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Keywords:

  • cyclical parthenogenesis;
  • density-dependent population growth;
  • life history evolution;
  • optimal allocation;
  • rotifers;
  • sexual reproduction

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

This work explores theoretical patterns of reproduction that maximize the production of resting eggs and the long-term fitness of genotypes in cyclical parthenogens. Our focus is on density-dependent reproduction as it influences the consequences of a trade-off between producing amictic daughters – which reproduce parthenogenetically and subitaneously – and producing mictic daughters – which undergo meiosis and bisexual reproduction. Amictic females increase competitive ability and allow the population to achieve a larger size; mictic females directly contribute to population survival through harsh periods by producing resting eggs. Although morphologically indistinguishable, the two types of females differ greatly in their ecological and reproductive roles. What factors underlie the differential allocation of resources to produce amictic and mictic females?

Using a demographic model based on readily accessible parameters we demonstrate the existence of a frequency of mictic females that will maximize the population's long-term fitness. This frequency, termed the optimal mictic ratio, mo, is 1 − (q/b)1/2, where q is the mortality rate and b is the maximum birth rate. Using computer simulation we compared the fitness of a population with this constant mictic ratio with populations having multiple switches from complete parthenogenetic growth to complete allocation in mixis (mictic ratio either 0 or 1). Two important conclusions for optimal mixis in density-dependent growth conditions are: (1) intermediate mictic ratios are optimal, and (2) optimal mictic ratios are higher when habitat conditions are better. Physiological cues responding to differences in birth and death rates are common so that it is possible that populations may adjust their relative rates of mictic and amictic female production in response to environmentally induced changes to the optimum mictic ratio. Our analysis demonstrates that different patterns of mixis are expected in different type of habitats. Since the optimal mictic ratio is sensitive to the effects of a variety of environmental challenges, our model makes possible a new means to evaluate life history evolution in cyclical parthenogens.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

Cyclical parthenogenesis is a mode of reproduction combining the advantages of rapid clonal propagation via diploid, ameiotic parthenogenesis with the advantages of sexual recombination. This mode of reproduction is particularly important in two very common groups of freshwater zooplankton: cladocerans and monogonont rotifers. It also occurs in a scattering of other animal groups, for instance in aphids. These groups tend to inhabit time-varying environments that become unsuitable for more or less predictable periods, and so are frequently re-colonized. Parthenogenetic reproduction is the predominant mode of reproduction in the initial stages of colonization because it produces rapid population growth. During the parthenogenetic growth phase reproduction is clonal and the genotype is copied with no recombination.

Sexual reproduction in cyclical parthenogens is initiated when amictic (parthenogenetic) females produce mictic (sexual) females that can be fertilized by males to produce resting eggs. The resting egg is the life cycle stage having the greatest capacity to disperse both in time and space. Because bisexual reproduction can be absent for extended periods, the factors inducing mictic female production are central to understanding the life-history evolution of these animals.

The results of empirical studies on this topic in rotifers are diverse. Both intrinsic and environmental factors as well as interactions between the two have been related to mixis induction in rotifers (see Gilbert, 1977, 1980; Pourriot & Clément, 1981; Pourriot & Snell, 1983). The factors inducing mixis vary among species, and different responses to putative cues have even been found among genotypes of the same taxonomic species (Hino & Hirano, 1977; Snell & Hoff, 1985; Lubzens, 1989; Carmona et al., 1994). Most empirical studies have been based on correlations and therefore fail to demonstrate a direct causal link between mixis and the studied factor. An association between population density and mixis has been found in field observations of several species (e.g. Wesenberg-Lund, 1930; Buchner, 1941; Ito & Iawi, 1958; Bogoslavsky, 1963; King & Snell, 1980), as well as in both laboratory and field experiments (e.g. Ito, 1960; Hino & Hirano, 1976; Miracle & Guiset, 1977; Pozuelo, 1977; Lubzens et al., 1985; Snell & Boyer, 1988; Carmona et al., 1993). A conditioning effect produced by animals reared in laboratory culture media on the proportion of mictic daughters of their conspecifics has been demonstrated in several species (e.g. Gilbert, 1963; Carmona et al., 1993). These results suggest that population density affects mixis induction through chemical modifications of the medium. Similarly, it has been found in cladocerans and other cyclical parthenogenetic zooplankters that crowding induces male production (e.g. Hobaek & Larsson, 1990).

Theoretical insights on the evolution of mixis patterns (i.e. when and how much bisexual reproduction occurs) in rotifers have been based on both verbal reasoning and more formal models. One approach to the problem was taken by Williams (1975) and King (1980) to address the consequences of mixis pattern on the quantity of genetic variation, while an alternative approach was adopted by Gilbert (1974) and Snell & Garman (1986) to consider effects related to the density dependence of sexual encounters. Another cost of mixis that has been considered is a reduction in population growth rate (e.g. Snell, 1987; Serra & Carmona, 1993; Aparici et al., 1996). Salinity, food and population density thresholds required for resting-egg production have been considered by Snell & Boyer (1988).

Serra & Carmona (1993) focused on temporary habitats, and found that the optimal reproductive pattern (i.e. that maximizing resting egg production) was to produce only amictic daughters until a short time before the birth rate was exceeded by the mortality rate, then switching to produce only mictic daughters. This result is analogous to that found for individuals with a so-called ‘bang–bang’ strategy (in which all available energy is initially allocated to somatic growth until a critical body size is achieved, and then is exclusively allocated to reproduction). Serra & Carmona (1993) noted that their results are puzzling when compared to empirical data. Large blooms of mixis that have been observed in the field (Miracle & Guiset, 1977; Carmona et al., 1995) suggest the existence of high mictic ratios, but in the laboratory, the maximum reported frequencies of production of mictic daughters by amictic females are about 50% (Gilbert, 1963).

Population cycles in cyclical parthenogens are initiated by the hatching of resting eggs and followed by high rates of reproduction. During this colonization phase density restraints are either absent or relatively unimportant. Following the colonization phase is a period of variable length during which growth rates are constrained by population density. Finally, as the environment deteriorates and population size decreases towards 0, density-independent forces once again play the major role. Thus it is during the middle phase that density-dependent effects are most important. In this paper we explore theoretical consequences of mixis patterns on fitness under density-dependent population growth. Among other things we are interested in determining whether such growth conditions affect the conclusion that a ‘bang–bang strategy’ for mixis induction is optimal, as found by Serra & Carmona (1993) in habitats with time-dependent, density-independent growth.

Model

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

Ecological and demographic context

Our focus is on fitness differences derived from different patterns of the timing of mixis in organisms reproducing by cyclical parthenogenesis. We assume the modelled population to be growing in a seasonal habitat that is only suitable for part of the annual cycle. Thus resting egg production is the ultimate measure of fitness because it is the resting egg that permits population survival when the habitat is unsuitable, and subsequent recolonization when environmental conditions improve. We will assume there is no variation between years, so that the reproductive pattern that produces highest fitness in one year also has the highest fitness in subsequent years. We further assume that the modelled population has density-dependent growth and that the relevant assemblage of rotifers is composed of one sexually isolated group that is monomorphic for the traits involved in mixis.

Demographic model

A generalized life cycle for monogonont rotifers is presented in Fig. 1. Bisexual reproduction is based on mictic females that produce haploid eggs by meiosis which, if unfertilized, develop into haploid males and, if fertilized, develop into thick-shelled, diploid resting eggs (Bell, 1982; Wallace & Snell, 1991). Two basic variables that describe the demography of a heterogonic rotifer population are the numbers of amictic females (A) and mictic females (M). Amictic females can parthenogenetically produce both amictic and mictic daughters, while mictic females produce haploid eggs that develop into haploid males or, if fertilized, diploid resting eggs. Resting eggs hatch producing amictic females, usually after a dormant period of variable length.

image

Figure 1.  General life cycle of monogonont rotifers (modified from King & Snell, 1977).

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Assuming that the numerical contribution of resting egg hatching to the number of amictic females is not important in the period under consideration, we can describe the dynamics of mictic and amictic females under density-dependent growth by

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where bA,M is the birth rate at density A + M, q is the mortality rate, assumed to be density-independent and equal for both types of females, and m is the mictic ratio, defined as the proportion of eggs from amictic females that develop into mictic females. Given the functional equivalence of amictic and mictic females except in the types of progeny they produce, m is a measure of the relative frequency of mictic females. We will assume m is constant for a given population, so that the value of m defines a mixis pattern. We will use a simple function to account for density effects on bA,M:

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in which b is the birth rate without density effects (i.e. the intrinsic birth rate), K is the carrying capacity and bA,M is constrained to be equal to or greater than 0. That is, bA,M varies linearly in relation to the total number of females, both types of females having the same effect. The maximum birth rates occur for values of A + M near 0, and bA,M equals q if A + M=K. Note that the condition bA,M = q does not imply or require that deaths equal births for population growth, since bA,M accounts for birth of both amictic and mictic females, but the latter do not contribute to current population growth. Thus, K is the population density at equilibrium only if m = 0, and consequently M at equilibrium is also 0.

Fitness measure

To compare the results of different mixis patterns, we define fitness (W) in its ecological context as the total number of resting eggs that are produced by a given pattern in a population growth cycle. That is

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where R is the rate of resting egg production, α is the onset and ω is the end of the period of resting-egg production.

The relationship between R and the variables in the demographic model (eqn 1) is affected by the following features of the rotifer life cycle: (1) only young mictic females can be fertilized by a male, (2) fertilized mictic females produce resting eggs, whereas (3) unfertilized mictic females parthenogenetically produce haploid males. The threshold age for fertilization is ≈4 h in the rotifer Brachionus plicatilis. Aparici et al. (1998) applied sex allocation theory to cyclical parthenogenetic rotifers. According to their theoretical analysis, the threshold age for fertilization is expected to evolve so that half of the mature mictic females would be fertilized and produce resting eggs, and half would be male producers, a result related to the common finding of even allocation in sexes. Field data, although scarce, seem to support this expectation (King & Snell, 1980). According to this result we assume from eqn 3 that

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Optimal constant mictic ratio in populations at equilibrium

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

We now ask what value of the mictic ratio maximizes resting-egg production in equilibrium populations? In Appendix A we show that an equilibrium exists for a population with a constant mictic ratio. The number of amictic and mictic females at equilibrium (A*,M*, respectively) are both functions of the mictic ratio, m:

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Equation 5a is a decreasing linear function of m, and eqn 5b is a nonmonotonic function of m (Fig. 2). From eqn 5 we obtain an expression for total population size:

image

Figure 2.  Relationship between number of females at equilibrium and mictic ratio, as predicted by eqn 5, for three different birth rates (b). The corresponding mo values (maximum of the mictic female curves) are 0.198 (for b=0.7 d−1), 0.329 (for b=1 d−1) and 0.452 (for b=1.5 d−1). (Assumed values: K=100, q=0.45 d−1.)

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Given that 0 ≤ m ≤ 1, A* + M* ≤ K. Note that the population size at equilibrium will be lower than K if M > 0. In addition, an equilibrium with (A + M) ≠ 0 exists if b(1 − m) > q. At low densities more amictic females are recruited than die.

Now that the equilibrium condition has been addressed, an analysis of the optimal mictic ratio of a monomorphic population at equilibrium is possible. At equilibrium M is constant and by eqn 4 it will be considered a measure of fitness. According to our derivation in Appendix B, the mictic ratio that maximizes M* is

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Equation 7 states that the optimal mictic ratio, mo, for the scenario assumed here is higher if the conditions are good (i.e. low mortality and high intrinsic birth rate). Relationships between mictic ratio and number of both mictic and amictic females are shown in Fig. 2 for three birth rates.

Optimal constant mictic ratio vs. multiple switches in mictic ratio

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

In the previous section we assumed a constant mictic ratio and demographic equilibrium. In order to compare the fitness of constant, intermediate mictic ratios, as studied above, with the fitness of switching mictic patterns, in this section we will analyse the effect on the number of mictic females of multiple switches in mictic ratio. We will assume that the mictic ratio switches from 0 to 1 at a given density (the ‘switch-on’ density for mixis), which will cause the population to decline after a time lag Subsequently, when the decreasing density achieves a lower threshold, the mictic ratio switches from 1 to 0 (the ‘switch-off’ density for mixis), allowing the population to grow again. Note that when mictic female production is switched-off, mictic females that were produced before the switch may still be present in the population. If this pattern is repeated, it causes periodic oscillations in population density (see Fig. 3). This approach allows us to compare the fitness of ‘bang–bang’ strategies, as described here, and intermediate strategies.

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Figure 3. A comparison of population growth form expected when the mictic ratio periodically switches between 0 and 1 (bang–bang reproduction) with that of a population producing a constant proportion of both mictic and amictic females after sexual reproduction is initiated.

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We consider populations having a growth season that is long enough to acquire a constant or periodically varying mictic ratio after the initial colonization phase. If the mixis pattern is switched on and off through time, the population density will oscillate (Fig. 3A). If m is constant after the initial growth phase, population size will stabilize (Fig. 3B). This permits us to compare (1) the average mictic female density in populations with multiple switches to mixis (‘bang–bang’ strategies) with (2) the mictic female density at equilibrium for a population having an optimal constant mictic ratio (intermediate mictic ratio), as given by eqn 7.

We have analysed the problem using simulated dynamics since eqn 1 cannot be integrated. Figure 4 shows some of our simulation results for a realistic set of parameters (carrying capacity, and birth and death rates). Very high – close to carrying capacity – and very low mixis initiation densities had negative effect on the average density of mictic females. Average density of mictic females tend to be higher when mixis switches off at intermediate population densities. For instance, if mixis switches on at 70 females L−1, the highest value in Fig. 4 is for a switch-off density that is 90% of the switch-on density (i.e. 63 females L−1). Two processes are acting to produce this type of response. First, mictic female recruitment tends to be high if mixis initiation occurs at high population density. That is, as the number of females at mixis initiation increases, the number of females available to produce mictic offspring also increases. Second, population growth after mixis switches off is faster if population density is close to K/2 (see Fig. 3), and so the population again quickly reaches the switch-on density. The result of these two processes is that the highest average density of mictic females is obtained if mixis initiation and mixis end occur at intermediate densities very close to each other. Detailed explorations were performed for the parameters assumed in Fig. 4 by stepping the switch-on and switch-off densities using increments of one female per litre. We found that the highest average density of mictic females (29.20 females L−1) was obtained if mixis switches on at 65 females L−1 and switches off at 64 females L−1. Under these conditions, the average value of mictic ratio (either 0 or 1) is 0.454. The corresponding figures for the same K, b and q if mixis is intermediate and constant and mictic ratio is optimal (according to eqn 7) are: population density at equilibrium = 64.61 females L−1, mictic females at equilibrium = 29.22 females L−1, mictic ratio = 0.454. Obviously, a constant, intermediate mixis pattern is a limit to which the switching mictic ratio converges if mixis switches on and switches off at densities that are progressively closer to each other. The remarkable result here is that the mixis switching pattern with highest fitness is the one that is the closest to a pattern with constant mixis and a mictic ratio given by eqn 7. Qualitatively similar results were found for other sets of parameters (K = 10, 100; b = 1, 2). We also simulated dynamics with mictic ratio showing multiple switches of the mictic ratio from 0 to a value lower than 1, and then to 0 again. In these simulations we again found that the average density of mictic females was lower than the density of mictic females in a population with the optimum mictic ratio given by eqn 7.

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Figure 4. Effect of several ‘bang–bang’ (switching mixis) patterns on the average number of mictic females as computed from simulated dynamics. The number on each curve is the density at which mixis switches-off expressed as percentage of the switch-on density (Assumed conditions: K = 100 females L−1, b = 1.5 d−1, q = 0.45 d−1, and mictic ratio – if induced – is 1. Under these conditions the equilibrium number of mictic females in a nonswitching population is 29.22).

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Our results demonstrate that under density-dependent growth, intermediate mictic ratios lead to the production of more resting eggs than any bang–bang pattern with multiple mixis initiations. Since our focus is on density-dependent effects we have not considered either the transient initial colonization phase or the transient terminal phase during which the environment deteriorates and population density drops to zero. Our conclusions regarding the optimality of intermediate mictic ratios do not apply to these phases since they are dominated by density-independent effects. Between these two phases there exists a density-dependent middle period of variable length during which the expected mixis pattern is an intermediate allocation of mictic and amictic daughters. The relative frequencies of mictic and amictic females expected during the middle period are those given by eqn 7.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

From demographic considerations, we have analytically shown that intermediate mixis levels can be optimal in a deterministic environment. When population growth is density-constrained, there is an intermediate mictic ratio that maximizes resting-egg production. In a constant environment, mo is a point of stable equilibrium. Populations producing an excess proportion of mictic females should therefore decrease their mictic ratio. The opposite should occur if mictic ratio is lower than mo.

Using a simulation model, Snell (1987) also detected advantages associated with intermediate mictic ratios. From a review of the literature, Snell found that the average mictic ratio reported for the rotifer Brachionusplicatilis is 21.2 ± 3.84% (range 0–50%). This average should be interpreted with caution as it is based on different strains and experimental conditions in the laboratory and includes some studies in which there was no mictic reproduction. For the conditions derived from the literature on B. plicatilis by Serra & Carmona (1993;b=0.95 d−1, q=0.4 d−1, = 0.55 d−1), mo is 35%. From eqn 7 we can estimate a realistic range for optimal mictic ratio. The lowest threshold for optimal mictic ratio is zero, a value corresponding to b=q (see eqn 7). Under favourable conditions in the laboratory, typical values for birth rate and mortaility rate are, respectively, 1.8 and 0.2 (Serra, 1987; Miracle & Serra, 1989). These values lead to an optimal mictic ratio of 0.67. In the field, however, lower birth rates and higher mortality are expected. Thus, a more realistic value for highest optimal mictic ratio in nature would be around 50%, as reported in the review by Snell (1987).

Our theoretical demonstration of the existence of an optimal mictic ratio is remarkable because it helps us to understand why amictic females are never totally absent from the progeny of amictic females in life table experiments. Clearly, rotifers are not ‘bang–bang’ strategists in which the population reaches its maximum size by producing only amictic females and then switches to an exclusive production of mictic females (Serra & Carmona, 1993). ‘Bang-bang’ strategies are most likely to arise when the temporal constraints for resting egg-production are more important than the resource constraints. By contrast, in a more stable and resource-limited environment, intermediate mictic ratios favour the maximal conversion of resources into resting eggs. Populations growing in a resource-limited environment would have advantages if they reproduce parthenogenetically until they attain their carrying capacity and then switch to an intermediate mictic ratio.

Intermediate mictic ratios may also evolve in uncertain habitats, where early mixis serves to ensure the production of at least some resting eggs, even in bad years. If the length of the suitable habitat period is unpredictable, mixis should start at a low rate as soon as resting eggs can be produced so that the critical effects of bad years in which the growth season is short can be diminished. Unhatched resting eggs in the ‘seed pool’ of former years also will have an effect on fitness, since they provide a means to re-establish the population following occasional failures of resting egg production. In several respects this problem is analogous to King & Roughgarden's (1982) demonstration that intermediate allocations of resources into growth and reproduction are optimal in randomly varying environments.

Attempting to unify and summarize some of the arguments presented above, we propose the following association between environmental features and mixis patterns:

1 density-independent growth in deterministically varying habitats: mixis is expected to be induced close to the time at which the habitat becomes unsuitable with a mictic ratio that is close to 1 (see Serra & Carmona, 1993; Aparici et al., 1996);

2 density-independent growth in randomly varying habitats: mixis is expected to be induced early in the growth season with a low mictic ratio;

3 density-dependent growth: mixis is expected to be induced when the population is close to its carrying capacity with an intermediate mictic ratio. In these habitats we expect that the better the growth conditions, the higher the mictic ratio with a maximum that is ≤ 0.5.

A density-dependent growth pattern becomes less likely as the growing season becomes shorter. Thus it is possible for a population to go directly from the initial colonization phase to the final phase during which the population is eliminated by environmental deterioration. In the absence of a density-dependent phase between population colonization and extinction, it is expected that the population will switch directly from a mictic ratio of 0 to a mictic ratio of 1 when the environment becomes unsuitable. Such a switch would occur in habitats characterized almost exclusively by density-independent constraints. However, we suspect these conditions are rather rare since populations living in these habitats would be subject to high extinction rates. In addition, the available evidence from the field studies suggests that mictic ratios greater than 0.5 are exceedingly uncommon (Gilbert, 1963; Snell, 1987; Carmona et al., 1995).

The association outlined above between habitat features and mixis patterns is based on the demographic effects of mixis. However, mixis patterns also have effects on the levels of genetic diversity in the sexual offspring – e.g. early mixis promotes higher levels of diversity among the progeny than late mixis (King, 1980) – and an increased genetic diversity in sexual offspring may confer fitness advantages to the parental clones. Nevertheless, it is important to distinguish whether the level of genetic variation is selected for directly, or is a by-product of selection for a mixis pattern that simply increases the number of resting eggs produced. For instance, the mictic patterns associated with randomly varying habitats should tend to preserve levels of variation as a side-effect of selection for resting-egg production.

Real habitats seldom fall into single, pure categories of the type described above. Moreover rotifers are subject to a suite of selective pressures that are dependent upon the characteristics of the environments they inhabit and the features of each population's genetic structure. Nevertheless, in spite of these complications there are clear associations that can be drawn between habitat characteristics and mixis patterns. For instance, the occurrence of mixis at high densities is predicted for conditions that are common in nature. Most studies of rotifers have been performed in relatively predictable habitats such as lakes and permanent ponds. Thus, it is not surprising that empiricists have found a positive relationship between mixis and population density (Wesenberg-Lund, 1930; Buchner, 1941; Ito & Iawi, 1958; Ito, 1960; Bogoslavsky, 1963; Hino & Hirano, 1976, 1977; Miracle & Guiset, 1977; Pozuelo, 1977; King & Snell, 1980; Lubzens et al., 1985; Snell & Boyer, 1988; Carmona et al., 1993). Carmona et al. (1995), studying rotifer populations in a nonpermanent pond, also reported an increase in mictic levels with density. However, in the less predictable habitats occupied by winter populations of the Brachionusplicatilis species complex, continuous mixis spanned the entire population growth period. By contrast, mixis was absent in populations inhabiting the pond in late spring and early summer until just before a population crash when high densities were achieved. Clones inhabiting the pond in spring and summer had a higher threshold population density for mixis induction than those inhabiting the pond in winter. Although more evidence is required, these empirical findings are in line with our predictions.

Empirical studies have shown that mixis levels increase under favourable conditions for growth in mass cultures (e.g. Snell, 1986). The experimental designs in these studies frequently fail to isolate or distinguish between the direct effects on mixis of conditions that promote rapid growth and the indirect effects that are mediated through the influence of high densities. Nevertheless, it has been proposed that the relationship between mixis and favourable environments for growth is not spurious, but instead reflects the high resource investment needed for resting egg production. According to this view mixis is likely to occur under conditions that are also favourable for parthenogenetic reproduction (Snell, 1986). A correlation between mixis levels and environmental quality is also predicted from dynamic models, despite their neglect of the argument that resting eggs are costly to produce. In a simulation study of the dynamics of a single population, Snell (1987) found a positive relationship between the mictic ratio yielding maximal resting egg production and the intrinsic growth rate. Our derivation of the optimal mictic ratio for a single population with density-dependent growth produced the same result, but is based on a more general and rigorous analysis. Equation 7 shows that optimal mictic ratio increases with birth rate and decreases with mortality rate.

The higher optimal mictic ratio in better conditions can be interpreted as due to the lower relative cost of mictic female production. That is, when the birth rate is high, the production of a mictic female rather that an amictic female will have a relatively small impact on the population growth rate. However, as b decreases, the diversion of resources to mictic female production results in a progressively greater loss in growth potential. This conclusion is based on the following logic: focusing on the investment of an amictic female at very low densities, the cost of mictic daughter production can be computed as b·m (the loss in per capita growth potential due to mixis), while the per capita growth rate without mixis would be b − q. Thus the relative cost of mictic female production would be b · m/(b − q), which, by taking derivatives with respect to b on one hand and q on the other, can be shown to decrease as conditions improve (Fig. 5).

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Figure 5. Relationship between the relative cost of mixis (evaluated in noncompetitive conditions) and the intrinsic birth rate for different death rates (values on the right side). Assumed mictic ratio: 0.33.

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In addition to variation in density, rotifer populations are subject to a variety of seasonal changes in their lakes and ponds (King & Serra, in press). One consequence of this latter form of environmental variation is that intrinsic rates of birth (b) and death (q) will also change. Note, however, that there are numerous physiological processes involved in birth and death rate determination that could signal these changes. This raises the intriguing possibility that rotifers may adaptively alter their relative rates of mictic and amictic female production in response to changes in the optimal mictic ratio.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

We thank to T. W. Snell, E. Aparici, J. M. Mazón and M. J. Carmona for their suggestions and comments on this work. The criticism on a previous version of this paper by an anonymous referee greatly improved our analysis. M.S. was the recipient of a fellowship of the Ministerio Español de Educación y Ciencia for a sabbatical visit to the Department of Zoology at Oregon State University, during which time the research for this paper was performed. Additional funds were received from the Universitat de València to support travel of C.E.K. to Spain where the final draft of this manuscript was prepared. We thank the hospitality of colleagues and friends in Corvallis and Valencia.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices
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  • 3
    Bell, G. 1982. The Masterpiece of Nature. Croom Helm, London.
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    Bogoslavsky, A.S. 1963. Materials to the study of the resting eggs of rotifers. Communication I. Bulletin Moskovskoe Obshchestvo Ispytatelei Prirody 68: 50 67.
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Appendices

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Optimal constant mictic ratio in populations at equilibrium
  6. Optimal constant mictic ratio vs. multiple switches in mictic ratio
  7. Discussion
  8. Acknowledgments
  9. References
  10. Appendices

Appendix A

From eqn 1a, if dA/dt=0 and A ≠ 0, and substituting eqn 2 for bA,M≠ 0, we obtain

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and from eqn 1b, if dM/dt=0 and ≠ 0,

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By combining eqns a1 and a2, and simplifying, we can obtain eqn 5.

Appendix B

According to eqn 5,

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whose derivative on m is

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which, setting equal to 0 and solving for m, gives

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where mo is the mictic ratio maximizing M*. Given that by definition 0 ≤ m ≤ 1, the square root above should be evaluated as negative, and it yields eqn 7.