Variance effective population size is affected by census size in sub‐structured populations

Measurement of allele frequency shifts between temporally spaced samples has long been used for assessment of effective population size (Ne), and this ‘temporal method’ provides estimates of Ne referred to as variance effective size (NeV). We show that NeV of a local population that belongs to a sub‐structured population (a metapopulation) is determined not only by genetic drift and migration rate (m), but also by the census size (Nc). The realized NeV of a local population can either increase or decrease with increasing m, depending on the relationship between Ne and Nc in isolation. This is shown by explicit mathematical expressions for the factors affecting NeV derived for an island model of migration. We verify analytical results using high‐resolution computer simulations, and show that the phenomenon is not restricted to the island model migration pattern. The effect of Nc on the realized NeV of a local subpopulation is most pronounced at high migration rates. We show that Nc only affects local NeV, whereas NeV for the metapopulation as a whole, inbreeding (NeI), and linkage disequilibrium (NeLD) effective size are all independent of Nc. Our results provide a possible explanation to the large variation of Ne/Nc ratios reported in the literature, where Ne is frequently estimated by NeV. They are also important for the interpretation of empirical Ne estimates in genetic management where local NeV is often used as a substitute for inbreeding effective size, and we suggest an increased focus on metapopulation NeV as a proxy for NeI.


| INTRODUC TI ON
The concept of genetically effective population size (N e ) plays a central role in population and evolutionary biology and has attracted particular interest in the field of conservation biology because of its relation to the rates of allele frequency change and loss of genetic diversity . Rapidly growing efforts have been devoted to developing and applying methods that are based on genetic markers for estimating contemporary N e in natural populations, and a series of papers discuss and compare the performance of various approaches (Gilbert & Whitlock, 2015;Luikart et al., 2010;Palstra & Ruzzante, 2008;Wang, 2005Wang, , 2016Waples, 2016). This is particularly relevant in light of the current policy focus on effective size as an indicator | 1335 RYMAN et al. to monitor genetic diversity (CBD, 2022;Frankham, 2021;Hoban et al., 2020).
Effective population size can be seen as the size of an idealized population with the same properties of genetic drift as the population at hand (Wright, 1931). There are many ways to describe and quantify genetic drift; however, several effective sizes have been identified following Wright's original work (Gilbert & Whitlock, 2015). Wright considered a single, isolated population and defined the inbreeding effective size (N eI , Table 1) that quantifies the expected increase in inbreeding over one generation as 1/(2N e ). Another commonly used measure is the variance effective size (N eV ) that relates to the amount of random allele frequency change from one generation to the next (genetic drift). More recently, increasing interest has also focused on the linkage equilibrium (LD) effective size (N eLD ;Waples & Do, 2010) because it can be estimated from a single sample, in contrast to N eV which requires two samples collected one or more generations apart.

N eGD
Gene diversity effective size (in general). This quantity reflects the rate at which gene diversity, that is, expected heterozygosity, declines N eI Inbreeding effective size (in general). N eI reflects the rate at which inbreeding (f) increases; inbreeding is the occurrence of homozygosity of alleles that are identical by descent, that is, alleles that can be traced back to the exact same allele copy in an ancestor (also known as the coalescent). N eI is not defined for situations where inbreeding decreases, and in a case where inbreeding stays constant we have N eI = ∞

N eE
Eigenvalue effective size (of the metapopulation as a whole). The global population will eventually reach a state where inbreeding increases at a constant rate, which results in the inbreeding effective size of the metapopulation to stay constant at a value indicated by N eE . In a metapopulation where each subpopulation exchanges migrants with the rest of the system (through one or more subpopulations), the rate of inbreeding will eventually be the same (1/(2N eE )) in all subpopulations as well as for the system as a whole N eIMeta Total (global) inbreeding effective size of the metapopulation as a whole. This quantity reflects the change of inbreeding of the metapopulation (f Meta ) from generation t to t + 1. N eIMeta can be viewed as a weighted average of N eIRx over all subpopulations, and it will eventually approach N eE N eIRx Inbreeding effective size of subpopulation x (under prevailing migration scheme)

N eLD
Linkage disequilibrium effective size (in general); it reflects the degree of linkage (gametic phase) disequilibrium. Mathematical treatment of N eLD is complicated and not yet fully resolved. Approximate equations for N eLD in a local population exist for the special case of an ideal (N ex = N cx ) island model (Ryman et al., 2019;Waples & England, 2011)  For an isolated population of constant size, all these versions of effective size are typically the same (see Appendix S5).
In many contexts such as conservation biology, N eI is the effective size of primary interest because it relates directly to the rate of inbreeding, which is important to quantify in conservation because it affects population fitness (Frankham, 2005 ulation is expected to approach that of the metapopulation as a whole (N eIMeta ), whereas N eV does not Ryman et al., 2014Ryman et al., , 2019.
The above observations in Ryman et al. (2014Ryman et al. ( , 2019 on different behaviours of various N e have been obtained when modelling and simulating idealized populations assuming that local census (N c ) and effective sizes are the same (N c = N e ). Such an assumption is highly unrealistic for most natural populations, however, and in this paper we remove this restriction. Using a finite island model of migration, we derive novel expressions for how local and total (global) N eV is expected to change as a function of local census size (N c ). We then employ computer simulations to verify our analytical results. We show that N eV is strongly affected by census size, particularly at high migration rates. The dynamics of local N eV is strikingly different from that of N eI when the metapopulation evolves towards migration-drift equilibrium. In contrast to the results for N eV , N eI is not affected by N c, and computer simulations suggest that there is little or no effect of census size on N eLD . The results are discussed in the context of assessment of effective size of natural populations.

| ME THODS AND THEORE TIC AL BACKG ROU N D
Using an island model of migration, we derive new explicit formulas for the influence of local census size (N cx ; Table 1) on local and global variance effective size (N eV ) when the latter quantifies genetic drift between consecutive generations. We also explore the corresponding influence on inbreeding (N eI ) and linkage disequilibrium (N eLD ) effective size, and compare the results with those from a linear stepping stone model. The analysis is based on our previously developed theory for several local and global effective sizes of an arbitrary metapopulation during the approach to migration-drift equilibrium (Hössjer et al., 2013(Hössjer et al., , 2015Ryman et al., 2014Ryman et al., , 2019, and the results are checked against computer simulations. We use the more general definition and define N eV of a population as the size of an ideal population with the same amount of allele frequency change from one generation to the next as the focal one, regardless of the evolutionary forces causing those changes. Throughout this paper, we ignore the effects of mutation and selection and focus exclusively on genetic drift and migration. We follow the terminology of Laikre et al. (2016) and Ryman et al. (2019) and make a distinction between N e of a local population 'x' under isolation (N ex , which is the same for all types of effective size; Table 1) and the realized (R) effective size of subpopulation x when the joint effects of drift and migration are considered (e.g. N eIRx or N eVRx ). Importantly, the realized effective size is the quantity being estimated when sampling from a local population under migration and applying an unbiased estimator.

| Definitions and terminology
Furthermore, the metapopulation as a whole is thought to be isolated without immigration from other sources, implying that 'realized N e ' only refers to local subpopulations, whereas migration between subpopulations is always included when considering the total metapopulation effective size (N eVMeta ).

| Migration
In an island model, where immigrants originate from the global population 'as a whole', some texts include the target population in 'as a whole', whereas others do not. In this paper, we let m and m′ signify situations that include and exclude the target popula-

| Population model
We consider an island model metapopulation of a non-selfing diploid organism with discrete generations and a population system of s subpopulations, each of effective size N ex (in isolation) and census size N cx (Table 1). Mating occurs after migration, and migration is stochastic such that migration rates reflect the binomial average. Immigrants are assumed to be drawn from an infinitely large migrant pool to which all the s subpopulations (including the focal one) have contributed equally. In the limiting case of m = 1, the global population corresponds to a homogeneous population with sN c individuals and 2sN c genes, which is also a Wright-Fisher model in the special case when the local effective and census sizes are the same in all subpopulations (N cx = N ex ). We refer to this situation (m = 1) as panmixia (see Appendix S1: Section 2 and Appendix S5).
In brief, the population model is demographically deterministic and genetically stochastic, where s, N cx , N ex and m are fixed quantities, whereas alleles are sampled binomially from a mixture of the focal subpopulation and the migrant gene pools, corresponding to the case of 'stochastic migration and fixed migration rate' discussed by Sved and Latter (1977). As detailed in Appendix S1, this migration scheme corresponds to a reproduction scheme with the following three steps (see also Appendix S2 and Figure S1).

| Analytical approach
We provide two different explanations of how N eVRx depends on N cx (Appendices S1 and S2). The derivations in Appendix S1 are mathematically oriented and deal with the more general situation where k out of s subpopulations are sampled in two consecutive generations.
The approach in Appendix S2 is less rigorous, more intuitive and only refers to the basic case where one and the same subpopulation is sampled in both generations (k = 1); it is summarized in the beginning of the Results section. The main text only includes a few important expressions and is meant to be possible to follow without examining the appendices.
In Appendix S1 (mathematically oriented), we derive the expected value for nearly unbiased estimates of N eV calculated from infinitely large samples collected in each of two consecutive generations from a single population (x) that belongs to an island model population system (E N eVRx ; Table 1). More precisely, we apply the so-called temporal method (Nei & Tajima, 1981;Ryman et al., 2014;Wang & Whitlock, 2003;Waples, 1989), where samples are collected at two points in time and the allele frequency change is used for assessment of N eV , and we focus on consecutive generations to estimate contemporary N eV by the method proposed by Jorde and Ryman (2007). This approach of assessing N eV will hereafter be referred to as 'expected estimate'.
When describing how N eI and N eV change as the metapopulation moves towards migration-drift equilibrium, we applied our recently derived recursion equations, where key expressions include equation (48) in Hössjer et al. (2015) and equation (26) in Hössjer et al. (2016). The two methods for computing N eV ('expected estimate' vs. recursion) result in almost the same value for samples collected at two consecutive generations (Appendix S1: section 3.3), but the approach using 'expected estimate' allows derivation of explicit expressions describing how N eV depends on the parameters characterizing the metapopulation (Appendix S1: section 2). In contrast, since the recursion approach makes use of matrix algebra, it extends to subdivided populations of arbitrary form and longer time periods between the samples (section 3 of Appendix S1). For some of the recursion computations of N eI , we used the GESP computer program (Olsson et al., 2017).
A general theory for the linkage disequilibrium effective size (N eLD ) is missing, most likely because LD between a pair of markers in a sub-structured population is due to a complicated balance between genetic drift, recombination and migration. We believe the results of Hössjer and Tyvand (2020) can be used to find analytical expressions for N eLDRx . In the absence of such analytical results, however, approximate expressions for the expected N eLD of subpopulations belonging to an ideal island model metapopulation with N cx = N ex have been derived by Waples and England (2011), subsequently generalized and modified by Ryman et al. (2019).
To our knowledge, there are no corresponding expressions for the global form of N eLD . We assess the influence of census size on subpopulation N eLD under an island model by computer simulations (below). We also checked the reliability of the simulations by comparing simulated results for ideal conditions (N cx = N ex ) with those expected analytically (equation (19) in Appendix of Ryman et al. (2019)).

| Simulation approach
A detailed description of the computer simulations is given in Appendix S3. In brief, we conducted simulations to verify our analytical results for N eV and for assessing the influence of census size on subpopulation N eLD (because of the abovementioned lack of theory). The Easypop software (Balloux, 2001) was used to simulate the drift process. We considered an island model metapopulation of s = 10 subpopulations with an effective size of N ex = 50 (in isolation) and a census size N cx ≥ N ex . Effective size was estimated from a minimum of 100 individuals from the output for the final generation(s).
We applied the unbiased estimator of Jorde and Ryman (2007) for estimation of N eV , and for N eLD we used the LD method of Waples and Do (2008) as implemented in the software NeEstimator V2 (Do et al., 2014). Ultimate estimates were taken as the harmonic mean from 100 replicate runs.

| Relationship between N eV , N eI and N eLD
Our focus in this paper refers to the variance effective size (N eV ), but we also compare the dynamics of N eV with that of the inbreeding (N eI ) and linkage disequilibrium (N eLD ) effective sizes. The local forms of all N e are typically the same in an isolated (m = m′ = 0) population.
Thus, under an island model, where the effective size of all subpopulations is expected to be the same, we have N eI =N eV =N eLD =N ex in isolation (see Appendix S5). this global average will change much slower than those of the separate subpopulations, and N eVMeta is therefore expected to be (much) larger than N eVRx . The magnitudes of N eIMeta and N eVMeta may be quite different before migration-drift equilibrium has been attained, but at equilibrium they are very similar. The reason is that N eVMeta at equilibrium behaves like a haploid version of N eIMeta , which, in turn, is very similar to the diploid version of N eIMeta considered here (Hössjer et al., 2015. The difference between the equilibrium values is typically negligible (as exemplified in Figure 1) and is ignored in the present paper.
F I G U R E 1 Theoretically expected (solid lines) and simulated (dots) subpopulation variance realized effective size (N eVRx ) at different migration rates (m) and census sizes (N cx ; indicated to the right of each curve) for an island model metapopulation with s = 10 subpopulations of effective size N ex = 50 in migration-drift equilibrium. Memory restrictions of the Easypop software precluded simulation of the largest census size N c = 1000,000 (black curve). The top dotted and solid blueish lines show the global effective sizes (N eVMeta and N eIMeta , respectively) which are not affected by census size. Expected values for the local and global N eV were calculated from equation (A17, Appendix S1) assuming that one and the same subpopulation is sampled in both generations (k = 1).
Note that under the island model N eVMeta ≈ N eIMeta = N eIRx .

| Influence of migration model and sampling strategy
The island model of migration is extreme with respect to connectivity as an individual can migrate from one subpopulation to any of the others in a single generation. To check that the basic results from the island model are applicable to other migration models, we compared with results from a linear stepping stone model, where migration only occurs between neighbouring populations.
In some situations, more than a single local population may be included in a sample. Such inclusion may be unintentional, reflecting a poorly understood population structure, difficulties to identify subpopulation boundaries or deliberate in an attempt to assess the global effective size (N eVMeta ). We examined, for a few basic situations, how the resulting estimates of N eLD and N eV are affected by sampling from multiple populations. We used both computer simulations and the analytical results of the present paper for these analyses.

| Sampling from one subpopulation
We derive analytical expressions for the expected estimate of variance effective size (N eV ) for a finite island model when sampling from k (out of a total of s) subpopulations in two consecutive generations (Appendix S1). We deal primarily with the situation where the k sampled subpopulations are the same in both generations (Appendix S1: section 2.3.2), but we also give expressions for the case where they are completely different (Appendix S1: section 2.3.3).
We start with the basic situation where one and the same subpopulation is sampled in both generations (k = 1). Application of equation  Table 1), and the graphs depict the expected N eVRx (equation (A18)) of a subpopulation at various migration rates for census sizes in the range 50 ≤ N cx ≤ 1000,000. A large census size relative to N ex tends to increase N eVRx , and the effect is most pronounced at high migration rates. In contrast, in the 'ideal' situation where census and effective size are the same (N cx = N ex = 50 in this example), the local variance effective size decreases from N eVRx = 50 to N eVRx = 26.3 as the migration rate goes from m = 0 (complete isolation) to m = 1 (panmixia).
For the very large N cx = 1000,000, the corresponding change is an increase from 50 to 500.
3.1.1 | Why is N eVRx affected by N cx ?
The derivations in Appendix S1 are mathematically oriented, and it may be difficult to see directly from those equations how different parameters interact to result in the pattern exemplified in Figure 1.
The less rigorous approach of Appendix S2 attempts to provide a more intuitive explanation, and a summary of this appendix that is meant to focus on the biologically most important results follows below.
We show (Appendix S2 (equation A1)) that an approximate relationship between local variance effective size (N eVRx ) and the basic parameters describing an island model metapopulation is a sum of three terms, such that These terms relate to the three steps, gamete formation, migration and fertilization, in the reproduction cycle (cf. Figure S1, Appendix S2 (the text below equation (A1)), and section 2.1 of Appendix S1) with F ST referring to the fixation index, that is, the amount of genetic differentiation between all subpopulations. Note that in the case of complete isolation (m′ = m = 0), equation (1) reduces to 1/(2N ex ), as it should.

The case N cx ≈N ex
To describe how N cx impacts N eVRx , we focus on the behaviour at high migration rates where the influence of N cx is most pronounced ( Figure 1). If we assume that N cx is close to N ex , then there is virtually no contribution from the first term because N cx ≈N ex .
Furthermore, and for the same reason, the second term is reduced to m � 2 (s∕(s − 1)) 2 F ST , and the last one can be replaced by 1/(2N ex ).
Thus, the total expression is larger than it would be in x alone in isolation (i.e. 1/(2N ex )), causing N eVRx to decrease compared to N ex .
The larger m′ is, the larger is the impact of the second term (immigration into x) causing N eVRx to get progressively smaller (cf. Figure 1).

The case N cx ≫ N ex
In contrast, if N cx is much larger than N ex (N cx ≫ N ex ), all the terms in Equation (1) that contain N cx can be ignored (because they become very small). Furthermore, in the limit of panmixia, the impact of subpopulation differentiation can be ignored, since F ST is negligible (Appendix S2). Therefore, by adding the two remaining terms of (1) that involve N ex , we find that the total expected squared allele frequency change within x is proportional to 1/(2sN ex ), and hence N eVRx is close to sN ex , which in Figure 1 corresponds to N eVRx ≈ 10 × 50 = 500.

| Approach to equilibrium
The approach to equilibrium of the local forms of N eI and N eV is shown in Figure 2a for an island model metapopulation with the same demographic characteristics as that in Figure 1, exemplified with m = 0.50. Clearly, the approach is very quick for N eVRx , and much quicker than for N eIRx , in line with previous results from ideal situations Ryman et al., 2019). Furthermore, it appears that the rate of approach of N eVRx is more or less independent of census size. The general relationships among the different forms of effective size are the same at other migration rates as exemplified in Figure 2b where migration is m = 0.10, and the approach to equilibrium is therefore slower. Here, the effect of census size on N eVRx is also much smaller, in line with the results in

| Effect of migration model
The effect of N cx on N eVRx is not confined to metapopulations following the migration pattern of an island model. As an example, Figure 3 shows the realized local variance effective size (N eVRx ) at equilib-

F I G U R E 3
Expected global (N eVMeta ) and realized local (N eVRx ) variance effective population sizes at equilibrium under island and linear stepping stone migration models for metapopulations with identical basic demographic parameters (s = 10, N ex = 50, and N cx = 50, 500 or 10,000 (red, green and blue lines, respectively)) but different migration patterns. Census size for local populations is shown to the right of the curve. Linear stepping stone: m′ = rate of immigration from neighbouring populations (0-0.9; Table 1), and each subpopulation receives an average proportion of m′/2 immigrants drawn at random from each of the neighbouring ones. Realized local effective size is only given for subpopulations 1 and 5 (ordering from left to right) as indicated after each specific N eVRx curve; note that the symmetry of the linear stepping stone model implies that pairwise identical N eVRx are expected for subpopulations 1 and 10, 2 and 9, etc. The abbreviated notation for local populations and N cx means, for example, that '5-10,000' corresponds to local subpopulation 5 and N cx = 10,000. The stepping stone calculations were performed using recursion equations (26)

| Sampling from multiple subpopulations
Up to this point, we have only considered samples drawn from a single subpopulation. Below we present results for a few basic situations of sampling, in equal proportions, from multiple subpopulations of an island metapopulation. The expected N eVRx is then calculated from the mixed sample.

| Sampling the same k subpopulations
It follows from equation (A21) of Appendix S1 that a sample from all the s local populations (k = s) at both occasions is expected to estimate N eVMeta regardless of the value of N c , because this strategy implies that the entire metapopulation is sampled representatively ( Figure 4). The situation becomes more complicated when less than s subpopulations are included in the sample (k < s), because the expected estimate is now also affected by k, m and N c (Appendix S1, equation (A17)).
As an example, we evaluate equation (A17) of Appendix S1 numerically for an island metapopulation with the same basic characteristics as in Figure 1

| Sampling from different subpopulations
We show analytically (Appendix S1) that the effect of sampling completely different subpopulations at two occasions is qualitatively the same regardless of the relation between N ex and N cx , which can be verified by numerical evaluation of equation (A26) in Appendix S1. This formula gives the expected value of the estimate of N eV when both samples are drawn from k completely different subpopulations (no overlap). At low migration rates, the expected estimate is very small and often less than unity. At the other extreme, when m = 1, the expected estimate coincides with that for sampling from the same k subpopulations at both occasions (as in equation (A17, Appendix S1) and Figures 1 and 2). That is, under panmixia the expected estimate is the same for all samples based on k subpopulations regardless of which they are (cf. Ryman et al., 2014).

| N eV vs. N eLD in mixed samples
Estimates of local N eV and N eLD are affected differently when obtained from mixed samples. As an example, Figure 5 shows results

F I G U R E 4 Expected estimate of
realized N eV at different migration rates (m) when sampling with equal probabilities the same k = 1, 2, 5, 9 or 10 subpopulations from an island model metapopulation with the same basic characteristics as in previous figures (s = 10 and N ex = 50). (a) N cx = 50, (b) N cx = 500. N eV was calculated using equation (A17) in Appendix S1.

| Relationships between N eV , N eI and N eLD
The dotted top graph in Figure 1

| DISCUSS ION
Using both analytical and simulation approaches, we have shown that census size (N cx ) influences the variance effective size (N eVRx ) of local F I G U R E 5 Simulated estimates of N eLD (rings and triangles) and N eV (squares and diamonds) at different migration rates when sampling, with equal probabilities, from a mixture of k = 2 subpopulations in an equilibrium island model metapopulation (s = 10, N ex = 50 and N cx = 50 or 500). The curves are theoretically expected estimates of N eVRx , and they correspond to those for k = 2 in Figure 4.
populations that are part of a metapopulation system, and we have examined both an island and a linear stepping stone model to exemplify this effect. Increasing local census size results in increased variance effective size (but the relationship is not linear), and the effect is most pronounced at high migration rates. We note that this effect of census size on local N eV has nothing to do with the simple fact that N ex is expected to increase with N cx if N ex /N cx is constant, because N ex stays constant when N cx increases (e.g. Figure 1). The behaviour of local N eV is strikingly different from that of the other forms of effective size discussed in this paper (local N eI and N eLD and global N eI and N eV ), which for all practical purposes are independent of census size.
It is important to realize that the effect of N c on N eV does not represent some kind of bias. The parametric value of local N eV under migration (N eVRx ) reflects the expected amount of contemporary allele frequency shift and is the quantity estimated when applying an unbiased estimator of N eV .
Our results may appear counter intuitive. It might be felt that once we know the effective size of a population in isolation we should be able to predict its genetic dynamics under migration. For example, the subpopulations of two metapopulations that differ only in local census size (N cx ), but have the same local effective size in isolation (N ex ), could be expected to evolve identically. As we have shown, however, they do not with respect to variance effective size.

| Interpretation of N eV estimates
Our study expands on some of the findings in a previous paper Applying the temporal method to samples collected from an island model subpopulation is expected to result in N eVR estimates similar to those exemplified in Figure 1. At low migration rates, say m < 0.10, those estimates should constitute relatively good assessments of the effective size in isolation (N ex ) regardless of the relationship between effective and census size. At higher migration rates, however, say m > 0.2, the local estimates are low or high relative to N ex , depending on N ex /N cx , and at panmixia they approach the global effective size as the census size goes towards infinity.
Interpreting such estimates in terms of local or global effective size is therefore difficult or impossible without some kind of information on m and/or N cx . Similar problems are expected also under other migration scenarios as suggested by Figure 3 for the linear stepping stone, which represents an extreme opposite to the island model with respect to connectivity.
Our new understanding of the dynamics of local N eV under migration and its dependence on census size should not be exclusively viewed as an obstacle, however. As discussed below, it presents the empirical investigator with novel opportunities for making inference about other forms of effective size that are difficult to assess directly in natural populations. One example, that is the main focus of Section 4.2, refers to the estimation of inbreeding effective size (N eI ), which is the primary genetic parameter for many workers in the fields of conservation biology and management.
F I G U R E 6 Meta-and subpopulation variance (N eVRx ), inbreeding (N eIRx ) and linkage disequilibrium (N eLDRx ) realized effective size at different migration rates (m) and census sizes (N cx = 50, 200, 500 or 1000) for an island model metapopulation in migration-drift equilibrium with s = 10 subpopulations of effective size N ex = 50. The line for N eLDRx was calculated using equation (19) in the appendix of Ryman et al. (2019) that assumes N cx = N ex (cf. Appendix S4). As detailed in the text, N eVRx is influenced by N cx , whereas N eIRx and N eLDRx are not (or nearly not).

| Management implications: Using N eV as a minimum estimate of N eI
In many situations where N eV is assessed by the temporal method from the data at hand, the investigator is primarily interested in another form of N e (e.g. N eI ) and uses N eV as a proxy. Such an approach is perfectly reasonable when dealing with completely isolated populations (no immigration) where all forms of N e are typically the same (see Appendix S5). In structured populations, however, various forms of N e may differ seriously, and using one as a substitute for another can often be misleading (Ryman et al., 2019). The '50/500 rule' of Franklin (1980) presents an example of such a situation.
The 50/500 rule has become widely established in conservation biology, suggesting that for a single isolated population N e ≥ 50 is needed for short-term conservation and N e ≥ 500 for long-term conservation Franklin, 1980). Here, the shortterm rule of N e ≥ 50 refers to an effective size quantifying the rate of inbreeding (inbreeding effective size, N eI ). The long-term N e ≥ 500 refers to an effective size relating to loss of additive genetic variation, which for local populations is usually similar to a haploid version of N eI referred to as gene diversity effective size, N eGD (Table 1; Hössjer et al., 2016;Ryman et al., 2019); the concern here is maintenance of sufficient levels of genetic variation for quantitative traits associated with fitness. Thus, in the context of genetic conservation, interest is often focused on N eI rather than N eV .
It is a major problem, however, that N eI is difficult or impossible to estimate in natural populations using current estimators, and those most frequently applied target N eV or N eLD (Gilbert & Whitlock, 2015;Ryman et al., 2019). In some situations, however, guidance may be obtained by comparing different forms of estimates. To exemplify we focus on equilibrium conditions and samples collected from a local population. The observed allele frequency change estimates contemporary N eV , and if the focal population is isolated (m = 0), we expect the estimates of N eV and N eLD to be similar and coincide with N eI . In contrast, for populations belonging to a metapopulation N eVRx is expected to severely underestimate the focal parameter N eIRx (e.g. Figures 1 and 6). Obtaining a low estimate of N eVRx (say N eVRx ≈150) raises the question: is this population at 'long-term risk', or is it part of a larger metapopulation, potentially implying that our estimate of N eVRx underestimates N eIRx and that the situation is probably (much) better than suggested by the N eVRx estimate? One way of approaching this issue would be to compare the N eVRx estimate with those of N eLDRx assessed from either or both of the same temporally spaced samples.
Empirical estimates of N ex /N cx ratios in natural populations are often found in the interval 0.05-0.30 (Frankham, 2021;Frankham et al., 2019;Hoban et al., 2020;Palstra & Ruzzante, 2008). Thus, under a metapopulation model and most realistic N ex /N cx ratios, we expect the estimate of N eVRx to be smaller than those of N eLDRx at high migration rates (say, m > 0.50; cf. Figure 6). Such an observation would speak in favour of N eVRx underestimating N eIRx , thus relaxing the impression of immediate threat and allowing time for more detailed investigations before launching a rescue program.
In contrast, if the focal population is isolated we expect the estimates of N eVRx and N eLDRx to be similar and coincide with N eIRx .
Thus, using the larger of the two estimates (N eVRx vs. N eLDRx ) as a lower limit for the targeted inbreeding effective size (N eIRx ) should in most cases be considered a safe approach.
Estimates of N eLDRx being smaller than that of N eVRx are only expected to occur when multiple populations have been sampled ( Figure 5) and would suggest that the assumption of dealing with a single subpopulation has been violated. We have recently observed such an empirical relationship between estimates of N eV and N eLD in brown trout (Salmo trutta) when comparing allele frequencies in natural populations sampled about 40 years apart. However, a detailed analysis of the population structure revealed the existence of multiple populations in the samples, and the conspicuously low estimates of N eLDRx turned into more 'normal' ones after adjusting for this previously unrecognized structuring (Andersson et al., 2022).
Comparisons of the above type must, of course, be made with great caution accounting for factors such as overlapping generations and statistical considerations when comparing different forms of N e estimates (Gompert et al., 2021;Jorde, 2012;Jorde & Ryman, 1995Waples et al., 2014;Waples & Yokota, 2007). We also note that our present observations refer to the island and linear stepping stone migration models and equal subpopulation sizes. Additional analysis may be needed for other migration models, and the assignment of weights to the data available from different subpopulations must be considered when their sizes vary.

| A proxy for N eI that is independent of N c
It appears that difficulties associated with translating local N eV (N eVRx ) into local N eI (N eIRx ) are inevitable because of the dependence on local N c (N cx ), particularly at high migration rates. An alternate strategy that should be applicable to metapopulations at, or reasonably close to, migration-drift equilibrium would be to focus on N eIMeta rather than on the elusive N eIRx of some particular subpopulations. The logic is as follows and is based on the characteristics of a metapopulation at equilibrium.
First, the only form of N eV that is independent of N c is global variance effective size (N eVMeta ), and unbiased estimates of this parameter can be obtained using the temporal method. Furthermore, under equilibrium conditions, N eVMeta can be safely used as a proxy for N eIMeta (e.g. Figure 1). Finally, estimating N eIMeta at equilibrium is equivalent to estimating N eIRx . This is so because, regardless of local effective size in isolation (N ex ), all subpopulations that both receive immigrants from and send emigrants to the rest of the system have the same rate of inbreeding as the system as a whole, that is all N eIRx = N eIMeta at equilibrium Laikre et al., 2016;Ryman et al., 2019).
Thus, an estimate of N eVMeta can be used as a proxy both for the rate of inbreeding in the metapopulation as a whole (N eIMeta ) and for that in each of the separate subpopulations (N eIRx ). Such a strategy is also in line with the view that the N eI > 500 criterion should refer to N eIMeta rather than to N eIRx (Jamieson & Allendorf, 2012;Laikre et al., 2013Laikre et al., , 2016 because they are expected to be the same at equilibrium. An unbiased estimate of N eVMeta requires representative sampling from all the subpopulations included in the metapopulation (Figure 4), and this may be difficult to accomplish for large metapopulations or when the structure is poorly known. We note, however, that an estimate based on a restricted number of subpopulations is always expected to result in an estimate that is closer to N eVMeta than for samples drawn from a single subpopulation, and this general relationship holds true regardless of migration rate and the N e /N c ratio.
Estimation of N eVMeta requires at least some crude information on the population structure such that the metapopulation and its subpopulations can be identified. Inclusion of only a portion of the subpopulations will result in an estimate that has a downward bias, but this may be acceptable in situations where this estimate suggests that N eMeta is substantially larger than what is considered required for, for example, long-term conservation (say, N eMeta > 500).
Furthermore, N eVMeta is typically (much) larger than N eVRx , and estimates of a large N e are generally associated with large standard errors, frequently resulting in upper confidence bounds approaching infinity. Even such estimates can be quite valuable, though, because in practice the lower bound is always finite, and in conservation interest is often focused on threshold criteria such as the 50/500 rule.
Using N eVMeta as a proxy for N eIMeta or N eIRx is based on the assumption of drift-migration equilibrium, and this conjecture can be difficult to test empirically. Here, theoretical modelling based on available data on the demographic characteristics of the metapopulation can provide information on whether it is realistic to assume conditions reasonably close to equilibrium. For example, N eV approaches equilibrium faster than N eI (Figure 2; Ryman et al., 2019), and the dynamics of this quantity can be modelled for arbitrary metapopulations using the GESP software (Olsson et al., 2017).
There is no corresponding expectation for N eLD estimates based on a mixture of samples from multiple subpopulations (lack of theory). Our simulation results suggest that such estimates tend to approach N eMeta as migration increases towards m = 1 (panmixia), and that this tendency appears independent of both the number of subpopulations sampled and N e /N c ( Figure 5 and Appendix S4). However, the disequilibrium due to mixing samples from differentiated subpopulations affects the N eLD estimates at intermediate and low migration rates, and N eLD cannot be used as an unbiased estimator of N eMeta , not even when all the subpopulations are included in a sample. As an example, we simulated the sampling of n = 50 individuals from each of the s = 10 subpopulations of the island metapopulation with N cx = 50 described in Figure 4, and estimated N eLD from the mixture. The resulting estimates were N eLD = 202 and N eLD = 342 at m = 0.1 and m = 0.2, respectively, that is much lower than the expected values of N eMeta ≥ 500 (Figures 1 and 6).

| The ratio N e /N c
Considerable interest has focused on the relationship between effective and census size (N e /N c ) in natural populations, and the reason is easy to see. N c is often easier to estimate than N e , and if the ratio is reasonably stable within species or species groups it would be possible to obtain crude estimates of N e through assessment of N c alone (Luikart et al., 2010;Palstra & Fraser, 2012). Empirical estimates of the N e /N c ratio show a wide range of variation, however, and a series of reasons for this variability have been suggested (Myhre et al., 2017;Nunney, 1993;Ruzzante et al., 2016;Waples, 2016).
Our present results suggest that the ratio of effective to census population size must be interpreted with great caution when N eV is used to estimate effective size in spatially structured populations.
N eVRx is highly dependent on migration rate (m); it may either increase or decrease as m grows larger. This dependence on m is expected to result in considerable variation of N e /N c ratios among conspecific populations under migration, which would exhibit similar ratios in isolation, just because of the potential for connectivity patterns to vary enormously both within and between metapopulations of the same species under natural conditions. Further thought on how to deal with this dependence on m is warranted, and it is possible that meaningful comparisons of local N eV /N c ratios should be confined to completely isolated populations (m = 0).

| Future use of the temporal method and N eV
The temporal method, which requires two temporally spaced samples, was traditionally the one most commonly applied for assessments of effective size. During the past decade, however, estimation procedures that only need a single sample, collected at one point in time, have become prevailing (Palstra & Fraser, 2012;Wang, 2016;Waples, 2016), particularly the one that is based on LD (Gilbert & Whitlock, 2015). In the light of those relaxed sampling requirements and our present observation that census size affects N eV , it may be appropriate to ask if the temporal method has had its day.
We do not think that the temporal era is over, though. Rather, we anticipate intensified use of the temporal method, for example in the context of genetic monitoring where much of the interest is focused on temporal genetic change in natural populations (Hoban et al., 2020). One-sample methods were originally designed to assess effective size over one or a few generations, describing conditions as they were at the time when the sample was collected. For N eLD , this is true when LD is estimated between unlinked markers, since the balance between recombination and genetic drift then reflects what happened in the very recent past. Although LD for closely linked markers can be used to trace historical changes of N e (Boitard et al., 2016;Santiago et al., 2020), the time frame of these changes is not a uniquely defined interval and typically larger than the time span that is of interest in conservation. In contrast, the temporal method more directly relates to the shift of allele frequencies between two well-defined timepoints at which genetic data are collected, and estimates of N eVRx can capture signals that would otherwise go unnoticed from, for example, unanticipated bottlenecks during the period between the sample collections (Frankham, 2021;Frankham et al., 2019). Furthermore, at low or moderate migration rates (e.g. m < 0.3) estimates obtained with the temporal method (N eVRx ) are typically good estimates of N ex , that is of any N e in isolation. Such estimates of N eVRx are fairly independent of m and only weakly affected by N cx (e.g. Figures 1 and 2). In addition, N eVRx never deviates much from its value under migration-drift equilibrium and approaches this equilibrium much faster than other effective sizes ( Figure 2). Thus, an estimate of N eVRx often provides a good assessment of local inbreeding effective size in isolation (N ex = N eI ), and this type of information can be extremely valuable for managers working with populations under risk to become isolated due to environmental changes.
Finally, a single effective size estimate of N eVRx obtained when sampling a local population under migration cannot capture, for example, a difference between N eIRx and N eVRx which can be critical in a conservation context. Estimating N eVMeta or comparing estimates of, say, N eVRx and N eLDx , may, however, provide guidance on how to interpret those estimates relative to inbreeding effective size (N eI ) and the degree of urgency for rescue actions.

AUTH O R CO NTR I B UTI O N S
The idea behind this paper developed through recurrent discussions among the authors on problems relating to estimation of effective population size in the context of conservation and monitoring of genetic diversity, including indicator developments. Calculations were completed by N.R. who also conducted the computer simulations and wrote the first version of the manuscript with input from both coauthors. O.H. derived the mathematical expressions, wrote the appendices S1, S2 and S5, and contributed theoretical input throughout the project. All the authors participated in discussions that developed the work, and they actively contributed to the entire writing process that was led by N.R.

ACK N O WLE D G E M ENTS
Three anonymous reviewers provided comments and suggestions on the text that improved the presentation considerably.

CO N FLI C T O F I NTE R E S T S TATE M E NT
None declared.

DATA AVA I L A B I L I T Y S TAT E M E N T
This study is theoretical and not based on empirical data. The software used are referenced and freely available. Analytical expressions have been published previously, except for those derived in the Supporting Information.