### Abstract

- Top of page
- Abstract
- Introduction
- Model
- Simulation results
- Discussion
- Acknowledgments
- References

The Red Queen hypothesis posits a promising way to explain the widespread existence of sexual reproduction despite the cost of producing males. The essence of the hypothesis is that coevolutionary interactions between hosts and parasites select for the genetic diversification of offspring via cross-fertilization. Here, I relax a common assumption of many Red Queen models that each host is exposed to one parasite. Instead, I assume that the number of propagules encountered by each host depends on the number of infected hosts in the previous generation, which leads to additional complexities. The results suggest that epidemiological feedbacks, combined with frequency-dependent selection, could lead to the long-term persistence of sex under biologically reasonable conditions.

### Introduction

- Top of page
- Abstract
- Introduction
- Model
- Simulation results
- Discussion
- Acknowledgments
- References

Sexual reproduction is widely regarded as enigmatic by evolutionary biologists (Williams, 1975; Maynard Smith, 1978; Bell, 1982). The reason is that an asexual clone would have a per capita reproductive advantage, because every individual can give birth. A clone would therefore be expected to replace competing sexual individuals, assuming all else is equal (Maynard Smith, 1978). One possible explanation for the persistence of sexual females in the face of such fierce competition is that the greater genetic diversity of sexual broods allows for escape from infection by coevolving parasites (Levin, 1975; Jaenike, 1978; Hamilton, 1980; Lloyd, 1980; Hamilton *et al.*, 1990). The basic tenets of this idea (the ‘parasite’ or ‘Red Queen’ hypothesis for sex) have received recent empirical support from a variety of different sources (e.g. Busch *et al.*, 2004; Decaestecker *et al.*, 2007; Jokela *et al.*, 2009; King *et al.*, 2009; Koskella & Lively, 2009; Wolinska & Spaak, 2009; Paterson *et al.*, 2010).

The Red Queen hypothesis has nonetheless been controversial since its inception. In a groundbreaking paper, W.D. Hamilton showed using a simulation model that sex could be favoured by host–parasite coevolution (Hamilton, 1980). However, as pointed out by May and Anderson, Hamilton’s model, which incorporated strong frequency-dependent selection, did not rely on parasites as the unique mechanism for the source of selection (May & Anderson, 1983). Specifically, Hamilton’s model did not incorporate any of the epidemiological details associated with parasites, such as a threshold density in susceptible host population and time-lagged responses by the parasites. In May and Anderson’s model, where such details were incorporated, the conditions under which host–parasite coevolution could select for sex required very high levels of parasite virulence, leading them to suggest that the parasite hypothesis had limited application to real-world situations (May & Anderson, 1983).

May and Anderson also pointed out, however, that the epidemiological details in their model were not ‘brought down from Mount Sinai,’ and that other assumptions might revive the theory. Since that time, however, very little theoretical effort has gone into expanding on the epidemiological details of infection. A notable exception comes from work by Galvani *et al.* (2001, 2003). Using epidemiological models, they showed that sex could be maintained in parasitic helminthes when interacting with the vertebrate immune system. In the present study, I consider some different epidemiological features of host–parasite interactions. Most importantly, I relax the assumption of many models that the number of parasite exposures per host is constant overtime (usually equal to one). Here, the mean number of propagules encountered by each host depends on the number of hosts infected in the previous generation, and, as such, it can change over time. The results suggest that host sex can persist over a broad range of parasite fecundities.

### Model

- Top of page
- Abstract
- Introduction
- Model
- Simulation results
- Discussion
- Acknowledgments
- References

Most models of the Red Queen hypothesis assume that the probability of infection for a particular host genotype depends only on the frequency of the matching parasite genotype in the parasite population. This, in turn, assumes that every host is exposed to exactly one parasite. As such, most models do not allow for numerical dynamics in the parasite population, which could change the probability of exposure or the number of exposures over time. For example, it is possible that when a clone spreads into a sexual population, the total number of infections would increase, thereby increasing the number of exposures to parasites. In the simulations that follow, I kept track of the number of infected individuals in each generation, and the number of infected hosts in one generation directly affected the number of exposures to parasites in the next host generation.

The basic simulation is similar in structure to a previous model (Lively, 2009). I assume a haploid host with two loci, each with three alleles, which determine susceptibility/resistance to parasites. Parasites were also treated as having three alleles at each of two loci, and they were assumed to be asexual. Parasites that matched the host at both loci infected the host; parasites that did not match the host were killed by the host’s innate immune system, resulting in strong selection on the parasite, which is a reasonable assumption for obligate parasites (Salathe *et al.*, 2008). This kind of genetic interface for infection genetics is known as the matching alleles model, and it serves as a framework for modelling the self–nonself recognition in invertebrates (Frank, 1993; Otto & Michalakis, 1998). The matching alleles model can be seen as being at one end of a continuum between ‘gene-for-gene’ models in plant pathology and matching allele models (Parker, 1994; Agrawal & Lively, 2002). Elsewhere, we have found that, over most of the continuum, the genetical dynamics are similar to the pure matching allele models (Agrawal & Lively, 2002).

Similarly, the number of asexual individuals having the *ij*th genotype at time *t *+* *1 is:

- (3)

where A_{ij} is the number of infected asexual clones having the *ij*th genotype at time *t*. Hence, the only differences between the recursions for sexual and asexual genotypes are that the sexuals pay the cost of producing males, and they have recombination between the resistance loci (The results presented below were remarkably robust to the rate of recombination as well as to the assumption that hosts and parasites were haploid).

The probability of infection for either a sexual or asexual individual having genotype *ij* at time *t *+* *1 is given as:

- (4)

where *β* is equal to the number of propagules produced by each infection that also make contact with a host; thus, *β* can be thought of as the realized fecundity of the parasite. On the right-hand side of eqn 4, *β* [*S*_{ijI} + *A*_{ijI}]/*N*′ gives the Poisson mean number of exposures for hosts having genotype *ij* to parasites having a matching (infective) genotype. The exponential term (exp (−*β* [*S*_{ijI} + *A*_{ijI}]/*N*′) gives the probability of the zero class, which here is the probability of not encountering a matching parasite. Equation 4 thus gives the probability of encountering one or more matching parasites. Hence, this formulation assumes that one exposure is sufficient for infection. I also assume that if there are two or more exposures to matching parasites, then only the first parasite establishes, meaning that there are no coinfections in the model. This latter assumption is conservative with respect to virulence whenever unrelated coinfections lead to an increase in virulence (van Baalen & Sabelis, 1995; May & Nowak, 1995; Frank, 1996).

The number of secondary infections, *R*_{0}, produced following the introduction of a single infected individual having genotype *ij* into a large host population is , where *g*_{ij} is the frequency of the *ij*th genotype in the host population (Lively, 2010c). Thus, the spread of infection by an asexual parasite strain that can infect the *ij*th host genotype requires that . The mean value for *R*_{0}, averaged over all parasite genotypes, is simply , where *G* gives the number of genotypes in the population (Lively, 2010c). The spread of the disease requires that *R*_{0 }>_{ }1, which in turn requires *β* > *G* if all host genotypes are equally common.

Virulence (*V*) was calculated in the simulation runs as

- (5)

where *W*_{U} gives the fitness of uninfected host individuals and *W*_{I} gives the fitness of infected individuals. In the present simulations, the fitnesses for the annual host were further defined (following Maynard Smith & Slatkin, 1973) as:

- (6)

and

- (7)

where *b*_{U} gives the number of offspring produced by uninfected hosts in the absence of competitors and *b*_{I} gives the number of offspring produced by infected hosts in the absence of competitors. Similarly, *a*_{U} scales the effect of total host density on offspring production in uninfected hosts, and *a*_{I} scales the effect of total host density on offspring production by infected hosts. Defined in this way, virulence potentially has both density-dependent and density-independent aspects, and it depends on the relative fitness of infected individuals (Lively, 2006). This formulation has two advantages that were not possible in the linear formulations that I used previously (Lively, 2009). First, the density-dependent effects are easily separated from the density-independent effects. For example, for *a*_{I} = *a*_{U}, virulence becomes simply 1 − (*b*_{I}/*b*_{U}) for all host densities. Similarly, for *b*_{I} = *b*_{U}, virulence depends only on host density. Second, for *x *=* *1, the population dynamics are stable for all values of *b* (see Doebeli & de Jong, 1999). In the present simulations, I set *a*_{I} = *a*_{U}, so virulence was constant over all densities, and *x *=* *1, so the numerical dynamics were inherently stable in the absence of infection.

The deterministic simulations all began with a ‘burn-in’ period of 1000 generations, during which the presence or absence of Red Queen dynamics in the sexual population was observed. (The oscillation of host/parasite genotype frequencies over time is known as ‘Red Queen dynamics.’) After the burn-in period, a single clonal individual (hence a single genotype) was introduced into the sexual population, and the simulation was run for another 1100 generations. The goal was to determine the effect of *β* on the genotype frequency changes in the sexual population during the burn-in period and on the fate of the clone. I assumed that males constituted half of the sexual population (*s *=* *0.5) and that the probability of recombination between loci in the sexual population was 0.20. The other parameter values used were as follows: *b*_{U} = either 10 or 20; *b*_{I} = 3; and *a*_{U} = *a*_{I} = 0.001. Thus virulence, *V*, was equal to 0.7 for *b*_{U} = 10 (i.e. *V *=* *1−(3/10) = 0.7), and virulence was equal to 0.85 for *b*_{U} = 20 (i.e. *V *=* *1−(3/20) = 0.85). Finally, I allowed migration into the host population. Infected migrants entered the host population stochastically with a probability of 0.02 for each genotype, and uninfected migrants entered the host population with a probability of 0.10 for each genotype.

### Discussion

- Top of page
- Abstract
- Introduction
- Model
- Simulation results
- Discussion
- Acknowledgments
- References

The model presented here differs from most previous models in several ways. The most important difference is that the number of exposures to parasites was not constant but rather depended on the number of infected hosts in the previous generation. Hence, the probability of infection relied on more than just the frequency of matching parasite genotypes, which implicitly assumes that every host is exposed to exactly one parasite. In addition, the present model allowed for variation in the number of exposures. The mean number of exposures is simply the average number of parasite propagules per host. The probability of exposure is calculated as one minus the probability of the zero class in a Poisson distribution (), where the mean (λ) is number of parasites per host. Hence, the possibility exits that some hosts are not exposed to parasites, unless the mean number of parasites per host is high. For example, if the mean number of parasites per host is 1, then about 37% of the hosts are not exposed to parasites; if the mean number of parasites per host is 4, then only 2% of hosts are not exposed to at least one parasite. Thus, small changes λ in can lead to big changes in the strength of parasite-mediated selection.

The results of this study show that host–parasite coevolution can lead to the stable coexistence of sexual and asexual (mixed) populations over a wide range of values for realized parasite fecundity, *β* (Fig. 1). They also show that oscillations in genotype frequencies over time (Red Queen dynamics) are not required for parasites to select for sex (see also Howard & Lively, 2003). In fact, the results suggest that the parameter space for which sexual genotypes are oscillating prior to the introduction of the clone can be quite narrow (Figs 1 and 2). As such, it would appear that Red Queen dynamics are neither necessary nor sufficient for the Red Queen hypothesis for persistence of sex. On the other hand, genetic drift in small populations might be sufficient to perturb an otherwise stable polymorphism to show oscillatory behaviour (Kouyos *et al.*, 2007).

The results also show that parasite-free sexual populations cannot be taken as evidence to suggest that parasites could not become an important source of selection. The reason is that the spread of a clone into a sexual population reduces the genetic diversity of the host population, thereby increasing *R*_{0} for the matching parasite genotype. If *R*_{0} for the matching parasite increases to be greater than one, then the infection will spread in what was a previously disease-free population (Fig. 3). In addition, the number of parasite exposures per host can increase dramatically as the infection spreads (Fig. 3), which increases the overall strength of selection against the clone. This combination of epidemiological feedbacks, along with the coevolutionary response of the parasite, can prevent fixation of the clone in the short term, allowing for the persistence of sexually reproducing females.

In this study, a single clone was placed in competition with a sexual population. Elsewhere, we have found that sequentially introducing different clones into the sexual population would lead to replacement of the sexual population by a genetically diverse clonal population, unless the reproductive advantage of clones was blunted over time by Muller’s ratchet (Howard & Lively, 1994; Lively & Howard, 1994). Similarly, models that initiate the clonal population as ‘fully saturated’ (meaning that the clonal population contained all the genotypes contained in the sexual population) tend to be less favourable to the parasite theory of sex (e.g. May & Anderson, 1983; Otto & Nuismer, 2004), presumably because it is difficult for the sexual population to move away from the clonal population in genotypic space (review in Lively, 2010a). For similar reasons, parasite-mediated selection may not favour sexual reproduction in species for which individuals could potentially produce both sexual and asexual offspring within the same brood, unless the parasites are transferred from the mother to her offspring (Agrawal, 2006). Finally, parasite-mediated selection is unlikely to prevent the invasion of alleles for self-fertilization, but it can prevent the fixation of such alleles, leading to stable mixed mating in the host population (Agrawal & Lively, 2001). In this study, I found that an obligately sexual population could coexist with an obligately asexual population, provided the asexual population consisted of a single resistance genotype.

The results presented here differ from those found in the epidemiological model by May & Anderson (1983), but the assumptions also differ. May and Anderson assumed that infected individuals either recovered or died. They found that host–parasite coevolution would only favour sexual reproduction if none of the infected hosts recovered, so virulence was extremely high. In the present model, I assumed that infection by parasites did not increase the mortality rate but rather reduced the birth rate of infected individuals. Under these assumptions, parasites selected for sexual reproduction when virulence was greater than 1/(2*f* − *g*), where *f* is the frequency of infection in the asexual population and *g* is the frequency of infection in the sexual population (see Lively, 2010b). It is possible that the frequency of infection in the asexual population was not high enough in May and Anderson’s model to select for sexual reproduction. May & Anderson (1983) also assumed that the clonal population started with the same genetic diversity as in the sexual population (i.e. ‘fully saturated’), which (as indicated above) reduces the advantage of sex. In any case, May & Anderson (1983) were correct to point out that different epidemiological assumptions might be more favourable to the Red Queen.