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Individual-Based Model of the Reproduction Cycle of Moina Macrocopa (Crustacea: Cladocera) Egor S

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Individual-Based Model of the Reproduction Cycle of Moina Macrocopa (Crustacea: Cladocera) Egor S 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.
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