## Introduction

To understand the evolutionary response to selection in a given population, it is of fundamental importance to know the magnitude of the additive genetic variances and the structure of genetic covariances among traits, summarized by the **G**-matrix (e.g. Lande, 1979; Lande & Arnold, 1983). For long-term predictions of selection to be accurate, we need to know whether the **G**-matrix is stable over time (e.g. Steppan *et al*., 2002). If the **G**-matrix is highly integrated, then the response to selection is directed towards the direction of most genetic variation (‘lines of least resistance’, Schluter, 1996; Björklund, 1996, 2003), rather than in the direction of selection. Thus, long-term stability of the **G**-matrix can help to explain the observation of long-term stasis (e.g. Lande, 1979), and macro-evolutionary changes in terms of size rather than shape (e.g. Björklund, 1994). If the **G**-matrix is robust over time, so will the constraints be and we will see this as a lack of variation among taxa for certain trait combinations, which indeed has been observed (e.g. Björklund & Merilä, 1993; Björklund, 1994). On the other hand, if the **G**-matrix can change quickly over time, then the constraints imposed by the structure of the **G**-matrix will probably be important on a short ecological time scales, but is then unlikely to account for long-term stasis.

A number of theoretical models have tried to understand the evolutionary dynamics of the **G**-matrix (e.g. Bohren *et al*., 1966; Bulmer, 1980; Steppan *et al*., 2002; Jones *et al*., 2003, 2007; Björklund, 2004; Revell, 2007; see review by Arnold *et al*., 2008). The precise model of the genetic background is important for the detailed predictions of change over time (Barton & Turelli, 1987), and the time scale of interest matters for the mechanism of change. Hence, on longer evolutionary time scales, and in large populations, mutational input and selection are the main factors of change due to changing allele frequencies (e.g. Barton & Turelli, 1989; Arnold *et al*., 2008). On the much shorter ecological time scale, mutations and changes in allele frequencies are of less importance and too slow in relation to transient factors such as build-up of linkage disequilibrium. On this shorter time scale, stabilizing and directional selection will reduce variance, whereas disruptive selection will increase the variance, but because this is due to the build-up of linkage disequilibrium, this will disappear after a few generations of recombination, assuming the infinitesimal model (Bulmer, 1980). Hence, a short bout of directional selection will temporarily decrease genetic variance, and this is well known empirically (e.g. Gibson & Thoday, 1963; Sorensen & Hill, 1982; Hill & Caballero, 1992). Directional and quadratic selection can also create, or increase, genetic covariances (Bohren *et al*., 1966; for empirical data see e.g. Thoday, 1959; Scharloo *et al*., 1967), whereas correlated selection can drive the covariances towards zero (Villanueva & Kennedy, 1992) or strengthen them (Roff & Fairbairn, 2012), hence changing the shape of the **G**-matrix. Thus, given the ubiquity of selection in nature, there are many reasons to believe that the **G**-matrix will change over time (Barton & Turelli, 1989). In addition, the **G**-matrix can change even in the absence of selection and drift due to changes in the variance in allocation and acquisition as this can quickly change the sign of the genetic covariances (Houle, 1991; de Jong & van Noordwijk, 1992; Björklund, 2004). Thus, fluctuations in the **G**-matrix can be expected on these grounds, but it is unclear how large and fast these changes will be in any given natural population.

The lack of long-term studies of the stability of the **G**-matrix in natural populations is not surprising because the number of long-term studies of natural populations (i.e. more than 10 years; Clutton-Brock & Sheldon, 2010) is relatively few. Furthermore, the estimation of the **G**-matrix requires data on a large number of individuals with known relatedness, which is a challenging task in most natural populations. There is an increasing number of studies where the **G**-matrix, or parts of it such as the genetic variances, has been estimated in different environments in different populations and species (e.g. Cano *et al*., 2004 in *Rana temporaria*, Caruso *et al*. (2005) in *Lobelia siphilitica* and *L. cardinalis*, Estes & Phillips, 2006 in *C. elegans*). A frequent result is that the **G**-matrices differ to some extent (for a review, see Arnold *et al*., 2008), suggesting that at least some components of the **G**-matrix are rather labile and subject to environmental influence, in accordance with theory. Even if these studies are not strictly speaking a test of the temporal stability of the **G**-matrix in the same population, it can be argued that because most natural populations live in a temporally fluctuating environment, any environmentally induced changes in elements of the **G**-matrix are likely to have an impact on the temporal stability of **G** as well.

The results from the few studies that have examined temporal stability of **G** so far are inconclusive. For example, Pfrender & Lynch (2000) found significant changes between generations in *Daphnia pulex*. Divergence of the **G**-matrix was found over a 20-year period in the soil nematode *Acrobeloides nanus* (Doroszuk *et al*., 2008). In contrast, a study of a great tit (*Parus major*) population over a 40-year period showed no obvious changes in the **G**-matrix for three important life-history traits (Garant *et al*., 2008). On shorter time scale (3 years), a significant difference in **G**-matrices between years was found in the collared flycatcher (Merilä & Gustafsson, 1996).

To address this lack of empirical knowledge on temporal changes in, and hence stability of, the **G**-matrix, we used data from a long-term individual-based study of almost 7000 breeding events over the last 25 years in the collared flycatcher (*Ficedula albicollis*) on the island of Gotland, Southern Sweden (e.g. Merilä & Gustafsson, 1996; Qvarnström *et al*., 2006). We used an animal model approach to estimate the **G-**matrix (Henderson, 1950; Kruuk, 2004) using four different morphological traits for the whole 25-year period and separate **G**-matrices for five different 5-year periods. We then compare these matrices in terms of the size, shape and orientation and test whether the differences observed are due to drift or selection. The main finding was that the **G**-matrix changes in size and structure over the time periods in an unpredictable way.