Diapause and maintenance of facultative sexual reproductive strategies

Abstract

Facultative sex combines sexual and asexual reproduction in the same individual (or clone) and allows for a large diversity of life-history patterns regarding the timing, frequency and intensity of sexual episodes. In addition, other life-history traits such as a diapause stage may become linked to sex. Here, we develop a matrix modelling framework for addressing the cost of sex in facultative sexuals, in constant, periodic and stochastically fluctuating environments. The model is parametrized using life-history data from Brachionus calyciflorus, a facultative sexual rotifer in which sex and diapause are linked. Sexual propensity was an important driver of costs in constant environments, in which high costs (always > onefold, and sometimes > twofold) indicated that asexuals should outcompete facultative sexuals. By contrast, stochastic environments with high temporal autocorrelation favoured facultative sex over obligate asex, in particular, if the penalty to fecundity in ‘bad’ environments was large. In such environments, obligate asexuals were constrained by their life cycle length (i.e. time from birth to last reproductive adult age class), which determined an upper limit to the number of consecutive bad periods they could tolerate. Nevertheless, when facultative asexuals with different sexual propensities competed simultaneously against each other and asex, the lowest sex propensity was the most successful in stochastic environments with positive autocorrelation. Our results suggest that a highly specific mechanism (i.e. diapause linked to sex) can alone stabilize facultative sex in these animals, and protect it from invasion of both asexual and pure sexual strategies.

This article is part of the themed issue ‘Weird sex: the underappreciated diversity of sexual reproduction’.

1. Introduction

All around us are plant and animal populations with both asexual and sexual reproduction. Can we seriously consider the quantitative apportionment of resources to these two processes is not subject to Darwinian selection?

—Williams 1975 [1, p. 11]

The ubiquity of sexual reproduction across eukaryotes presents a long-standing and still unsolved puzzle to evolutionary biologists [1–5]. Theoretically, sexual populations should be prone to invasions by asexual mutants, because several costs are associated with sex [6–8]. However, the exact magnitude of such costs and the compensatory benefits required for resistance to asexual invasions in each species are still to be elucidated. The same applies to the question of whether such benefits are due to universal advantages of sex, e.g. those related to recombination, or whether they arise merely from lineage-specific traits that prevent ‘loss of sex’ by means of mechanisms that are highly contingent to the life history of a species, or taxon [9].

The ‘paradox of sex’ is perhaps most striking in an obligate sexual organism with separate sexes that lacks paternal care. In this case, the cost of sex should amount to twofold [2], meaning that an asexual mutant could produce twice as many female offspring per generation. Such conditions are not fulfilled in a large number of organisms that have mixed sexual and asexual reproductive modes, which include taxa from all kingdoms, from bacteria to animals and plants. Yet any serious universal explanation on the maintenance of sex needs to be compatible with these cases of facultative sex. In fact, Williams [1] used his ‘balance argument’ [2], which is quoted above, to postulate the existence of a universal, or at least a very broadly applicable mechanism that could counter-balance in the short term the costs of sexual reproduction in facultative sexual organisms. But are universal explanations required at all in such cases or can lineage-specific mechanisms explain maintenance of intermediate rates of sex?

In this study, we examine whether a lineage-specific mechanism alone (a diapause stage linked to sex; see below ‘Traits linked to sex in facultative sexuals’) can prevent loss of sex in a facultative sexual organism. To this end, we develop a matrix model based on age-specific survival and fecundity schedules of the monogonont rotifer Brachionus calyciflorus. Importantly, our model does not include any other mechanisms that might relate to universal benefits to sex (e.g. production of recombined and genetically variable offspring).

(a). The diversity of facultative sexual life cycles

The terminology for describing facultative sexual systems is quite diverse. Sometimes it refers to the exact mechanism of asexuality (e.g. ‘facultative parthenogenesis’ or ‘facultative apomixis’), or to the frequency and duration of sexual episode (e.g. ‘occasional sex’, ‘cryptic sex’). Another common term is ‘cyclical parthenogenesis’, which refers to periodic alternations of sex and asex (often in a seasonal context), and the life cycles of aphids, cladocerans and rotifers are typical examples of cyclical parthenogenesis. However, there is no intrinsic cyclical component in most reproductive systems in the sense that the number of asexual generations would be fixed until engaging in sex. Rather, the cyclical pattern arises from extrinsic seasonal variables and associated ecological factors (e.g. photoperiod, population density and food abundance) which act as the causative triggers for the initiation of sex [10–14]. In this study, we will refer to all those cases as facultative sex, as they encompass mixed sexual and asexual reproductive systems.

Facultative sex allows for a much wider range of phenotypic variation than obligate sex or obligate asex, partly because sex and asex can be combined in so many ways, and because associations with other phenotypic traits are possible. Individual genotypes of a facultative sexual species may vary in the proportion of sexual or asexual offspring (sexual propensity), or in the timing and duration of sexual episodes. Indeed such variation seems to be present in many organisms, as can be seen in studies demonstrating variation in sex propensity between and within geographical populations (e.g. [15–17]). Sexual or asexual episodes in a facultative sexual may also be correlated to environmental states. For example, sexual episodes often coincide with deterioration of the environment [18]. The mechanistic basis for such correlations is provided by studies demonstrating that sexual episodes can be directly triggered by changes in day length, food deprivation, high population density or by combinations thereof [10,12,19]. Likewise, the absence of certain stimuli can act as trigger for asexual reproduction (e.g. the lack of mating opportunities can elicit parthenogenesis in Komodo dragons [20]). Facultative sexuality also allows for sexual stages to become linked to other life-history traits. In many facultative sexuals, for example, in aphids and rotifers, a stage resistant to environment (e.g. a diapause stage) is linked to the sexual phase [13,21].

(b). Costs and benefits of facultative sex

Theoretically, facultative sex should enable organisms to benefit from the advantages of sex and recombination, while avoiding paying the full costs of sex. In that sense facultative sex may even be regarded as superior to the pure forms, obligate sex and obligate asex. Interestingly, facultative sex might actually be much more common than usually appreciated, especially if we do not only consider the textbook cases, such as cyclical parthenogenesis. For example, most protists divide by asexual means but are believed to engage in sex occasionally [22–24]. Likewise, most plants are capable of clonal (vegetative) reproduction in addition to sexually produced seeds [25]. There are two assumptions connected to the proposition that facultative sex might be superior to obligate sex: (i) low rates of sex, or occasional sex, might already provide most of the benefits, e.g. through recombination [26–28], and (ii) costs of sex should be indeed reduced in facultative sexuals. By allowing some proportion of females to reproduce asexually, facultative asexuals should at least have a lower cost of males. Nevertheless, the exact pattern of how the costs of sex decrease along a gradient of sexual propensity will depend on characteristics of the reproductive system of the studied organism, as there might be interactions with costs of sex other than those of male production [7,9].

(c). Transitions to obligate asex

Are transitions to obligate asex more common in facultative sexuals than in obligate sexuals? There is wide agreement in the literature that sex is ancestral and asexuality is a derived state in eukaryotes [29,30]. This is because the basic meiotic machinery is shared by all eukaryotes suggesting that it derives from a common ancestor [22]. On the other hand, there are plenty of cytological mechanisms by which organisms can transition from sex to asex, involving a variety of modifications, from altered meiotic processes to render unisexual reproduction to complete short-cuts avoiding meiosis [29,31]. Because facultative sexuals are already capable of unisexual or clonal reproduction, transitions to obligate asex are more easily accomplished in these taxa, because they require merely a loss of the sexual function. For instance, several studies have demonstrated that obligate asexual variants are simply homozygous for recessive loss-of-function alleles [32–34]. Another example by which transition to asex is mediated by simple genetic means is contagious asexuality, where dominant meiosis-suppressing genes are transferred from asexual populations to sexuals via males [35,36]. A recent review that also includes other cases and examples is provided by Neiman et al. [31].

Apart from cases where asexuality is mediated by single genes, studies on facultative sexuals have shown that there is abundant genetic and phenotypic variation in the propensity for sex (e.g. [15,16]). Furthermore, it has been shown that populations can sometimes be gradually selected for increased/decreased rates of sex [37,38]. This is more compatible with a quantitative trait model, in which multiple genes determine the propensity for sex in an individual. Thus, in addition to an obligate asexual mutant, which can appear instantly in a sexual population, there could be gradual selection for decreased rates of sex until the first obligate asexuals begin to appear.

(d). Traits linked to sex in facultative sexuals

Facultative sexuals are important models for understanding how high rates of sex are maintained in the presence of variants with low or no sexual reproduction at all. The ‘balance argument’ is one of the first propositions in this direction [1]. It states that there must be universal benefits of sex that act on short timescales, otherwise, facultative sexuality would not be evolutionarily stable. However, in many facultative sexual organisms, the balance argument is not straightforward to apply, due to confounding differences between sexual and asexual stages [2]. A very common observation is that sexual stages are more resistant against environmental influences and often associated with some form of dormancy, for example, the resting eggs (REs) of daphnids and rotifers. These desiccation-resistant stages may also facilitate spatial dispersal across non-aquatic boundaries, in addition to the dispersal in time (via so-called ‘RE banks’). Associations between sex and other life-history traits can create additional benefits, but also additional costs to sex, as we will explore in detail in this study. Even though for individual genotypes these links are ‘hard-wired’, the existence of exceptions (e.g. asexual diapause stages) shows that on a population level, this covariation is less than perfect. It is nevertheless interesting to elucidate the ultimate factors causing covariation between sex and other life-history traits.

In this study, we focus on the association between dormancy and sex. This association can be found in many cyclical parthenogens, such as aphids or rotifers [13,21]. We develop a matrix model tailored to the life cycle of the monogonont rotifer B. calyciflorus in order to identify the conditions in the environment that would maintain sex, despite the costs that sex and diapause can cause. Specifically, we run our model in different environments, constant, periodic and stochastic environments with varying degrees of temporal autocorrelation. Using our model, we address the following questions related to the maintenance of sex in facultative sexuals:

• (1) Can loss-of sex be prevented solely through the beneficial effects of diapause?

• (2) Which are the main drivers (i.e. combinations between life-history features and environmental variation patterns) for the maintenance of sex?

• (3) Can the mechanism implied in (1) explain intermediate frequencies of sex, as observed in facultative sexuals?

We thus strive to establish a modelling framework that may answer these questions, as well as serve as a starting point for future studies in explaining why sex is stabilized at a certain frequency in facultative sexuals. We envision that this framework could be useful for disentangling the contributions of ‘universal mechanisms’ versus ‘lineage-specific’ mechanisms for the maintenance of sex in facultative sexual organisms.

2. Model

(a). Biological background of the model system

The rotifer B. calyciflorus exhibits the typical life cycle of a cyclical parthenogen [39]. Females can reproduce by ameiotic parthenogenesis via unreduced, diploid egg cells for many generations, which enables fast population growth and colonization of new habitats by only one or few founding females. Sexual reproduction is triggered at higher population densities, usually around 0.1 females ml⁻¹ [40], by the release of the so-called mixis-inducing protein (MIP) into the surrounding water [41]. In a process analogous to quorum sensing in bacteria, the MIP elicits asexual females to produce mictic female offspring, whose egg cells are haploid [42]. The transition to mixis is typically not 100%, thus even at high population densities a considerable proportion of offspring will remain asexual [43]. If the eggs of mictic females are not fertilized, they develop into (haploid) dwarf males. If they are fertilized, they develop into REs, in which the zygote will undergo a couple of divisions until it arrests its development (diapause). Diapause can last from a few days to several decades [21,44,45]. Previous studies have demonstrated that there is abundant variation with regard to variables in this life cycle, for example, variation in mixis propensity. Even obligate asexuality has been found, which appears to be mediated by a Mendelian allele that causes irreversible loss of sex in individuals homozygous for that allele [32,46]. Obligate parthenogens may still produce the MIP, however, they have lost the ability to sense or respond to this chemical cue [47].

(b). The matrix model

Our mathematical model is a hybrid of age- and stage-dependent matrix models, parametrized with life-history data from B. calyciflorus (electronic supplementary material, table S1). It is represented by a matrix of 41 × 41 elements (electronic supplementary material, table S2). In this model, we incorporated important aspects of the life cycle mentioned above, such as a switch between sexual and asexual stages or developmental delays due to diapause (figure 1). The life histories of amictic and mictic B. calyciflorus females were represented by their respective Leslie matrices, reflecting the age-dependent transition probabilities in terms of survival and offspring production. The Leslie matrices were established from cohort life table data ([45], electronic supplementary material, table S1) of amictic and mictic B. calyciflorus females using birth pulse approximation with postbreeding census, i.e. reproductive events were considered to occur synchronously at the end of each projection interval [48]. The sampling interval of the life tables was identical to the projection interval of the matrix model (12 h). In accordance with the empirical data, our model set-up incorporates overlapping generations, a non-reproductive juvenile stage and an age of last reproduction after which reproduction is no longer possible. Survival is still possible for a short period of time after age of last reproduction, until maximum lifespan is reached (figure 1 and electronic supplementary material, tables S1 and S2, for further details). In the matrix model, the two Leslie matrices were linked via two coefficients representing the trait ‘mixis propensity’. More specifically, one coefficient G_2,1 = p represented the probability that an amictic egg would give rise to an amictic juvenile, while another coefficient C_18,1 = 1 − p represented the probability that it would give rise to a mictic female (figure 1 and table 1).

Figure 1.

Life cycle graph of facultative sexual B. calyciflorus. Circles indicate stages, and the arrows indicate possible transitions: surviving and growing/hatching into a subsequent stage (G_i,j), surviving and staying in the same stage (P_i), or fertility (F_i). The following stages are represented in the model. Stage 1: amictic eggs; stages 2–17: amictic females; stages 18–34: mictic, RE-producing females; 35–41: REs. For simplicity, some stages are not directly displayed in this graph (i.e. adult stages of amictic females 5–16, adult stages of mictic RE-producing females 22–33, and dummy stages of REs 36–39, produce the appropriate delay for short-diapausing REs). The corresponding transition matrix consisting of 41 × 41 elements is given in the electronic supplementary material, table S2.

Table 1.

Parameters of the matrix model and their values. RE, resting egg.

trait	meaning	variable	matrix coefficient 1	matrix coefficient 2	default value	range explored
sex (mixis) propensity	proportion of mictic offspring	p	G2,1 = 1 − p	G18,1 = p	0.5	0–1
dormancy duration	proportion of long-diapausing REs	q	G41,40 = q	G2,40 = 1 − q	1	0–1
sex allocation	proportion of mictic females that will become RE producers	m	(F35,21 … F35,29) × m		0.5
reduced fecundity	fraction of fecundity remaining in the bad environment	k	(F1,4 … F1,15) × k	(F35,21 … F35,29) × k	0	0–0.1
environmentally determined hatching	fraction of hatching rate remaining in bad period	s	(F2,41) × s		1	0–1

Mictic (sexual) females were represented only as RE-producing females in our model. Thus, we did not include separate stages for unfertilized mictic females (which are male-producing) or males, nor did we explicitly include fertilization probabilities in this model. Nevertheless, we accounted indirectly for such processes by adjusting the fertilities of RE-producing females such that only 50% of mictic females in a population would become RE-producers, which is suggested by theoretical models [49] and supported by empirical observations [50]. In other words, sexual costs due to limitation by fertilization opportunities were accounted for by multiplying the fertility coefficients of RE-producing females by 0.5, which is represented by the parameter m in table 1.

RE development and hatching was modelled by taking into account several recent empirical studies on variability and duration of diapause in rotifers of the genus Brachionus. First, we assumed that RE development lasts for a minimum of 3 days, i.e. six projection intervals [45]. After this time, a certain proportion of REs could hatch (i.e. short diapause), which was reflected by the coefficient G_2,40 = q. The remaining proportion consisted of long-diapausing REs, represented by the coefficient G_41,40 = 1 − q (figure 1 and table 1). These long-diapausing REs stayed dormant for a much longer time, on average, as only a small fraction was allowed to hatch in each projection interval (default value: 1/200, corresponding to 50% REs hatched after approx. 73 d). Note that the value of q was set to 0 in our model by default, implying that all REs were long-diapausing, which is the case most often found in textbooks. However, in our analysis of constant environments (see below), we also considered variation in the proportion of REs that are short diapausing. Hatching of REs was represented in our model by two coefficients that contributed to the first juvenile stage of amictic females. These coefficients were G_2,40 and G_2,41 for short-diapausing and long-diapausing REs, respectively. This agrees with the observation that hatchlings of REs in rotifers are almost always amictic ([51]; for one notable exception from this pattern, see [52]).

Hatching rates of REs might be influenced by environmental conditions such that ‘good’ environments are associated with higher hatching probabilities. This strategy could be advantageous in periodic environments or in stochastic environments with high levels of temporal autocorrelation. We explored this possibility in our model by multiplying F_2,41 (i.e. the hatching rate) with the coefficient s which determines the degree to which the ‘bad’ environment suppresses hatching. Under default conditions, the value was s = 1 in the bad environment (table 1), which means that hatching is completely independent of the environmental conditions (i.e. a pattern which might result in a diversified bet-hedging strategy). However, if s = 0 in the bad environment, REs would only hatch during times of benign conditions, since in good environments s was always 1.

(c). Quantifying the cost of sex

The ‘cost of sex’ can be quantified as the N-fold increase in population size of an obligate asexual relative to a facultative sexual within one generation. This quantity can be calculated for all those cases where it is possible to obtain asymptotic growth rates, such as for constant or periodic environments (see below, ‘Periodic environments’). For stochastic environments, this cannot be easily done, since the population growth in stochastic environments is determined by the geometric mean given the individual trajectories of environmental states. Thus, in stochastic environments, it is more convenient to focus on the probabilistic outcome of competition between facultative sexuals and obligate asexuals. To this end, we may quantify the fraction of trials where one of the two reproductive modes prevails in the end. There is no contradiction in using these two measures for quantifying the cost of sex interchangeably, since the former definition (the N-fold cost of sex) can be regarded a strong predictor of the outcome of competition.

To quantify the ‘N-fold cost of sex’, we first calculated the difference between population growth of a facultative sexual and an obligate asexual. Under constant environmental conditions, the population growth rate r can be conveniently calculated as the logarithm of the dominant eigenvalue of the matrix [48], and with a few modifications this can also be done for periodically fluctuating environments (see below, ‘Periodic environments'). Obligate asex was introduced into our model by setting the parameter p = 0 (table 1), thus setting the coefficients G_2,1 and G_18,1 to 1 and 0, which excludes all stages beyond the last amictic stage (stage 17). The ‘N-fold cost’ was calculated as N-fold increase in population size of an obligate asexual relative to a facultative sexual within the average generation time of B. calyciflorus, as determined by life table data from a previous study [45]. There are several alternative definitions for generation time T in the context of age-structured populations [53]. The definition we used is the mean age at reproduction of a cohort of females [48,53]. This same definition was used in Stelzer [54]:

where l_x is the survival of individuals to age x, and m_x is the number of offspring born to a female of the same age. The N-fold increase in an obligate asexual is then

where Δr is the difference in the growth rates of obligate asexuals and facultative sexuals.

(d). Constant environments

Our constant environment consisted of only good environmental conditions. This matches the conditions under which the cohort life table data was obtained, i.e. ad libitum food, benign temperature (23.5°C), and individually cultured females with frequent transfers to fresh medium (electronic supplementary material, table S1). In the constant environment, we explored life-history variation regarding two components of the matrix model: first, we calculated population growth rates for different mixis propensities (range: 0–1). In these calculations, we also modified our matrix model to suppress recruitment from REs into the growing population, which was done by exclusion of all sexual stages (stages 18–41). This allowed us to compare our matrix model to earlier calculations done by Stelzer [54]. Stelzer [54] used an alternative method, the Euler–Lotka equation, to calculate growth rates; however, this is mathematically equivalent to computing the leading eigenvalue of the corresponding Leslie matrix [48], making the results directly comparable. Second, we simultaneously modified coefficients related to the proportion of early hatching REs (range: 0–1) and coefficients related to mixis propensity (20, 40, 60, 80, 100%), to elucidate whether early hatching can ameliorate the costs of sex in constant environments.

(e). Periodic environments

Periodic environments consisted of regular alterations of good and bad environments (e.g. GGGBBBGGGBBB…). In the bad environment, all fertility coefficients of amictic and RE-producing females were assumed to be reduced, which we modelled by multiplying them with a factor in the range [0,1] (default value: 0) while survival was not affected. That is, with a default value of zero, females that experienced a transition from good to bad environments immediately stopped reproduction, but exhibited survival patterns identical to females in good environments. This approximation may not be realistic in the extreme case of complete starvation, but should hold for a wide range of food concentrations. Indeed, empirical studies suggest that rotifers rather exhibit extended lifetime than increased mortality under regimes of periodical starvation or under caloric restriction [55–58].

We explored different scenarios in periodic environments. First, we examined the effect of period length on population growth of a facultative sexual (50% mixis propensity) versus an obligate asexual. Calculating the growth rate for a periodic environment is simple in principle. If the matrices for good and bad environments are G and B, respectively, then the matrix corresponding to a full cycle in an environment with period 3 is found simply by the matrix multiplication G × G × G × B × B × B [48]. Computing the leading eigenvalue for this yields the growth rate over a full period; for one time step, we take the 6th root corresponding to three good and three bad time steps (or divide by 6 if working with the logarithmic growth rate). This allowed us to extract the ‘critical period length’ at which obligate asexuals just become inferior to facultative sexuals. Second, we recalculated the growth rates for this ‘critical period length’ for values of the reduced fecundity parameter larger than the default value of zero (table 1). Thus, we were interested in finding the reduced fecundity larger than zero at which the competitive edge would tilt back to the asexuals.

(f). Stochastic environments

Environmental stochasticity was modelled with a two-state Markov chain model, which allowed us to generate stochastic sequences of good and bad environments with varying degrees of autocorrelation [48]. We used a standard autocorrelation transition matrix procedure that generates a series of environments with any given autocorrelation parameter ρ and any given overall probability of a good or bad environment [48]. The autocorrelation parameter ρ ranged between −1 and 1, with positive values indicating positive autocorrelations and negative values indicating negative autocorrelations. The value of zero corresponds to a purely random sequence, in which the environmental state at each time was drawn from a fixed distribution, independent of the previous states [48].

For the simulations in stochastic environments, we modified our model to include density-dependent regulation of fecundity, using the Ricker density dependence function [48]:

where N is the number of all females and c determines the strength of density dependence. The fecundity of all reproductive females was then multiplied by g(N). The parameter c was set to 5 × 10⁻⁸, which roughly corresponds to a carrying capacity in the order of 500 million females. This number was chosen based on an estimation of the total water volume of the pond from which the B. calyciflorus strain was initially sampled, and based on a hypothesized maximum density of 1000 females l^–1 (cf. [39]). Dimensions of the pond were 30 × 60 m, with a depth of approximately 0.5 m. Volume was approximated by the shape of an elliptic cylinder.

Stochastic simulations were initiated with a population vector consisting of 100 individuals in each age class. Simulations were run for a duration of 14 600 projection intervals (12 h), which corresponds to roughly 20 years. We allowed each age class (including REs) to ‘die out’ if their abundances declined below 1. As the dependent variables, we extracted the abundances of obligate asexuals, facultative sexuals, and REs at the end of the duration of the simulation. All three variables were calculated as the means from the last 50 projection intervals. In addition, we scored the incidents when facultative sexuals did persist until the end of a simulation run.

Simulations in stochastic environments were run mostly at the default parameter values (table 1). But we also explored additional parameter space with regard to mixis propensity (0.25 and 0.75), and we varied the coefficient s, which adjusts environmental determination of RE hatching from 0 (i.e. full environmental determination; no hatching in the bad environment) to 0.5 (i.e. intermediate environmental determination) to 1 (i.e. independent of environmental state). Additional simulations were carried out to examine the maintenance of facultative sexual reproduction, i.e. to study competition between facultative sexuals differing in mixis propensity. To this end, we carried out simulations in which all the mixis propensities 0, 25, 50, 75 and 100% were present simultaneously and at equal frequencies at the beginning of each run. For each combination of model parameters, we used 150 independent runs, and the displayed results are the average of these runs.

In the simulations with stochastic environments, we limited population growth with density dependence. One potential pitfall here is that density dependence might have an independent effect on the success of facultative sex, because in growing populations early births have a higher contribution to fitness than late ones (and vice-versa in a decreasing population; [59]). Therefore, if density dependence renders population size stationary, costs of diapause might be significantly reduced or eliminated entirely. To account for such effects, we ran simulation trials with periodic environments with density dependence, to complement our analytical results of periodic environments (above). We may then expect the facultative sexual strategy to be slightly more competitive under density dependence, although this depends on the relative effect of density dependence on the different stages.

In the simulation trials of periodic environments, we allowed populations of facultative sexuals and obligate asexuals to grow with and without density dependence for 14 600 projection intervals, and plotted the proportion of facultative sexuals at the end of the simulation for each period length. Even in the treatment without density dependence, the population was not allowed to grow without bounds, as this would have resulted in computational problems. Instead, all stages were reduced by the same proportion after exceeding the carrying capacity, which retained the stage-structure and hence the relative competitiveness of asexuals and facultative sexuals.

3. Model results

(a). Constant environments

Theoretical predictions of the cost of sex in the rotifer B. calyciflorus in a constant environment are shown in figure 2. In general, costs of sex showed a gradual increase with increasing mixis propensity, which is in agreement with an earlier model of the same rotifer [54]. Over a wide range of mixis propensities (0–0.8), both model types yielded similar estimates of the cost of sex; the estimates began to diverge only at very high mixis propensities (more than 0.8), with higher costs observed in the model without recruitment from REs (electronic supplementary material, figure S1). Fast RE development (i.e. 3 days from egg production until hatching), i.e. higher proportions of early hatching REs, decreased the cost of sex in constant environments (figure 2). However, this effect was pronounced only at very high mixis propensities (more than 0.8).

Figure 2.

Effect of mixis propensity and the proportion of fast (3-day) RE development on the cost of sex in a constant environment. Isolines show the cost of sex for different mixis propensities. The fraction of REs that was not early hatching (i.e. after 3 days) was assumed to hatch with a rate of 0.01 d⁻¹. The cost of sex was defined as the N-fold increase in population size of a hypothetical obligate asexual relative to a facultative sexual (within the average generation time of B. calyciflorus as determined by life table data used in this study, i.e. 3.8 days). Population growth rates were calculated as the logarithms of the dominant eigenvalues of the matrix model. Note that because the fold cost of sex is a comparative measure, a value of 1 means equal rates and no cost.

(b). Periodic environments

Theoretical predictions of the cost of sex for different period lengths of good and bad environments are displayed in figure 3. Population growth rates decreased both in facultative sexuals and obligate asexuals with increasing period length. However, the decrease was more pronounced in obligate asexuals, which caused the growth rates to cross at a period length between 6.5 and 7 days (figure 3). In fact, at period lengths above 7 days, obligate asexuals could not reproduce at all, because at such high period lengths all females of the youngest cohort died before they could leave any offspring. This is caused by the fact that the life cycle length (LCL) in B. calyciflorus, i.e. the time from first juvenile stage to last fertile adult stage, was exactly 7 days (cf. transition matrix in electronic supplementary material, table S2). To probe further into the mechanisms contributing to the pattern in growth rates observed in figure 3, we fixed the period length at 7 days (i.e. the first period length at which the growth rate of asexuals was lower than that of facultative sexuals) and examined how growth rates of both reproductive modes changed, if milder forms of the bad environment were used (electronic supplementary material, figure S2), for different mixis propensities of the facultative sexual (figure 4). These calculations showed that the growth rate advantage of facultative sexuals at the 7-day period length quickly diminished for increasing values of the parameter ‘fecundity reduction’. In other words, if the fecundity remaining in the bad environment was more than 0.02% of maximum fecundity (good environment), the growth rate of the obligate asexuals is higher than that of facultative sexuals. By contrast, the parameter ‘mixis propensity’ did not affect the crossing point of the growth rates at all (figure 4).

Figure 3.

Theoretical predictions of the cost of sex in B. calyciflorus in periodic environments with runs of good and bad environments of different period lengths. The good environment was identical to the constant environment; in the bad environment fecundity of all females was assumed to be zero, while survival was unaffected by the environment (see main text for rationale of this assumption). (a) The growth rates of facultative sexuals and obligate asexuals based on the logarithms of the dominant eigenvalues of the matrix model (see main text for details of calculation of eigenvalues in the periodic environment). The cost of sex displayed in (b) was defined as the N-fold increase in population size of a hypothetical obligate asexual relative to a facultative sexual (within the average generation time of B. calyciflorus as determined by life table data used in this study, i.e. 3.8 days). Note that because the fold cost of sex is a comparative measure, a value of 1 means equal rates and no cost.

Figure 4.

The cost of sex in B. calyciflorus in a periodic environment with good and bad environments alternating in fixed 7-day periods. Graphs show the growth rates of facultative sexuals (solid line) and obligate asexuals (dashed line) based on the logarithmized dominant eigenvalues of the matrix model. ‘Reduced fecundity’ (x-axis) refers to the fraction of fecundity remaining in the bad environment. The different panels refer to different mixis propensities in the facultative sexual (values indicated on the top of each panel).

(c). Stochastic environments

In our simulations of stochastic environments, we analysed the performance of facultative sexual and obligate asexual populations in environments consisting of two states (good versus bad environment). We examined time series with varying degrees of temporal autocorrelation specified by the autocorrelation parameter ρ. Each simulation covered a period of 20 years and 150 simulations were run for each parameter combination. In these stochastic simulations, we modified the matrix model by including dependent population regulation. Thus, populations could not grow infinitely but reached equilibrium population sizes after some time (examples of such simulation runs are shown in the electronic supplementary material, figure S3).

Our simulations showed that a high temporal autocorrelation (autocorrelation parameter ρ > 0.3) favoured facultative sexuals, whereas at lower or negative values of ρ obligate asexuals seemed to be favoured. These patterns were evident in terms of persistence of facultative sexuals at the end of the 20-year simulation period (figure 5) and in terms of the abundances of facultative sexual females and REs (electronic supplementary material, figures S4 and S5). In addition to this overall pattern, we found that mixis propensity and environmental determination of RE hatching had some influence on the outcome when the autocorrelation parameter ρ ranged between 0.1 and 0.3. Both a lower mixis propensity (25%) and environmental determination or RE hatching seemed to increase the chance of facultative sexuals being present at the end of the simulation run (figure 5). Interestingly, both parameters also influenced the RE bank, as the size of the RE bank was the highest for high mixis propensities (75%), or in the case of full environmentally determined hatching (electronic supplementary material, figures S4 and S5). In the simulation trials where all the mixis propensities 0, 25, 50, 75 and 100% were present at equal frequencies at the beginning of each run, asexuals (0% mixis) were the winners at low autocorrelation (figure 6). However, as autocorrelation increased, the facultative sexuals with the lowest sex propensity (25%) were dominating at the end.

Figure 5.

Effect of mixis propensity and environmentally determined hatching on the proportion of runs with sexuals present at the end of 20-year simulations for stochastic environments with different autocorrelation patterns. (a) Effect of environmental autocorrelation and mixis propensity. (b) Effect of environmental autocorrelation and environmentally determined hatching. Each data point corresponds to an average of 150 repetitions of the simulations. For details on the simulation runs, see main text. Although we simulated environmental autocorrelation in the range [−1 to 1], we only present here the range where there was variation in the results, i.e. range [−0.2 to 0.6].

Figure 6.

Abundances (females) with different sexual propensities at the end of 20-year simulations for stochastic environments with different autocorrelation patterns. At the beginning of the run all the mixis propensities 0, 25, 50, 75 and 100% were present at equal frequencies; the last two are not shown because they were outcompeted (abundance = 0) over the entire range. Each data point corresponds to an average of 150 repetitions of the simulations.

In the simulation trials of periodic environments, with density dependence versus without density dependence, asexuals won when the period length was less than 7 days, as expected based on figure 3. Density dependence made the facultative sexuals slightly more competitive, so that a period length of 5.5 days was sufficient for facultative sex to prevail (electronic supplementary material, figure S6).

4. Discussion

One of our aims was to make sure the results remain comparable with an earlier model on the cost of sex in B. calyciflorus [54], while significantly extending the versatility of the model. Although the methods (Euler–Lotka equation in Stelzer [54] versus matrix models here) are superficially different, they are in fact mathematically equivalent in the simple, age-structured case [48]. In contrast to Stelzer [54], our model incorporates more biological detail, such as sexual females, REs with variable durations of diapause, and recruitment into a growing population via RE hatching. Many of our modelling results made intuitive sense and confirmed the expectations about which reproductive mode should be superior in which environment. We will summarize those aspects very briefly, and will then focus on the less intuitive observations and model behaviour for each of the environments, and we will suggest future additions to our modelling framework, which might aid to disentangle costs of sex in other model organisms.

(a). Constant environments

The constant environments model consisted of a constant series of good environments, allowing high fecundity for both facultative sexuals as well as obligate asexuals. Not surprisingly, the obligate asexuals had higher fitness under these conditions, as the population growth rate of sexual females was reduced by (i) the production of males and by (ii) the longer development time needed by REs (at least 72 h, but up to many years). These results of our matrix model agree with a previous model on the cost of sex in B. calyciflorus [54] (see the electronic supplementary material, figure S1), which made the simplifying assumption that REs would never hatch (i.e. diapause lasts infinitely). Altogether this is analogous to the ‘twofold cost of sex’ [2], even though in the case of facultative sexual rotifers, the exact value of this ‘N-fold cost’ can take values higher or lower than 2, depending on variables like the mixis rate and development time of REs (figure 2).

Compared to the previous model [54], our model included much more biological detail. For example, we implemented a higher diversity of diapause duration, from an extremely fast 3-day RE development [45] to ‘long diapause’, in which only a very small proportion of REs could hatch per time interval. Surprisingly, this modification had only a minor effect on the cost of sex in this system, unless the mixis rate was very high (figure 2). Such observations stress the need for explicit quantitative modelling, in addition to verbal arguments about the effects of life-history variables on fitness. We did not vary RE mortality in our model. However, it could be implemented in future studies by adjusting the coefficient P₄₁, to account for losses due to mortality in stage 41 (i.e. long-diapausing REs). RE mortality is known to be high in natural systems, often reaching values of 90% [60], and including RE mortality should further increase the cost of sex in this system.

(b). Periodic environments

In the periodic environments model, good and bad environments alternated with different period lengths. Facultative sexuality was only better in those cases where period length approached LCL (7 days in figure 3) or exceeded LCL (more than 7 days in figure 3). In the latter case, obligate asexuals could not produce enough offspring, meaning that the population went extinct, while facultative sexuals laid REs and could thus survive the adverse conditions. In retrospect, this result may not seem too surprising. Nevertheless, our model stresses that LCL is an important proximate mechanism that determines whether asexuals can persist at all in an environment (and in principle, this conclusion holds for any fluctuating environment). We are not aware of many empirical studies quantifying LCLs of organisms in the context of the maintenance of sex, and thus suggest that this could be a promising avenue in future studies, especially if LCL is contrasted to the possible ‘life extension’ provided by a sexual diapause stage.

It is important to realize that the LCL is not a constant in nature. In our case, the value of 7 days imported into the model was the result of the experimental conditions at which our B. calyciflorus strain was cultured (ad libitum food, temperature 23.5°C). It is known that temperature can strongly affect metabolic rates in ectothermic organisms and can thus influence biological times, such as development time and lifespan [61]. For instance, a study on B. calyciflorus [62] has shown that LCL amounted to 5.5, 10 and 13.6 days at temperatures of 25°C, 20°C and 15°C, respectively (note: we calculated life cycle length as ‘LCL = lifespan − senile period’ from the data reported in Halbach [62]). Thus, low temperatures may decrease the probability of extinction in stochastic environments by increasing LCL, which would allow obligate asexuals to tolerate longer periods of bad environmental conditions. In fact, the existence of ‘overwintering clones’ in Daphnia and aphids found in temperate zones suggests that this mechanism could be important in nature [63,64].

In addition to temperature, LCL might be affected by food quantity, although the resulting patterns are less straightforward. Complete starvation unequivocally leads to an earlier death, which has also been shown in rotifers [65]. Nevertheless, starvation times depend on species-specific life-history patterns, such as whether a species still invests in reproduction during starvation [65,66]. Rotifers of the genus Brachionus are known to reduce their reproductive output during starvation and can thus tolerate starvation better than species which continue to reproduce [65]. Moreover, dietary restriction, e.g. intermittent starvation or reduced food concentrations, have been shown to increase lifespan in Brachionus [55,58] and in many other organisms (reviewed in [67]). Altogether, this suggests that LCL might show a unimodal relationship with food quantity.

An unexpected observation in our model was the rather extreme values of fecundity reduction in the bad environment, which were required for ensuring a competitive advantage of facultative sexuals. Even the slightest increases in fecundity in the bad environment already tilted the competitive advantage back to obligate asexuals (figure 4; electronic supplementary material, figure S2). In our model, the crossing point, i.e. the point where growth rates of sexuals and asexuals were equal, required bad environments in which fecundity was reduced to 0.02% of the fecundity in the good environment. Another surprising observation was that the crossing point itself was virtually uninfluenced by sex propensity in the facultative sexual (figure 4), which is in contrast to the strong influence of this variable in determining the cost of sex in constant environments. Altogether this suggests that prolonged phases of extremely bad conditions are necessary to favour facultative sex over obligate asex in this system. It remains an empirical question to evaluate how often such extreme conditions hold in natural systems (apart from cases of complete desiccation, which may occur in non-permanent aquatic habitats).

(c). Stochastic environments

In stochastic environments, facultative sexuality was favoured when there was high temporal autocorrelation. This makes sense if one considers that obligate asexuals only lose in those cases where several bad environments followed on from each other—which is a likely scenario at high temporal autocorrelation, but very unlikely in low autocorrelation. Sex could win in those cases because diapausing eggs could survive the adverse conditions. However, there are several technical points to consider, if we try to generalize this result.

First, in our model we considered environments that were fluctuating between an optimal and an extremely harsh state. What would happen in a more continuous environment? Like in the periodic environments, the outcome of competition between facultative sexuals and obligate asexuals is contingent on a series of extremely harsh bad conditions lasting longer than the LCL. Theoretically, continuous environments could be implemented in our model by allowing the parameter fecundity reduction k (table 1) to vary on a continuous scale and temporal autocorrelation could be modelled using an ARMA (autoregressive-moving-average) process [48]. We envision that such a modification would favour asexuality, because it should be less likely in a continuously varying environment to obtain very long runs with fecundity reductions below 0.02%. In the long run, such a situation would be inevitable, but at low autocorrelation it should be rather unlikely.

Second, it is important to realize that even those endpoints in our simulations that may look like coexistence should not be interpreted as stable coexistence, as these endpoints are the means of 150 independent runs and thus represent the probabilistic outcome of competition between facultative sexuals and obligate asexuals. The timescale of our simulations (20 years) equates to roughly 2000 generations in our model organism. As our model does not include any evolutionary mechanisms, mutations from facultative sex to obligate asex (e.g. through alleles coding for a loss of the sexual function) are not considered. In real rotifer populations, however, such mutations might contribute to the evolutionary dynamics, given the large population sizes rotifers can reach in their natural habitats [39]. So, it is plausible that invasions by asexuals are repeatedly initiated on relatively short timescales, which makes the competitive interactions that occur on such timescales very relevant for understanding the overall patterns of facultative sexual versus obligate asexual reproduction.

Third, density dependence alone can decrease the cost of sex, which was shown by our simulations in the periodic environments (electronic supplementary material, figure S6) and is consistent with the fact that early reproduction loses its advantage when the population is not growing [59]. As our stochastic simulations contained both mechanisms, density dependence and stochasticity, it is likely that density dependence partly influenced competition between facultative sexuals and obligate asexuals. But it should have done so equally for the whole range of environmental autocorrelation, and thus not have changed the general pattern, because most of the time populations were close to their carrying capacity (see example, simulation runs; electronic supplementary material, figure S3).

Our simulations have shown that other life-history variables can influence this simple outcome in autocorrelated stochastic environments, for example, environmentally determined hatching and mixis propensity. In our simulations, both variables influenced the outcome in a narrow zone of environmental autocorrelations ranging from 0.1 < ρ < 0.3. Interestingly, low mixis rates (25%) appeared to be more favourable than high ones (75%), such that they increased the chance that facultative sexuals were present at the end of the 20-year simulation run (figure 5). Additionally, in the simulations where all the mixis propensities 0, 25, 50, 75 and 100% were present at equal frequencies at the beginning of each run, the facultative sexuals with the lowest sex propensity (25%) were dominating at the end (figure 6). Both observations suggest that a small investment into sex (diapause) can already provide most of its benefits, an argument that has been previously made in terms of the beneficial effects of recombination [26,27]. Environmentally determined hatching appeared to be favourable for facultative sexuals. This makes intuitive sense in environments with high autocorrelation, where current environmental conditions are predictive of future environmental conditions. In such situations, ‘unproductive hatching’ (hatching into an environment where no growth is possible) can be reduced.

There are several more life-history features not explicitly considered in our model, which could be implemented in future models. For example, in many monogonont rotifers, sexual reproduction is induced by quorum sensing chemicals, exuded by females into the water once population densities rise above a certain threshold level (often in the magnitude of approx. 0.1 females l^–1 [40]). Such a mechanism could be implemented in our model by defining mixis propensity p (table 1) as a variable that depends on current population density. As a result, facultative sexuals would reproduce entirely asexually below this threshold population density and would thus not suffer from a growth disadvantage compared to obligate asexuals (but neither would they produce REs). It seems unlikely that density-dependent sex induction would have a major influence in our results in the stochastic environments, as these populations were mostly fluctuating around the carrying capacity of the good environment, which is far above typical values for the threshold for mixis induction. However, a pattern of predominantly asexual reproduction at low population densities may be common in natural habitats, e.g. during colonization of a new habitat, at the start of a growing season, or if population densities are kept at low levels due to predation or limiting resources. One could speculate that it is perhaps such flexibility, to behave at times like an asexual and at other times like a sexual, that gives facultative sex the decisive advantage over obligate asex in a highly variable environment.

5. Conclusion

Our matrix model allowed calculation of the ‘N-fold’ cost of sex in facultative sexuals (in our case, the rotifer B. calyciflorus), and it predicted the probabilistic outcome of competition between facultative sexuals and obligate asexuals in fluctuating and stochastic environments. Our model confirmed and clarified several intuitions about such reproductive systems: (i) that there are large costs of sex in constant benign environments, often larger than twofold since the time required for diapause can substantially delay reproduction in the generation following a sexual event; (ii) that if sex is linked to a diapause stage, costs of sex diminish in periodic environments or in stochastic environments with strong temporal autocorrelation; and (iii) that the latter is strongly influenced by the LCL of the obligate asexual. Specifically, LCL predicted whether obligate asexuals could persist at all in a given environment (in which case they displaced the sexuals), or whether they faced a high likelihood of going extinct (in which case facultative sexuals remained). The pivotal role of LCL calls for further comparative studies on this life-history variable.

When several facultative sexuals with different sexual propensities were allowed to compete with each other, those with a low sex propensity were superior. Thus, facultative sex was superior to obligate sex, even in stochastic environments with high temporal autocorrelation. The somewhat unexpected result that extremely bad conditions were required for favouring facultative sex in stochastic environments suggests that sex may be more difficult to maintain if the environment fluctuates gradually and less severely, and that in such situations additional stabilizing mechanisms might be required to prevent facultative sexuals from being displaced by obligate asexuals. In conclusion, our model suggests that loss of sexual reproduction in these facultative sexuals can be prevented, in principle, by a ‘lineage-specific’ life-history feature (sex linked to RE production), which is irrespective of any ‘universal’ genetic benefits of recombination. We thus advocate an explicit prior analysis of the costs of sex and possible ‘lineage-specific’ stabilizing mechanisms in facultative sexuals, before drawing inferences about ‘universal’ mechanisms for the maintenance of sex.

Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

Authors' contributions

Both authors made substantial contributions to this paper's conception and design, acquisition and analysis of data and drafting and revisions.

Competing interests

We have no competing interests.

Funding

Funding was provided by an Austrian Science Fund (FWF) grant P20735 to C.P.S. J.L. was funded by a University of New South Wales Vice-Chancellor's Postdoctoral Research Fellowship and a University of New South Wales Early Career Research Grant.