## Introduction

There is current concern over accelerating rates of loss of genetic variation in natural populations due to declining population size, potentially resulting in reduced capacity for adaptive change and future evolution (Frankel 1974; Traill *et al*. 2010). To obtain quantitative assessments of present rates of loss of genetic diversity, estimating the genetically effective size of natural populations has become increasingly common in the fields of conservation and evolutionary genetics (Palstra & Ruzzante 2008).

The expected rate of loss of heterozygosity of a population is 1/(2*N*_{eI}) per generation, where *N*_{eI} is the inbreeding effective size of the population. There is also a variance effective size (*N*_{eV}) that reflects the amount of random gene frequency change from one generation to the next (genetic drift), but the two quantities *N*_{eI} and *N*_{eV} are identical when population size is constant (Crow & Kimura 1970). Many simplified models of population structure only use the concept ‘effective size’ (*N*_{e}) without making the distinction between *N*_{eI} and *N*_{eV}.

When assessing the effective size of natural populations, many studies apply the so-called temporal method that provides an estimate of the variance effective size (*N*_{eV}) through the amount of allele frequency change over one or more generations (Waples 1989). The temporal method assumes that the study population is completely isolated and that any observed genetic change is entirely due to genetic drift caused by restricted effective size. In the real world, however, many or most populations are only partially isolated, that is, they belong to a population system and are connected to neighbouring ones through migration. Violation of the assumption of complete isolation constitutes a common source of bias, the direction and magnitude of which is generally unknown, and investigators applying the temporal method tend to ignore or minimize the effect of migration in the discussion of their results (Wang & Whitlock 2003; Leberg 2005; Wang 2005; Luikart *et al*. 2010; Waples & England 2011).

There are several challenges involved in the estimation of effective size of a population that belongs to a population system. First, allele frequency shifts caused by immigration into the focal population may erroneously be interpreted as genetic drift and thus bias the estimate of *N*_{eV} (Wang & Whitlock 2003). Further, when genetic differentiation is weak between the subpopulations that constitute the population system (the global population), it may be difficult to identify population boundaries and target the focal population for sampling. Many marine organisms, for example, are characterized by high migration rates and low levels of divergence between populations (Ward *et al*. 1994). A sample from the wild may easily include multiple populations, and samples collected at different occasions may consist of individuals from more or less disjunct population segments. The present study was actually prompted by empirical results on brown trout (*Salmo trutta*) where the temporal method provided strikingly small estimates of *N*_{eV} that appeared incompatible with a seemingly very large and genetically homogeneous population (Palm *et al*. 2003).

Finally, there are two effective sizes to be considered: the local population and the population system as a whole (*N*_{eV,tot}). In the absence of extensive studies designed to delineate genetic population structure, it is difficult to tell the difference between a subdivided population and a randomly mating one (Ryman *et al*. 2006; Waples & Gaggiotti 2006). An investigator sampling from a seemingly large and genetically homogeneous population may actually be dealing with a population system without being aware of it. In such situations, an estimate of *N*_{eV} can be strongly misleading because, depending on the subpopulations included in the sample, it may refer to a local subpopulation affected by immigration, the global population or something in between.

The problem of using the temporal method for estimating effective size of a population under migration has been addressed by Wang & Whitlock (2003). They devised an approach for simultaneous estimation of *N*_{eV} and the immigration rate (*m*) that can be applied to situations where allele frequency estimates are available for the immigrant gene pool as well as for the focal population. They also discuss some of the general effects of ignoring migration when estimating *N*_{eV} for the special case of a local population receiving immigrants from an infinitely large donor population.

There is, however, no theory that quantifies the expected bias in the estimate of *N*_{eV} in terms of the characteristics of the global population and the number of subpopulations included in the sample. In this study, we provide expressions for the expected value of the estimate of *N*_{eV} when applying the temporal method to samples from a population system where the component local populations (subpopulations) are connected by migration, focusing on the traditional island model of migration (Wright 1965). Rather than estimating migration rates, our main interest is to quantify the amount of bias of local and global *N*_{eV} estimates when disregarding migration. We pay special attention to situations of high gene flow where population structure may be difficult to detect or delineate, and where the same or different subpopulations may be included in the samples used for measuring the temporal change of allele frequencies.