temporally-varying habitats. Manuel Serra & Maria Jose Carmona. Area de Ecologia, Universitat de Valencia, E46100-Burjassot (Valencia), Spain. Key words: ...
Hydrobiologia 255/256 : 117-126, 1993 . J. J. Gilbert, E . Lubzens & M. R . Miracle (eds), Rotifer Symposium VI. © 1993 Kluwer Academic Publishers. Printed in Belgium .
117
Mixis strategies and resting eeg production of rotifers living in temporally-varying habitats Manuel Serra & Maria Jose Carmona Area de Ecologia, Universitat de Valencia, E46100-Burjassot (Valencia), Spain
rotifers, cyclic parthenogenesis, mixis induction, sex, population growth, resting eggs, dynamic model, simulation, temporally-varying habitats Key words :
Abstract A dynamic model based on six differential equations has been developed to explore the control of mixis of rotifers living in temporally-varying habitats . The equations give variation rates of amictic females, three stages of mictic females, males and resting eggs . The model takes into account some constraints on mixis (e .g ., male-female encounter probability and effort involved in resting egg production) and its predictions have been generated by computer simulation using parameter values from the literature . For simulation, a time-dependent birth rate function was assumed to account for changes in the environment, and several mixis patterns (i.e ., moment of mixis induction and mictic rate if induced) were presumed, followed by computing resting egg production . The results were very insensitive to changes in those model parameters associated with mixis ; however, several assumptions could quantitatively affect the conclusions . When compared with the results reported in the literature, results from our simulations suggest that the optimal mixis strategy could be very dependent on the ecological features of the habitat . Moreover, in temporal habitats high mictic rates should be expected when the mixis is induced, and the optimal moment of mixis induction would be few days before the mortality rate overcompensates the birth rate of amictic females . Furthermore, the optimal strategy could be affected by informational constraints, and some degree of variability in the moment of mixis induction, whether or not genetically determined, could be maintained in the population .
Introduction The initiation of sexual reproduction in cyclic parthenogenetic rotifers - mixis - is a process that has been related to a wide range of external factors (for review, see Gilbert, 1977, 1980 ; Pourriot & Clement, 1981 ; Pourriot & Snell, 1983 ; see also Hino & Hirano, 1984; Lubzens et al., 1980, 1985 ; Snell, 1986; Lubzens & Minkoff, 1988) . Since rotifers frequently live in temporal or ephemeral, isolated habitats, some empirical studies have associated sexual reproduction with
environmental changes (e .g ., Ruttner-Kolisko, 1964 ; Halbach & Halbach-Keup, 1972 ; Lubzens et al., 1980 ; Hino & Hirano, 1984) . Mixis is interpreted as a response to deterioration in the habitat (Thane, 1974 ; for discussion see RuttnerKolisko, 1974) since the sexual resting egg is the stage that can disperse in space and time . In contrast, the `balance hypothesis' (Williams & Mitton, 1973 ; Williams, 1975), a more genetic approach, addresses the timing of mixis as a result of the balance between the increase in genetic variability promoted by recombination and the
118 cost of sex . However, the two approaches are not mutually exclusive because the timing of mixis predicted by the `balance hypothesis' could be modified in temporal habitats when correlation between parental and offspring environments is low and sexual reproduction is more advantageous (Williams, 1975) . Besides these approaches, which are focussed on the adaptive significance of sex in rotifers, it has been noted that mixis requires resources, and therefore should be induced in good environmental conditions (Gilbert, 1980 ; Pourriot & Snell, 1983 ; Snell, 1986 ; Snell & Boyer, 1988) . Other constraints on resting egg production arise from required densities of sexual females (e .g ., Snell, 1987) and male-female encounter probabilities (Birky & Gilbert, 1971 ; Gilbert, 1980 ; Snell & Garman, 1986) . We have developed a model that accounts for these constraints and describes the dynamics of a rotifer population inhabiting temporally-varying habitats . The ecological context of our model stresses the adaptive significance of dispersal (Johnson & Gaines, 1990), as well as relaxes the relevance of the `balance hypothesis' to the study of the timing of mixis . Our aim was to address the optimal rate and timing of mixis . We also studied the sensitivity of our results to the values of model parameters and asked if the optimal strategy for mixis was evolutionarily stable .
Model Two types of females can be distinguished in monogonont rotifers: amictic and mictic (sexual) . The former produces eggs mitotically and the latter meiotically (Birky & Gilbert, 1971 ; Gilbert, 1977 ; King, 1977). Mitotic eggs develop into females, but the fate of meiotic eggs depends on other factors . The fertilization of a mictic female by a male is only possible during her early life, when her tegument is thin . If a mictic female becomes fertilized, her meiotic eggs develop into resting eggs, but if she remains unfertilized, her eggs develop into haploid males . The resting eggs have a minimum period of required dormancy
and develop into amictic females (Gilbert, 1974 ; Pourriot & Snell, 1983) . Therefore, at least six state variables should be considered to describe the dynamics of a rotifer population : number of amictic females (FQ), fertilizable mictic females (Ff), male producing mictic females (F,,,), resting egg producing mictic females (F,), males (M), and resting eggs (E). If resting eggs remain dormant during a population growth cycle, all recruitment is direct from amictic to amictic female through mitotic eggs, so the current population growth is supported by this type of female . In contrast, survival through unfavorable periods depends on resting egg production because it is the primary dispersal stage of the life cycle . Population density is a critical factor for resting egg production since male-female encounter is improbable at low densities and because more mictic females can be produced at high densities . However, the substitution of mictic for amictic females depresses the current population growth rate and thus affects population density . Consequently, a complex compromise exists between population growth and resting egg production (Snell, 1987) . Moreover, other factors must be considered to understand the constraints affecting mixis ; for instance, sexual reproduction demands resources and suitable environmental conditions to allow the survival of mictic females and males, as well as the allocation of resources into resting eggs . Our model accounts for these compromises and constraints (Table 1) and identifies the strategy among several alternative ones that produces the highest number of resting eggs . The model focusses on the population dynamics of rotifers living in temporal, time-variable habitats . It assumes that the potential, instantaneous birth rate [b(t)] is time-dependent and accounts for the variable environmental factors affecting reproduction . Therefore, b(t) appears in the equations describing the growth of amictic females, unfertilized (i .e ., new-born) mictic females, males, and resting eggs . The mictic rate (m(t), defined as the proportion of mitotic eggs that develop into mictic females) is also considered to be time-dependent .
1 19 Table 1 . Differential equation model for the dynamics of a rotifer population . Values of constant parameters used in the simulations are also indicated . Variable
Growth rate
Amictic females, Fa Fertilizable mictic females, Ff Male-producing mictic females, F,,, Resting egg producing mictic females, F, Males, M Resting eggs, E
dFa (t)/dt = b(t)[ 1 - m(t)]F,(t) - qF,(t) dFf(t)/dt = b(t)m(t)Fa (t) - gFf (t) - sFf(t) - eF(t)M(t) dF,,,(t)/dt = sFf (t) - qF,(t) dF,(t)/dt = eFf (t)M(t) - qF/t) dM(t)/dt = pb(t)F, ,(t) - W(t) dE(t)/dt = gb(t)F,(t)
Parameter b(t) m(t) • s • • p •
potential, instantaneous birth rate for amictic females at time t mictic rate (proportion of mitotic eggs that develop into mictic females) at time t instantaneous mortality rate for females instantaneous rate at which mictic females become unfertilizable instantaneous male-female encounter rate b instantaneous mortality rate for males ratio between male and female production rates ratio between resting egg and female production rates
Value
Source a
0 .4 d - ' 8 d-' 0 .004 Id "' 0 .7 d - ' 1 .5 0 .2
Serra, 1987 Snell & Childress, 1987 Snell & Garman, 1986 Snell & Garman, 1986 Serra, 1987 Snell & Garman, 1987 ; Serra, 1987
a The indicated sources have been used to obtain data or criteria to derive the parameter values, but assumptions not contained
in the sources have been added . • Note that, due to the units of e, the results of simulations are in individuals or eggs per liter .
Consequently, b(t)[ 1-m(t)] and b(t)m(t) are respectively the recruitment rates of amictic and mictic females from the amictic females . Amictic females disappear due to death (described by q) . Fertilizable mictic females disappear due to one of three processes : death (described by q), somatic growth (involving the pass to unfertilizable females and described by s), and fertilization (described by e and proportional to male density) . The model supposes that fertilization involves an outcome from this type of female because further fertilizations are considered improbable or irrelevant to resting egg production . The equations proposed for other variables (Table 1) are a straightforward consequence of the relationships described above . The same instantaneous mortality rate (q) is assumed for every type of female . Another parameter (u) is proposed for the male loss ; this parameter accounts for both death and passage to the post-reproductive stage . Finally, our model presumes that both instantaneous male production rate and instantaneous rate of resting egg production are pro-
portional to b(t), the ratios being p and g respectively . Table 1 shows the values of the constant parameters used in the simulations . These values were estimated from the available information for Brachionus plicatilis, although frequently additional assumptions were necessary . For instance, an estimation of q from a laboratory population was 0 .22 d - t (from Serra, 1987), but this value gave unrealistic densities in simulations ; thus a higher value, more realistic for field populations where ecological risks would be greater, was used . The value of u was calculated assuming a male reproductive life of two days (see Snell & Garman, 1986) ; thus, a set of males with symmetrical age-distribution (ranging from 0 to 2 d) and without recruitment would become half-reduced in one day, and consequently, from (M/ 2)=M. e - "(` - '), u= 0.7 d - '. The same argument was used for estimation of s, assuming that the mictic female is in the fertilizable stage for 4 h after her birth (Snell & Childress, 1987) . The male-female encounter rate, e, was derived from
120 Snell & Garman (1986), assuming that the encounter frequency observed by these authors (1 .49 encounters/min for one male and 500 females per ml) could be linearly extrapolated to one day and one female per liter . The ratio between 'femaleto-male' and 'female-to-female' generation times was used for estimation of p (Serra, 1987) . A value for g was roughly calculated assuming that the rate of resting egg production is 1/5 of that of mitotic egg production . This is consistent with values usually reported for total egg production per lifetime (e .g., 2 or 3 resting eggs vs . 10-15 mitotic eggs ; see, for instance, Snell & Garman, 1986 ; Serra, 1987). The model lacks explicit functional form, therefore sensitivity analyses were carried out to evaluate the effect of changes in the parameters on our main results (Gladstein et al ., 1991). Two types of functions were used for b(t) : sinusoid-like functions and linearly declining functions, both constrained to be non-negative . The first type (Fig . 1) was used to study the complete growth phase of a population starting from one amictic female per liter, b(t = 0) - q being positive, near to 0 . This function was assumed as a simple way to describe the annual cycle phase with available conditions for a particular rotifer strain, presuming that only one b(t) maximum occurs during this phase . We assumed that available conditions from b(t) - q > 0 to b(t) = 0, occur for 30 days ; a value that is consistent with the re-
W_jRWL___ 0
5
10
15
b(t)-q
20 25 day
30 35
40
+ amictic females
Fig. 1 . Simulated dynamics of a rotifer population without
mixis, beginning with one female per liter, and with instantaneous birth rate following the indicated sinusoid-like function (maximum b(t) = 0 .95) .
ported data for some populations (e .g ., Carlin, 1943 ; Miracle et al., 1988) or clones of the same species (King, 1972 ; King, 1977 ; King & Zhao, 1987). The second type of function was used to study the terminal phase of the population dynamics regardless of the previous events . In both types the values of b(t) were maintained constant within a day to average the effect of daily oscillations . Moreover, the maximum value of this parameter was constrained to observe b(t) - q = < 1 d -1 , ca . the maximum intrinsic growth rate reported in rotifers . The model accounts for changes in mictic rate supposing that m(t) can switch once from 0, t < D, to m, t > = D, where D is the first day of mixis (i.e ., constant mictic rate involves D = 0) . We simulated 41 days of population dynamics because of the time assumed for available conditions (see above) . Simulations were computed equaling the differential equations to difference equations (see e.g ., Jeffers, 1978) with short time increments (0 .042 d, i.e ., 1 h). Differential equations are more consistent with the continuous nature of the described processes and enable to have directly estimations of parameters from literature . Equaling these equations to differences equations we assumed an approximation error that becomes biologically negligible if sufficiently short time increments are used . Some simplifications assumed in the model are : (1) in general, age-structure of the population, as well as other physiological differences, are disregarded ; (2) density effects on b(t) are not explicitly considered; (3) time-lags are disregarded ; (4) mictic response is synchronic and irreversible ; (5) male fertility is not affected by previous copulations and every encounter is fertile ; (6) neither environmental nor intrinsic stochasticity is considered ; (7) several parameters are not timedependent ; (8) mortality rates do not depend on female type ; (9) all the resting eggs are considered equivalent regardless of the time of their production ; thus, differences in genetic variability, viability and hatching moment are neglected. Moreover, except for stability analysis, our simulations do not consider genetic variability in mixis strat-
1 21 egy. Other simplications arise from the functions assumed for b(t) .
Results A first set of simulations was carried out with b(t) following sinusoid-like functions . In the first run, the maximum b(t) was equal to 1 . Other runs were conducted by scaling the b(t) values with various factors . The resultant functions have different maximum b(t) values, causing different average values of b(t), and this sometimes results in b(t) - q becoming negative at different times . Figure 1 shows a b(t) function with maximum equal to 0 .95 and the dynamics of a rotifer population growing without mixis at these instantaneous birth rates . As expected from our assumptions, maximum female density (1864 females/1) was achieved when b(t) - q became negative (21S t day), not when the environment was most favorable (maximum b(t), 10th day) . Figure 2 shows the relationship between mictic rate and resting egg production when mictic rate was constrained to be constant . One of the simulations (b(t) with a maximum equal to 1 .2) gave maximum Fa densities improbably high, but despite this, a pattern arises : the optimal mictic rate that maximizes resting egg production, lower than
10000
10% in all these cases, decreased as the average b(t) increased.
Results obtained when the mictic rate was allowed to switch once from 0 to a different value are in Fig . 3 (maximum b(t) = 0.95). In this case the optimal mictic rate for resting egg production was 1 . Moreover, mixis should be induced at the start of day 18, nine days after the maximum b(t) and three days before b(t) - q became negative . Mixis caused a population decline, and thus the maximum record of total females (1301 females/1) was achieved on day 19 . Figure 3 shows also that the lower the mictic rate, the earlier the optimal day of mixis induction . We also explored the effect of different average values of b(t) by using b(t) functions with maxima equal to 0 .9, 1, 1 .1, and 1 .2 . The results were very similar to those found when maximum b(t) was 0.95 . Optimal mictic rate was 1 in each case, and the optimal day for mixis induction was day 17 (maximum b(t) = 0 .9), day 18 (maximum b(t) = 1 and 1 .1), or day 19 (maximum b(t) = 1 .2) . As the
resting eggs/I
resting eggs/1
1000 100 10 1 0.1 0
0 .05
Maximum b(t): - 1.2
0.1 mictic rate - 1 .1
0.15
-1.0
0 .2 -0.95
Fig . 2 . Effect of several constant mictic rates on total resting
egg production for populations that growth with sinusoid-like b(t) functions achiving the indicated maxima . The values on each curve indicate the optimal mictic rate that was recorded and, in parentheses, the maximum Fa (1 - ') for this mictic rate . .(0) = 1 1 - ' ) . (F
0 0 2 4 6 8
10 12 14 16 18 20 22 day
Fig. 3 . Effect of the day of mixis induction and the mictic rate
on total resting egg production . (b(t) function : sinusoid-like with maximum b(t) = 0.95 ; FQ (0) = 1 1 - ' ).
122 average b(t) increased, the day when b(t) - q became negative shifted as follows : maximum b(t) = 0 .9, 21s t day ; maximum b(t) = 1 or 1 . 1, 22 nd day ; maximum b(t) = 1 .2, 23r d day . Results from these simulations suggested that the optimal timing of mixis induction is mainly related to the moment when b(t) - q becomes negative . To explore this phase in detail, we performed a set of simulations using linearly declining b(t) with different slopes (Fig . 4a) ; b(t) - q became negative at the start of day 7 in all cases . We assumed several arbitrary, but realistic, initial numbers of amictic females, thus disregarding how the initial number of females was achieved . The optimal day of mixis induction (Fig . 4b, c) was day 3 regardless of the initial number of females and the b(t) slope . As the beginning of the simulation was arbitrary, our result is more correctly expressed by saying that mixis should be induced four days before b(t) - q became negative to maximize resting egg production . Note also in Fig . 4c that when mixis occurs later than optimum, resting egg production declined more sharply for b(t) with high negative slope than with intermediate negative slope . To evaluate the sensitivity of our results to the values assigned to the constant parameters, we determined the optimal time of mixis using the combinations of the following values : s = 2, 16 ; e = 0 .00 1, 0 .08 ; Fa(0) = 100, 1000, 2400 . We explored the terminal growth phase using the two extreme slopes of b(t) (see Fig . 4a) . Some of these values could be considered unrealistic, but they are useful for evaluating the robustness of our results . Each simulation gave the same optimal day of mixis induction (day 3) . As expected, resting egg production increased with the male-female encounter rate (e) and with the initial number of amictic females . The most general recorded pattern was the increase of resting egg production with the decrease of s (i .e ., when the fertilizable period of mictic females is extended) . The exceptions to this pattem were observed for the following set of conditions : (1) e = 0 .08, Fa(0) = 1000, 2400, b(t) with high negative slope ; and (2) e = 0 .08, FQ(0) = 2400, b(t) with low negative slope . We also found that a mictic rate equal to
1 2 b(t) A
a
1 a 0
0.8
a 0
0
0.8
0 0
A
a A
A
0.4
a e
0.2
s
a
0
0
o
0
0 0
1 2 3 4 5 6 7 8 910111213141518 day
b 2400 (5526) 1800 (3685) 1200 (2763) 1000(2303)
0
1
2
10000 resting eggs/I
C 1000 100
2400 (7121)
10
1000 (2987) n 2400 (15048)
1
1000 (6270)
0 .1
100 (297)
0 .01
100 (627)
0 .001 0
1
2
3
4
6 6 day
7
8
9
10
Fig. 4 . Study of the last phase of growth (b(t) functions as indicated in upper pannel, a) : b and c, effect of the day of mixis induction on total resting egg production . Symbols in the two lower pannels indicate the slope used in the corresponding simulation . The values on the right side of each curve are the initial number of amictic females per liter, and, in parentheses, the maximum Fa recorded for the optimal strategy of mixis induction . (Mictic rate if induced : 1 .)
0 .95 (s = 2,16 ; e = 0 .001, 0 .08 ; Fa (0) = 1000 ; b(t) with low negative slope) produced fewer resting eggs than a mictic rate equal to 1 .
The simulations previously described assumed a population without variability in mixis strategy . Consequently, the resulting optimal strategies are restricted to a comparison among several populations, each one homogeneous in its mixis timing . To determine whether a population with such optimal timing of mixis could be invaded by an alternative strategy, we performed a set of simulations assuming that two strategies were present in the initial population . Therefore, by comparing initial allele frequency (in amictic females) with the allele frequency in resting eggs, the initial selection for the alternative strategy can be evaluated . For simplification, we assumed that the alternative strategy was determined by a dominant allele carried by heterozygous amictic females . These simulations were carried out using b(t) with low negative slope (Fig . 4a) and Fa (0) = 1000 (see Table 1 for other parameters ; m(t) switching once for each phenotype from 0 to 1) . Our results (Fig. 5a) show that an alternative strategy with mixis earlier than the optimal moment for homogeneous populations (day 2 vs . day 3) could be initially selected, although the selection would be against the alternative strategy when its initial allelic frequency was higher than aprox . 0 .35 (i.e ., 70% of amictic females with the alternive phenotype) . Moreover, although some mixed populations enhanced resting egg production, the selection is not entirely dependent on this improvement, since when the frequency of the alternative allele was 0 .35, it would be selected, but the resting egg production would be lower than in a homogeneous population with the optimal strategy (day of mixis induction : 3 rd ) In contrast (Fig . 5b), an alternative strategy with delayed mixis (day of mixis induction : 4t'') would not be selected, although an improvement in resting egg production would be achieved in a mixed population of `optimal' and `delayed' strategists . We also found that the optimal strategy for homogeneous populations, if determined by a dominant allele carried by heterozygous females, would invade both populations with delayed (day 4) and early mixis (day 2) . Finally, our simulation showed that a population inducing mixis
1 23
b
-0.00 0
0.1
0.2 0.3 initial allelic frequency
'-"'- relative increment
0.4
70 0.6
-- resting eggs
Fig. 5 . Relative increment (from heterozygotic amictic females to resting eggs) of a dominant allele affecting the day of mixis induction, the rest of females developing the optimal strategy for a homogeneous population (i .e ., day of mixis induction : day 3) ; a: allele determining that mixis induction occurs at day 2 ; b : allele determining that mixis induction occurs at day 4 . Total resting egg production in the mixed populations and in the optimal homogeneus population (on the right in parentheses) are also indicated .
on day 2 could be invaded by a strategist inducing mixis on day 1, assuming that the latter phenotype was determined by a dominant allele carried by heterozygotes .
Discussion Simulation results showed that when mictic rate was constrained to be constant, resting egg production was maximized by low proportions of sexual daughters (< 10%) . This result is similar to that found by Snell (1987) when either densityindependent or density-dependent growth in a
124 constant environment was assumed . Snell (1987) also reported that the optimal proportion of sexual daughters increased with r and decreased with the carrying capacity, while we found that an increase in the average birth rate caused a decrease in optimal proportion of sexual daughters. However, this divergence may simply reflect differences in assumptions since in our model higher birth rates involve both higher r values and larger maximal population densities, and the latter could strongly affect mixis dynamics . In contrast, when mixis was allowed to switch once from zero to a different value, optimal mictic rate was 1 . Similar values have not been reported in laboratory studies, these ranging from 0 to 50 in B . plicatilis (Hino & Hirano, 1977 ; Pourriot & Rougier, 1979) . Laboratory evaluations of mixis may underestimate mictic rates in natural environments, since laboratory conditions could produce selection for low mictic rate (Hino & Hirano, 1976 ; Ruttner-Kolisko, 1985 ; Buchner, 1987) . Higher proportions of sexual daughters are strongly suggested by observations in field samples (Amrem, 1964 ; Pourriot, 1965), as well as by the population dynamics observed in enclosures (Miracle & Guiset, 1977) . However, optimal mictic rate equal to 1 must be considered as a result dependent on the temporal variation assumed for birth rate, because if potential r (i .e ., r in absence of mixis) does not fall below zero, the loss of population would not be adaptive . Of course, natural habitats are frequently typified by temporal heterogeneity and studies of the genetic structure of rotifers suggest that many strains are adapted to a narrow set of environmental conditions (King, 1977; Snell, 1979 ; Serra & Miracle, 1985 ; King & Zhao, 1987 ; Carmona et al., 1989) . In this ecological context high mictic rates should be expected . Our model probably overestimates the optimal mictic rate when mixis is induced, since our results could be affected by the shape of the birth rate function . A birth rate with several alternative declining and elevating phases would probably produce an optimal mictic rate below 1 . It seems likely that such a reduction would be adaptive, since parthenogenetic daughters could quickly
contribute to population growth when birth rates rise . Moreover, Fig. 3 shows that resting egg production does not sharply fall when mictic rate slightly departs from 1 . Note also that birth rate functions with several maxima should be expected if birth rate is affected by environmental stochasticity . Frequently a 'bet-hedging' strategy (here, the production of both sexual and asexual daughters) may be optimal in unpredictable habitats (see, e.g ., Begon et al., 1990) . Our results stress that the timing of mixis that maximizes resting egg production is mainly related to the moment at which potential growth rate becomes negative - i .e ., when the habitat becomes unsuitable . However, mixis should occur before that moment because resources and suitable conditions are needed for bisexual reproduction (Snell, 1986 ; Snell & Boyer, 1988) . Our results are consistent with the frequentlyreported association between mixis and high population densities (Wesenberg-Lund, 1930 ; Carlin, 1943 ; Nauwerck, 1963 ; Amren, 1964 ; King & Snell, 1980) . Mixis causes a population decrease and, in our simulations, optimal mixis induction can be related to the potential maximum of population density that would be achieved, if mixis had not been induced, when growth rate became negative (see Fig . 1). Invasibility analysis suggests that a genetic polymorphism for the moment of mixis could be maintained in the population, this polymorphism causing an earlier average mixis than in an optimal homogeneous population . Moreover, this analysis also suggests that resting egg production would increase if mixis does not steeply switch from 0 to 1, but changes more gradually, either for each individual or for the population as a whole . Observe in Fig . 5 that the highest resting egg production was achieved when two different phenotypes at the moment of mixis induction were present in the population, no matter if phenotypic differences were determined by the genotype, plasticity or environmental heterogeneity . These considerations, along with the lack of lag-times in our model, also suggest that the optimal time of mixis induction could be slightly earlier than the one resulting in simulations .
125 Finally, an interesting result from our simulations is that the optimal moment of mixis induction is dependent on a future condition ; i.e ., negative potential growth rate. Hence, that condition, its consequences, or its temporally proximate correlates cannot serve as a cue for mixis induction . That is, the cue should be correlated with, but earlier than, the negative condition . It seems unlikely that this habitat-dependent correlation will be equal to one, hence, the evolution of an optimal strategy may be seriously affected by this type of `informational' constraint .
Acknowledgements We thank Dr Charles King and Dr Terry W. Snell for reading and improving our manuscript . We also thank Miss Veerle Van Meerhaeghe and Miss Veerle Peeters for their language advice in preparing the manuscript .
References Amrem, H ., 1964. Ecological and taxonomical studies on zooplankton from Spitsbergen . Zool . Bidr . Upps . 36 : 209276 . Begon, M ., J . L. Harper & C . R . Townsend, 1990 . Ecology: individuals, populations and communities . Blackwell Scientific Publications, Oxford, 945 pp . Birky, C . W . & J . J . Gilbert, 1971 . Parthenogenesis in rotifers : The control of sexual and asexual reproduction . Am . Zool. 11 :245-266 . Buchner, H ., 1987 . Studies on the control of heterogonous reproduction in Rotifers . III . The loss of mictic potential in Brachionus urceolaris . Arch . Hydrobiol . 109 : 333-354 . Carlin, B ., 1943 . Die Planktonrotatorien des Motalastrdm . Medd. Lunds Univ. Limnol . Inst 17 : 1-163 . Carmona, M . J ., M . Serra & M . R . Miracle, 1989 . Total protein analysis in rotifer populations . Biochem . Syst . Ecol . 17: 409-415 . Gilbert, J . J ., 1974 . Dormancy in rotifers . Trans . Am . Micros. Soc . 93 : 490-513 . Gilbert, J . J ., 1977 . Mictic female production in monogonont rotifers . Arch . Hydrobiol. Beih . 8 : 142-155 . Gilbert, J . J ., 1980 . Female polymorphism and sexual reproduction in the rotifer Asplanchna : Evolution of their relationship and control by dietary tocopherol . Am . Nat. 116: 409-431 . Gladstein, D . S ., N . F . Carlin, S . N . Anstad & W . H . Bos-
sert, 1991 . The need for sensitivity analyses of dynamic optimization models . Oikos 60 : 121-126 . Halbach, U . & G. Halbach-Keup, 1972. Einfluss von Aussenfaktoren auf den Fortpflazungsmodus hetergoner Rotatorien . Oecologia 9 : 203-214 . Hino, A . & R . Hirano, 1976 . Ecological studies on the mechanism of bisexual reproduction in the rotifer Brachionus plicatilis . I . General aspects of bisexual reproduction inducing factors . Bull . Jap . Soc . Sci. Fish . 42 : 1093-1094 . Hino, A . & R . Hirano, 1977 . Ecological studies on the mechanism of bisexual reproduction in the rotifer Brachionus plicatilis. II . Effects of cumulative parthenogenetic generation . Bull. Jap . Soc . Sci . Fish . 43 : 1147-1155 . Hino, A . & R . Hirano, 1984 . Relationships between water temperature and bisexual reproduction rate in the rotifer Brachionus plicatilis. Bull. Jap . Soc . Sci . Fish. 50 : 148 11485 . Jeffers, J . N . R ., 1978 . An Introduction to Systems Analysis : with ecological applications . Arnold, London, 198 pp. Johnson, M . L . & M . S . Gaines, 1990 . Evolution of dispersal . Annu . Rev . Ecol. Syst. 21 : 449-480 . King, C . E ., 1972 . Adaptation of rotifers to seasonal variation . Ecology 53 : 408-418 . King, C. E ., 1977 . Genetics of reproduction, variation, and adaptation in rotifers . Arch . Hydrobiol . Beih . 8 : 187-201 . King, C. E. & T . W. Snell, 1980 . Density-dependent sexual reproduction in natural populations of the rotifers Asplanchna girodi . In H . J . Dumont & J . Green (eds), Rotatonia . Developments in Hydrobiology I . Dr W . Junk Publishers, The Hague : 149-152 . Reprinted from Hydrobiologia 73 . King, C. E . & Y . Zhao, 1987 . Coexistence of rotifer (Brachionus plicatilis) clones in Soda Lake, Nevada . In L . May, R . Wallace & A. Herzig (eds), Rotifer Symposium IV . Developments in Hydrobiology 42 . Dr W. Junk Publishers, Dordrecht : 57-64 . Reprinted from Hydrobiologia 147 . Lubzens, E ., R . Fishier & V . Berdugo-White, 1980 . Induction of sexual reproduction and resting egg production in Brachionus plicatilis reared in sea water. In H. J . Dumont & J . Green (eds), Rotatoria. Developments in Hydrobiology I . Dr W. Junk Publishers, The Hague : 55-58 . Reprinted from Hydrobiologia 73 . Lubzens, E., G. Minkoff & S . Marom, 1985 . Salinity dependence of sexual and asexual reproduction in the rotifer Brachionus plicatilis . Mar. Biol . (Berl .) 85 : 123-126 . Lubzens, E . & G. Minkoff, 1988 . Influence of algae fed to rotifers (Brachionus plicatilis O . F . Muller) on the expression of mixis in their progenies . Oecologia 75 : 430-435 . Miracle, M . R ., M . Serra, R . Oltra & E . Vicente, 1988 . Differential distributions of Brachionus plicatilis species in three coastal lagoons . Verh . int. Ver . Limnol . 23 : 20062015 . Miracle, M . R . & A . Guiset, 1977 . Some effects of enclosure on congeneric species of rotifers . Arch. Hydrobiol . Beih . 8 : 94-97 . Nauwerck, A ., 1963 . Die Beziehungen zwischen Zooplankton
126 and Phytoplankton im See Erken . Symb . Bot . Upps . 1-163 . Pourriot, R ., 1965 . Sur le determinisme du mode de reproduction chez les rotiferes . Schweiz Z . Hydrol . 27 : 76-87 . Pourriot, R. & P . Clement, 1981 . Action de facteurs externes sur la reproduction et le cycle reproducteur des rotiferes . Acta. Oecol . Gen . 2 : 135-151 . Pourriot, R . & C. Rougier, 1979 . Influences conjuguees du groupement et de la qualite de la nourriture sur la reproduction de Brachionus plicatilis O . F . Mitller (Rotifere) . Neth . J . Zool. 29 : 242-264 . Pourriot, R. & W . Snell, 1983 . Resting eggs of rotifers . In B . Pejler, R. Starkweather & Th . Nogrady (eds), Biology of Rotifers . Developments in Hydrobiology 14 . Dr W . Junk Publishers, The Hague : 213-224 . Reprinted from Hydrobiolgia 104 . Ruttner-Kolisko, A ., 1964 . Ober die labile Periode in Fortpflanzungszyklus der Radertiere . Int. Revue ges . Hydrobiol . 49 :473-482 . Ruttner-Kolisko, A ., 1974 . Plankton Rotifers . Biology and taxonomy . Binnengewasser 26 : 146 pp . Ruttner-Kolisko, A ., 1985 . Results of individual cross-mating experiments in three distinct strains of Brachionus plicatilis (Rotatoria) . Verh . int . Ver . Limnol . 22 : 2979-2981 . Serra, M ., 1987 . Variaci6n morfometrica, isoenzimatica y demografica en poblaciones de Brachionus plicatilis : diferenciaci6n genetica y plasticidad fenotipica . Ph . D . Universitat de Valencia, Valencia . Serra, M . & M . R. Miracle, 1985 . Enzyme polymorphism in Brachionus plicatilis populations from several Spanish lagoons . Verh . int . Ver . Limnol . 22 : 2991-2996 .
Snell, T . W ., 1979 . Intraspecific competition and population structure in rotifers. Ecology 60 : 494-502 . Snell, T . W ., 1986 . Effect of temperature, salinity and food level on sexual and asexual reproduction in Brachionus plicatilis (Rotifera). Mar . Biol . 92 : 157-162 . Snell, T. W ., 1987 . Sex, population dynamics and resting eggs production in rotifers . Hydrobiologia 144 : 105111 . Snell, T . W . & E . M . Boyer, 1988 . Thresholds for mictic female production in the rotifer Brachionus plicatilis (Maller) . J . exp. mar . Biol. Ecol . 124 : 73-85 . Snell, T . W. & M . Childress, 1987 . Aging and loss of fertility in male and female Brachionus plicatilis (Rotifera) . Inter . J . Invertebr . Reprod . Dev . 12 : 103-110 . Snell, T . W . & B . L . Garman, 1986 . Encounter probabilities between male and female rotifers . J . exp . mar . Biol. Ecol . 97 :221-230 . Wesemberg-Lund, C., 1930 . Contributions to the biology of the Rotifera. Part II . The periodicity and sexual periods . Mem . Acad . Roy . Sci . et Lett . de Danmark . Sec . Sci . Serie 9 . 2 : 1-230 . Williams, G . C ., 1975 . Sex and Evolution . Princeton University Press, Princeton . Williams, G . C. & J . B . Mitton, 1973 . Why reproduce sexually? J . Theoret . Biol. 39 : 545-554 . Thane, A ., 1974 . Rotifera . In A . C. Giese & J . S . Pearse (eds), Reproduction of Marine Invertebrates . Vol . I : Acoelomate and Pseudocoelomate Metazoans. Academic Press New York, London . pp . 471-484 .
