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A neutral model can explain geographic patterns of sexual and asexual individuals

Many species reproduce both sexually and asexually producing genetically unique and clonal recruits. The relative contribution of sexual and asexual reproduction is mostly considered a consequence of natural selection, for example, disfavouring sexual propagules during invasions. Here, we present a novel model for the invasion of species into a new habitat. The model is fully neutral with respect to the dispersal and the survival of sexual and asexual recruits. In addition to local dispersal through sexual and asexual reproduction, long-distance dispersal is implemented through a non-zero probability for long-range relocations. We parameterized our model using empirical data on the distribution of asexually and sexually recruited individuals of the recently established macroalga Fucus radicans over its current area of distribution, the 8000 year old Baltic Sea. As a solely stochastic alternative mechanism to natural selection, our simulation results suggest local abundance of one sex, that during a persistent phase of establishment will result in a heterogeneous distribution of sexual and asexual recruits. Specifically, the model suggests an initial general dominance of one or a few clones, and this pattern is observed in Fucus radicans as well as in several other species with recent colonization histories.


Introduction
The potential to reproduce both sexually and asexually, producing genetically unique or clonal recruits, is widespread in protists (including macroalgae) and aquatic and terrestrial plants, and also found in some species of social and colonial animals such as aphids, wasps, ants and corals [1][2][3]. Although sexual reproduction implies extra individual costs it is likely to be beneficial from an evolutionary perspective [4,5] for the simple reason that more than 99% of all species reproduce sexually.
Fisher and Muller showed genetic disadvantages of clonal reproduction in large populations being due to its weak evolutionary potential [6][7][8]. This effect is particularly distinct in non-equilibrium systems where adaptation to changing conditions is favourable [4,9]. However, asexual reproduction tends to dominate under certain circumstances, for example, during invasions [10,11], and at the edges of species' geographic ranges [12,13] resulting in sexual individuals and asexual clones showing spatial separation (termed geographic parthenogenesis) [14,15].
Explanations of clonal advantage include better colonizing ability at low densities [16], adaptive mechanisms like the frozen niche variation model [17] and insusceptibility to inbreeding [18]. In some species, asexual reproduction may simply be the consequence of species formation by hybridization [15,19]. Peck et al. modelled range expansion of competing sexual and asexual individuals into a habitat with a strong environmental gradient and a severe selection pressure for local adaptation [20]. The simulations confirmed the considerable advantage of clonal reproduction in marginal habitats, under these conditions. Other models show that asexual individuals may be favoured in marginal populations because the relative cost of homozygosity will be lower in already inbred, small and marginal populations [18,21]. Although these models shed light on possible mechanisms favouring an asexual reproduction strategy, they do not address the more general question if facultative uniparental reproduction per se may produce observed patterns of spatial separation between sexual and clonal individuals. Hence, novel and neutral models would be valuable for further investigations of spatial population structures as a function of the interplay between sexual and asexual reproduction.
The present contribution offers a novel approach to the problem of understanding the role of asexual reproduction during invasions and the consequences this may have on the population genetic structure of species. We present a model and simulation results on invasions of a species reproducing both sexually and asexually. A particular focus is on the spatial distribution of sexes during invasion, which has not explicitly been taken into account in this context before. As study species we focus on the macroalga Fucus radicans, a recently formed species that has established in the 8000 y old Baltic sea [22,23] for which there is comprehensive population genetic data [24]. We constructed an elementary model that takes into account sexual and asexual reproduction, and characteristics of long-range dispersal. Based on this model we suggest a neutral interpretation of patterns of population genetic structure that is a consequence of occasional long range dispersal and an advantage of uniparental reproduction at low population densities. More specifically, we show that the model can largely explain spatial population structures of sexual and asexual individuals observed in Fucus radicans, and we argue that the general framework is applicable to various species of terrestrial and aquatic organisms. Since our model does not depend on natural selection it may serve as a null-model in studies of geographic parthenogenesis.

Fucus radicans in the Baltic sea
The distribution of F. radicans, as currently observed, is in the northern Baltic Sea (Gulf of Bothnia) along the Swedish coast 60 • 20 ′ to 63 • 25 ′ N, and on the Finnish coast 62 • 19 ′ to 63 • 25 ′ N, and in addition, the species is found in two localities in Estonia (around 58 • 35 ′ N) [24]. Further distribution south and east of the present distribution around the Finnish coast, has been suggested, but is not confirmed. The species lives in stands of densities from a few to tens of individuals per m 2 . The species was formed no more than a few thousand years ago inside the 8000 y old postglacial Baltic Sea and has undergone a recent (and perhaps, still ongoing) process of establishment in the Baltic Sea [22]. As generation time is relatively long in this species, the time since species formation corresponds to no more than some hundreds of generations [22].
Fucus radicans has separate sexes and gametes of both sexes are released into the water during late spring to autumn, peaking at calm days and full moon. Eggs are short-lived, have a negative buoyancy and sink to the bottom close to the female individual. Indeed, preliminary experiments in a closely related species in the field have confirmed that sexual recruitment may only result in new attached individuals within some meters distance of the mother individual [25]. Alternatively, new individuals may arise from budding off small branches (adventitious branches) that are formed from the main branches, and these branches reattach to the substratum through the formation of rhizoids [26]. Adventitious branches are only some 10 mm long and have a negative buoyancy. Notably, also individuals recruited asexually produce gametes and are sexually active (as males or females).
Genotyping a large number of individuals using 5-9 microsatellite loci has generated a rather comprehensive picture of the distribution of individual multi-copy clones (hereafter clones), as well as unique multilocus genotypes (hereafter uniques) the latter recruited by sexual reproduction and zygote formation [24, 26, unpublished data]. The genetic structure is strongly heterogeneous; the Estonian populations being completely sexually recruited, some populations on the Swedish coast being almost completely dominated by a widespread female clone, while other populations in Finland and Sweden have a mixture of clones and uniques ( Figure 1). Although we here use data on distribution of clones in F. radicans as our empirical example, there are many other examples of species with both sexual and asexual recruitment in which the pattern of distribution of clones looks very similar to that of F. radicans, namely a mosaic pattern of asexually and sexually dominant populations and an overall dominance of one or a few clones (see Table 1 Figure 3). In the panels the respective 12 sampling locations are indicated on maps of the Baltic Sea (A) and an artificial circular habitat (B), respectively. The genetic composition of the sampled populations is illustrated graphically, where each square box in the respective populations refers to an individual. Boxes remain white if the individuals are unique, for instance since they directly originate from sexual reproduction or belong to a rare clone sampled only once in the entire sampling campaign. Clones encountered at several sampling locations (11 pairwise different for the empirical sampling, panel A, and 2 for the simulated data, panel B) are indicated by a unique color. Clones unique to respective sampling locations are colored grey and numbered, with each number referring to a different clone. Please mind that local clones by definition are different between locations. Core features of the empirical data, such as the existence of local clones, of global clones at distant locations, and of unique individuals are reproduced with the simulations.

An elementary model involving sexual and asexual reproduction and long range dispersal
The purpose of this study is to formulate and investigate an elementary model of species invasion where individuals may reproduce either sexually or asexually. Both fertilisation of short-lived eggs in the water and the negative buoyancy of adventitious branches make Fucus radicans a suitable species for efficient numerical modelling, since we may assume that reproduction is a rather local process. We intend explicitly not to involve adaptation to the conditions of new environments during spread into marginal areas, since the role of adaptation is not yet clear in Fucus radicans as in many other species. That is, genetic differences among individuals are assumed to be neutral. In this respect our approach offers an alternative to existing models for biological invasions in marginal environments (see e.g. [19,20]). From these basic considerations a model was constructed as follows: We model the invasion of a species into a habitat consisting of N patches of equal quality, where each patch can either be occupied by one individual of known sex or be empty. The model is explicit in space in a sense that the patches are arranged linearly on a circle with periodic boundary conditions, where the distance d(i, j) between two patches i and j is defined as their shortest distance on the circle (ranging from 0 to N/2). Besides from being convenient for technical reasons this configuration actually bears a resemblance to the situation of F. radicans in the Baltic Sea (see below), which primarily is located at a depth of 2 to 8 m in coastal shallow waters around the deeper basins of the northern Baltic Sea [27]. In this habitat we investigate a population invasion model which involves two effects: 1. occasional long distance dispersal of individuals through relocations 1 of whole organisms, which is known to have important effects during invasions [12,28,29], and 2. a competition between sexual and asexual reproduction, implemented in a way implying as little bias as possible towards one of the modes of reproduction.
Getting a bit more specific we take advantage of the particular features of reproduction in Fucus radicans. 2 Empty sites are colonised at rates depending on the status of near-by sites. While the potential of colonisation through asexual reproduction depends generally on the local density of individuals only, the potential for colonisation through sexual reproduction depends on the distribution of sexes in the neighbourhood of site i. We assume that the local densities of female and male individuals around patch i are representative for the potential of colonisation through sexual reproduction at this site. As an intermediate step for each empty patch i the local densities of female, p f (i), and male, p m (i), individuals are calculated as Here, K(d) is a kernel that models the potential of individuals to contribute to the colonisation of a site at distance d. For reasons of clarity and due to a lack of reliable empirical data on Fucus radicans we at this stage assume that the dispersal capabilities of gametes are sex-independent and equal to the dispersal capabilities of asexual recruits through adventitious branches. For the kernel K we use a Gaussian-like localized function characterized by a scale factor α, From the densities (1) per-patch colonisation rates for sexual reproduction, r s (i), and asexual reproduction, r v (i), at site i are defined as r s (i) = 2 min(p m (i), p f (i)) and (3a) Equation (3a) mirrors the fact, that for fertilisation both female and male gametes need to be present at the same place at sufficient probability. Here we assume that the limiting factor for sexual reproduction is the minimum of p m and p f , weighted by a factor 2 in order to equate the potential for sexual with the potential for asexual reproduction at V = 1 in the case p f (i) = p m (i) (see also [12]). The constant V for the asexual reproduction accounts for the differences in costs per asexual propagule compared to the production of spawn, which can result in different dispersal capabilities (for Fucus radicans we assume V < 1, since the large amount of biomass of adventitious branches suggests higher per recruit costs for asexual reproduction).
Technically V can be interpreted as the rate of successful asexual reproduction of one individual in an otherwise empty habitat. Individuals go extinct at a per individual extinction rate e. Long distance dispersal is included through relocations of individuals at a per-individual relocation rate, r. At the beginning of a relocation the source patch becomes unoccupied. Then the destination patch is drawn from a power-law kernel K r (d) = d −β with β ≥ 0 normalised to the size of the habitat accordant with Equation (2). In particular for small values of β K r (d) models long-range dispersal [30]. If the destination patch is already occupied, the dispersing individual is disregarded. Otherwise it occupies the destination patch.

Numerical investigation of the invasion process
We investigated the model numerically with the intention to identify its key features. Instead of a complete characterisation of the model, representative parameter values were selected for discussion. The model was implemented in continuous time using Gillespie's algorithm for stochastic simulations [31]. For characterisation of the genetic structure during simulation each individual was labelled with a unique ID, which was copied upon relocation and asexual reproduction. Individuals originating from sexual reproduction were given a new ID. Our main analysis builds on simulations performed on a habitat consisting of N = 100 000 patches, which initially were all empty. At simulation time t = 0 a number of 100 patches arranged side by side were populated with unique individuals of alternating sex, 50 females and 50 males. In order to be able to discuss the impact of this starting configuration on the simulation results, additional simulations initiated from 100 individuals (50 of each sex) distributed randomly on the entire habitat were performed, which are described in detail in the Supplementary material (available on request) and which henceforth is referred to as the random habitat case.
Simulations for three different per-individual rates of asexual reproduction (V = 0.002, 0.020 and 0.200) and two different extinction rates (e = 0.001 and 0.000) were performed. The different parameter values for V open up for a discussion of the ratio of asexual to sexual reproduction, which is unknown in many relevant systems. Testing e = 0 and e > 0 enables for an estimation of the impact of extinction events on genetic patterns. A combination of a starting configuration with a certain set of parameters V and e henceforth will be called a scenario. For the local dispersal and the relocation kernel α = 10 and β = 1.25 were used, which resulted in a well-balanced proportion of local reproduction and relocation of individuals to long-range relocation events. The relocation rate was fixed at r = 0.000001 and simulations were advanced for 100 000 time units, which corresponds to 1 000 mean life times of individuals in the scenarios with e = 0.001.
For comparison with the empirical sampling results on the brown alga F. radicans an artificial sampling procedure was performed on the results of the numerical simulations: for this purpose at several simulation times 12 samples of 50 individuals each were taken randomly (without replacement) from distinct areas of the habitat (see white circles in the centre panel of Figure 3, which indicate sampling time and locations) each consisting of 100 neighbouring patches. The state of the model is fully specified through the states of the individual patches, the sexes of the individuals and their IDs, which determine transition rates for the individual patches (e.g. for extinction or relocation). Gillespie's approach implements the individual transition rates in a stochastic simulation algorithm, where events with higher rates in a self-consistent way occur at higher probability in a certain period of time (and, hence, at a higher frequency). As a consequence simulation results can differ significantly between individual simulations. For each scenario therefore 10 independent stochastic simulations (henceforth called realisations) were performed in order to be able to quantify their stochastic component. For determining the significances of differences the results were subjected to an analyses of variance (ANOVAs) with a significance level 0.05. For the main analysis N = 100 000 patches were used. In order to assure that our results are not biased by the selected combination of parameters with this particular habitat size additional simulations were performed for the habitat dimensions N = 1 000, 10 000, 25 000, 50 000, and 1 000 000.

Figure 2 depicts model realisations of two scenarios of the invasion processes for intermediate asexual cost
(V = 0.020), that differ in extinction rate. In the simulation shown in the left panel, extinction of individuals was not considered explicitly (i.e. e = 0). Starting from 100 occupied locations at the origin the invasion initially is dominated by local sexual reproduction (see the initial increase to 25 000 sexually reproducing individuals). Within short, however, new colonies are founded at distant patches due to the long distance relocation dispersal kernel. New clonal colonies started from a single individual consist of one sex only (left upper panel of Figure 2) since they were established by a single asexually reproducing individual. In the absence of extinctions this can result in a permanent separation of sexes. Although at V = 0.020 the speed of local dispersal through sexual reproduction exceeds the dispersal by means of asexual reproduction by more than one order of magnitude, long-distance relocations followed by asexual dispersal seem to be more effective in the long run in establishing populations in new areas. After a short time colonies originating from relocations and reproducing asexually prevent the sexually producing part of the population from spreading through sexual reproduction by competition for space. After the entire habitat is invaded at about simulation time 40 000 the situation remains almost static and the sexes remain separated until the end of the simulation. If a finite extinction rate is considered as in the right panels, the genetic patterns decay in the long run after peaking at about simulation time 35 000. Instead, sexual reproduction slowly erodes the genetic patterns generated by the invasion process. The increase in individuals originating from sexual reproduction is almost linear between simulation time 10 000 and 80 000. Note that the presence of a finite extinction rate not necessarily seems to have a strong influence on the maximum spread of the dominant clone in the early phase of the invasion until simulation time 30 000 which is indicated by the dashed purple lines in 2). Figure 3 demonstrates the impact of the per-individual cost for asexual reproduction, V , on the course of invasions by comparing simulation results for three different scenarios with different V . In the first case (V = 0.002, left panel) the potential for asexual reproduction is too low for playing a relevant role in the invasion process. Interestingly the invasion process is most efficient in this case with the entire habitat being colonized in less than 20 000 time units. Large values for V such as V = 0.200 (right panel) result in a big advantage of asexual reproduction in range extension with a large potential for only a few clones to dominate the entire population. In the long run sexual reproduction, however, might be able to take over, as the lower right panel suggests. An intermediate value of V = 0.020 (centre panels) allowed for a coexistence of sexual and asexual reproduction during invasion. Simulations were performed by a stochastic simulation algorithm implemented in continuous time, and this resulted in realizations of the same scenario differing from one another. The centre panels of Figure 3 illustrate the impact of stochasticity on the outcome of simulations by comparing the course of invasions of three independent realizations performed for the same scenario. These replicates differ significantly in the general shape of the clonal fraction of the population in space and time, in dominating sex and in the spread of the dominant clone. Figure 4 contains a statistical analysis of the maximal spread of the dominant clone in the six scenarios (left panel) and of the impact of habitat size on the maximum relative spread of the dominant clone (right panel). Generally the spreading potential of the dominant clone increases with V , which confirm the main observations from the Figure: for V = 0.020 and 0.200. Notably, the presence of an extinction term e = 0.001 has no significant impact on the maximum spread of the dominant clone. Small habitat sizes reduce the potential of a dominant clone to spread, which probably is due to the advantage of dispersal through sexual reproduction in the early phase of the invasion process (right panel of Figure 4). Between an habitat of size 100 000 and an habitat consisting of 1 000 000 locations, however, no significant difference is observed with respect to the potential of the dominant clone to spread in the population.
Empirical data on genetic structure for F. radicans populations show large similarities with artificial samples from the model performed at simulation time 30 000 such as dominance of a few clones, and a strongly heterogeneous distribution of sexual and asexual individuals (Figure 1).

Discussion
Our elementary invasion model can reproduce key features of spatial genetic structure in F. radicans without invoking adaptations to geographically separated environments. This is in contrast to earlier attempts to explain observed patterns of spatial genetic structure, e.g. geographic parthenogenesis, where most models have focused on explaining the relative advantage of clonal reproduction in marginal environments. Adaptive explanations of clonal advantage include better average exploitation due to a diversity of frozen genotypes (Vrijenhoek 1984), insusceptibility to inbreeding (Haag andEbert 2004, Pujul et al. 2009), avoidance of genetic recombination with the central population (Peck et al. 1998) or as an indirect effect of hybridity preceding asexual reproduction (Handl 2009).
We here propose the present model as a null-model which any adaptation-based hypothesis could be tested against. Our model demonstrates that complex genetic patterns like the one observed, e.g. in F. radicans could stem from rather basic effects during range expansion in species for which both sexual and asexual reproduction is possible. Standard genetic models, as well as models for invasion processes, do not explicitly include aspects of the sexual structure of populations. As a consequence it is difficult to consider both sexual and asexual reproduction, since these routes of dispersal generally depend on the sexual structure of local populations.
We addressed this requirement by developing a self-consistent basic model that takes into account both sexes and their interactions for modelling dispersal and invasion processes. Our modelling results suggest that the genetic patterns observed in F. radicans might stem from unlinked stochastic processes of the two sexes during the establishment of this new species in the Baltic Sea. A similar modelling approach is also applicable for a number of other species with spatial genetic patterns resembling that of F. radicans (see Table 1 of the Supplementary Material, available on request). Consequently, the dominance of a few clones needs not necessarily imply that these clones are particularly well adapted but could instead originate from the (partly stochastic) sexual structure of the population. For dioecious organisms or outcrossing hermaphrodites sexual reproduction is a rather local process. If an alternative route of dispersal through asexual reproduction exists and occasionally individuals are relocated to distant sites, asexual reproduction can outperform sexual reproduction in the short run (see Figure 3). In this case, genetic patterns during invasion strongly depend on random effects of dispersal (compare the replicates depicted in the centre panel of Figure 3).
In particular, the dominant clone and its sex is a random phenomenon varying strongly between realisations. If a finite extinction rate is considered, in the long term sexual reproduction is able to take over, if it is less costly for the individuals and, hence, more efficient ( Figure 2). If we view the competition between sexual and asexual recruits for space as a competition for resources, the short term handicap but long term benefit of sexual reproduction is consistent with basic results from evolutionary game theory [32] and with a recent modelling approach by Song et al. [33].
Results of a statistical analysis presented in Figure 4 indicate, that small extinction rates and also the limited size of the habitat used for the main analysis do not have an impact on the simulation results. Concerning the spatial patterns of populations after invasion we observe a local clustering of sexes and clones (Figures 2 and 3), which is consistent with previous reports on the role of long-distance dispersal and local reproduction [10,34,35].
The structure observed significantly differs between realisations ( Figure 3, centre panels) and is a result of stochastic effects during invasion. Just as in empirical observations on F. radicans a single clone can spread through a great fraction of the habitat and dominate the entire population (Figure 1), which in case of the simulations is a matter of chance rather than an individual advantage of the dominant clone. Often the early phases of an invasion is of particular importance for the long term genetic patterns, as these are associated with priority or founder effects [10,36,37]. We therefore tested two different starting configurations: In addition to the simulations shown in Figures 1 to 4 a random habitat case initiated from 100 individuals randomly distributed all over the habitat. Detailed results prepared accordingly are available in the Supplementary information (available on request). As a general trend, the random habitat starting conditions result in a much faster invasion of the species, since the colonisation starts from a number of different spots in the habitat. The potential for a single clone to cover great parts of the habitat is strongly reduced. For this reason we assume, that complex genetic structures with only a few dominating clones as observed in F. radicans are likely to originate from an invasion process (or speciation event) that started from a few spots instead of from a great number of spots distributed over the habitat.
If we assume that the genetic structure in F. radicans indeed is driven by the effects included in the simulation model, what patterns should be expected in the future Baltic population of F. radicans? Based on our model we expect that sexual reproduction will take over in the long run, and the trend towards sexual dominance should start when the total available habitat is colonized. When the rate of long-range relocations is low the number of interfaces between areas dominated by sexual reproduction is rather constant over time.
Since we assumed local Gaussian dispersal of sexual and asexual recruits, these interfaces propagate at constant speed, resulting in a more or less linear increase of the fraction of sexually reproducing individuals with time. The linear relationship, however, is not likely to be found in nature, since dispersal and relocation processes are more complex and the restriction to one spatial dimension may be an unrealistic simplification of many natural habitats. The qualitative effects of a long-term success of sexual reproduction, however, can be expected to hold. One single observation supports the basic mechanism of a population being able to change sex-ratio rather dramatically over a short period of time; a population of F. radicans sampled in 1996 outside Umeå (Drivan) was found to be strongly female-biased at that time (80% females, N=82 [38]), but re-sampling the exact same locality in 2010 showed an almost equal sex ratio (56% females, N=193, Pereyra et al., unpubl.).
The main effects of our model are likely to apply also to the broad class of harsh marginal habitats subjected to periodic extinctions and re-colonisations, e.g. during interglacial recolonizations [15,24]. Based on these simulations we believe that it is important to explicitly include the distribution of sexes in future attempts to model the spread of species reproducing both sexually and asexually. Moreover the fundamental mechanisms implemented in this modelling approach should be investigated in detail in future studies both in the laboratory and in field experiments.