1The aim of the present study is to model the stochastic variation in the size of five populations of great tit Parus major in the Netherlands, using a combination of individual-based demographic data and time series of population fluctuations. We will examine relative contribution of density-dependent effects, and variation in climate and winter food on local dynamics as well as on number of immigrants.
2Annual changes in population size were strongly affected by temporal variation in number of recruits produced locally as well as by the number of immigrants. The number of individuals recruited from one breeding season to the next was mainly determined by the population size in year t, the beech crop index (BCI) in year t and the temperature during March–April in year t. The number of immigrating females in year t + 1 was also explained by the number of females present in the population in year t, the BCI in autumn year t and the temperature during April–May in year t.
3By comparing predictions of the population model with the recorded number of females, the simultaneous modelling of local recruitment and immigration explained a large proportion of the annual variation in recorded population growth rates.
4Environmental stochasticity especially caused by spring temperature and BCI did in general contribute more to annual fluctuations in population size than density-dependent effects. Similar effects of climate on local recruitment and immigration also caused covariation in temporal fluctuations of immigration and local production of recruits.
5The effects of various variables in explaining fluctuations in population size were not independent, and the combined effect of the variables were generally non-additive. Thus, the effects of variables causing fluctuations in population size should not be considered separately because the total effect will be influenced by covariances among the explanatory variables.
6Our results show that fluctuations in the environment affect local recruitment as well as annual fluctuations in the number of immigrants. This effect of environment on the interchange of individuals among populations is important for predicting effects of global climate change on the pattern of population fluctuations.
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A recurrent theme in population ecology has been how density-dependent processes interact with stochastic variation in the environment to cause fluctuations in population size (Andrewartha & Birch 1954; Lack 1966; Turchin 1995). This problem has recently received increased attention because we need to understand how the expected changes in climate will affect population fluctuations. The necessary decomposition of the population dynamics into density-dependent and density-independent components requires that the effects of environmental variation must be estimated and modelled and that other factors that affect population fluctuations, such as the form of density regulation and demographic stochasticity, are properly accounted for. Furthermore, immigration among populations that are separated in space will also influence local population dynamics. Theoretical studies have demonstrated that immigration of individuals will strongly affect the population dynamical responses to temporal variation in the environment and may in turn affect the spatial autocorrelations in population fluctuations (Kendall et al. 2000; Engen, Lande & Sæther 2002a,b; Ranta, Lundberg & Kaitala 2006). Thus, accounting for density dependence and immigrations is a prerequisite for reliable modelling of global climate change on avian population dynamics.
Two important theoretical advances over the past decades have considerably improved our basic understanding of factors affecting the patterns in natural population fluctuations. One important advancement was provided by May's (1976) analyses of simple deterministic population models that showed that small variation in critical parameters could strongly affect dynamical characteristics of the populations. In particular, varying the form and strength of density regulation can change the system from stable fluctuations around an equilibrium into a chaotic behaviour (May 1976; Royama 1992). The other important advancement was provided by the development of stochastic population models (May 1973; Turelli 1977) that also included environmental stochasticity, that is, random variation that affects the whole or parts of the population (Lande, Engen & Sæther 2003), and later demographic stochasticity (Engen, Bakke & Islam 1998), that is, random variation among individuals in their fitness contributions (Lande et al. 2003). This provided an analytical framework for decomposing population fluctuations into deterministic and stochastic components. Such a partitioning is important when assessing the impact of changes in environmental conditions on fluctuations in population size. As a consequence, during the last decade of the previous century, a large number of studies (e.g. Bjørnstad, Falck & Stenseth 1995; e.g. Turchin 1990) of temporal variation in population fluctuations appeared that were based on statistical time-series analyses techniques.
In studies of population dynamics of natural populations, the geographical origin of new recruits is often ignored because this requires that a large proportion of the nestlings are individually marked and breeding birds identified. The lack of such data makes it impossible to distinguish the relative contribution of local recruitment and immigration to changes in population size. This can seriously restrict our ability to assess the importance of the various processes underlying population dynamics. For instance, theoretical analyses (Engen et al. 2002a,b) and empirical evidence indicate that migration can strongly affect density regulation processes (Ives et al. 2004) and viability of small populations (Arcese & Marr 2006). Thus, a proper understanding of regulatory processes in populations requires that the effects of both local recruitment and immigration on the annual fluctuations in population size are accounted for.
Here we will use an approach that allows us to partition population fluctuations into their deterministic and stochastic components by developing a stochastic demographic model for analysing long-term fluctuations in the size of five great tit Parus major populations in Netherlands (Table 1). We separate variation in local recruitment and immigration into components due to density dependence, environmental stochasticity and demographic stochasticity (e.g. Engen et al. 2005). This enables us to examine how different environmental covariates affect local recruitment as well as immigration. This approach allows us also to evaluate whether the effects of environmental covariates shows nonlinear relationship to population size (Grenfell et al. 1998; Stenseth et al. 2004), causing non-additive effects on the population fluctuations (Coulson et al. 2001). The proportion of the variance in the annual fluctuations in population size explained by the model can then be decomposed into contributions from density dependence and environmental variables. In this way, we can model how demographic effects of different variables translate into changes in population size due to local recruitment and immigration.
Table 1. Characteristics of the five study populations. The size of the areas, and the number of boxes that are put up varied between years. The Hoge Veluwe area was damaged in a severe storm in the winter 1972/73. HV1 included a large block of pure pine plantation, while HV2 consist only of mixed coniferous–deciduous woodland. For ‘No. of years included in selection’, I represents the modelling of local recruitment, II is the modelling of immigration, and III is the prediction of number of females. The selection criteria for including years are described in the Methods section
No. of Years incl. in selection
Area size (ha)
Median no. of nestboxes (min–max)
Mean no. of females (min–max)
Hoge Veluwe (HV1)
Mixed + pure pine
Part of larger area
Hoge Veluwe (HV2)
Part of larger area
Mature oak wood
Part of larger area
Mature oak wood
Study areas and field methods
Data were collected (1955–2003) at four different forested localities in the Netherlands (Table 1): Vlieland (VL), Hoge Veluwe (HV), Oosterhout (OH) and Liesbos (LB). These study sites were chosen along two axes: mature oak vs. mixed woodland and isolated vs. part of a larger area (Table 1). VL is an island in the Dutch Waddensea, and consists of mixed pine–deciduous woodland. HV (HV1) included mixed coniferous–deciduous woodland and a large block of pure pine plantation until 1972. A severe storm damaged the pine plantation in the winter of 1972/73, and from 1973, the study included only mixed coniferous–deciduous woodland (HV2). Thus, we treated HV1 and HV2 as two independent populations. Mature oaks dominate in OH, an isolated forest, and LB. For more details, see Table 1, van Balen & Potting (1990) and Verhulst & van Eck (1995).
The average number of breeding birds per hectare was high in the oak woods and low in the mixed woodlands. Throughout the study period, the population size increased in VL, HV1 and OH. This increase in population size in HV1 and VL was probably caused by maturation of the forest. Selective felling of conifers on VL also took place (Kluijver 1971), thereby increasing the proportion of deciduous trees. In OH, the population growth is probably partly due to an increase in the number of nest boxes. In all areas there was a large surplus of nest boxes as the number of females relative to the number of nest boxes was 0·41(0·26,0·67), 0·33(0·15,0·51), 0·43(0·19,0·79), 0·23(0·09,0·43) and 0·34(0·11,0·59) [mean(min,max)] in areas HV1, HV2, LB, OH and VL, respectively. The study areas remained the same size throughout the years included in the study.
The great tit data were collected during the period 1955–2003 (see Table 1 for details). In all areas, nest boxes were visited at least once every week. The number of females present in an area in a given year was defined as the number of first clutches. The number of eggs or young present was counted. When the nestlings were 7–10 days old, the parents were caught on the nest using a spring trap. Parents already ringed were identified and unringed birds were given a metal ring with a unique number. Young were ringed either at day 7 (HV, OH, LB) or day 10–15 (VL). As all fledglings from nest boxes were ringed and very few birds bred in natural cavities, we classified all adults caught unringed or ringed outside the study area as immigrants. If the number of caught females were lower than the number of first clutches, the number of immigrant females and local females (including surviving adults and locally produced recruits) were calculated by multiplying the proportion of the respective categories with the number of first clutches. We thus assume that there is no difference in the probability of identification of breeding birds that have immigrated to the area and of local recruits/surviving adults. Note that birds are only classified as immigrants in the year of their arrival. For the subsequent year(s), they are classified as local birds.
We excluded years if large-scale experiments were carried out that affected parental survival (breeding birds were removed), recruitment probability (removal of second broods or an overall reduction of brood size), or the number of fledglings produced (overall reduction of brood size). For some years in LB and OH, parents were not identified, and these year/area combinations were therefore not included in the model of individual fitness contributions (Table 1).
Writing Nt for the population size at time t, we have Nt+1 = ΛtNt + It+1, where It + 1 is the number of immigrants in year t + 1 and Λt is the stochastic multiplication rate of the population in absence of migration. In general, the bivariate distribution of (Λt, It+1) will depend on Nt as well as an observed vector of covariates. The mean and the variance of the population size the next year is then
We model the local recruitment by considering annual variation in the contribution to the next generation by a female, that is, the number of female offspring she gives birth to during this year that survive for at least 1 year, plus 1 if the female herself survives to the next year (Sæther et al. 1998). Considering the emigrants as dead individuals, we get where wi is the contribution of female i in year t.
Let Z denote the vector of environmental covariates. We assume that (ΛtNt | Z) is approximately Poisson distributed (see Appendix S1) with parameter λNt, where λ = E(w | Z, Nt). The parameter λ is density dependent, characterized by the strength of density dependence γ, as well as linear functions of different environmental covariates Zi,
where the time index has been omitted for simplicity. Here, ɛ denotes the residual environmental variance not accounted for by any of the specified environmental covariates Zi. We assume, following Lebreton (1990), a logistic model (May 1981) of density regulation. It is shown in the Appendix S2 how the variance of Λ can be partitioned into a demographic and environmental component.
We further show in Appendix S2 that
where β is the vector of effect sizes of environmental covariates Z on the change in population size. Thus, the environmental variance in the local recruitment is decomposed into a component due to contribution from specific environmental covariates Z as well as a component from residual, unobserved variation in the environment U (see Appendix S2). We also see that in this model, both become density dependent.
Our next step was to model the immigration process. Very few birds have been observed breeding in natural cavities (see Methods). Thus, the number of nest boxes b was considered to represent an upper limit of the number of possible immigrants. We assume that the number of female immigrants I to the population is determined by the number of available boxes (because b in all areas is always larger than N) and follows a beta-binominal distribution with probability density,
where B(α1,α2) denotes the beta function with parameters α1 and α2. The variable I can take values from 0 to b, and its distribution shows overdispersion relative to the binomial distribution. The overdispersion parameter is defined as φ = 1/( α1 + α2 + 1). We assume that (α1 + α2) is constant, whereas α1/α2 fluctuates among years due to their dependence on Z, U and N (see Appendix S2).
Following the same approach as when decomposing the variance in local recruitment, the variance in the number of immigrants var(I) has a component E var(I | Z, U, N) due to demographic stochasticity and a component var E(I | Z, U, N) due to environmental stochasticity so that (see Appendix S2)
In Appendix S2, we give the precise expressions showing how depend on Z, U and N.
The final term in the population model (equation 1) is the covariance between the local dynamics and the immigration cov(NΛ, I), which also can be partitioned into a demographic component E cov(NΛ, I | Z, U, N), and an environmental component cov(E(NΛ, I | Z, U, N), E(I | Z, U, N) of the covariance (for details of this partitioning, see Appendix S2).
We will now use this model to examine how variation in climate and density dependence affects local recruitment and immigration.
To examine extrinsic influences on the population fluctuations, we included environmental covariates that are known to affect the demography of the dynamics of great tit populations. Beech mast is an important source of food for great tits, especially during winter when other resources are scarce (Betts 1955; Ulfstrand 1962). It is also indicative of seed production of other tree species (Perdeck et al. 2000). There is no continuous series of beech-crop data for the whole study period. For the years 1955–76, a beech-crop index (BCI) is available that was collected on a 10-point scale throughout the Netherlands, whereas for 1976–2003, we have measurements on nuts per square metre for the Hoge Veluwe study area. As these indices cannot be mixed, Perdeck et al. (2000) defined a three-level index to combine the series, which we will also use here. The relationship between the index and the amount of food supply could be far from linear, so we included BCI as a factor in the models.
Winter weather is an important determinant of mortality and hence changes in population size in tits (Kluijver (1951), Gibb (1960), Perrins (1965), van Balen (1980) and Tinbergen, van Balen & van Eck (1985). We characterized winter severity by the monthly mean temperatures of the months December, January, February and March from the De Bilt meteorological station of the Royal Netherlands Meteorological Institute (KNMI), which is situated in the centre of the Netherlands. Spring temperature determines the date of peak caterpillar biomass (Visser, Holleman & Gienapp 2006), and although great tits are phenotypically plastic in their reproduction, laying earlier in warm springs (van Balen 1973; Nussey et al. 2005), there are regional differences with respect to phenological changes as a response of the recent climate change (Visser et al. 1998, 2003), causing a mistimed reproduction in relation to the food peak (Visser, Both & Lambrechts 2004). Thus, temperatures during spring (February to May) the year of breeding was therefore included as candidate explanatory variables. All temperature variables were tested singly, and as 2, 3, and 4 monthly averages.
estimation of parameters
The yearly estimates of demographic stochasticity in the local dynamics were calculated according to Sæther et al. (2000a).
We modelled individual fitness w using the function glm (R Development Core Team 2007) with a Poisson error distribution (see Appendix S1) and a log-link function. Candidate explanatory variables entered were area, number of females, BCI and one temperature variable. We also allowed the possibility for interaction between area and the other explanatory variables. The environmental variance was calculated as
The parameters in the model describing the number of female immigrants in year t + 1 was estimated by the beta-binominal generalized linear model implemented in function betabin in package aod using a logit link function (Lesnott & Lancelot 2005). Area, number of females in year t, BCI in year t and one temperature variable were used as candidate explanatory variables. In addition, possible interactions between the area and the other explanatory variables were also included during model selection. The overdispersion was also allowed to have an interaction with area.
Model selection was based on the Akaike information criterion (AIC) and corrected AIC (AICc) (Burnham & Anderson 1998) for local recruitment and immigration respectively. Modelling local recruitment was based on data from 7604 breeding females, whereas immigration was modelled by analyses of 146 area × year combinations. The model set consisted of all possible combinations of main effects of area, N, BCI and one temperature covariate. In addition, we also tested models including combinations of interactions between area and the other explanatory variables. All models ranging from one single main effect (e.g. area) to a full model including main effects of area, N, BCI and a temperature covariate and interactions area × N, area × BCI and area × temperature covariate were therefore included in the model set subject to selection by the information criterion.
Engen and colleagues (Engen, Sæther & Møller 2001; Sæther & Engen 2002a) have advocated the use of the concept population prediction interval (PPI) when making predictions about future population fluctuations. A PPI is defined as the stochastic interval that includes the population size with a fixed probability (1 − α). The advantage of this approach is that the width of the PPI accounts for varying influences of demographic stochasticity with varying population sizes and uncertainties in parameter estimates. For instance, uncertainty in parameter estimates will over a given time period decrease the precision of population predictions (Sæther et al. 2000a; Engen et al. 2001; Sæther & Engen 2002a,b; Asbjørnsen et al. 2005).
Here, our goal is to study whether the population model could produce PPI's that included the observed population sizes within a given coverage. This was examined by simulating the stochastic population process defined by the population model, the estimated parameters (Table 2) and measured environmental variables (temperature and BCI). If we denote the vector of parameter estimates as , uncertainty in parameter estimates was accounted for by simulating sets of parameters from a multivariate normal distribution with mean and the estimation covariance matrix V̂θ (see Gelman & Hill 2007 for a similar approach). The sets of simulated parameter values were thereafter used when simulating the population sizes. Thus, this approach not only account for uncertainty in parameter estimates, but also for dependencies among parameters in uncertainty.
Table 2. Estimated parameters in the selected model for (a) local recruitment in year t (equation 2), and (b) number of immigrating females in year t + 1 (equation 6). Model selection was based on the Akaike information criterion (AIC) and corrected AIC (AICc) for local recruitment and immigration, respectively. In (a) γ is the effect of number of females in year t, βTemp is the effect of beech crop index (BCI) in autumn year t, and βTemp is the effect of mean temperature during March–April in year t. In (b) δ is the effect of number of females in year t, ζBCI is the effect of BCI during autumn year t and, and ζTemp is the effect of mean temperature during April–May in year t. Different intercepts for each area were included in both models, but the parameter estimates are omitted here. Both selected models for (a) and (b) included interaction effects between area and the number of females (γ and δ) and between area and temperature. Confidence intervals (95%) are enclosed in brackets
a) Local recruitment
γ × HV1
γ × HV2
γ × LB
γ × OH
γ × VL
–0·0015 [–0·0039, 0·0009]
–0·0102 [–0·0132, –0·0072]
–0·0188 [–0·0284, –0·0091]
0·0013 [–0·0065, 0·0092]
–0·0011 [–0·0038, 0·0016]
0·32 [0·25, 0·39]
0·41 [0·33, 0·50]
–0·22 [–0·29, –0·15]
–0·06 [–0·15, 0·02]
–0·10 [–0·20, 0·00]
–0·08 [–0·19, 0·02]
–0·01 [–0·10, 0·07]
δ × HV1
δ × HV2
δ × LB
δ × OH
δ × VL
–0·0051 [–0·0100, –0·0001]
–0·0025 [–0·0086, 0·0035]
–0·0138 [–0·0250, –0·0027]
0·0042 [–0·0097, 0·0182]
0·0046 [–0·0012, 0·0104]
0·33 [0·2, 0·46]
0·44 [0·3, 0·59]
0·34 [0·13, 0·54]
0·01 [–0·21, 0·24]
0·10 [–0·12, 0·33]
0·17 [–0·08, 0·43]
0·14 [–0·11, 0·40]
To simulate the first annual change in population sizes, we used the observed population sizes the first year in each study area, but at other time-steps, we substituted the observed population sizes with the simulated population sizes. Similarly, the effects of environmental variables (temperature and BCI) were incorporated by calculating yearly effect sizes based on observations of the environmental variables and parameter estimates.
We used two different approaches to evaluate the predictive power of our model. First, we estimated parameters based on the complete data set and calculated the PPI for all years with observations. Second, we estimated the parameters based on data before 1985 and simulated the stochastic fluctuations in population sizes from 1985 and onwards. The prediction intervals were only constructed for the areas HV2, LB and OH, as the numbers of years with experimental manipulations in areas HV1 and VL were large.
calculations of relative contributions to annual changes in population size
Based on the fitted models (equations 1, 2, 6), we calculated the relative contributions from the explanatory variables to variation in local recruitment and immigration. Generally, the contribution of a component θ1 varying with time was calculated as
( eqn 9)
and the interaction between two components θ1 and θ2 was calculated as
( eqn 10)
Thus, a negative interaction indicates a positive covariation between the temporal effects of the components and the fraction of the explained variance that could be attributed to the combined effect of the components is less than sum obtained when calculating fractions separately.
Based on linear regression, the annual point estimates of demographic variance (Sæther et al. 2000a) decreased with population size in the two populations (Fig. 1) HV2 (P < 0·01, n = 30) and LB (P < 0·05, n = 37). For locality LB, the fit was still significant after removing an outlier (Fig. 1). By taking all available years into account and weighting for the number of individual contributions (Sæther et al. 2000a), the estimates of demographic variance was = 0·496, = 0·447, = 0·329, = 0·608 and = 0·839 for HV1, HV2, LB, OH and VL, respectively.
The most parsimonious model (Appendix S3) for the expectation of local individual fitness contributions in year t (equations 2, 3) included the effect of number of females in year t, BCI in the autumn of year t and mean temperature during March–April in year t, that is, few birds were recruited after years with high spring temperatures and after years with no beech crop (Table 2a). In addition, all areas had different intercepts, and interactions with area were also present between number of females and temperature (Appendix S3). In contrast, the positive effect of beech crop on local recruitment did not differ among areas (likelihood-ratio test, P > 0·1). Population size had a negative effect on local recruitment in all areas except OH. The ratio of residual deviance to degrees of freedom was 0·94, indicating slight underdispersion. The environmental variance in the local individual fitness contributions (equation 8) were = 0·108, = 0·048, = 0·038, = 0·051 and = 0·145 in HV1, HV2, LB, OH and VL, respectively.
Based on the fitted model, the largest contribution to variation in local recruitment was in general provided by BCI (Table 3a). However, density explained the highest proportion of the annual variation in the localities HV2 and LB. Temperature during March–April had also a strong effect in localities HV1 and LB. In all areas except OH, the temporal effects of the explanatory variables were correlated. Thus, the combined effects of the explanatory variables were generally not additive due to covariation among variables in the yearly effects on local recruitment (equation 10).
Table 3. The proportion of explained variance in the model for (a) individual fitness contributions, and (b) number of immigrating females attributed to number of females (N), beech crop index (BCI), mean spring temperature (temp) during March–April for (a), and April–May for (b), and all possible combinations of these variables (equation 13). In columns with combinations of variables, a negative value means that the calculated proportion of the variance explained by these variables is less than the sum of variances explained by entering them separately, indicating that the temporal effects of the variables are positively correlated
N BCI Temp
a) Local recruitment
The most parsimonious model explaining the number of immigrant females in year t + 1 was a model including number of females in year t, BCI in the autumn of year t and mean temperature during April–May in year t (Table 2b). In addition to separate intercepts for each area, interaction between area and number of females as well as interaction between area and temperature were included, but no interaction between area and BCI. However, differences in AICc values indicate that support for the area × temperature interaction was weak (Appendix S3). A common overdispersion parameter among areas f = 0·0054 was included in the most parsimonious model.
The environmental factors affected the immigration in a similar way in all areas. Immigration increased with mean temperature during April–May (Table 2b). As was the case for local recruitment, high immigration rates were also found for BCI = 2 and BCI = 3. Accordingly, BCI was in general important in explaining the variance in the number of immigrants (Table 3b). In contrast, the density-dependent effect on immigration differed in sign among areas. The number of immigrants decreased with number of females in HV1, HV2 and LB, whereas a positive density-dependent relationship was found in OH and VL. However, the effects of N and BCI were not independent (Table 3b), because the temporal contributions from N and BCI to population growth rates were positively correlated in HV1, HV2 and LB, and negatively correlated in OH and VL. The effect of temperature during April–May was large in HV1, OH and VL, but the effects of temperature were in these localities correlated to the temporal effects of BCI and N. Thus, as was the case for local recruitment, the effects of various variables in explaining fluctuations in population size were not independent, and the combined effect of the variables were generally not additive. This illustrates that estimating effects of variables causing fluctuations in population size should not be carried out separately because the total effect will be influenced by covariances among the explanatory variables.
relative importance of local recruitment and immigration
Following the same approach as for separate models of local recruitment and immigration, we estimated the contributions from the local recruitment, immigration and the effects of the explanatory environmental variables for both local recruitment and immigration. Both local recruitment and immigration contributed to temporal variation in population growth rates. Local recruitment was more important than immigration in areas HV2, OH and VL (OH and VL are isolated woods, see Table 1), whereas immigration was more important in area LB (Table 4). In area HV1, there were equal contributions from local recruitment and immigration. However, the effects of local recruitment and immigration were not additive due to correlated effects of local recruitment and immigration within years. The combined effect of BCI from local recruitment and immigration was the most important explanatory variable in all areas except in VL where beeches are absent. Furthermore, there was a covariation between local recruitment and immigration in the temporal effects of BCI (Table 4). However, in area VL, population size was the explanatory variable explaining the largest proportion of the variance in the model predictions.
Table 4. The proportion of the explained variance in total population size attributed to the local recruitment and the immigration in the model (equation 1), the interaction effect of local recruitment and immigration, the proportion of the explained variance attributed to the explanatory variables N (number of females), BCI (beech crop index), and spring temperature (local recruitment: March–May; immigration: April–May) and the interactions between local recruitment and immigration for N, BCI and temperature (equation 13). Please note that the rows do not add up to 1 (100% of the explained variance) as interactions such as, for example, N (local recruitment) × BCI (immigration) are omitted
Local × immigration
N × N
BCI × BCI
Temp × Temp
predictions of population fluctuations
As expected, the prediction intervals (Figs 2, 3) were generally wide due to demographic stochasticity and parameter uncertainty causing large variation among simulated realizations of the stochastic processes. However, effects of variation in environmental variables and density dependence are still so influential that there is large year-to-year variation in population sizes covered by the PPI. The observed processes did in general fluctuate within the 95% intervals (Figs 2, 3). However, the number of observations outside the 95% prediction interval was larger than expected for area OH, possibly indicating that some important processes or effects of climate, especially during the 1970s (Fig. 2), were not accounted for in the model for this area.
Our analyses of the dynamics of the Dutch great tit populations reveal that they are influenced by density-dependent feedback mechanisms as well as stochastic variation in the environment, affecting both local recruitment and immigration from surrounding areas. An implication of this is that we can predict the population responses to changes in the temporal variation of key ecological variables. For instance, this modelling framework can be used, provided availability of long-term individual-based demographic data, to quantitatively predict the effects of global climate change on the pattern of population fluctuations.
Although our model includes several parameters, it is still based on several simplifying assumptions. First, we assume a logistic model of density regulation, that is, that the density regulation is linear in N. This model seems to give a reasonable description of the density regulation in great tit populations (Lebreton 1990), as well as for small passerines in general (Sæther & Engen 2002b). Second, we assume no age-specific effects in the adult segment of the populations because the age composition of the populations is unknown due to several of the recruits having unknown origin. Age dependency in demographic traits can easily induce autocorrelations in the population fluctuations that can be wrongly interpreted as density dependence (Lande, Engen & Sæther 2002; Lande et al. 2006). However, age-specific demographic effects seem to be small in small passerines such as the great tit compared to other species (Clobert et al. 1988; Sæther 1990). Accordingly, fluctuations in the age structure had only a minor influence on the dynamics of a willow tit Parus montanus population (Ekman 1984). It also seems to be a tendency that changes in size of different age classes of tit populations are highly correlated (Sæther, Engen & Matthysen 2002c). As a consequence, an examination of the residuals after fitting the density model (equation 1) did not reveal any significant lagged autocorrelations. Third, we assume that the immigrants have demographic characteristics that do not differ from the residents, which may not necessarily be true (Clobert et al. 1988; Altwegg, Ringsby & Sæther 2000; Postma & van Noordwijk 2005). Fourth, we assume that the density regulation acts only on the number of females present. If total population size (i.e. including both males and females) is important, this may strongly affect the estimates of both the population parameters as well as the stochastic components (Sæther et al. 2004a). Fifth, we assume no census error that is known to introduce biases especially in the estimates of the environmental variance (Freckleton et al. 2006). However, in present study areas almost all great tits nests in nest boxes. In addition, the environmental stochasticity is relatively large (see Results section), so the bias introduced by observation error in the population estimates is likely to be neglectable. Finally, we assumed that the fraction of nest boxes occupied by female immigrants were unaffected by the number of local females present. The numbers of nest boxes relative to the numbers of local and immigrant females were large (see Study Areas section) in all areas and years. Thus, competition for nest boxes should be quite limited. Despite all these simplifying assumptions, our model was still able to describe the actual population fluctuations quite accurately (Fig. 2). This indicates that the model (equation 1) captures some basic features of the underlying dynamics of these populations.
relative contribution of density dependence and environmental stochasticity
In this study, we modelled both the demographic and environmental stochasticity as density-dependent functions (equations 3, 4 and 7). Significant effects of population size were found on the demographic variance (Fig. 1) in the areas HV2 and LB. Previously, a similar density-dependent decrease in demographic variance was found for the great tits in Wytham Wood (Sæther et al. 1998). Such a decrease was interpreted that relatively fewer females were able to produce a large number of recruits at higher population size that will result in a density-dependent decrease in the among-female fitness-variation. Although we used a slightly different model to estimate the stochastic components, our estimates of (see Results section) were quite similar to previously published estimates (Sæther et al. 2003) for the same populations. With Poisson-distributed individual contributions, the demographic variance should theoretically be equal to 1 at the carrying capacity (Engen et al. 1998), and this theoretical expectation differs from our estimates of the demographic variance that were < 1 in all populations. However, in our modelling of the local recruitment process, we have considered fledglings that were not found in subsequent years as dead. Some of these individuals could have migrated out of the study area. In addition, there is an influx of individuals into the study area from other populations. Hence, at carrying capacity, the mean contribution from local females is expected to be < 1, and estimates of demographic variance < 1 could thus be explained by the properties of the Poisson distribution.
It is now generally accepted in population ecology that the magnitude of population fluctuations is determined by a combination of density-dependent processes and the effects of environmental stochasticity (Turchin 1995), as originally suggested by Lack (1954, 1966). Our results (Tables 2, 3, 4) show that fluctuations in the size of great tit populations are more influenced by stochastic variations in the environmental variables such as the beech mast and temperature than by density-dependent feedback mechanisms. Annual fluctuations in the size of great tit populations are generally dependent on two mechanisms that both were identified more than half a century ago (Lack 1947; Kluijver 1951). First, density-independent factors affect the fledgling production (Both & Visser 2000) and the number of juveniles and adults surviving, especially during winter (Tables 2a, 3a). Second, density dependence affects the number of individuals surviving from one breeding season to the next (Clobert et al. 1988) or the number of new individuals that enable themselves to establish in the breeding population (Kluijver & Tinbergen 1953; Krebs 1971; Klomp 1972). Our quantitative evaluation of the relative contribution of these two mechanisms to fluctuations in the size of Dutch great tit populations showed that, in general, density-independent effects contributed more to the variance in population growth rates than density-dependent effects of the number of females present the previous year (Tables 3a, 4).
The strong effects of beech mast on the fluctuations in the size of great tit populations operate mainly through an effect on the survival of both juveniles and adults (Perrins 1965a; Perrins 1979; McCleery & Perrins 1985; Clobert et al. 1988; Perdeck et al. 2000). Similarly, the influence on the dynamics of climate during late winter and spring was also found by Slagsvold (1975) in an analysis of the dynamics of great tits in Wytham Wood outside Oxford, UK. This suggests that climate during this period may also affect juvenile survival or the probability that especially juveniles are able to establish themselves in the population (Krebs 1971). The effects of temperature during spring could possibly be explained by the mistiming hypothesis (Visser et al. 1998, 2004, 2006). High temperatures during late spring, after the birds have started laying, leads to a faster development of vegetation and prey species causing a mismatch with the offsprings’ needs and the peak date in food availability, which in turn would lead to a lower reproductive output. Accordingly, the effects of temperature during spring were included in the selected models (Table 2) both for local recruitment and immigration. However, while for the local recruitment we find the predicted negative relationship between spring temperature and population growth rate, this estimate is positive for the immigration term. One possible explanation for this is that in cold springs, which are favourable for local recruitment, increased competition will reduce the number of immigrants that can establish themselves in the population. Accordingly, in several passerine species (Sutherland, Gill & Norris 2002; Wilson & Arcese 2008) as well as in great tit in particular (Tufto et al. 2005), there is a decrease in immigration rates with local population size.
An important conclusion of our analyses is the presence of an interaction between the effects on the population fluctuations of density dependence and environmental stochasticity so these two processes cannot be considered as independent. Royama (1992, p. 38) has previously suggested that variation in beech crop could cause a nonlinear perturbation on the curve relating population growth rate to population density. Similar interactions among density-dependent and density-independent effects have been found in other studies on vertebrate population dynamics as well (Grenfell et al. 1998; Coulson et al. 2001; Stenseth et al. 2002; Stenseth et al. 2004; Haugen et al. 2007). This could imply that proper modelling of the effects of changes in the environment, for example, due to climate on great tit dynamics must take into account density-dependent processes affecting local recruitment as well as immigration in a nonlinear way. However, it is important to note that our findings are based on analyzing contributions to annual changes in population sizes (equations 9, 10). This implies that non-additive contributions could also be caused by correlations among explanatory variables (Altwegg et al. 2003). Irrespective of the detailed mechanism causing non-additive contributions, the findings underline the importance of modelling and estimating effects of important factors influencing variations in population size simultaneously rather than as the effects of single variables separately.
the relationship between local recruitment and immigration
Variation in the size of natural populations is caused by a combination of demographic processes that act locally and influx of individuals from outside. The interaction between those two processes is, however, poorly known. For instance, few examples have been provided for the presence of density-dependent migration rates (Sutherland et al. 2002), even though such relationships can have strong dynamical consequences (Sæther, Engen & Lande 1999; Andreassen, Stenseth & Ims 2002). Density dependence as well as stochastic environmental effects, such as weather and beech mast, affected local recruitment as well as immigration of new birds (Table 4). Thus, immigration represents a key process for the dynamics of these tit populations.
This large effect of immigration on the local dynamics may be more typical for most passerine populations than the more limited influence found in more geographically isolated populations (e.g. Smith et al. 2006). However, our results show that local variation in demography generally contributed more to annual changes in population size than the number of immigrants (Table 4) except that immigration was more important than local recruitment in area LB (Table 4). This study area is located within a larger area of suitable habitat for the great tit (Table 1). This suggests that the degree of isolation may influence the relative importance of immigration in the local dynamics of great tit populations. Accordingly, the two most isolated populations VL and OH (Table 1) were less influenced by immigration than the other populations. Furthermore, the reduced importance of immigration relative to local recruitment after the reduction of the study area in HV due to a severe storm damaging the pine plantation in winter of 1972/73 suggests that the effect of immigration was not dependent upon size of the study area.
A problem in studies of immigration on the dynamics of natural population is that the effects will depend on the spatial scale of the study area. If the study includes localities that are located far away, the effects of immigration are likely to be far less than among populations located within close distances (Harrison & Taylor 1997). The distance between the study sites ranged from 17 km between HV and OH to 169 km between VL and LB. Over such short distances, some interchange of individuals may be possible (Tufto et al. 2005), which may induce independence among the populations in their dynamics. However, movements of individuals among the study sites were rare. During the period from 1955 to 2006, we have 298 991 catches of 123 312 individual great tits in the four study populations. Of these, there were only eight individuals that have been recorded at two different localities: three birds were ringed as a nestling in one area and bred in another area while the other five cases are on winter catches and hence do not affect the number of breeding birds. Thus, our best estimate is that 3 out of 123 312 individuals moved between sites (0·0024% of the birds). This indicates that our localities represent geographically separated populations, in which the dynamics are not influenced by movements of individuals among the four study populations.
Still there was a positive association in all study populations between local recruitment and the number of immigrants (Table 4). This positive association supports the suggestion that the population dynamics of tits are affected by processes that affect all classes in the population similarly (Sæther et al. 2002c) over larger geographical areas (Sæther et al. 2007). The short spatial scaling of dispersal in the great tit (Tufto et al. 2005) also supports this explanation. Accordingly, the same environmental variables (BCI and temperature during spring) were important for predicting temporal variation in local recruitment and the number of immigrants from other areas (Table 2).
We thank Jan Visser and Huybert van Eck (†) for all their efforts in the field, and Jan Visser for careful management of the great tit data that form the basis of this paper. This study was supported by the EU 5th framework (METABIRD project) and the Research Council of Norway (NORKLIMA). We thank the board of the National Park ‘de Hoge Veluwe’, the State Forestry Service at Vlieland and Breda (Liesbos), and Barones van Boetzelaer van Oosterhout for permission to work in their woodlands for all these years. We thank M. Lima for comments on a previous version of this paper.