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From this video (21'15"), the speaker gives the following formulae in order to calculate the between and among populations genetic variance from the $F_{ST}$:

$$V_{Among Pop} = 2 F_{ST}V_G$$

$$V_{Within Pop} = (1-F_{ST})V_G$$

, where $V_G$ is the total genetic variance of this population if it was well mixing. $V_{Among Pop}$ is the variance among populations and $V_{Within Pop}$ is the variance within populations.

In this same video, the speaker defines $F_{ST}$ as:

$$F_{ST} = frac{ ext{Var}(p)}{ar p (1-ar p)}$$

, where $p$ is a vector of frequencies of a given allele and $ar p$ and $ ext{Var}(p)$ are the mean and variance of this vector.

For example, consider a metapopulation made of 4 subpopulations. The allele frequencies in these 4 subpopulations are p=[0.2, 0.5, 0.8, 0.3]. $ar p$ is the mean of $p$ ($ar p = 0.45$) and $ ext{Var}(p)$ is the variance of $p$ ($ ext{Var}(p )=0.07 space$).

**Can you help me to make sense of the formulas for $V_{Among Pop}$ and $V_{Within Pop}$?**

Can you prove these formulas? I would expect that $V_{Among Pop} + V_{Within Pop} = V_G$ but it doesn't! Maybe the issue has to do with the fact that $V_{Within Pop}$ as defined above fit to haploid population and not diploid populations. Then, I would expect that for diploid populations $V_{Within Pop}$ becomes $(1-2F_{ST})V_G$. Is that correct? Just in order to make it more general, how do you extrapolate these definitions for a tetraploid population? Also maybe I misunderstand the meaning of $V_G$. Is the variance within population the sum of the within populations variance or is it the average (or something else)? Thanks for your help!

## CHAPTER 13 - Genetic Approaches to Understanding Marine Metapopulation Dynamics

This chapter demonstrates how genetic approaches can help define three fundamental aspects of marine metapopulation dynamics: what a population is, how populations are connected, and what impact population extinction recolonization events have. Genetic markers can contribute to the recognition and understanding of metapopulations in the sea in a number of ways. Much work has gone into measuring subdivisions among populations, detecting phylogeographic breaks and exposing cryptic species, all of which has helped to define the spatial scale of populations for different species. The population turnover characteristic of strictly defined metapopulations should be detectable by genetic means, but supporting demographic data and temporal sampling regimes greatly strengthen any conclusions that can be drawn. The genetic impact of population turnover hinges on whether colonists are small groups of related individuals or large groups drawn from a larger geographically mixed pool. Progress toward a better understanding of marine population dynamics will come from both new analyses and new types of data. All these new analytical approaches are data hungry, and increasingly the data they will analyze will come from nuclear gene sequences. The greatest progress in understanding the dynamics of marine populations will come from coupling new genetic approaches with other sources of data. Even sophisticated genetic analyses can have a difficult time detecting certain population dynamics without additional information and some new analyses depend on intensive temporal sampling.

## Background

While gene flow decreases the differentiation among populations, it may increase genetic diversity within them. Population connectivity is therefore important to maintain overall genetic diversity across small local populations that would otherwise "erode" because of drift [1]. The effects of isolation by distance and reduced local population sizes tend to be most visible at the edges of species ranges, as these fringes go through periods of expansion with founder effects and contraction with bottlenecks [2]. Empirical studies on butterflies show that peripheral populations are indeed less diverse than central populations [3], and experience larger population fluctuations due to less favourable conditions [4]. In addition the breeding system will also affect within-population genetic diversity, with asexual species being least diverse and sexual systems being variably affected by deviations from random mating, which may affect effective population size independent of drift [1].

Endangered species often occur in small isolated populations where demographic and environmental stochasticity impose additional risks of local extinction. This has made some researchers question the role that genetic factors play in driving population extinction [5], because genetic factors are likely to be negligible when population decline occurs rapidly. However, when effective population sizes remain moderate, inbreeding over many generations may have marked fitness effects due to increasing disease susceptibility and inbreeding depression [2, 6–8]. This is because purging tends to remove primarily the few deleterious recessive alleles with large negative effects, and hardly affects the more numerous slightly deleterious alleles [6]. Theory indicates [9] and comparative studies across 170 species have shown [10] that a significant proportion of endangered populations/species have reduced levels of genetic variation compared to related non-endangered species, suggesting that genetic factors often play a role in population extinctions [11].

Researchers have traditionally been forced to evaluate present day diversity of endangered populations against other contemporary populations of the same or closely related species. Such studies are valuable, but since populations rarely have identical demographic and environmental histories, precise identification of the factors that caused extant genetic differences remains impossible. Recent technical advances in the extraction and amplification of old DNA have made the large resources of natural history collections (NHC) available for population genetic studies, providing direct and highly relevant reference points for studies of genetic diversity in endangered populations. Particularly taxa with long histories of collection by entomologists, such as beetles, butterflies and hoverflies, have thus become very useful for long term population studies of genetic change over time.

The number of studies utilizing NHC material for evolutionary genetic studies is increasing, and many focus on past and present genetic diversity in endangered populations [12]. Despite the promises these methods hold for conservation genetics, there are also limitations to the use of historical DNA, and special precautions are required in the experimental and the analytical phase of such work. The highly degraded nature of DNA extracted from historical samples, which increases with age, temperature and water content [13, 14], generally restricts PCR amplification to short fragments (< 200 bp) thus limiting the choice of genetic markers. Nuclear microsatellite markers have proven useful in this context, as they have short and highly polymorphic amplicons [12]. However, historical DNA is not only of low quality but also occurs in very low quantity, increasing the risk of genotype errors caused by cross contamination, allelic dropout or false alleles. The importance of following standard protocols when working with historical samples can therefore not be stressed enough, and the assessment and reporting of genotype error rates is indispensable in order to validate such datasets [15–17].

The large blue butterfly, *Maculinea arion*, is one of many butterfly species that have declined in Europe during the last century, both in terms of population numbers and population connectivity [18]. As a result, many extant populations are considered endangered and only exist because they are actively managed [19]. The physical attractiveness and fascinating biology of *M. arion* has made the species popular among amateur collectors, so that many European natural history museums hold large collections often with good numbers of specimens collected in particular years that together form attractive time series for single localities. When these series coincide with periods of population decline they provide an outstanding opportunity to analyse how isolation and demographic fluctuations may have affected genetic variation in the past. Such time series are common for *M. arion* and we exploit such collection material in this study. In particular, we investigated whether/how a recent, severe reduction in population census size and a long history of isolation by distance has affected genetic diversity in the last extant Danish population of *M. arion*, on the island of Møn (Figure 1). As contemporary reference points we used a cluster of six *M. arion* populations in south and central Sweden approximately 100-600 km away [20] and as historical reference points we used NHC specimens from the Møn population covering the time period 1930-1975.

*Maculinea arion***in Scandinavia**. *a*) Count data of *M. arion* imagos on Møn, Denmark. Maximum (solid line) and minimum (dashed line) counts from the best day during the flight season. Imago counts were converted into approximate population census size (# imagos × 3.5) according to Thomas *et al*. [39]. *b*) Distribution of *M. arion* in Denmark and southern Sweden before (open symbols) and after (closed symbols) 1990, 10 km 2 UTM grid. Records have been compiled since 1900 by the Atlas Project of Danish Butterflies and the Swedish ArtDatabankens fynddatabas. Danish populations marked by an asterisk went extinct in the late 1990s.

Chen, J., Wang, Y., Wang, R., Lou, A., Lei, G., and Xu, R. (2002). The effects of patch quality on the metapopulation dynamics of two fritillary butterfly species. Manuscript in preparation.

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Demarais, E., Lanneluc, I., and Lagnel, J. (1998). Direct amplification of length polymorphism (DALP), or how to get and characterize new genetic markers in many species. *Nucleic Acids Res*. **26**(6): 1458–1465.

Ehrlich, P. R. (1992). Population biology of checkerspot butterflies and the preservation of global diversity. *Oikos* **63**:6–12.

Excoffier, L., Smouse, P. E., and Quattro, J. M. (1992). Analysis of molecular variance inferred from metric distances among DNA haplotypes: Application to human mitochondrial DNA restriction data. *Genetics* **131**:470–491.

Hanski, I. (1999). *Metapopulation Ecology*, Oxford University Press, New York.

Hanski, I., and Gilpin, M. E. (1997). *Metapopulation Biology: Ecology, Genetics, and Evolution*, Academic Press, San Diego.

Hanski, I., and Kuussaari, M. (1995). Butterfly metapopulation dynamics. In Cappuccino, N., and Price, P. (eds.), *Population Dynamics: New Approaches and Synthesis*, Academic Press, London, pp. 149–171.

Hanski, I., Kuussaari, M., and Nieminen, M. (1994). Metapopulation structure and migration in the butterfly *Melitaea cinxia*. *Ecology* **75**(3):747–762.

Hanski, I., Pakkala, T., Kuussaari, M., and Lei, G. (1995). Metapopulation persistence of an endangered butterfly in a fragmented landscape. *Oikos* **72**:21–28.

Hanski, I., and Simberloff, D. (1997). The metapopulation approach, its history, conceptual domain, and application to conservation. In Hanski, I., and Gilpin, M. E. (eds.), *Metapopulation Biology: Ecology, Genetics, and Evolution*, Academic Press, San Diego, pp. 5–26.

Hedrick, P. W., and Gilpin, M. E. (1997). Genetic effective size of a metapopulation. In Hanski, I., Gilpin, M. E. (eds.), *Metapopulation Biology: Ecology, Genetics, and Evolution*, Academic Press, San Diego, pp. 165–181.

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Lewis, O. T., Thomas, C. D., Hill, J. K., Brookes, M. I., Crane, T. P. K., Granear, Y. A., Mallet, J. L. B., and Rose, O. C. (1997). Three ways of assessing metapopulation structure in the butterfly *Plebejus argus*. *Ecol*.*Entomol*. **22**:283–293.

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Meglécz, E., Nève, G., Pecsenye, K., and Varga, Z. (1999). Genetic variations in space and time in *Parnassius mnemosyne* (Lepidoptera) populations in North-east Hungary: Implications for conservation. *Biol*.*Conserv*. **89**:251–259.

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Petit, S., Moilanen, A., Hanski, I., and Baguette, M. (2001). Metapopulation dynamics of the bog fritillary butterfly: Movements between habitat patches. *Oikos* **92**(3):491–500.

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Wang, R., Wang, Y., Lei, G., Xu, R., and Painter, J. N. (2002). Genetic differentation within metapopulations of *Euphydryas aurinia and Melitaea phoebe* in northern China. Manuscript in preparation.

Wang, Y., Wang, R., Chen, J., Lei, G., and Xu, R. (2002). Metapopulation structure and dynamics of two fritillary butterflies in northern China. Manuscript in preparation.

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## Climate variables explain neutral and adaptive variation within salmonid metapopulations: the importance of replication in landscape genetics

Understanding how environmental variation influences population genetic structure is important for conservation management because it can reveal how human stressors influence population connectivity, genetic diversity and persistence. We used riverscape genetics modelling to assess whether climatic and habitat variables were related to neutral and adaptive patterns of genetic differentiation (population-specific and pairwise *F*_{ST}) within five metapopulations (79 populations, 4583 individuals) of steelhead trout (*Oncorhynchus mykiss*) in the Columbia River Basin, USA. Using 151 putatively neutral and 29 candidate adaptive SNP loci, we found that climate-related variables (winter precipitation, summer maximum temperature, winter highest 5% flow events and summer mean flow) best explained neutral and adaptive patterns of genetic differentiation within metapopulations, suggesting that climatic variation likely influences both demography (neutral variation) and local adaptation (adaptive variation). However, we did not observe consistent relationships between climate variables and *F*_{ST} across all metapopulations, underscoring the need for replication when extrapolating results from one scale to another (e.g. basin-wide to the metapopulation scale). Sensitivity analysis (leave-one-population-out) revealed consistent relationships between climate variables and *F*_{ST} *within* three metapopulations however, these patterns were not consistent in two metapopulations likely due to small sample sizes (*N* = 10). These results provide correlative evidence that climatic variation has shaped the genetic structure of steelhead populations and highlight the need for replication and sensitivity analyses in land and riverscape genetics.

**Fig. S1** Box plots showing the distribution of population specific measures of differentiation (*F*_{ST}) calculated by GESTE for sets of populations within each of five major steelhead metapopulations in the Columbia River Basin, USA.

**Table S1** Sample data metadata and variables data used in analysis for each population.

**Table S2** Pairwise correlation coefficients (*r*) reported from the DISTLM forward program for each of the 5 steelhead metapopulations.

Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

## $F_{ST}$ and the genetic variance in metapopulations - Biology

__1. Multi-population systems__

In recent times more than before, human populations expand and require larger areas for agriculture and other land-use goals, often at the expense of natural ecosystems. Many organisms have to face large-scale habitat destruction and fragmentation of larger populations. New approaches have been developed to deal with this problem theoretically and practically. These are based on the idea that empty fragments (patches) are colonized by the organism, and that populations persist for a while but eventually become extinct. Thus, metapopulations persist as long as colonization takes place despite extinctions. The notion of meta-populations (populations of populations) was not the first to address this question: Island Biogeography (MacArthur and Wilson 1967) already dealt with immigration of species into islands, from a mainland.

One of Island Biogeography's important contributions was the idea that each island has an equilibrium number of species on it, which is a function of __local extinction__ rate and of __colonization__ (or immigration) rate: extinction increases as more species colonize the island, while colonization decreases as more species already had colonized. Since immigration also depends on the distance between the mainland (the source of immigrants) and the island (the sink, in a broad sense), nearby coastal islands should have more species of animals and plants than remote oceanic islands. Furthermore, large islands should also have more species than smaller ones.

However, the presence of a large mainland was not realistic enough to deal with sets of more or less isolated populations. In fragmented habitats extinctions take place, but colonization is not automatic since there is no infinite reservoir of immigrants. According to the metapopulation approach, all immigrants have to come from other fragments. In its "classical" form (Levins 1969), a metapopulation consists of isolated patches (islands) within an unsuitable matrix (sea), without a mainland. Thus, all local populations can become extinct, as can the entire metapopulation (i.e., when the longest-lived population becomes extinct).

__2. The metapopulation approach__

Metapopulations persist due to the balance between extinction and colonization of local populations. In contrast to the IFD, each population in a patch has its own local dynamics, which are not dependent on the state of other patches as in structured populations, nor on immigration from other patches as sink-populations do. In "true" metapopulations, immigration is just sufficient to recolonize unoccupied patches (where there may have been a population before). We can view this as an intermediate case of a continuum of connections between patches (Fig. 1), including 1) continuous populations with approximately instantaneous immigration among its sub-populations, 2) structured populations, with sub-populations, whose dynamics depend on the others in the landscape, as in sink populations and populations behaving more or less according to the Ideal Free Distribution, 3) metapopulations whose populations receive some immigrants, including island - mainland systems, and 4) isolated populations too far apart for effective immigration.

Fig. 1. Spatial distributions of (sub)-populations, according to the ratio of dispersal and between-patch distances.

The distances of between patches are crucial in the metapopulation view of immigration, and are determined relative to the dispersal distance the organisms can travel. This is obviously not a single value, but a probability distribution of travel distances. This depends not only on the abilities of the propagules, but also on the properties of the terrain between the patches (matrix) and in passive dispersal on the behavior of the dispersal agents or vectors. Of special importance are corridors, enabling movement among hostile or adverse terrain, and stepping stones.

There are a number of criteria to decide whether a collection of populations of the same species in distinct patches form a "classical" metapopulation:

1. The organisms spend most of their life within a single patch If the patches are ephemeral aggregates, we are dealing with forage patches or something equivalent.

2. Even the largest population can become extinct Otherwise, the metapopulation persists simply due to the persistent population (mainland in Island Biogeography).

3. Patches are not too isolated to prevent colonization

The classical metapopulation model predicts that metapopulations persist as long as there is a balance between extinction and colonization, with an equilibrium proportion of occupied patches. However, we should not assume all collections of populations are in equilibrium. Many may be non-equilibrium metapopulations, where colonization is just a little too low to have an equilibrium, and the metapopulation eventually goes extinct.

__3. Metapopulation models__

In Levins' (1969) model, there are many small, equal patches. They are either occupied or not. If they are, they reach a single carrying capacity K. Levins (1969) worked with fast-growing insect populations, where density rapidly becomes resource or site-limited. The variable of interest is P, the fraction of occupied patches (or incidence, with 0 £ P £ 1). The rate of change of P, dP/dt, determines whether P will increase, decrease or stay the same (if dP/dt >0, <0 or =0). The rate dP/dt is given by the difference between colonization rate C and extinction rate E. This is analogous to population growth rate as the difference between birth and death rates. Colonization rate is the number of successful colonizations of unoccupied patches as a proportion of all available patches, and extinction rate is the proportion of patches becoming empty. Thus,

dP/dt = C - E, or dP/dt = cP(1 - P) - eP,

where c and e are the local immigration and extinction probabilities per patch. In this model, colonization rate depends positively on the number of occupied patches (as sources of immigrants), but also on the number of empty patches (1 - P). If this is low, there are few patches available for colonization. Extinction rate only depends on P, the proportion of populations that can become extinct. At equilibrium dP/dt = C - E = 0, or P' = 1 - e/c. For persistence of the metapopulation, it is required that P' >0, which is true if e/c < 1 or c > e. In other words, as long as the probability of a patch to be colonized exceeds its probability to go extinct, the metapopulation exists, with a single stable equilibrium incidence P'. The metapopulation is stable at the point in Fig. 2A where dP/dt = 0, at the intersection of the C and the E lines. Where P' is depends on the shape of these lines. If actual observed P lies in the area where the difference between the lines is positive, the number of occupied patches P increases. If P is in the area where the difference is negative, P decreases. If colonization exceeds extinction (C > E) then there is an equilibrium value for incidence (0 < P' < 1). If extinction exceeds colonization (C < E) then there is no equilibrium (P' = 0). This is represented by the lower C curve in Fig. 2A, which lies completely below the E line, except at P=0.

Colonization C and extinction E are functions of P with coefficients c and e, the per-patch probabilities of colonization and extinction. These parameters represent demographic processes, and are therefore dependent on the biology and life-cycle of the organism studied, and on the particular environment the populations experience. Thus, for each real situation (organism x environment) a different set of curves in the models represents the relationships between P, C and E.

Hanski (1982) developed the metapopulation approach further, by assuming that extinction rate is a quadratic function of P. As in the previous model, E or the number of patches becoming empty, increases if more patches are already occupied when there are few occupied patches. In this model, E also decreases when there is an increase at high incidence. As more patches are occupied, the chance to go extinct decreases (Fig. 2B). This is due to the "rescue effect", as immigration from many populated patches tends to prevent local extinction by instantaneous recolonization. The rescue effect is an important and probably realistic addition. The result is that P' = 0 (implying metapopulation extinction) if C < E, like in Levins' model, while in contrast P' = 1 (all patches are occupied) if C > E. Intermediate values are transient, and the metapopulation either grows or disappears.

Many real collections of distinct populations are probably non-equilibrium metapopulations, going to extinction. However, variation in local conditions and in those governing the entire landscape occupied by the metapopulation, may result in considerable constancy. In addition, Hanski's model predicts that species assemblages should have a bimodal distribution of incidence, with many rare and sparse species and many very common dense species. This is known as the core - satellite hypothesis (Hanski 1982), which is not always found (Gaston 1994), as in the annual plant communities in the northern Negev shrublands (Boeken and Shachak 1998).

There are a number of recent additions to the development of a metapopulation theory, making it increasingly realistic. One of them is the incorporation of variation in population size. This makes these deterministic models stochastic, which results in a stable equilibrium 0 < P' < 1, in stead of 0 or 1. This also changes the behavior of E, as small populations have a larger extinction probability than larger ones, due to demographic stochasticity.

Another valuable addition to metapopulation theory is the assumption that immigration is not from local populations, but from a constant propagule rain (Gotelli 1991), or dormant seedbank. This makes colonization rate independent of P, as in island - mainland systems. The combinations with the two previous models are shown in Fig. 3: If both C and E are linear functions of P (Fig. 3A), then P' = c /(c + e). If c = e then P' = 0.5 (instead of 0 in Levins' model). A propagule rain combined with Hanski's model (Fig. 3B), P also has a positive equilibrium, P' = c/e. If e is very low, P' = 1.

Using these models to predict how many populations a metapopulation will contain, requires a great amount of information on the relationships of incidence P, with C and E, the immigration and extinction rates, which is often hard to obtain. Experiments are necessary, but are rarely feasible in practical situations. For instance, problems of conserving a rare species of plant or animal in the face of habitat fragmentation seldom allow experiments. This is especially true in cases of rapid habitat destruction. Empirical research on model species that are rare, but not yet face extinction, may be helpful.

__4. Practical applications__

The models, together with observational studies on metapopulation dynamics in the field, have generated some very useful insights. One of the central questions in practical problem situations is whether there is a minimum viable metapopulation size, and whether this MVM, expressed as the number of occupied habitat patches, can be estimated (Hanski et al. 1996). This is, within the metapopulation context, equivalent to the "minimum viable population" concept (MVP, Soulé 1980). MVP is usually expressed as the density that has a 95% chance to persist for at least 100 years. Considerations of MVM also involve MASH, the minimum available suitable habitat.

Using Levins' (1969) model is not helpful, because it a) is deterministic, b) assumes large networks of patches, and c) c and e are very small relative to the number of patches. Nisbet and Gurney (1982) proposed an expression based on a stochastic version of this model, assuming metapopulations in small networks of patches. The method defines the time to metapopulation extinction T_{M}, and allows estimating equilibrium (or stochastic steady-state) incidence P':

T_{M} = T_{L} e x , with x = (H P' 2 )/(2(1 - P')),

where T_{M} = the time to metapopulation extinction,

T_{L} = the time to local extinction,

e = extinction probability per patch,

H = the number of patches with suitable habitat, and

P' the stochastic steady-state incidence as a fraction of H.

If we want T_{M} > 100 T_{L}, then P' ÷ H £ 3. In that case, if H = 50, colonization and extinction rates C and E should be such that P > 0.43, or P * H = 21 patches, in order for the metapopulation to persist longer than 100 times T_{L}.

Such analytical models are useful, but have limitations. They do not deal well with questions like: what would happen to metapopulation persistence if some patches are removed (due to habitat destruction) or reduced in size (ongoing fragmentation) or quality? This is because they cannot - at this point - incorporate variation in patch quality and spatial location and configuration (distances). In these cases, simulation models based on the analytical ideas are helpful, but they yield inherently weak predictions. Spatially explicit models (such as percolation models and cellular automata) are also useful. There are more detailed "spatially realistic" models being developed too, based on local population models with assumptions on immigration rates, in combination with explicit mapping and GIS.

In order to have some ecologically sound answers to counter dangers of metapopulation extinctions by fragmentation and habitat destruction, Hanski (1997) and others have suggested a number of rules-of-thumb, or questions to consider in conservation and management:

1. If habitat destruction continues, the metapopulation will surely become extinct.

2. This may take a long time, depending on the largest population there may be time to do something, such as facilitating recolonization.

3. Equilibrium conditions may never arise.

4. As many fragments as possible should be preserved (In Britain more than 10 isolated populations of butterfly species have gone extinct within 20 years, including 3 species completely).

5. Large numbers of suitable patches in not sufficient, if distances are too large, preventing recolonization and the rescue effect.

6. Distance is not the only factor affecting immigration probabilities: the properties of the terrain are crucial, including corridors and stepping stones.

7. Large numbers of suitable patches are not sufficient if they are very close together, due to possible synchronous dynamics.

8. There should be as much variance in local patch quality (different habitats within the range of the organism) as possible to prevent synchronous dynamics. (Not only the "best" patches.)

9. Recolonization has to be observed within a few generations for metapopulations to have a chance.

10. Sizes of suitable patches are important, because demographic stochasticity can lead to extinction, especially in organisms with low reproductive output.

11. Large patches are desirable, for they have large populations, with many potential immigrants, and have high internal variation in habitat quality.

12. Patch sizes can be deceiving if negative edge effects reduce effective patch size.

** References** Boeken, B. and Shachak, M. 1998. The dynamics of abundance and incidence of annual plant species during colonization in a desert.

__Ecography__21: 63-73.

Gaston, K. J. 1994. __Rarity__. London, Chapman and Hall.

* Gotelli, N. J. 1991. Metapopulation models: the rescue effects, the propagule rain, and the core-satellite hypothesis. __American Naturalist__ 138: 768-776.

Hanski, I. 1997. Metapopulation dynamics: From concept and observations to predictive models. Pp. 69-91. In Hanski, I. A. and Gilpin, M. E., Eds. __Metapopulation biology__. San Diego, USA, Academic Press.

Hanski, I., Moilanen, A. and Gyllenberg, M. 1996. Minimum viable metapopulation size. __American Naturalist__ 147: 527-541.

* Hanski, I. 1982. Dynamics of regional distribution: the core and satellite species hypothesis. __Oikos__ 38: 210-221.

Levins, R. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. __Bull. Entomol. Soc. Am.__ 15: 237-240.

MacArthur, R. H. and Wilson, E. O. 1967. __The theory of island biogeography__. Princeton, New Jersey, Princeton University Press.

Nisbet, R. M. and Gurney, W. S. C. 1982. __Modelling fluctuating populations__. New York, USA, Wiley.

## Materials and methods

### Study species

*S. austriacum* (jeweled rocket) (Brassicaceae) is a small, biennial or perennial, diploid (2*n*=14) species originally occurring in the mountainous regions of South and Mid-Europe and the Pyrenees (Hegi, 1986). In its original distribution area, *S. austriacum* is a typical pioneer plant species that colonizes organic disposals on rocky soils, often in shady places or under inclining rocks. Plant height varies between 30 and 60 (80) cm. The plant is biennial, but may survive for longer time periods (more than 5 years) (K Van Looy, personal observation). In this case, old individuals can easily be recognized in the field by their large rooting system (more than 2 cm in diameter) and more than eight flower branches. Flowers are self-compatible and strictly pollinated by insects (bees and syrphid flies). The species flowers from June till September and seeds (mean seed weight: 0.3 mg) are dispersed by the wind (anemochory) or fall directly on the ground (barochory). In addition, seeds may be transported by water through the transfer of flowering branches.

### Study system and study populations

The study area is situated in the Meuse River valley situated on the border between Belgium and the Netherlands. The Meuse River is for the largest part of its course a regulated river, but in the study area, an ongoing project aims at restoring the natural flow regime of the river. In Belgium, *S. austriacum* was first observed in 1824 by Lejeune on the shores of the Vesder River, one of the Ardennes' tributaries of the Meuse, where it was most likely introduced as a result of the transport of sheep wool (Van Landuyt et al, 2006). Probably, seeds germinated along the river and the species slowly expanded its range northwards. The first *S. austriacum* populations have been observed in the Meuse River some 30 years ago, although at that time populations were not persisting. However, during the last 10 years, several permanent populations have been established (see below). Along the Meuse River, the species mainly occurs in warm and sunny places on sand or gravel.

During the last 25 years, at least five (1982, 1993, 1995, 2000 and 2002) major water flushes (more than 2.200 m 3 /s) occurred, resulting in overbank depositions of sand and gravel banks. Colonization of these sediment zones resulted in more or less permanent populations, in contrast to populations occurring on riverbanks in the immediate neighborhood of the river, where populations are rather short-lived and subject to yearly fluctuations of water levels. In 2004, 14 populations were found in this dynamic river system (Figure 1), five of which were located outside the regular influence of the river, were more than 5 years old and considered persisting populations and nine that were located in the immediate neighborhood of the river and were less than 5 years old. For each population, we determined population size by counting the number of flowering individuals. Most populations were small and consisted of less than 100 flowering individuals. The straight-line distance between populations was determined directly on the basis of site coordinates (mean distance: 7.23 km, range: 0.53–16.95 km) (Figure 1). Besides, distances between populations were also measured along the river using GIS (ArcView 3.2). In this case, distances between populations ranged from 0.56 to 25.60 km (mean: 9.77 km).

Study area and location of the 14 study populations of *S. austriacum* along the Meuse River. The arrow depicts the direction of the water flow.

### Sampling scheme for genetic analyses and AFLP protocol

In Spring 2004, a total of 242 individuals was sampled from the 14 populations. Individuals were sampled from the entire area occupied by the population in order to avoid the effects of population substructure. Young leaf material was collected and immediately frozen in liquid nitrogen. Before DNA extraction, leaf material was freeze-dried and homogenized with a mill (Retsch MM 200) to a fine powder. Total DNA was extracted from 20 mg of lyophilized leaf material using Dneasy Plant Mini Kit (Qiagen). After extraction, DNA concentrations were estimated on 1.0% (w/v) agarose gels.

AFLP analysis was carried out according to Vos et al (1995). The enzymes *Eco*RI and *Mse*I were used for DNA digestion. Each individual plant was fingerprinted with four combinations: *Eco*RI-AAG/*Mse*I-CAT, *Eco*RI-ATC/*Mse*I-CTA, *Eco*RI-AAG/*Mse*I-CAG and *Eco*RI-ATC/*Mse*I-CCA. Fragment separation and detection took place on a Nen IR 2 DNA analyzer (Licor) using 36 cm denaturing gels with 6.5% polyacrylamide. IRDye size standards (50–700 bp) were included for sizing of the fragments. AFLP patterns were scored using the SAGAmx.software from Licor. We scored the presence or absence of each marker in each individual plant. Twenty randomly selected samples where loaded three times to infer reproducibility of the AFLP protocol. Average similarity between replicated samples was very high (more than 95%).

### Data analysis

Population genetic structure was analysed using the Bayesian methods proposed by Holsinger et al (2002) and Holsinger and Wallace (2004). These methods allow direct estimates of *F*_{ST} from dominant markers without previous knowledge of the degree of within-population inbreeding and they do not assume that genotypes are in Hardy–Weinberg equilibrium. Although model outcomes for *f* (an estimate of *F*_{IS}) may not always be very accurate, results for *θ* B (an estimate of *F*_{ST}) are mostly very informative (Holsinger et al, 2002). We tested several models using Hickory version 1.0: (i) a full model with noninformative priors for *f* and *θ* B , (ii) a model in which *f*=0 and (iii) a model in which *θ* B =0. The three models were compared using the deviance information criterion (DIC) (Holsinger and Wallace, 2004). The model with the smallest DIC value was chosen. We also calculated *I*_{E}(*x*), an often used measure of the information provided about a parameter in Bayesian inference (Holsinger and Wallace, 2004), for *f* and *θ* B . Analyses were performed for all populations, and for young, dynamic and old, persisting populations separately. Several runs were conducted with default sampling parameters (burn-in= 50 000, sample=250 000, thin=50) to ensure that the results were consistent.

To investigate the relative importance of spatial and temporal dynamics on genetic structure, we used AMOVA (Excoffier et al, 1992). Total genetic diversity was partitioned among groups of populations, among populations within groups and within populations by carrying out a hierarchical AMOVA on Euclidean pairwise distances among individuals using GenAlEx v. 6 (Peakall and Smouse, 2005). Three different models were tested: a model without any structure, a model in which an age structure was incorporated and a model in which spatial structure was included. In the second model, two subgroups were defined, old, persisting populations, and young dynamic populations. Spatial groups were defined based on their geographical position along the river: upper (Elsloo, Meers 1, Meers 2 and Meers 3), middle (Maasband, Mazenhoven, Meeswijk, Kerkeweerd 1, Kerkeweerd 2 and Kerkeweerd 3) and lower course populations (Elereert, Heppeneert, Roosteren 1 en Roosteren 2) (Figure 1). Significance of the three models was determined using a permutation test (*n*=9999 permutations).

Pairwise genetic distances (Φ_{ST}) among the 14 *S. austriacum* populations and their levels of significance were also determined from the AMOVA using GenAlEx v. 6 (Peakall and Smouse, 2005). To illustrate the relationship among populations based on their pairwise genetic distances, an UPGMA dendrogram between all populations was constructed using Nei's unbiased genetic distance. Phylip 3.6.15 (Felsenstein, 1993) was used to construct the dendrogram. Pairwise genetic distances were plotted against geographic distances to test for regional equilibrium and to evaluate the relative influences of gene flow and drift on genetic structure (Hutchison and Templeton, 1999). Significance of the observed relationships was obtained by using a Mantel test (Mantel, 1967). A total of 5000 random permutations were performed.

Three measures of within-population genetic diversity were estimated: the number of polymorphic loci, Nei's gene diversity and the Bayesian estimate of gene diversity. Nei's gene diversity was calculated using AFLPsurv v.1.0 (Vekemans et al, 2002). Estimates of allelic frequencies at AFLP loci were calculated using the square root method. Although this method may lead to biased results of averaged estimates of heterozygosity and genetic differentiation when the number of polymorphic loci is low (Lynch and Milligan, 1994 Zhivotovsky, 1999, Krauss (2000) has shown that in highly polymorphic data sets, no statistical difference between methods was found. After estimation of allele frequencies, statistics of gene diversity and population genetic structure were computed according to Lynch and Milligan (1994). For each population, we calculated the number and proportion of polymorphic loci at the 5% level and Nei's gene diversity (*H*_{j}). Bayesian estimates of gene diversity (*H*_{eH}) were calculated using Hickory v.1.0. Differences among young and old populations in the percentage of polymorphic loci and both measures of gene diversity were investigated using *t*-tests. To investigate whether population size was related to the percentage polymorphic loci and both measures of gene diversity, Pearson's product moment correlations were used.

## Introduction

The movement of individuals and genes in space affects many important ecological and evolutionary properties of populations (Hanski & Gilpin, 1997). For example, it is well known that the extent of gene flow affects species integrity, because gene flow counters divergence which can lead to the evolution of reproductive isolation. The rate of movement of genes from one population to another helps to determine the possibility of local adaptation and of adaptive evolution on complex landscapes. Furthermore, dispersal affects the persistence of local populations, species extinction rates, the evolution of species ranges, synchrony of population size changes, and many other important ecological properties. These genetic and ecological issues have taken new urgency in the wake of the rapid loss of biodiversity, since developing effective species conservation strategies depends on knowing the genetic and ecological relationships among populations. Population biologists would very much like to be able to measure the rate at which migration among populations occurs and have collectively devoted a great deal of effort towards measuring gene flow, migration, and their consequences in a large number of species.

Unfortunately, direct measures of migration are fraught with difficulty. Marking and following individual organisms is at the least very time-consuming and expensive, and often technically very difficult. Mark and recapture techniques are prone to biases: long-distance dispersal may be very hard to observe but very important biologically. Estimates of migration are limited in time and do not accurately reflect rare but important events, such as the dramatic gene flow which may accompany storms or climatological shifts. Finally, direct measures of dispersal do not necessarily reflect the movement of genes, because the migrant must reproduce effectively in the new location for gene flow to have occurred.

As a result of these problems, methods have been developed that attempt to use gene frequency data to infer the extent of gene flow in natural populations indirectly (Slatkin, 1985, 1987). Most famously, Sewall Wright's island model of population structure predicts that, if a long list of assumptions is true, the variance in gene frequencies among different populations should be related to the number of migrants which come into each population each generation. With the advent of molecular biology, it has become easy to measure the distribution of alleles within and among populations and therefore tempting to use these data to study gene flow. A number of recent papers have addressed the estimation of gene flow (Milligan et al., 1994 Neigel, 1997 Bossart & Prowell, 1998a), but there is controversy about the usefulness of these estimates (see Bohonak et al., 1998, Bossart & Prowell, 1998b).

These indirect estimates of gene flow have the advantage that the data necessary to make such estimates are relatively easy to gather. Further, such estimates reflect migration rates averaged among numerous populations through time. However, indirect estimates of gene flow are not without their own problems. In particular, since those estimates rely on a mathematical relationship between genetic structure and the rate of gene flow, such estimates implicitly assume that the ecological properties of the populations from which the genetic data are taken match the often unrealistic assumptions of the theoretical model upon which that mathematical relationship is based. Even when such an estimate is warranted, the estimate is subject to sampling error, which can be very large. The central theses of this paper are that these real deviations from the artificial assumptions of the models undermine the reliability of indirect measures of gene flow and that these measures have a high degree of statistical uncertainty. We suggest that, for many applications, measures of genetic structure are valuable in their own right, but that transformations of these measures to quantitative estimates of gene flow or dispersal are at best not needed and, at worst, misleading.

### Underlying theory

Wright's *F*-statistics are a set of hierarchical measures of the correlations of alleles within individuals and within populations. The *F*-statistic most relevant to the study of gene flow is *F*_{ST}, which has various interpretations most famously it is the variance in allele frequencies among populations, σ 2 _{p}, standardized by the mean allele frequency (*p*) at that locus:

See Slatkin (1985) for details concerning its derivation and Weir (1996) concerning its estimation. Wright (1931) introduced a simple model of population structure, called the island model, which predicts a simple relationship between the number of migrants a population receives per generation and *F*_{ST} (Fig. 1). Under the assumptions of the island model,

The island model. Each population receives and gives migrants to each of the other populations at the same rate *m*. Each population is also composed of the same number of individuals, *N*.

where *N* is the effective population size of each population and *m* is the migration rate between populations. Since *F*_{ST} can be estimated readily from data gathered with molecular techniques, we would seem to have a way to quickly measure the number of migrants coming into a population per generation, *Nm*. The promise of such easy information has led to a minor cottage industry of estimating *Nm* from *F*_{ST}. For example there were 13 papers in this journal which have done this in 1997 alone. (Note that there are several methods for deriving a measure of differentiation from genetic data, such as *G*_{ST}, Φ_{ST}, AMOVA , private alleles, etc., but the estimates of gene flow derived from each of these make fundamentally the same assumptions as *F*_{ST}, and we will be referring to these measures collectively in the following section.)

The island model, however, makes a large number of simplifying assumptions. It assumes an infinite number of populations, each always with *N* diploid individuals, and that each of these populations gives and receives a fraction *m* of its individuals into and from a migrant pool each generation. The individuals which do migrate are randomized and dispersed back to the populations without respect to any geographical structure, such that all populations are equally likely to give and receive migrants from all other populations. Furthermore the island model assumes that there is no selection or mutation and that each population persists indefinitely and has reached an equilibrium between migration and drift. Each of these assumptions is unlikely to be true in any particular case sometimes this will not matter very much at all with regard to estimating *Nm*, but in some cases it will matter tremendously. One intention of this review is to investigate the common ways in which natural systems violate the assumptions of the island model and to explore the effects these deviations from the simple model will have on the quantitative and qualitative conclusions from indirect studies of gene flow.

## DEFINITIONS AND THEORY

We consider *d* demes of *N* diploid individuals each and *n* diallelic loci (with alleles denoted by + and –) contributing additively to a quantitative trait.

We first define some necessary quantities. When discussing *F*_{ST} and *F*_{IS}, the underlying variable is an indicator variable for the + allele at locus *i* thus, if an allele is + and if the allele is –. We let *p _{ik}* denote the mean of in deme

*k*,

*i.e.*, the fraction of + alleles at locus

*i*in deme

*k*we let

*p*denote the overall fraction of + alleles at locus

_{i}*i*in the population (composed of two or more demes). Thus (1) We let

*q*= 1 −

*p*with any subscript. We let (2) denote the total variance of in the population, (3) the between-deme variance, and (4) the average within-deme variance of expected in a set of random-mating demes with + allele frequencies

*p*. (In Equation 4, the middle term is obtained as the sum of squared deviations of from the mean for each haplotype, weighted by the frequencies of the haplotypes.) The fixation index

_{ik}*F*

_{STi}at locus

*i*and the overall multilocus fixation index

*F*

_{ST}are then defined as (5) We note that our definition of

*F*

_{ST}, which averages variances over all loci before taking a ratio, is analogous to the preferred estimator of

*F*

_{ST}proposed by W eir and C ockerham (1984).

We let *p*_{++ik} denote the frequency of individuals in deme *k* with two + alleles at locus *i*, with *p*_{+−ik} and *p*_{−−ik} defined analogously. We define the inbreeding coefficient *F*_{ISik} as the correlation in deme *k* between the indicator variables for homologous alleles at locus *i* within an individual: (6) To discuss *Q*_{ST}, we must define trait means and variances. We denote the additive effect of locus *i* by *a _{i}* and the phenotypic value at the

*i*th locus in the

*j*th individual in deme

*k*by

*x*. We let

_{ijk}*p*denote the fraction of + alleles at locus

_{ijk}*i*in that individual, so that

*p*can take on the genotypic values 0, , or 1. Under additivity, therefore, we find that the phenotypic value of the

_{ijk}*j*th individual in deme

*k*is given by (7) We let (8) denote the between-deme component of the (additive) genetic variance contributed by locus

*i*. The between-deme component of the total trait variance is given by (9) where (10) is the covariance between loci

*i*and

*i*′.

The between-deme trait variance can be partitioned into two components: one comprising covariances between loci contributing to the trait (quantitative trait loci, QTL) and one comprising variances at individual loci. We write the ratio of these two components as (11) so that (12) As noted by L e C orre and K remer (2003), the quantity is a measure of gametic disequilibrium among QTL contributing to the trait. If the trait is neutral, then it should average to zero across replicate evolutionary histories (R ogers and H arpending 1983). We see below that writing *Q*_{ST} in terms of and its within-deme analog facilitates comparison of *Q*_{ST} and *F*_{ST}.

Analogously to (4), we define the within-deme genetic variance that would be expected in a random-mating population with + allele frequencies *p _{ik}*, (13) where is the ratio of average within-deme covariances between indicator variables for different loci to average within-deme additive genetic variance. In other words, (14) where (15) and (16) As with (defined above), is the ratio of the component of within-deme trait variance that is due to covariances between loci to the component due to individual loci and is expected to be zero for a neutral trait (R ogers and H arpending 1983). Also, it is straightforward to check that within a single population if Hardy–Weinberg equilibrium holds.

In the notation we have now established, the within-deme component of total trait variance is . Also, (17) so if *F*_{ISik} = *F*_{IS} for all *i* and *k*, then (18) Analogously to (5), and in keeping with the usage of L e C orre and K remer (2003), we finally define (19) This is the same as if mating is random (so *F*_{IS} = 0).

We now consider the relationship between *F*_{ST} and *Q*_{ST}. First, from (8), (12), (13), and (19) we obtain (20) while from (3)–(5) we obtain (21) Comparison of (20) and (21) shows that if , and if in addition all QTL have equal effects on the trait (*i.e*., for all *i*), then *Q*_{ST} = *F*_{ST}. This was observed by L e C orre and K remer (2003), although they did not examine the case of unequal *a _{i}*. As L e C orre and K remer (2003, p. 1207) noted, the condition that means that “linkage disequilibrium among QTL contributes equally to the within- and between-deme variances for the trait.” This condition does not seem to have an intuitive biological interpretation, except when both and are zero, as would be expected (on average across evolutionary replicates with random mating) for a neutral trait (R ogers and H arpending 1983).

Alternatively, using (3)–(5) one can show that (22) then from (22) and (20) it eventually follows that (23) Thus if and if (24) we have . However, we do not have *Q*_{ST} = *F*_{ST} in general.

If testing a null hypothesis of neutral evolution is the goal, then we must ascertain whether the expectations and , taken over replicate populations (*i.e.*, replicate evolutionary histories), are equal. If (as would be true if both are equal to zero), we have (25) We see from (22) and (25) [using (1)] that both *Q*_{ST} and *F*_{ST} are nonlinear functions of the random variables *p _{ik}*. Thus even if the ratio of the expected values of the numerator and denominator of

*Q*

_{ST}does equal the expected value of

*F*

_{ST}, there is no reason to anticipate that will also equal .

If and are not constrained to be equal, then *Q*_{ST} and *F*_{ST} will be nonlinear functions of these two quantities as well as of the allele frequencies *p _{ik}*. It is conceivable that in this case and could vary across evolutionary replicates in such a way as to make . However, we are not aware of any biological mechanism that could plausibly produce such a phenomenon, except as a rare coincidence.

## General Overviews

There are several edited volumes of metapopulation biology that cover metapopulation ecology as well as genetics and evolutionary biology and that review the development of the field both in terms of theory and empirical studies. Hanski and Gaggiotti 2004 is the latest and most comprehensive of the edited volumes. Hanski 1999 is a monograph on metapopulation ecology, with a focus on classic metapopulations with significant population turnover, local extinctions, and recolonizations, and the relevant models (stochastic patch occupancy models, see also Metapopulation Models). Empirical research on metapopulation biology has been largely restricted to animals because the habitat of many animals has a well-defined patchy distribution, and it is often relatively easy to define what constitutes a suitable habitat independent of the occurrence of the species the latter is critical for the study of recolonization of currently unoccupied habitats. Husband and Barrett 1996 presents a thorough overview of the application of the metapopulation approach to plants, for which the delimitation of a suitable habitat is often problematic. Over the past decade, the unified neutral theory of biodiversity has received much attention. Rosindell, et al. 2011 reviews the achievements, challenges, and potential of the neutral theory, which is a well-defined theory about spatial dynamics. Thompson 2005 has developed a theory of coevolutionary dynamics, which needs to be mentioned in this context because it assumes that the spatial structure and dynamics of populations play a critical role in their evolution and that the ecological metapopulation dynamics and evolutionary dynamics may influence each other.

Hanski, Ilkka. 1999. *Metapopulation ecology*. New York: Oxford Univ. Press.

A monograph of metapopulation ecology, covering both theory and empirical studies, with a section on the Glanville fritillary butterfly as a model system. Written for advanced students and researchers.

Hanski, Ilkka, and Oscar Gaggiotti, eds. 2004. *Ecology, genetics, and evolution of metapopulations*. Amsterdam: Elsevier Academic.

This is the most recent edited volume on metapopulation ecology, genetics, and evolution, with twenty-three chapters covering both metapopulation theory and empirical studies and a wide range of taxa and specific topics. Several chapters are quite technical, but other chapters are appropriate for advanced students.

Husband, Brian, and Spencer Barrett. 1996. A metapopulation perspective in plant population biology. *Journal of Ecology* 84.3: 461–469.

A pioneering overview of the application of the metapopulation concept and approaches to plants, discussing the particular features of plants that may make a difference in this context: seed dormancy, restricted dispersal, and local adaptation.

Rosindell, J., S. P. Hubbell, and R. S. Etienne. 2011. The unified neutral theory of biodiversity and biogeography at age ten. *Trends in Ecology & Evolution* 26.7: 340–348.

Reviews the “unified neutral theory of biodiversity” at the age of ten years. The neutral theory presents one approach to the study of spatial dynamics. Many researchers are enthusiastic about it, but others are doubtful about the power of the neutral theory to explain the spatial dynamics of species in the wild.

Thompson, John. 2005. *The geographic mosaic of coevolution*. Chicago: Univ. of Chicago Press.

Outlines a conceptual framework for coevolution in the spatial context, and thereby this book has much to offer to students and researchers interested in the evolution of metapopulations.

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