Individual-based model of the reproduction cycle of Moina macrocopa (Crustacea: Cladocera)

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 Individual-based modelling is a rapidly developing method of mathematical analysis of population dynamics. A single animal with its individual characteristics acts as a modelling unit in individual-based models. This approach describes properties of the population through the sum and interaction of its parts. Generally, development of such models employs the theory of discrete models (Bolker et al., 1997). Wide application of discrete models to describe the population dynamics of aquatic animals is determined by the ∗ Corresponding author. Tel.: +7-3912-495839; fax: +7-3912-433400. E-mail address: [email protected] (E.S. Zadereev).

ability of this kind of models to describe populations with pronounced time periodicity of life cycles. Most efficiently discrete models are used to describe fish populations (Grimm, 1999). Cladocera are very interesting for the investigation of population dynamics and application of discrete modelling. These organisms are the key trophic elements of aquatic ecosystems. Cladocera populations (in particular those inhabiting ephemeral water bodies) develop in cycles determined by their ability to alternate parthenogenetic and gamogenetic modes of reproduction. Gamogenetic progeny enter the diapause in the embryonic stage of development and in this state are capable of surviving drying and freezing (Makrushin, 1996). Thus, depending on conditions, Cladocera females form either parthenogenetic eggs (which develop into

0304-3800/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 3 4 8 - 4

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females or into males without fertilisation) or gamogenetic latent eggs (which require fertilisation and develop into females after the stage of rest). The ecological significance of these different modes of reproduction is as follows: parthenogenesis allows the population to increase rapidly to use favourable environmental conditions with maximum efficiency. Gamogenesis, does not increase the growth rate of a population (as parthenogenesis does), but expands the genetic base of populations to form multiple genotypes; the ability of gamogenetic progeny to enter diapause allows a population to survive under adverse environmental conditions. Also, if the number of produced latent eggs is important for the short-term success of a population in the next cycle of its development, then the bank of latent eggs in the sediments of a water body “passes” the genetic material through the time (Marcus et al., 1994) and can ensure the long-term success of the population. The major factors that directly influence induction of gamogenesis are photoperiod, temperature, trophic conditions and effect of population density (Alekseev, 1990). For inhabitants of temporary water bodies, the effect of density-dependent factors (food concentration and metabolic by-products) is of high ecological significance (Zadereev and Gubanov, 1996). Even though such complex nature of development of Cladocera species has been known for quite a time, the existing mathematical models of Cladocera populations usually either do not incorporate switching to sexual reproduction (e.g. De Roos et al., 1997; Koh et al., 1997) or use population level approaches to control reproductive switching (Acevedo et al., 1995; Acevedo and Waller, 2000). As individual-based models are able to describe populations with pronounced time periodicity of life cycle and as there are a number of evolution and ecological questions connected with the presence of the reproductive switching and diapause in the life cycle of a population, we consider the development of an individual-based model of cyclic reproduction of Cladocera to be absolutely necessary. Previously, we experimentally investigated (Zadereev, 1997; Zadereev and Gubanov, 1996, 1999; Zadereev et al., 1998) the effect of density-dependent factors (food concentration and concentration of metabolic by-products) on the induction of gamogenesis, the survival, fecundity, sex ratio in parthenogenetic progeny, the period between clutches in individual

females of M. macrocopa. These results allowed a detailed parameterisation of gamogenesis induction by these factors. It is safe to conclude that an almost complete data set pertaining to the M. macrocopa life cycle is available. It makes possible the use of discrete modelling to analyse cyclic development of this species. The aim of the presented work was to develop an individual-based model of cyclic development of Cladocera population and on the basis of this model: (1) based on comparative analysis to select the mechanism of control of gamogenesis induction by density-dependent factors, and (2) to analyse the role of reproductive switching in population dynamics.

2. Description of the mathematical model 2.1. The general structure of the model As a rule, discrete modelling divides a model population into classes, by age, weight, or sex criteria. In our case, the population of M. macrocopa is divided into four classes of living animals (juvenile, gamogenetic and parthenogenetic females and males) and ephippial eggs (Fig. 1). Each class is described by a single characteristic— the number of animals NjX in the class X at time j. Each i-th animal at time j of development of a population is described by its weight Wi,j (mg of dry weight), age ti,j (days), index gi,j and by a set of discrete variables. According to the index gi,j = 1, 2, 3, 4, the i-th animal belongs at time j to juvenile, gamogenetic or parthenogenetic females or males. The model takes into account the following processes describing the development of an i-th animal maturity (Fig. 2): maturation (Pi,j ), transition into other change

G→P ), selection of , Pi,j reproductive classes (Pi,j the reproduction mode (parthenogenetic or gamogeP or P G ), release of parthenogenetic progeny netic, Pi,j i,j birth ) and death (P death ), at time j of population (Pi,j i,j development. P , P G , P death describing the Discrete variables Pi,j i,j i,j development of an animal are determined through comparison of the value of the function describing the given process with random number PR ∈ [0, 1] maturity (Eqs. (8), (9) and (13)). Discrete variables Pi,j , change

Pi,j

birth , P G→P are connected to the steady pe, Pi,j i,j

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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.

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riodicity of the animal’s life cycle, i.e. the values of these characteristics depend on the animal’s age ti,j and are not stochastically determined (Eqs. (6), (10), (18) and (19)). Each of the discrete variables can take only two values: 0 or 1. When the value of a variable is 1, an animal enters the process described by this variable, with value 0 this process is not realised. For death , are always zero. males all variables, except for Pi,j Neonates have initial zero values for all variables. Thus, a model life cycle of an animal is variation of values ti,j , Wi,j (growth of an animal) and ∗∗∗ , g Pi,j i,j (development of an animal) through discrete time j. Dynamics of different classes is stochastic and is based on individual characteristics of animals, death , P birth , P change , (Eqs. (1)–(4)). When variables Pi,j i,j i,j G→P take value 1, the appropriate population classes Pi,j change their size at time j + 1 of population development. The following mathematical model describes dynamics of different classes in a population: NP

J Nj+1

=

NjJ

+

j


i=1

NJ

birth Di Ei Pi,j

−

j


i=1

change

Pi,j

G = NjG + Nj+1

i=1

change

G Pi,j Pi,j

NjG

−


i=1

NG

G→P Pi,j

−

j


i=1

death Pi,j

(2)


i=1


i=1

2.2.1. Maturation As the developed model was to be verified by population experiments, the model was initially adjusted to the data obtained in laboratory conditions. Maturation can be determined by visual observation when the length of an animal is 0.7 mm. Therefore, the age of an animal at which it becomes mature (Ti ) is determined as:

(5)

NjP

G→P Pi,j −


i=1

death Pi,j

(3)

where W0.7 is the weight of an animal with body length 0.7 mm. Consequently the maturation of the i-th animal is defined as: maturity

NjP

M = NjM + Nj+1


i=1

⊃ (Wi,j−1 < W0.7 ) ∩ (Wi,j ≥ W0.7 )

change

P Pi,j Pi,j

NjG

+

Table 1 gives a brief description of all parameters used in the equations, their values and sources of values.

Ti = ti,j ,

NjJ

P = NjP + Nj+1

2.2. Growth and development of a model animal

(1)

NJ

j


parthenogenetic female, Di is the proportion of females in progeny, released by the i-th parthenogenetic female. The model allows the number of ephippial eggs to be tracked. However, at this stage of development of the model they are supposed to be in diapause indefinitely; that is why we excluded the equation of the dynamics of ephippial eggs from the description of the model. The proposed model applies to a laboratory culture with artificially controlled environmental conditions. The population dynamics was simulated for the following experimental conditions: temperature, 26 ◦ C; photoperiod, 16 h light, 8 h dark; volume of experimental flow-through system with non-growing food, 400 ml; flow-through rate, 1200 ml per day.

Pi,X

NjM

birth (1 − Di )Ei Pi,j −


i=1

death Pi,j

= 1,

⊃ X ≥ Ti ,

gi,X = 1

(6)

where X is the current time of population development. (4)

where NjJ , NjG , NjP , NjM are the number of juvenile, gamogenetic, parthenogenetic females or males at time j, Ei is the number of neonates released by the i-th

2.2.2. The change of reproduction mode of a mature female The concentration of algae contained in the culture medium and metabolic by-products of a population were selected as factors that determine the

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Table 1 Summary of parameters Parameter

Description

Ks Y µmax sh W

S1

Half-saturation food concentration for growth rate Efficiency of food consumption Maximum specific growth rate of an animal Model count step Weight difference between animals with body sizes 0.34 and 1 mm Time interval required for animal to reach body length of 1mm with optimal food requirement Half-saturation total food quantity for life span

S2

Half-saturation total food quantity for juvenile growth

td tmin Dmax

Difference between maximal and minimal animal age Minimal age when an animal can die Coefficient determining the maximal age of animals of opposite sex Age when females start to hatch males Moulting period Experimental chamber volume Sequence of natural numbers

T

τ DEX V N

Value (units) 0.00035 (mg/ml) 0.3 (adimensional) 0.019 (1/h) 0.2 (h) 0.01096 (mg) 48 (h)

Reference Gladyshev et al. (1993) Jorgensen et al. (1978) Gladyshev et al. (1993)

Zadereev (1998)

0.06 (mg) 0.15 (mg) 11 (days) 2 (days) 1 (female), 0.76 (male) 4 (days) 28 (h) 400 (ml) 1, 2, 3, . . . , N

Experimentally determined constant Experimentally determined constant Zadereev et al. (1998) Zadereev et al. (1998) Korpelainen (1989) Zadereev (1998) Zadereev et al. (1998)

reproduction mode. Previous experiments with individual females established the relationship between these two factors and the proportion of females switching from parthenogenesis to gamogenesis (PSj ) at time j (Zadereev and Gubanov, 1996):

The i-th juvenile female starts a gamogenetic reproG = 1) when conditions described in duction (Pi,X Eq. (8) are realised, or alternatively, the i-th juvenile female starts a parthenogenetic reproduction (Eq. (9)).

PSj = 0.72 + 0.0013 × CHj

2.2.3. Transition of an animal into the different reproductive class The reproduction mode can be determined by visual observation methods when the length of a female body is not less than 0.9 mm. A marker of gamogenetic reproduction is albescent bars in the brood pouch (Makrushin, 1971). For parthenogenetic reproduction, the marker is the absence of bars in the brood pouch and formation of parthenogenetic embryos. This reproduction mode is visually determined when the length of a female body is about 1 mm. Therefore, the variables that determine the given process are the G weight of an animal Wi,j and discrete variables Pi,j P . The i-th female transits into the different and Pi,j reproductive class when:

−

0.007 × Xjreal − 0.000015 × CHj × Xjreal V (7)

Xjreal

where is total quantity of algae in the culture volume (mg dry weight), CHj is the concentration of metabolic by-products (mg/ml) and V is the volume of the experimental chamber (ml). This relationship is used in the model as the function determining reproduction mode of the i-th mature female. Reproduction mode is determined at the moment of maturation Ti . maturity

G Pi,X = 1, ⊃ PSX > PR , Pi,X

= 1, gi,X = 1 (8)

P Pi,X = 1, ⊃ PSX ≤ PR ,

maturity

Pi,X

= 1, gi,X = 1 (9)

change

Pi,X

G = 1, ⊃ Wi,X ≥ W0.9 , Pi,X = 1, gi,X = 1 P ⊃ Wi,X ≥ W1 , Pi,X = 1, gi,X = 1

(10)

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where W0.9 and W1 are the weights of animals with body length 0.9 and 1 mm, respectively. 2.2.4. Death Previous experiments have shown that maximal life span (tmax ) of M. macrocopa in favourable conditions is equal to 11–13 days (Zadereev et al., 1998). The model assumes that tmax depends on food availability as: tmax = tmin +

td × Xjreal

(11)

S1 + Xjreal

where tmin is the minimal age at which an animal can die, td is the difference between an animal’s maximal and minimal ages, S1 is the experimentally selected constant—value of the total food quantity at which the life span is half of its maximum. Taking into account that according to Korpelainen (1989) the male life span is 76% of the female life span, the proportion of animals of the same age which will die till the age j (FjD , Fig. 2) is defined by: FjD =

  0,

⊃ ti,j < tmin

ti,j − tmin  , tmax × Dmax − tmin

⊃ ti,j ≥ tmin

(12)

where Dmax is the factor determining the maximal age of animals of different sexes. The i-th animal dies when: death = 1, Pi,X

D ⊃ Fi,X > PR ,

gi,X = 2, 3, 4

(13)

2.2.5. Realisation of a parthenogenetic clutch A unisex clutch is typical for M. macrocopa. A mixed clutch is an infrequent, but not unique, event. The probability of a male clutch has been found to increase with the life span of a female (Zadereev et al., 1998). Also it was noticed that the first clutch usually consists of females. To mimic these observations, it was assumed that starting at age τ the proportion of females in successive clutches decreases linearly. Thus, the proportion of females Di , in a released progeny, depends on age ti,j of an animal as follows:   ⊃ ti,j < τ  1, (14) Di = (ti,j − τ)  , ⊃ ti,j ≥ τ 1− tmax

where τ is the age at which a female begins to release males. After achieving a body length of 1 mm, a female is assumed to gain all weight Wi,j to form a clutch. A similar approach to calculate fecundity is not new in discrete models (Uchmanski, 1999). The average body length of a neonate is 0.34 mm. The number of neonates (Ei ) in a clutch is determined as:   Wi,j − W1 , ⊃ gi,j = 3 (15) Ei = W0.34 where W0.34 is the weight of an animal with body length 0.34 mm.    Wi,j − W1   ≥ 8, gi,j = 2 8, ⊃   W0.34 Ei = (16)    Wi,j − W1   < 8, gi,j = 2  0, ⊃ W0.34 Eq. (15) determines the number of neonates released by a parthenogenetic female. A gamogenetic female releases two ephippial eggs. The formation of two ephippial eggs is energetically equivalent to the formation of approximately eight parthenogenetic embryos (Lynch, 1983). Therefore, a gamogenetic female releases eggs if Wi,j ≥ W1 + 8 × W0.34 . The hatching of gamogenetic eggs is determined by Eq. (16). An animal’s weight was calculated according to the Lebedeva and Vorojun (1983) regression: W = 0.114L3.027

(17)

where W is an animal’s wet weight (mg), L is an animal’s body length (mm). After producing ephippial eggs, the gamogenetic female changes the way of reproduction—goes into the class of parthenogenetic females. This process is defined as: G→P Pi,X = 1,

birth ⊃ Pi,X = 1,

Ei = 8,

gi,X = 2 (18)

Hatching of either a parthenogenetic or a gamogenetic brood coincides with moulting. The period between successive moults (DEX ) is considered to be stable over a broad range of environmental conditions (Chmeleva, 1988). DEX for M. macrocopa ranges from

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1 to 1.5 days (Zadereev et al., 1998). Therefore, the i-th female hatches neonates when: ti,j birth Pi,X = 1, ⊃ ∈ N, gi,X = 2, 3 (19) DEX where N is the set of natural numbers. Depending on values of Di and Ei all hatched progeny go into classes of either males or juvenile females. 2.2.6. Growth An equation describing dynamics of an animal’s weight can be written as: Wi,j+1 =

birth Wi,j + dWi,j − Pi,j

× Ei × W0.34

(20)

Eq. (20) takes into account the gain of an animal’s weight at each calculation step dWi,j and the reduction of an animal’s weight due to hatching of progeny (for mature females only). Our experimental observations have shown that under favourable conditions it takes 2 days (48 h) for a female of M. macrocopa to grow from size 0.34 mm (size of hatched parthenogenetic progeny) to size 1 mm (the female begins to hatch progeny). The animal with body length more than 1 mm grows exponentially.  W × Xjreal    , ⊃ Wi,j < W1 dWi,j = (T/sh) × (S2 + Xjreal )    Wi,j × (esh×µi,j − 1), ⊃ Wi,j ≥ W1 (21) where W is the weight difference between animals with body sizes 0.34 and 1 mm, T is the time interval required for an animal to reach the body length of 1 mm with optimal food requirements, sh is the time interval between model steps, S2 is the experimentally selected constant—value for the total food quantity at which an animal grows to maturation in 4 days, µi,j is the specific growth rate. The specific growth rate of weight of the i-th animal at time j (µi,j ) is defined according to the classical Monod function: µi,j = µmax

Xjreal /V Ks + Xjreal /V

(22)

where Ks is the half-saturation constant, µmax is the maximal specific growth rate of an animal’s weight.

21

It is common to use the Monod function (Eq. (22)) to describe the specific growth rate of a population. However, this function can be used to describe the specific growth rate of a separate animal with the following assumptions: compared to other animals neither external conditions nor individual (genetically caused) features of animals give them any advantages in feeding and growth. In this case, all animals with body length more than 1 mm are similar in terms of food consumption and growth and this application of the Monod function is possible. 2.3. The description of culture medium The model simulates artificially controlled food supply and environmental conditions. The culture medium where the population develops is described by the quantity of food Xjreal (mg dry weight) (in our case, the source of food is unicellular green algae C. vulgaris) and the concentration of metabolic by-products CHj of a population (mg/ml) at time j. The food supply is modelled as a manipulated and controlled variable. The quantity of food Xjreal contained in the system at time j is defined as:  Xj , ⊃ Xj > 0 real Xj = (23) 0, ⊃ Xj ≤ 0 The quantity of food present in the system when all animals satisfy their food requirements (Xi ) depends on the quantity of food at the previous step of calculations, the quantity of supplied food, the quantity of the food consumed by the animals and the exchange rate of the medium (the “reduction” of the quantity of food due to the exchange of the medium is not incorporated into the presented equation as this is a technical procedure, which is not essential to the understanding of the model): NjJ ,NjG ,NjP ,NjM

Xj =

real Xj−1

+X − 0




Ii,j

(24)

i=1

where X0 is the quantity of algae supplied per time interval (mg dry weight). The quantity of food consumed by the i-th animal at time j (Ii,j ) is determined as: Ii,j =

dWi,j Y

(25)

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where Y is the efficiency of food consumption. The regression describing the effect of food concentration and metabolic by-products on the induction of gamogenesis (Eq. (7)) was derived in experiments with individual females of M. macrocopa (Zadereev and Gubanov, 1996). The major difficulty of scaling these data to the population level is related to the coefficient responsible for the concentration of metabolic by-products. The effect of metabolic by-products on the induction of gamogenesis in individual females was studied under the assumption that the concentration of metabolic by-products in the medium is proportional to the population density and the duration of the development of a population in the medium. However, only proportionality to the population density was tested in the experiments (Zadereev and Gubanov, 1996). The influence of time when a population was present in the culture medium on accumulation of metabolic by-products was not estimated. In addition, not only the rate and intensity of excretion of metabolic by-products, but also the rate of their destruction remain unknown. Thus, the effect of metabolic by-products on the induction of gamogenesis in a population may differ from the effect of crowded water as a simulator of a population. In the model, the concentration of metabolic by-products in the medium was described in two ways: (a) In conformity with experiments on individual females: the concentration of metabolic by-products of a population CHj at time j is proportional to the population density and is inversely proportional to the quantity of water passing through the cultivator. (b) Assuming that the metabolic by-products accumulate in the medium with proportionality coefficient PC (the “reduction” of the concentration of metabolic by-products due to the exchange of the medium was accounted for during calculations but is not described in the text). NjJ ,NjG ,NjP ,NjM

CHj =


i=1

Wi,j V

(26a)

NjJ ,NjG ,NjP ,NjM

CHj = CHj−1 + PC ×


i=1

Wi,j V

(26b)

3. Simulations of development of M. macrocopa populations (the model calibration) The development of a M. macrocopa population was modelled with five food concentrations in the culture medium (100, 200, 400, 800, 1600×103 cells/ml) by both methods of calculating concentration of metabolic by-products in the medium (Eqs. (26a) and (26b)). For each food concentration, we performed 30 model runs. The average of these model runs was compared with results of previously performed laboratory experiments (Zadereev and Gubanov, 1999; Zadereev, 1998). As not all these results were published in the international scientific literature, a brief description of the laboratory set-up will follow. A flow-through system was used in the population experiments. The volume of the flow-through chambers was 400 ml. The chambers were refreshed at a rate of 1200 ml per day with the new medium containing a tested concentration of C. vulgaris. Aged tapwater was used as the culture medium. The flow-through chambers were placed in a thermostat with temperature 26 ◦ C and photoperiod 16 h light and 8 h dark. The population experiments were conducted at five food concentrations—100, 200, 400, 800 and 1600 × 103 cells/ml. The population experiments started with a single parthenogenetic female with the eggs of her first or second brood. All individuals in the population were counted and measured daily. The size of the population, age and sex structures were determined on the basis of these measurements. One of the main purposes of the developed model was to test different hypotheses about the mechanisms of gamogenesis control and to investigate the effect of diapause on population dynamics. Consequently, the start of gamogenetic reproduction in a population was selected as the first calibration characteristic. The comparison of model runs with laboratory results demonstrated that, with low food concentration (100 × 103 cells/ml) there was a good correspondence between them when gamogenesis induction in the model was controlled by the food concentration (PC = 0) (Table 2). With food concentrations of 200, 400, 800 and 1600 × 103 cells/ml, a good correspondence between the model and laboratory results was achieved when we took into account the accumulation of metabolic by-products with proportionality coefficient PC ranging from 0.5 to 1. In this case, the time

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Table 2 Effect of different methods of describing metabolic by-product concentrations and proportionality coefficient on emergence of gamogenetic females Food concentration (103 cells/ml)

Method of describing the concentration of metabolic by-products

100

Eq. (26a) Eq. (26b)

Proportionality coefficient (PC)

Emergence of gamogenetic females (days)

1 0.6 0.5 0.1 0

4.5 3.5 4.0 3.8 4.6 5.5 6

± ± ± ± ± ±

1.0 0.7 0.9 1.1 1.2 1.4

1 0.6 0.5 0.1 0

6.8 4.7 5.3 5.4 6.8 6.8 6

± ± ± ± ± ±

0.2 0.6 0.1 0.4 0.2 0.2

1 0.6 0.5 0.1 0

6.5 5.1 5.2 5.5 6.5 7.0 5

± ± ± ± ± ±

0.1 0.1 0.3 0.4 0.1 0.8

1 0.6 0.5 0.1 0

8.0 5.1 5.4 5.7 7.1 8.0 5

± ± ± ± ± ±

0.3 0.1 0.5 0.4 0.7 0.3

1 0.6 0.5 0.1 0

7.7 5.1 5.4 5.8 7.5 8.1 6

± ± ± ± ± ±

0.2 0.1 0.3 0.4 0.5 0.6

Experimental 200

Eq. (26a) Eq. (26b)

Experimental 400

Eq. (26a) Eq. (26b)

Experimentala 800

Eq. (26a) Eq. (26b)

Experimentalb 1600

Eq. (26a) Eq. (26b)

Experimentala a b

Experimental data till day 6. Experimental data till day 5.

difference in the emergence of gamogenetic females between the model and the natural population does not exceed 12 h (Table 2). This two-factorial control of the reproduction mode can be easily explained on the basis of the theory of limiting factors. When the food concentration is low, it limits the development of the population and controls gamogenesis induction. With the increase in food concentration, this factor does not

limit the development of the population (at least before gamogenesis induction) and diapause is controlled by the infochemicals. Out of the five analysed variants of food supplies, only one limits the development of population. Thus, all subsequent calculations were performed taking into account the accumulation of metabolic by-products with proportionality coefficient PC = 0.5 (Eq. (26b)).

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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 five regimes of food supply is represented in Fig. 3. The results of modelling are in good agreement with the laboratory data. With food concentration 100×103 cells/ml the maximum population numbers (∼400 animals) and the time of peak population numbers (days 9–10) in the model and experiment coincide; with food concentrations 400, 800 and 1600 × 103 cells/ml, the model is close to the experimental population numbers till day 7 of the development (Fig. 3). A more detailed comparison of the modelling and experimental results is given in Fig. 4. With food

concentration 200 × 103 cells/ml the population stabilises on day 9. Until stabilisation juvenile animals constitute about 60–80% of the population. During the next several days, these juvenile cohorts get mature and a majority of them reproduce by gamogenesis. After this, the proportion of juvenile females in the population drops down to 50% and the proportion of gamogenetic females increases. At the end of experiment, the proportion of males in the population stabilises at about 10%. These key aspects of population development are reflected in the results of 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 The examples presented in the previous section show that the individual-based model of cyclical development of M. macrocopa developed on the basis of experiments with individual females can be used to obtain population simulations close to ex-

perimental data. However, the model was verified on the set of data received in standardised experimental set-up. Simulation by the model verified on a limited set of parameters may not adequately predict the development of a population with different regulating parameters. A verification experiment was performed to estimate the reliability of the developed model.

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A brief introduction to the methodology of verification experiment is needed. Change of the reproduction mode is controlled by the co-operative effect of food concentration and the concentration of metabolic by-products. These are density-dependent factors. Consequently, it is very difficult to separately measure the effect of each factor in population experiments. The following experimental set-up could separate the effect of food concentration and metabolic by-products on the induction of gamogenesis. The population is maintained in a cultivator supplied with two flows of medium. The first supplies the cultivator with food, the second dilutes the culture medium in order to decrease the concentration of metabolic by-products. The outflow goes through a filter that keeps algae in the cultivator. The rate of the second flow is varied to estimate the effect of different concentrations of metabolic by-products on the induction of gamogenesis with the same food concentration. However, such an experiment is difficult to realise. We can propose the following simplification of this scheme. The batch culture of a model organism develops in cultivators of different volumes. The cultivators are supplied with the same amount of food daily. Thus, in cultivators with different concentrations of metabolic by-products in the medium the same total quantity of food is achieved. It is clear that this experiment distinctly differs from the previously described discriminatory experiment. The key difference is that the experiment with batch culture will be performed under the effect of different food concentrations, as the food density will vary with cultivator volume. However, as it was discussed in Section 3 and demonstrated in Table 2, the effect of food concentration on gamogenesis induction prevails over the effect of metabolic by-products only when the food availability is low (under the conditions of food limitation). As Cladocera species are filter feeders, they can satisfy their food requirements by varying the filtering rate at different food concentrations. Thus, if the food concentration does not limit the development of the population, animals will be able to satisfy their food requirements within a relatively wide range of food concentrations. Taking into account this argument, the experiment with batch cultures should be performed with non-limiting food concentrations.

The experimental set-up satisfying the aforementioned requirements was selected by the model. Model simulations demonstrated that in cultivators with volumes 200 and 800 ml (total food quantity 32 × 107 cells per day is the same in both cases, food concentrations in cultivators are 1600 × 103 cells/ml and 400 × 103 cells/ml, respectively), the reproduction mode in the 200 ml cultivator (day 4.7) changed 1.5 days earlier than in the 800 ml cultivator (day 6.2). After this step, we conducted the laboratory experiment with the same conditions. Experimental populations started with five juvenile females (body length 0.3–0.5 mm). The experiment was performed in three replicates. The experiment demonstrated that in the vessel with volume 200 ml gamogenetic females were found at day 4 which is 2 days earlier than in the vessel with volume 800 ml (day 6). It should be mentioned that both the time differences in the induction of gamogenesis between different volumes and the time of gamogenesis induction in both vessels obtained in the experiment are very close to the model prognosis. Results of these experiments suggest two important conclusions. First, the time difference found in the induction of gamogenesis in different volumes with the same total food quantity lends support to the following hypotheses: (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. Second, a good agreement between the experimental data and the model prognosis indicates that the mathematical model was constructed with adequate conceptions about cyclical development of M. macrocopa. It is worth mentioning that the experimental design suitable for the detection of time difference in gamogenesis induction was selected by model calculations. Comparison of model simulations with experimental data and successful verification experiment provide grounds for the statement that the mathematical model developed reflects the main features of M. macrocopa cyclic development. However, as mentioned in the review on individual-based modelling: “Blind faith in what a model predicts is not the purpose of the modelling” (Grimm, 1999). The mathematical model of M. macrocopa cyclic development can serve to analyse theoretically several aspects of population dynamics of this species.

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5. The role of the change of reproduction mode in population dynamics of M. macrocopa: theoretical analysis First, we analyse the effect of reproductive switching on the development of population. Model simulation of the population development of M. macrocopa with food concentration in the medium 400 × 103 cells/ml is shown in Fig. 5. Three versions of population development are shown: (A) the sole way of reproduction is parthenogenesis, (B) the change of the reproduction mode is determined by food availability, and (C) the change of reproduction mode is determined by the co-operative effect of food availability and metabolic by-products. During the first days of the development of the population “without gamogenesis” the proportion of juvenile females (Fig. 5A) is more than 90%. After the first peak (1465 animals), the population declines and stabilises at 1092 animals on days 13–14 of the development. Size and age structures of the population also stabilise after the population reaches its plateau. The proportion of juvenile females in the population is about 70%. The proportion of males fluctuates within 20–28%. The proportion of parthenogenetic females periodically fluctuates from 1 to 11%. When the change of the reproduction mode is regulated by food availability (Fig. 5B), the first population peak is the same as in the previous case both in numbers (1465 animals) and time (days 8–9). However, after day 14, an average population, when it reaches the plateau, is less and equals 1004 animals. The size and age structures are also different. The proportion of juvenile females declines and fluctuates near 50%. The proportion of males in the population is stable—about 20%. The proportion of parthenogenetic females fluctuates periodically from 5 to 10%. The gamogenetic females emerge in the population on day 7. The proportion of gamogenetic females reaches the maximum in day 13 (50% of total population), then declines and stabilises at 30%. When the change of the reproduction mode is regulated by the combined effect of food availability and metabolic by-products (Fig. 5C), the population dynamics and the size–age structures (after the population is stabilised) differ from the previous runs. The maximum population (1357 animals) and the average population after day 14 (943 animals) are the least

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among all three tested variants. The proportion of juvenile females after a population has stabilised is less than in previous variants and fluctuates periodically within the range of 20–40%. The proportion of males is stable and equals 30% of the population. The proportion of parthenogenetic females fluctuates from 0 to 20%. The first gamogenetic females emerge on day 5. The proportion of gamogenetic females reaches the maximum on day 11 (77%), decreases and performs regular fluctuations from 20 to 60% in antiphase with the proportion of juvenile females. Thus, because of the introduction of reproductive switching and subsequent complication of the mechanism of regulation of this processes, the population declines, changes its size and age structures (decline in the proportion of juvenile females, increase in the proportions of males and gamogenetic females). Diapause allows the population to survive in the adverse environment. The first step of cyclic reproduction is the induction of diapause that is determined by the environmental factors. When favourable environmental conditions are established, the diapausing organisms are reactivated and the population cycle repeats again. At the reactivation stage, a population with the maximum number of diapausing organisms has an advantage in competition. It means that the effect of regulating reproductive switching factors should synchronise the development of a population with the change of environmental conditions in order to ensure the production of the maximum number of diapausing eggs. In this case, combinations of regulating factors that maximise the production of diapausing eggs are selected and stabilised in the process of evolution. Similar considerations have been developed for some social insects using Pontryagin’s maximum principle (Oster and Rocklin, 1979). The developed model of cyclic reproduction can be used to examine this hypothesis in Cladocera at least for two factors regulating gamogenesis—food concentration and metabolic by-products. The dynamics of accumulation of diapausing eggs during the development of M. macrocopa populations with different regulation mechanisms of gamogenesis induction is shown in Fig. 6. When the change of reproduction mode is controlled by the sole effect of food concentration, the number of diapausing eggs is minimal for all tested food concentrations. The number of diapausing eggs increases with the coefficient responsible for

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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.

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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 number of diapausing eggs stabilises when the coefficient reaches 1. Mathematically, the coefficient might have any value. However, the increase of coefficient (PC = 2) that follows does not increase the production of the diapausing eggs with any tested food concentration (Fig. 6). With the same quantity of energy available for the development of population, the number of produced ephippial eggs will depend on the population structure. Depending on factors controlling the development of a population there is a wide range of possible population structures (e.g. Fig. 5A–C). However, in the ideal case, within the whole range of all possible population structures there is only one that ensures the production of the maximal number of ephippial eggs. That is why the increase in PC does not lead to the increase in the number of diapausing eggs. It should be clear that optimisation was tested for the effect of only two factors and within the limited set of environmental conditions (such physical factors as photoperiod and temperature were fixed at the level favourable for parthenogenesis). However, the main purpose of the analysis was to evaluate the feasibility of the selected mechanism of the two-factorial control of gamogenesis induction. It is noteworthy that the maximal number of diapausing eggs is produced when the coefficient of accumulation of metabolic

by-products is more than 0.5 and close to 1. The same range was selected for this coefficient on the basis of different principles during the verification of the model. Thus, now it is possible, within the above-mentioned limits, to give a biological justification for the regression with the selected coefficient: the derived regression equation with selected coefficients describes the effect of selected factors and ensures the formation of the maximum number of diapausing eggs.

6. Conclusion We have tried to apply the integrated approach to the individual-based modelling of such a natural phenomenon as cyclic reproduction of Cladocera. First, the individual-based model, was developed on the basis of the experiments with individual animals. In addition to our data and conceptions, various literature data and approaches to the Cladocera modelling were synthesised in the model. In the presented version of the model, the main emphasis was laid on the effect of density-dependent factors on gamogenesis induction in Cladocera. This choice was stimulated by the fact that density-dependent factors and especially infochemicals traditionally received less attention in

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Cladocera modelling than physical factors, such as photoperiod and temperature. Second, the model was calibrated on the set of preliminarily received, but independent population data. As we focused our attention on the gamogenesis induction and the role of the combined effect of food concentration and metabolic by-products, we did not try to vary all possible model parameters until the model runs would ideally simulate experimental data. On the contrary, we tried to introduce into the model constants experimentally determined as precisely as possible, in order to manipulate with variables responsible for the control of gamogenesis induction. Third, the verification experiment was planned on the basis of model runs and later the same experiment was conducted to verify the model. The realisation of the second and third steps allow a conclusion that during population growth, metabolic by-products build up in the medium and the effect of metabolic by-products on gamogenesis induction is concentration dependent. These conclusions are not self-evident. The nature of chemical substances involved in the control of population dynamics and the functional relationship between the effect of chemical substance and particular physiological response might differ significantly. A more detailed discussion on the effect and possible nature of chemical substances in aquatic ecosystems can be found in Larsson and Dodson (1993) and Zadereev (2002). The last step was to test some theoretical assumptions regarding the control of gamogenesis induction in order to have a deeper insight into the population cycle of Cladocera. What we consider to be most important at the current stage of research is that we have tried to perform an integrated experimental and theoretical investigation of the effect of density factors on cyclic reproduction of Cladocera. It was emphasised before (De Roos et al., 1997) that density dependence is very important for the understanding of population dynamics. In order to contribute to Cladocera population modelling we performed two important steps: (1) introduced gamogenesis induction into population dynamics, and (2) realised density-dependent control of this process. However, this is not the final stage of the research. If we consider the understanding of mechanisms responsible for the control of population dynamics as the ultimate goal of population modelling, at the next step of the model development it will be necessary to combine environmental and

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