same quantity of energy available the derived regression equation with selected coef- for the development of population, the number of pro- ficients describes the effect of selected factors and duced ephippial eggs will depend on the population ensures the formation of the maximum number of structure. Depending on factors controlling the devel- diapausing eggs. opment of a population there is a wide range of possi- ble population structures (e.g. Fig. 5A–C). However, in the ideal case, within the whole range of all possi- 6. Conclusion ble population structures there is only one that ensures the production of the maximal number of ephippial We have tried to apply the integrated approach to eggs. That is why the increase in PC does not lead to the individual-based modelling of such a natural phe- the increase in the number of diapausing eggs. nomenon as cyclic reproduction of Cladocera. First, It should be clear that optimisation was tested for the individual-based model, was developed on the the effect of only two factors and within the limited basis of the experiments with individual animals. In set of environmental conditions (such physical factors addition to our data and conceptions, various litera- as photoperiod and temperature were fixed at the level ture data and approaches to the Cladocera modelling favourable for parthenogenesis). However, the main were synthesised in the model. In the presented ver- purpose of the analysis was to evaluate the feasibility sion of the model, the main emphasis was laid on the of the selected mechanism of the two-factorial con- effect of density-dependent factors on gamogenesis trol of gamogenesis induction. It is noteworthy that induction in Cladocera. This choice was stimulated by the maximal number of diapausing eggs is produced the fact that density-dependent factors and especially when the coefficient of accumulation of metabolic infochemicals traditionally received less attention in 30 E.S. Zadereev et al. / Ecological Modelling 162 (2003) 15–31 Cladocera modelling than physical factors, such as density-dependent control of reproductive switching in photoperiod and temperature. Second, the model was Cladocera. calibrated on the set of preliminarily received, but in- dependent population data. As we focused our atten- tion on the gamogenesis induction and the role of the Acknowledgements combined effect of food concentration and metabolic by-products, we did not try to vary all possible model The work was supported by KRSF grant No. parameters until the model runs would ideally simulate 11F0029M and partially supported by RFBR grant experimental data. On the contrary, we tried to intro- No. 95-04-11794. Two anonymous reviewers are duce into the model constants experimentally deter- kindly acknowledged for valuable suggestions, crit- mined as precisely as possible, in order to manipulate icism and linguistic improvements. We also wish to with variables responsible for the control of gamo- thank Dr. Miguel Acevedo for the productive prelim- genesis induction. Third, the verification experiment inary discussions on the topic. was planned on the basis of model runs and later the same experiment was conducted to verify the model. References The realisation of the second and third steps allow a conclusion that during population growth, metabolic Acevedo, M.F., Waller, W.T., 2000. Modelling and control of by-products build up in the medium and the effect of simple trophic aquatic ecosystem. Ecol. Modell. 131, 269–284. metabolic by-products on gamogenesis induction is Acevedo, M.F., Waller, W.T., Smith, D.P., Poage, D.W., McIntyre, concentration dependent. These conclusions are not P.B., 1995. Modeling cladoceran population responses to stress self-evident. The nature of chemical substances in- with particular reference to sexual reproduction. Nonlin. 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