2 M. J. KEELING AND D. A. RAND (Lively et al. 1990, Jokela & Lively 1995, Weeks 1996), as well as more general accounts (Ghiselin 1974, Tooby 1982, La

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1 SPATILA HETEROGENEITY AND THE MAINTENANCE OF SEX M. J. KEELING AND D. A. RAND Abstract. The persistence of sexual reproduction is one of the outstanding problems in evolution. Two main explanations have arisen: mutation clearance and the enhanced spread of advantageous traits, including resistance to parasites. Eective parasite resistance sucient to maintain sex usually requires complex dynamics, temporal genetic cycles and is sensitive to parameters. Here we introduce both a probabilistic cellular automaton (PCA) model and a more conventional continuum model which utilise emergent spatial heterogeneities rather than such cycles. Not only do obligate sexuals reliably outcompete asexuals in this model, but we are also able to address the evolution of partial sexuality S. Although most models predict a low value of S which is not compatible with observed obligate sex, ours demonstrates evolution to a highly sexual state. In addition, we also observe bistability where the asexual state is also evolutionary stable casting a new light on the debate about the existence of persistent asexuals. We also consider the eects of dispersion and geographic parthogenesis and highlight the dierences with current theories. 1. Introduction Despite its long history - the problem was recognised by Darwin as early as the question of why sex is so stable and does not more easily give way to parthogenesis is still hotly debated (e.g. Maynard Smith 1978, Hamilton 1980, Bremermann 1980, Hurst & Peck 1996). Since sex carries a high cost (the two-fold reproductive disadvantage whereby a mutant gene causing parthogenesis is passed on to twice as many grandchildren as the sexual alternative), there must be powerful forces supporting it. However, it still is unclear what these forces are or how we can identify them. Current theories either assert that sex enables the creation and spread of advantageous traits or alternatively removes deleterious genes. The rst class contains those theories that claim that sex is associated with host resistance to parasites (Hamilton 1980, Bell 1982, Hamilton et al. 1990). Because of their high turnover rates, the time-scale of parasite evolution is relatively so fast that they have a great evolutionary advantage over their hosts because they can constantly side-step antiparasite adaptions and make them obsolete. Sex, on the other hand, by continually creating novel genotypes, can respond to this pressure. The possibility of this perpetual `arms-race' has become known as the Red Queen hypothesis. This line of explanation has received much of the attention from both theoretical (Rice 1983, Weinshall & Eshel 1987, Roughgarden 1991) and experimental and eld researchers Key words and phrases. sexual reproduction, persistence, cellular automaton, genetic variation, parthogenisis. 1

2 2 M. J. KEELING AND D. A. RAND (Lively et al. 1990, Jokela & Lively 1995, Weeks 1996), as well as more general accounts (Ghiselin 1974, Tooby 1982, Ladle et al. 1993, Judson 1995, Hurst & Peck 1996). In a series of early papers Hamilton and coworkers developed a range of coevolutionary models of this process which depend upon intrinsic temporal cycling of genotypes in which common (and hence highly parasitised) genotypes are constantly replaced by currently novel genotypes reconstructed by recombination (Hamilton 1980 & 1982). Although support for this viewpoint has come from a number of directions, including studies on selection on the immune system (Hughes et al. 1994), it is largely indirect and correlative and there are fundamental open questions. Amongst these are the following: Why is obligate sex so common when apparently a little sex appears to be the optimal balance between the two-fold advantage of asexuality and the benets of sex? As Hurst and Peck (1996) point out, to ask why mammals are sexual when there is a two-fold cost to sex and to simply compare obligate sexuality and asexuality is to address the wrong question. They claim that, for the most part, analysis of typical evolution-of-sex theories suggests that most of the benets of sex accrue when only a small fraction of the ospring are produced sexually (Green and Noakes 1995). Does the fact that almost all species have problems with parasites imply that there should be no persistently asexual species (e.g. Judson 1996)? As has been stated, apparent asexuals are something of an \evolutionary scandal" (Maynard Smith 1986) and the persistence of asexuality poses a powerful challenge for theories of sex. How are the evolutionary conclusions altered if instead of just competing sexuals against asexuals one allows for gradual evolution of the sexuality ratio S (i.e. the proportion of ospring that are produced sexually)? This question has obvious connections with the above two but has not been treated in any parasite models to our knowledge. In this paper we describe a spatial and stochastic approach to this problem which allows us to obtain some new insights on these questions as well as the the general problem of calculating the strength of parasite challenge and the eectiveness of sex in combating it. We also consider geographic parthogenesis and the eects of dispersion. It is not just space that is important in our model but also the fact that individuals are discrete entities and that local populations sizes are relatively small. This is important because, unlike previous models, in ours the evolutionary dynamics is uctuation-driven. For sexual hosts, the host-parasite dynamics cause large uctuations in the local spatial genetic structure at any one time and these deny the rapidly mutating parasites an evolutionary target. In contrast, asexual reproduction produces spatial clumps of genetically identical individuals which can be eciently exploited by the rapidly evolving parasites. We believe that this is the rst model which uses spatial eects to explain the persistence of sex, although others have previously observed in general terms that space may play some such role (Ghiselin 1974, Bell 1982, Hamilton et al. 1990, Hurst and Peck 1996). It has some similarities to the Tangled Bank model (Bell 1982) which stresses that sexual organisms benet from having diverse progeny so that a few may be well suited for a varying environment. However, our mechanism

3 MAINTAINING SEXUAL REPRODUCTION 3 is very dierent because it works at the individual level by providing them with a spatial buer of genetic variation. There are other signicant dierences between it and both the Tangle Bank and Hamilton's models involving temporal cycling. For example, the latter will be sensitive to the ratio of parasite and host turnover times. If this is very small as it is for viral or bacterial parasites of mammals then the parasite has time to evolve before a signicant proportion of the host population has been replenished and the protection oered by the cycling is greatly reduced. By contrast, since our approach is spatial rather than temporal this produces only a very weak eect which we can estimate by studying the variation with the mutation rate (see Section 3). Moreover, another dierence can be understood if one thinks of the interaction between this spatial mechanism and immune systems. The spatial uctuations can be seen as providing a way of covering epitope space which is a cheap alternative to providing a too-expensive `Rolls Royce' immune system. In this way individuals can be much less than 100% ecient at covering epitope space and can use the coverings of their neighbours who will provide them with a spatial buer for the parasites they would not recognise. We conjecture that, at least in part, this is behind the diversity of some immune system components such as the MHC. The temporally cycling model does not seem to have comparable immunitystrengthening eects. Finally, the obvious dierence is that patterns of infection will be very dierent. Temporal cycling will produce lower diversity and more violent epidemics in which the currently dominant parasite infects the dominant host type whereas spatial uctuations imply high diversity of hosts and parasites with a smouldering infection. We extend the model in Keeling & Rand (1996) and, in addition address the above questions. Moreover, we give much better treatment of the evolution of sexuality S. The treatment in this previous paper of the allocation of S to the ospring had an unnatural dependence upon the parents sexuality and here we consider a more natural mechanism. Moreover, that paper depended too heavily upon simulations which were so computationally intensive that extensive explorations of parameter space was impossible. Although we did include some simple continuum models there to indicate the ubiquity in parameter space, we believe that this is a valid criticism and therefore, together with simulations, present here an analytically tractable model which overcomes most of these objections. Briey, our conclusions are that spatial uctuations provide a robust mechanism for sexual stability in which obligate sexuals can evolutionarily out-compete asexuals. Moreover, for partial sexuals the sexuality ratio S evolves as follows. Asexuality (S = 0) is evolutionarily stable so that near-asexuals will evolve to asexuality. This is because there are too few sexuals to provide an adequate buer of diversity for their own protection and thus the parasite resistance obtained is not enough to overcome their twofold disadvantage. For a wide range of parameters moderately sexual populations evolve to nearobligate sexuality. Thus our model contains multiple evolutionary attractors. Clearly the evolutionary stability of asexuality gives a new answer to the above question about ancient asexuals because it suggests that their existence is perfectly natural even in the presence of parasites. On the other hand, the model suggests that a reasonable amount of sex will lead to near-obligate sex. This addresses the question above about the optimality of a little bit of sex. We also nd that dispersion helps asexuals not because it enables them to outrun the parasites as has been previously suggested (Bell 1982, Koella 1993) but because it helps them to average out the

4 4 M. J. KEELING AND D. A. RAND local spatial genetic uctuations giving the parasites more of a mean genotype to evolve towards. Finally, low quality habitats favour asexuals because their is a lower density of hosts found there and consequently lower numbers of parasites. 2. The individual-based spatial model The spatial model is a synchronously updated probabilistic cellular automaton (PCA) and is an extension of the system used by Rand, Keeling and Wilson (1995). Each site is either vacant or occupied by a host each of which also has associated to it an 8-bit genotype GH (from 0 to 255) and a sexuality level S which is the proportion of attempted reproductions that are sexual. So an asexual has S = 0 and a obligate sexual has S = 1. In addition, each host can be infected by a multiplicity of parasites, each of which has its own 8-bit genotype GP. Both hosts and parasites are assumed to be haploid, so that we are concerned only with a single 8-bit string. The eect of a parasite on its host is determined by the match M(GH ; GP ), which is the proportion of bits where the host and parasite genotypes are identical. The match between the host and parasite genotypes aects the host's growth rate g, the transmissibility T of the parasite to that host, the virulence V and recovery rate R. The eect of the match is non-linear (M is squared), so that parasites which match host genotypes closely do far better. We assume the following functional forms for all our models discussed here: g =?? Q (2? S) 1 + gg GH 255?g0 + gm M(GH ; GP i ) 2 parasites T = T0 + TM M(GH ; GP ) 2 (1) V =? V0 + VM M(GH ; GP ) 2 R = R0 + RM M(GH ; GP ) 2 The growth rate, transmissibility, virulence and recovery rate are respectively the probability per unit time (i.e. per iteration) of a host colonising an adjacent empty site, a parasitised host infecting a neighbouring host, a parasitised host dying from the infection and recovery from the infection. Hence the greater the match between host and parasite genotypes the greater the risk of infection and death from the parasite (TM ; VM > 0), and the lower the risk of recovery and growth while parasitised (g0 < 1 and gm ; RM < 0). There is also a low constant natural death rate d of both the healthy and infected hosts. The way in which a host colonises neighbouring empty site with ospring requires some further explanation. The (2?S) term in the growth rate ensures that asexuals receive the correct reproductive advantage (twofold if S = 0) over sexuals, and that this advantage decreases linearly with the rate of sexual reproduction. If a host is going to reproduce asexually into a neighbouring site then the ospring inherits the parent's genotype and sexuality level. However, if a host is going to reproduce sexually then it searches within a small radius (10 cells) for another individual which also wishes to breed sexually. If no mate is found the host fails to breed; otherwise the ospring's genotype is obtained by free recombination among the 8 loci of the two parents and its sexuality is chosen randomly to lie somewhere between the parents' values. We believe that sexual organisms have an advantage due to the high genetic diversity in the population. The diversity would be less surprising if all the hosts were equally t. Therefore, in order to determine whether the presence of parasites

5 MAINTAINING SEXUAL REPRODUCTION 5 is sucient to maintain genetic diversity, we force the growth rate of the host to be an increasing function of its genotype (controlled by a parameter gg). Thus in the absence of parasites a single (maximum) genotype dominates. Of course this feature is not necessary for the workings of the sexual-asexual competition part of the model; in fact, sexual organisms usually prot from setting gg = 0 as this leads to even greater genetic diversity of the sexual population. The genotype of the parasite is subject to slow reversible mutation, the probability of a change occurring at any one bit is per iteration. This will necessarily be much larger than the mutation rate per replication observed in the natural world ( 10?6 ) because a single time step of the model corresponds to many parasite generations. A more natural measure may be the number of parasite mutations expected per host generation; as an approximate guide, for a host with a lifespan of one year, a mutation rate = 0:1 roughly corresponds to a parasite which can produce thousands of ospring per day. After mutation, both the new mutant and old resident parasite are assumed to be present in the host, and as closer genetic matches are favoured we tend only to observe mutation in that direction. With such a computationally intensive simulation and several parameters a comprehensive sweep of parameter space is not feasible. Instead for most of this discussion the parameters used will be those in table 1, although the results have been found to be qualitatively similar for a wide range of parameters about these values. To explore parameter space we have constructed a simple continuum model of this individual based system which is discussed in section Dynamics of the spatial system We will rst consider the dynamics in the absence of competition between different levels of sexuality. When all individuals are fully sexual (S = 1), despite the strong selective pressure towards the higher growth rate of higher genotypes, controlled by the parameter gg, parasitism is able to maintain great genetic diversity (Fig. 1). The main graph shows the proportion of each of the 256 genotypes. There is a clear trend towards high genotypes with the higher intrinsic growth rates, although these in no way dominate. Some of the irregularity can be prescribed to the binary nature of the genetic string; 255 and 127 only dier at one position and yet have very dierent eects on the growth rate. The inset gure shows the density of ones at each position in the genotype, all lie above the 50% level. The parasite genotypes display a more skewed and more stochastic distribution; the greater number of parasites with high genotypes balancing the growth advantage possessed by hosts with high genotypes. After short transient dynamics, there is little change in the distribution of genotypes over time, and this appears to be consistent in all the simulations examined. This demonstrates that globally although the precise pattern is in ux, we quickly achieve statistical xation. We now return to our main goal, understanding the competition between sexual and asexual organisms. A typical spatial pattern for competition between sexuals and asexuals taken from the early stages of a simulation is shown in Fig 2. This simulation used the parameters given in table 1. It clearly shows large homogeneous asexual patches in a sea of genetically diverse sexual organisms. It is the local genetic heterogeneity that gives the sexual individuals an advantage because it denies the parasites the evolutionary target they have for asexuals.

6 6 M. J. KEELING AND D. A. RAND We wish to consider the competition between dierent levels of sexuality at a range of parasite mutation rates, although initially only competition between purely sexual (S = 1) and purely asexual (S = 0) organisms is considered (Fig. 3). The simulations are started with a low number of both sexuals and asexuals of random genotypes, randomly distributed on the lattice. The parasite mutation rate was varied between simulations and the long-term population densities recorded. Each host's sexuality is held constant and this is passed to its ospring, thus sexual and asexual population remain genetically distinct. The results can be described as follows. For large values of in the range 0.5 to 1 the parasites can adapt so fast that they can take advantage of any individuals in their local environment. This leads to a low number of hosts of both types and highly stochastic results. For lower values of (between 0.04 and 0.5) the parasites are only able to fully specialise if there are areas of genetically identical hosts, as only then is there sucient time for the necessary number of mutations. Otherwise, they have no evolutionary target. This means that only the patches of identical hosts formed by asexual clones are badly aected by parasites and these are soon wiped out. As the value of is decreased still further, larger and larger patches of identical hosts are required for parasite specialisation. Hence small patches of asexuals are relatively immune from the worst eects, so low densities of asexuals can persist in the population although it is dominated by sexuals. Finally, for very small mutation rates, the size of patch necessary for specialisation becomes comparable with the size of the lattice and the initial double growth rate of the asexuals forces out the sexuals before the parasites can have a detrimental eect. The size of the lattice and the initial proportions of sexuals and asexuals has a slight eect on the quantitative values for small but the general behaviour of the system remains unchanged. These simulations predict that, with simple competition between sexual and asexual individuals, intermediate values for the number of parasite mutations per host generation (as captured by the parameter ) favour sex despite asexuality's two-fold reproductive advantage. 4. A deterministic continuum model From the PCA we have observed that because ospring are always positioned close to their parents, asexual clones produce patches of genetically uniform individuals whereas sexual individuals give rise to more heterogeneous habitats. We will attempt to formulate a simple model of this phenomenon by considering a population of a given sexuality S in isolation. The spatial aspects are captured because we assume that a parasite can only spread between adjacent hosts, and that there is a given correlation between the genotypes of adjacent hosts. Let ES, HS, and PS be the density of empty, healthy and parasitised hosts of sexuality S. In this model a host can only be infected by at most one strain of parasite. We will dene DS(q) to be the proportion of the parasitised hosts where there are matches at q sites between the host and parasite genotypes. Other parameters are the growth rate of healthy hosts gh, the growth rate of parasitised hosts gp and the natural death rate d. To keep the equations simple, the growth rates are not dependent on the genotype of the host (as they were in the PCA), this means that each bit of the genome behaves identically. The virulence of the parasite V (q) and the transmissibility T (q) of a parasite both depend on the match

7 MAINTAINING SEXUAL REPRODUCTION 7 (q) between host and parasite genotypes, as before. Remaining parameters are size of the genome G and the probability m of a match at a given loci between adjacent hosts. It is only through the use of m that we are able to include some form of local spatial correlations in this mean-eld approach. As m! 0:5 all local correlations are lost and therefore there is no dierence between the spatial environments of the sexual and asexual populations. Our equations are derived for the stochastic process underlying the PCA. An advantage is that they do not depend upon the lattice structure of the PCA. One obtains that the empty, healthy and parasitised populations obey the following equations: _ ES = d [HS + PS] + X q PS DS(q) V (q)? ES (2? S)[gH HS + gp PS] _ HS = ES (2? S)[gH HS + gp PS]? X q;r HS PS DS(q) T (r) P(rjq; m)? d H S (2) X X _ PS = HS PS DS(q) T (r) P(rjq; m)? q;r q P S D S (q) V(q)? d P S where P(rjq; m) is the probability that a parasite and a neighbouring host match at r loci, given the parasite matches its host at q loci and that the probability of a match between adjacent X hosts is m. Using simple probabilistic arguments this is, i r? i P(rjq; m) = m i (1? m) q?i (1? m) r?i m G+i?q?r q G? q i We now need to consider the behaviour of DS, which tells us about how specialised the parasite has become. This equation is calculated by considering the rate of parasitism between hosts and parasites which match at q loci: d dt D S(q) = X r HS PS DS(r) T (q) P(qjr; m)? D S (q) [V(q) + d] + DS(q? 1) [G + 1? q]? DS(q) [G? q]? Nq (3) where Nq is a normalising factor such that the DS consistently sum to 1, and is the mutation rate per loci. It should be noted that, as with the PCA, only mutation to a greater match is considered due to within-host competition. The virulence and transmissibility of the parasite are functions of the match between host and parasite genotypes, as they were in the cellular automata. V (q) = V0 + q 2 VM T (q) = T0 + G q 2 TM G The non-linearity in the match q is necessary so that sexuals, whose average match is at least 0:5, can out-compete the asexual, whose average match tends to 1. All that remains is to approximate the match between neighbours m in terms of the host's sexuality S and the number of neighbouring sites N bd (Appendix 1). For all the simulations we will take m = ? S S(2? S) 8 N bd which is a good estimate for purely sexual or purely asexual populations In this form of deterministic, continuum model the question of competition can be simply assessed as follows. Starting from mainly hosts and a low density of

8 8 M. J. KEELING AND D. A. RAND parasites, two populations are allowed to proceed in isolation until they reach the equilibrium xed point. After which, a small amount of coupling " is introduced corresponding to competition across a small boundary. This coupling allows hosts from one population to colonise empty sites from the other and for parasites to be transmitted between the two populations. For small couplings there will be little change from the proportions at the independent equilibrium. Hence whichever population has the greatest initial growth rate (after coupling is introduced) will eventually dominate. We shall dene the competitive pressure to be initial rate of change from isolated equilibrium = lim "!0 " Figure 4 shows the competitive pressure for sexual individuals competing against asexuals which possess the full two-fold reproductive advantage (parameters from table 2). These results agree with the behaviour of the PCA; for small parasite mutation rates () the sexual species has a clear advantage, whereas for large the asexuals dominate. As expected from such a simple set of equations competitive coexistence is impossible. Note that in the PCA model for very small the asexuals again do well, however this is due to the nite size of the lattice and the discreteness of individuals. With the dierential equation model there are always some small number of parasites (perhaps a fraction of an individual) which exactly match any given genotype and can grow rapidly destroying the hosts. The inclusion of a small noise term and the removal of negative values for the specialisation DS(q) can be used to address this issue, allowing asexual individuals an initial advantage. However, this modication was not included as the level of noise is fairly arbitrary, depending on the population sizes and large scale spatial patterns we wish to model. The four inset graphs show DS=1 for the sexual population in isolation; that is the proportion of infected hosts with a given match between host and parasite genotypes. As the mutation rate increases we observe a greater match between host and parasite genotypes, which in turn leads to the sexual species' downfall. It is clear that when the competitive advantage is zero, the distribution DS=1 is heavily skewed showing that the mutation rate of the parasite is rapidly compensating for the heterogeneous environment presented by the sexuals. This simple model uses the spatial heterogeneity of the sexual population (as captured by m) to give the sexuals a competitive advantage against genetically homogeneous asexuals. This set of dierential equations has allowed us to examine the robustness of this phenomenon - something that was not possible with the PCA model. Sexuality was found to be favoured by low mutation rates of parasites where a close match has a pronounced eect (i.e. VM and TM are large). This agrees with our ndings for the full spatial simulations. 5. Gradual evolution of sexuality S The PCA model shall now be extended to the situation where the initial conditions contain hosts with a range of sexualities S (Fig. 5). The hosts sexuality S is now also subject to small random mutations to allow its evolution. The mutations occur at each generation and are uniformly distributed in the range?10?3 to 10?3. Instead of two isolated communities (one sexual, one asexual) the partially sexual hosts can now inter-breed and we frequently observe evolution to an optimal sexuality S. This set of simulations has been performed using both the parameters from table 1 (Fig. 5) and for parameters where the eect of an exact match of

9 MAINTAINING SEXUAL REPRODUCTION 9 genotypes is far less. Both simulations show very similar results, and the same qualitative behaviour as seen previously in the pure competition model. The conclusions are shown in Fig. 5. For high parasite mutation rates all hosts are subjected to extreme parasitism so obligate asexual individuals do very well. As the mutation rate decreases so the optimal sexuality S increases to a maximum of about 0.8; it is clear from the spread of sexualities (which is small and scales with the mutation rate in sexuality) that in this region the optimal sexuality is strongly attracting. For still smaller mutation rates the average sexuality declines and the standard deviation of the sexualities present increases. This is partly attributable to an increasing number of completely asexual individuals. If the mutation rate is decreased even further then eventually pure asexuality once again dominates, but the results are obscured due to the frequent extinction and subsequent reintroduction of the parasites, the highly stochastic nature of the system and the increasingly long transients. Longer simulations showed a more marked decline in sexuality at the lowest mutation rates. This varied sexuality model was also tested for a variety of initial conditions. Starting with a uniform mix of sexualities then evolution towards S is observed. When starting with mainly pure asexuality then S remains close to zero demonstrating that asexuality is stable. If the system was started from pure sexuality however, then when S is large the decline in sexuality was found to be very slow, changing by less than 2% in a hundred thousand time steps. Thus parameter regimes can be found where either sexuality or asexuality is relatively stable to slow mutation as well as direct competition. We have postulated throughout that it is the spatial heterogeneities in the system that can lead to the selective pressure toward high sexuality; this can be demonstrated in the following way. If the parasites could be transmitted over a larger range (while the transmissibility was correspondingly reduced) then, for large ranges, the parasites should experience some form of global average and the sexual advantage of spatial heterogeneity is lost. By increasing the range of the parasite we move from a spatial to a mean-eld model (cf Keeling and Rand 1997) and this has a marked eect of the average sexuality S (Fig. 6a). For each neighbourhood type of the parasite the average dispersal distance was calculated as the root-mean-square distance to every cell in the neighbourhood. There is a sharp decline in sexuality once the dispersal of the parasites has exceeded 4 cells and for dispersal much larger than 6 cells then a pure asexual population is the inevitable outcome. Once again the initial drop in sexuality is associated with an increase in the standard deviation and an increased proportion of purely asexual individuals. If the mutation rate was varied at these larger dispersal ranges then similar results to gure 5 are obtained but with the average sexuality substantially reduced. This predicts that systems which are well mixed (closer to the mean-eld) should have comparatively more asexual species than their heterogeneous counterparts. For example there should be more asexuals within aquatic systems than terrestrial, and more in fresh-water than marine (Bell 1982). This result is also in strong agreement with the result that plants with long distance dispersal (preventing the formation of large clonal patches) have lower recombination rates and therefore behave more like asexual organisms (Koella 1993). Finally we turn attention to a common spatial feature of many species where both sexual and asexual types are found. The phenomena termed geographic parthenogenesis (Vandel 1928) describes the fact that for many species where both asexuals

10 10 M. J. KEELING AND D. A. RAND and sexuals exists, asexuals are almost always located at the extreme limits of the species' habitat (Glesener 1978, Bierzychudek 1985). On a lattice we forced a 50 cell border to be an increasingly hostile environment by reducing the growth rate of organisms linearly to zero at the extreme outer edges. The average density of hosts and the average sexuality as a function of distance from the centre of the lattice was recorded (Fig. 6b). It is clear that asexuality is favoured near the borders and this is due to the lower numbers of parasites that exist at the lower density of hosts found at the edges. Although the growth rate declines over a 50 cell region, very little aect is noticed until nearly half way through this zone and then the change is dramatic. The slight increase in sexuality (from the center to a maximum at 75 cells out), may be due to slightly higher parasite levels which percolate through from the asexual border. 6. Conclusion and discussion Despite its simple generic formulation, the PCA model has demonstrated that spatial heterogeneity can replace complex temporal cycles as a mechanism for the maintenance of sex. It is a robust mechanism, powerful enough to overcome the twofold disadvantage for a wide range of parameters. When we consider competition between obligate sexuals and asexuals then we observe the following results. If the level of parasite mutation per host generation is very low, corresponding to relatively short-lived hosts, it is advantageous to be asexual. However, for hosts with longer life-spans, where the mutation rate per generation is higher, sexuality was strongly favoured. For very long lived creatures, asexuality is again predicted, however at this extreme scenario mutation clearance may begin to play a more signicant role. We nd that in our model greater dispersion favours asexuals not because it enables then to outrun parasites but because it enables them to average out the spatial uctuations in the host genotype and thus have a clear evolutionary target. We also observe geographic parthogenesis because in poorer habitats there is a lower density of host and hence the parasite load is aectively decreased and this favours asexuals. When the level of sexuality S was allowed to gradually evolve, although pure sexuality was never favoured, near-obligate sexual reproduction were predicted, overcoming the dilemma that \a little sex can go a long way" (Hurst and Peck 1996). It was found that in general sexuality is favoured by a fairly high mutation rate (per generation) of deleterious parasites, low levels of spatial mixing and high densities of hosts and parasites, which is in agreement with observations. The possible evolutionary bistability whereby both asexuality and near-obligate sex can be stable with the same parameter values opens a new aspect to the current discussion about persistently asexual species. The simple deterministic model (equations 2 and 3) allows us to consider a wide range of parameters and gives clear insights into the mechanisms involved. The results of this system are in good agreement with those of the PCA. Although they fail to predict asexuality for low mutation rates as this is a feature of the discrete, nite nature of the spatial model. It is because of the robust nature of the results from this deterministic model that we claim that the behaviour of the PCA model is not an artifact of the exact system used, but is a plausible explanation for the persistence of sexual reproduction. The deterministic model has highlighted many features not apparent in the PCA; the size of the genome has very little eect on

11 MAINTAINING SEXUAL REPRODUCTION 11 the dynamics, and the model is robust to changes in birth rates, transmissibility and virulence. The main parameters which inuence the qualitative behaviour of the system are the mutation rate of the parasite and the match between adjacent host m. As m is determined by the spatial arrangement of hosts, this upholds our assumptions that it is spatial eects which are driving the PCA model. Many more questions still remain to be answered: the eects of diploidy and sexually reproducing parasites have been ignored; a more realistic genotype could be included; and parameters could be chosen to match situations observed in the natural world. However, we believe that our results clearly indicate that spatial heterogeneity in the genotypes of sexual hosts to be one of the most important factors in the persistence of sexual reproduction. A fundamental problem is to determine experimental signatures to distinguish theories such as this from the alternatives. We believe that the dierences discussed in the introduction will provide a basis for this. ACKNOWLEDGEMENTS We would like to thank Joel Peck and especially Laurence Hurst for their invaluable comments and suggestions. We are grateful to the Wellcome Trust for their support of M.J. Keeling and thank UK EPSRC and NERC for their support of D.A. Rand. APPENDIX Let R be the relatedness of neighbouring hosts, such that clones have R = 1 and unrelated individuals have R = 0. If we assume that the genotype of a mate is completely random then parents and their sexually reproduced ospring will have of relatedness of 0:5. We can now formulate the probability of a match m in terms of the relatedness R m = 1 2 (1? R) + R = 1 (1 + R): 2 Consider two neighbours A and B; if A was born last then A and B can only be parent and ospring if B gave birth to A, this occurs with probability Nbd 1. A similar argument holds if B is younger than A. Hence with probability Nbd S one is the sexual ospring of the other and the relatedness R = 1 2, and with probability 1?S Nbd they are clones and the relatedness R = 1. Taking the worst possible case for the sexuals, we assume that if two neighbouring hosts aren't parent and ospring, then they are only separated by two generations. This leads to, R 1 S 2 + (1? S) + N bd? 1 N bd N bd 2 1? 2 S + S(2? S) 4N bd S 2 + (1? S) 2 this is the equation used for throughout the deterministic model. Hence, when S = 0 (asexuality) R = 1 and m = 1, whereas for the sexual case (S = 1) R = Nbd 1 = 3 8 and so m = Nbd 1 = This is very close to the results from the model which predict m = 0:672. It should be noticed that the parasites will act to reduce m as closely matching host pairs should suer greater parasitism and therefore greater mortality, however this eect is ignored.

12 12 M. J. KEELING AND D. A. RAND References [1] Bell, G The masterpiece of nature: the evolution and genetics of sexuality. University of California Press. [2] Bierzychudek, P Patterns in plant parthenogenesis. Experientia [3] Bremermann, H. J Sex and polymorphism as strategies in host-pathogen interactions. J. Theo. Biol [4] Green, R. F. and Noakes, D. L. G Is a Little Bit of Sex As Good As a Lot. Journal of Theoretical Biology [5] Ghiselin, M. T., 1974 The Economy of Nature and the Evolution of Sex. Berkeley UCP. [6] Glesener, R. R. and Tilman, D Sexuality and the components of environmental uncertainty: Clues from geographic parthenogenesis in terrestrial animals. Am. Nat [7] Hamilton, W. D Sex versus non-sex versus parasite. Oikos [8] Hamilton, W. D., Axelrod, R. and Tanese, R Sexual reproduction as an adaptation to resist parasites (A Review). Proc. Natl. Acad. Sci [9] Howard, R. S Selection Against Deleterious Mutations and the Maintenance of Biparental Sex. Theoretical Population Biology [10] Hughes, A. L., Hughes, M. K., Howell, C. Y. and Nei, M Natural selection at the class II major histocompatibility complex loci of mammals. Phil. Trans. R. Soc. Lond [11] Hurst, L. D. and Peck, J. R Recent advances in understanding of the evolution and maintenance of sex. TREE [12] Jokela, J. and Lively, C. M Parasites, Sex, and Early Reproduction in a Mixed Population Oresh-water Snails Evolution [13] Judson, O. P Preserving genes - a model of the maintenance of genetic-variation in a meta-population under frequency-dependent selection. Genet. Res. 65, [14] Keeling, M. J. and Rand, D. A A spatial mechanism for the evolution and maintenance of sexual reproduction. Oikos [15] Keeling, M. J. and Rand, D. A., 1995 Spatial Correlations and Local Fluctuations in Host- Parasite Systems in From Finite to Innite Dimensional Dynamical Systems (ed. Glendinning, P. ) Klewer, Amsterdam [16] Koella, J. C Ecological correlates of Chiasma Frequency and Recombination Index in Plants. Biol. J. Linn. Soc. 48, [17] Ladle, R. J. Jonnson, R. A. & Judson, O. P. Coevolutionary dynamics of sex in a metapopulation: escaping the Red Queen. Proc. R. Soc. Lond. Ser. B 253, [18] Lively, C. M., Craddock, C. and Vrijenhoek, R. C Red Queen Hypothesis Supported by Parasitism in Sexual and Clonal Fish Nature [19] Maynard Smith, J The evolution of sex. Cambridge University Press. [20] Maynard Smith, J. Contemplating life without sex. Nature 324, [21] Peck, J. R A Ruby in the Rubbish - Benecial Mutations, Deleterious Mutations and the Evolution of Sex Genetics [22] Rand, D. A., Keeling, M. J. and Wilson, H. B Invasion, stability and evolution to criticality in spatially extended, articial host-pathogen ecologies. Proc. R. Soc. Lond. B [23] Rice, W.R Parent-ospring Pathogen Transmission - a Selective Agent Promoting Sexual Reproduction American Naturalist [24] Roughgarden, J The Evolution of Sex American Naturalist [25] Tooby, J Pathogens, polymorphism and the evolution of sex. J. Theor. Biol. 97, [26] Vandel, A La Parthenogenese geographique I. Bull. Biol. France Belg. 62, [27] Weeks, S. C A Reevaluation of the Red-queen Model for the Maintenance of Sex in a Clonal-sexual Fish Complex (poeciliidae, Poeciliopsis) Canadian Journal of Fisheries and Aquatic Sciences [28] Weinshall, D. and Eshel, I On the Evolution of an Optimal Rate of Sexual Reproduction American Naturalist

13 MAINTAINING SEXUAL REPRODUCTION 13 TABLES g0 gg gm T0 TM V0 VM R0 RM d 0:01 1:0?0:007 0:2 0:6 0:0 0:6 1:0?0:9 0:009 0:05 Table 1. Standard parameter values for the PCA model gh gp T0 TM V0 VM d G 0:01 0:005 0:5 2:5 0:1 0:0 0: Table 2. Standard parameter values for the deterministic model

14 14 M. J. KEELING AND D. A. RAND FIGURE LEGENDS Figure 1. The percentage abundance of each of the 256 genotypes shows a marked increase with genotype and exhibits little temporal variation. The results are taken from a population consisting entirely of sexual hosts and represent the average over 500 iterations after transient dynamics have died out. The inset graph shows the frequency with which each of the eight bits has the value one (the dashed line is 50%). Figure 2. Large uniform patches of asexuals (dark grey) are observed in a `sea' of heterogeneous sexuals (light grey) when starting for an initial condition consisting of purely sexual and asexual hosts. The parasites (shown in black) have recently evolved to specialise on a group of asexuals in the top left corner of the lattice and have eradicated them, leaving empty (white) sites ready for colonisation. Figure 3. The average density of sexual and asexual hosts and parasites at a range of parasite mutation rates,. The results are from a lattice and are an average of 500 iterations after 500 iterations have been ignored to remove transients. Figure 4. The competitive pressure of sexual individuals against asexuals, for this simulation " = 10?4. The inset graphs show the specialisation by the parasite DS=1 at mutation rates, = 10?4, 0:02, 0:1 and 0:9. The degree of parasite specialisation increases with mutation rate. Figure 5. The density of sexuality levels for a range of mutation rates, both the average sexuality S and the spread are clear. As we are no longer dealing with simple competition the dynamics are far slower and hence the results are taken as a 500 iteration average after 4500 iterations have passed. Figure 6a. Instead of using the simple 4 cell Von Neumann neighbourhood, larger neighbourhoods can be considered. Diamond shaped Von Neumann type neighbourhoods are marked as crosses, square Moore type neighbourhoods are marked with circles. A solid line shows the mean sexuality, the standard deviation about the mean is represented by the dotted lines. This results are for = 0:1 where sexual reproduction has a large advantage. Figure 6b. The growth rate of hosts in forced to decline from normal at 50 cells distance to zero at 100 cells. For each distance from the centre of the lattice, the average sexuality is shown by the solid line, and the density of hosts by a dashed one. There is a sharp decline in sexuality between 80 and 90 cells distance. (M. J. Keeling) Zoology Department, Downing Street, Cambridge CB3 3EJ, UK address, M. J. Keeling: matt@zoo.camb.ac.uk (D. A. Rand) Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. address, D. A. Rand: dar@maths.warwick.ac.uk