Interspecific differences in stochastic population dynamics explains variation in Taylor s temporal power law

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1 Oikos 1: , 013 doi: /j x 013 The Authors. Oikos 013 Nordic Society Oikos Subject Editor: Thorsten Wiegand. Accepted 11 December 01 Interspecific differences in stochastic population dynamics explains variation in Taylor s temporal power law Marit Linnerud, Bernt-Erik Sæther, Vidar Grøtan, Steinar Engen, David G. Noble and Robert P. Freckleton M. Linnerud (marit.linnerud@bio.ntnu.no), B.-E. Sæther and V. Grøtan, Centre for Conservation Biology, Dept of Biology, Norwegian Univ. of Science and Technology, NO-7491 Trondheim, Norway. S. Engen, Centre for Conservation Biology, Dept of Mathematical Sciences, Norwegian Univ. of Science and Technology, NO-7491 Trondheim, Norway. D. G. Noble, British Trust for Ornithology, Thetford, Suffolk, IP4 PU, UK. R. P. Freckleton, Dept of Animal and Plant Sciences, Univ. of Sheffield, Sheffield, S10 TN, UK. Taylor s power law, i.e. that the slope for the increase in variance with mean population size is between 1 and at a logarithmic scale, provides one of the few quantitative relationships in population ecology, yet the underlying ecological mechanisms are only poorly understood. Stochastic theory of population dynamics predicts that demographic and environmental stochasticity will affect the slope differently. In a stable environment under the influence of demographic stochasticity alone the slope will be equal to 1. In large populations in which demographic variance will have a negligible effect on the dynamics the slope will approach. In addition, the slope will also be influenced by how the strength of density dependence is related to mean population size. To disentangle the relative contribution of these processes we estimate the mean-variance relationship for a large number of populations of British birds. The variance in population size of most species decreased with the mean due to decreased influence of demographic stochasticity at larger population sizes. Interspecific differences in demographic stochasticity was the main factor influencing variation in slopes of Taylor s power law among species through a significant negative relationship between the slope and demographic variance. In addition, slopes were influenced by interspecific variation in life history parameters such as adult survival and clutch size. These analyses show that Taylor s power law is generated from an interplay between stochastic and density dependent factors, modulated by life history. Numerous studies covering a wide range of taxa have demonstrated that there is a strikingly consistent relationship between variability in population size and average population size both in time and space (Taylor and Woiwod 1978, Taylor 1980, Hanski 198, Taylor 1986). Taylor (1961) proposed based on an empirical analysis of spatial data, that this relationship could be described by a simple power law in which the variance of the population fluctuations was related to mean population size. On a logarithmic scale this relationship becomes a classic linear model with intercept a and slope b. These analyses were later extended to include temporal data, which indicated a similar relationship between the log variance of the population fluctuations at a single locality and the log mean size of the population. Thus, temporal variability of population size increases exponentially with increasing population size. Empirically the temporal slope b t, of the log variance versus the log mean has been found to be between 1 and (Taylor and Woiwod 198, Hanski 198, Kendal 004). In particular, an important source of information for examination of these relationships has been analyses of fluctuations in the size of bird populations across the United Kingdom (Taylor 1980, Taylor and Woiwod 198, Hanski and Tiainen 1989, Maurer and Taper 00). A general pattern emerging from these studies is that the slopes show large interspecific variation but in general are less than. This lead Taylor and coworkers (Taylor 1977, 1986, Taylor and Woiwod 1978, 198, Taylor et al. 1980) to suggest that these slopes represent species-specific characteristics, related to the movement behaviour and the ecology of the species. Others have linked the slope to interspecies interactions (Kilpatrick and Ives 003) and reproductive correlation among individuals (Ballantyne and Kerkhoff 007). The presence of such a general functional relationship will be of great importance in developing population ecology as a predictive science because it provides a tool to derive population characteristics from some simple set of variables and to explain general macroecological patterns in species abundances both in space and time (Brown 1995). Analyses of Taylor s power laws for the relationship between variance and mean population size have focused on a statistical estimation of the relationship without any foundation in population dynamics theory. This approach has faced two major problems. Firstly, sampling will bias the estimates of the variance in the population size (McArdle et al. 1990). In general, ignoring observation errors in the population estimates produce overestimates of the variability in the population fluctuations (Freckleton et al. 006). 107

2 Accordingly, analyses of insect data have shown that the influence of sampling variation strongly influences the slopes between the temporal variance and average population size (Hanski and Woiwod 1993). Secondly, the estimates of the variance in the population fluctuations increase with the number of years the population has been studied (Pimm and Redfearn 1988, Hanski 1990, Ariño and Pimm 1995, Curnutt et al. 1996). In particular, this becomes especially important if the data sets include density independent populations because in absence of density regulation the variance of the logarithm of the population size increases linearly with time (Lande et al. 003). These problems have precluded the development of predictive models to explain interspecific differences in the mean-variance slopes because the variance components affecting those relationships have not previously been estimated from time series of population fluctuations. Recent developments of stochastic population theories have shown that annual changes in population size are determined by a combination of deterministic and stochastic factors (Lande et al. 003), which all can influence the slope of the mean-variance relationship. A central challenge is therefore to evaluate how different factors known to affect the population dynamics influence the slope. The two major stochastic components affecting annual changes of populations in a fluctuating environment are demographic stochasticity, i.e. random variation in individual fitness contributions to future generations, and environmental stochasticity, i.e. random variation in the environment affecting the whole population or groups of individuals in a similar way (Lande et al. 003). In addition, the strength of density regulation, i.e. the inverse of the return time to equilibrium (May 1974), and the form of the density regulation (Sæther and Engen 00) will also affect the population variability. These different processes affect the slope of the regression between the variance in the population fluctuations and the mean population size differently. To see this, let us consider a model for a change in a population of size N t at time t that in a given area fluctuates around an equilibrium population size K (carrying capacity). For simplicity, we assume that the density regulation is determined by some increasing function of N/K ΔN t rn t [1 h(n)/h(k )] 1 s e NU t, (1) where s e is the environmental variance, r is the intrinsic growth rate of the population and U t is a sequence of independent variables with mean 0 and variance 1. The stochastic term is realistic for populations that are large enough to ignore demographic stochasticity. If the function h is the same among different populations with different carrying capacities K, this is equivalent to assuming that the dynamics of the relative population size N t /K is independent of K. The diffusion approximation for this model has infinitesimal mean and variance rn[1 h(n )/h(k )] and s e N, respectively (Lande et al. 003). Using the linearization of h at N K, we obtain the familiar logistic model with infinitesimal mean rn(1 N/K ), which has previously been used to model the dynamics of British birds (Maurer and Taper 00). The mean and variance of the stationary distribution in this model obtained from the diffusion approximation are σe E( N ) K 1 r, and e e var( N ) K σ σ 1 r r. From this we see that σe var( N ) E( N ) r σ (a) (b) [ ], (c) e which gives b t log[var(n )]/ log[e(n )]. (3) Thus slopes b t of is typical of populations under environmental stochasticity alone (Fig. 1). For small populations the demographic variance term s d N must be added to the infinitesimal variance. Formally, this means that the process is then no longer stationary as extinctions may occur (Lande 1998). However, for populations with positive growth rates and large carrying capacities, fluctuations around the mean may still continue for a long time, producing a quasi-stationary distribution (Lande et al. 003). A rough approximation to this is found by approximating the demographic term s d by s e N /K (Lande et al. 003). With moderate fluctuations around K the contribution from the demographic term is relative constant. The mean and variance is given by E( N) e d K K 1 σ σ / r and σe σd/ K var( N ) E( N ) r σ σ / K e d (4a) [ ]. (4b) This line of arguing illustrates how demographic stochasticity will produce slopes smaller than for the relationship between the variance and mean population size on a logarithmic scale (Fig. 1). Thus, for a similar range of variation in mean population size, the slope is expected to decrease with increasing demographic stochasticity. In a stable environment under the influence of only demographic stochasticity the slope will be equal to 1 (Desharnais et al. 006). We see by inserting s e 0 into Eq. 4a and 4b that b t log[var(n )]/ log[e(n )] 1, (5) which means demographic stochasticity alone generates slopes b t of 1 (Fig. 1). This indicates that estimating the variance components due to stochastic effects and density dependence is necessary for deriving the relationship between the variance and the mean population size. The model for analysing the relative contribution of demographic and environmental stochasticity to the relationship between the variance and mean population size assumes that the strength of density regulation is independent of K. This may not necessarily be true. For instance, in species which are strongly influenced by competition over some 108

3 Figure 1. The mean variance relationship known as Taylor s power law depends on the effect of demographic and environmental stochasticity. Here we show the predicted theoretical effect of demographic stochasticity on the relationship between the variance and mean population size at logarithmic scale and the corresponding slope. (a) the relationship between the mean and the variance on log scale for different values of the demographic variance, s d. The other parameters (K, s e and r) are kept constant. (b) Illustrates how the slope of the regression on log scale between the mean and the variance changes with log mean for different values of s d. The stipulated lines gives the predicted limits of 1 and for the slope. limited resources the strength of density dependence may be stronger at higher population densities than in a population living in a similar sized area with a smaller equilibrium population size. In fact, Hanski and Tiainen (1989) found that territorial bird species had smaller slope values of the variance-mean relationship at a logarithmic scale than nonterritorial species. Thus, considering the effect of K on the strength of density regulation is important when estimating Taylor s temporal power law. While there has been few studies attempting to link differences in life history to the exponents of the temporal Taylor s power law, Hanski and Tiainen (1989) found that of 13 life history parameters, namely territoriality and survival, showed a significant relationship with the slope. Life history can influence the population dynamic through s d, s e and the strength of density regulation (Sæther et al. 00, 004, 005) and may thus generate differences in the slope b t. In this paper, we will estimate the relationship between the stationary variance of the population fluctuations and average population size for 40 bird species censused in fixed plots scattered over larger parts of the UK as part of the Common Bird Census scheme (Marchant et al. 1990). Data from this scheme have been analysed in this context previously (Taylor and Woiwod 198, Hanski and Tiainen 1989, Maurer and Taper 00) but here we use a more restrictive selection criteria by calculating the stationary variance of time series in which there are no temporal trends. Since we are not able to separate the effects of environmental and demographic stochasticity we will estimate the process variance which includes components due to demographic and environmental variance. However, based on theoretical knowledge of how demographic stochasticity contribute to the overall process variance we may derive interspecific estimates of demographic variance. We will then relate the estimates of two parameters affecting the magnitude of the population fluctuations, the process variance and strength of density regulation, to the among plot variation in the carrying capacity. Finally, we will relate the interspecific differences in slopes to different ecological and life history characteristics. This will provide one of the first examples for how large-scale macro-ecological patterns can be derived from simple models characterizing the local dynamics. Material and methods Data We used data from the Common Birds Census (CBC) of British birds between 196 and 001 based on annual censuses by the mapping method of breeding birds in fixed study plots distributed over larger parts of the UK in both farmland and woodland habitats (Marchant et al. 1990). This provides an annual estimate of the number of breeding pairs of birds in each plot. We included only time series from study plots that were censused for 10 or more years and if there were missing values in the time series we selected the longest continuous time series. First we excluded time series from study plots which could not be considered stationary by an insignificant regression of the change in population size N t 1 /N t on N t. 109

4 However, density independent time series may appear stationary in presence of observation error (Freckleton et al. 006). Thus, after parameter estimation we selected species which had at least 10 estimated time series in which density dependence was significant after accounting for observation error. The species are listed in Supplementary material Appendix 1 Table A1. Data on body mass and life history characteristics for each species, such as clutch size and survival of adults, were extracted from interspecific comparative studies (Dobson 1990, Peach et al. 001, Wernham et al. 00). For estimates of ecological species characteristics, see Supplementary material Appendix 1 Table A1. Models In addition to the underlying variance in the stochastic population model, known as the process variance, observed time series of population counts usually include some variance due to observation error. The observation error is a property of how the data have been collected and has been found to be present in the CBC data (O Connor and Marchant 1981). To account for the effect of observation error when estimating populations parameters from time series we used a Bayesian state-space framework, which combines a model for population growth with an observation model (Clark and Bjørnstad 004, Clark 007, Sæther et al. 009). Population model We used the logistic model of density regulation which has previously been found to give a reliable description of the density dependence of British bird populations (Sæther et al. 009). We now write the stochastic model (Eq. 1) as N t 1 1 N t 1 r t N t r(n t /K ) 1 s P N t U t, (6) where r(n t /K ) defines the density regulation in a logistic model and r t is the density-independent growth rate varying stochastically between years. s P is the process variance which includes components due to demographic and environmental variance so that s P s e 1 s d /N. (7) With a positive s d this process is not stationary, but the quasi-stationary distribution is close to the stationary var(δx t X t ) s P. (8b) Here b r K and define the logistic form of density dependence. The strength of the density dependence, e.g. the inverse of the return time to the equilibrium population size (May 1981), is then g r 1 s P, (9) which equals the stochastic growth rate in a logistic model. Observation model The observation model provides a description of the relationship between the observed and the true population size. We let X t be the true, but unobserved log population size at time t, whereas Y t denotes the observed log population size at time t. We assumed that the observed number of birds, Y t on a logarithmic scale at time t was normally distributed with mean X t and variance s Y (Dennis et al. 006). Estimation of parameters Observation error (s Y ) was included in the models by a Bayesian approach in combination with Markov chain Monte Carlo (MCMC) methods (Calder et al. 003, Gross et al. 005, Clark 007). To obtain posterior distributions of parameters we need to define full probability functions P 1 and P for all observed (Y 1,,Y T ) and unobserved (X 1,, X T ) population sizes, given by the population and observation model. We then chose independent and vague prior probability distributions for parameters not conditioned on other parameters or data as follows: g ~ N (0,10 6 ) b ~ N(0,10 6 ) s P ~ Inv-Gamma(0.001,0.001) s Y ~ Inv-Gamma(0.001,0.001) for p 1 (g), p (b), p 3 (s P ) and p 4 (s Y ), respectively. The initial values X 1 has a normal prior distribution, P 0, with mean Y 1 and variance 0.. The posterior distribution were insensitive to other specification of priors. Following Bayes theorem the joint posterior distribution conditioned on the observations, denoted as p(g, b, s P, s Y, X 1,,X T Y 1,,Y T ), is proportional to T 1 π1( γ) π( β) π3( σp) π4( σy ) P0 ( Y1 X1) P1( Xt 1 Xt, γβσ,, P ) P ( Yt X t, σ Y ) t 1 T t 1 distribution by approximating s d /N by s e /N (Lande et al. 003). The carrying capacity K is assumed constant over time. Writing the population size on a natural logarithmic form, X t log(n t ), a simple first-order approximation then gives the mean and variance in ΔX t X t 1 1 X t for a logistic model E(ΔX t X t ) r b exp(x t ) 1 s P and (8a) While this distribution is analytical intractable, it can be estimated by successive simulations from the proposed distributions using the program WinBUGS (Lunn et al. 000). We used a burn-in period of updates followed by at least updates until visual inspection of chain patterns indicated non-convergence in the MCMC algorithm. MCMC chains that had not converged after iterations were removed. The potential scale reduction factor for 110

5 the Gelman Rubin diagnostics (Gelman and Rubin 199) were close to 1 for all parameters (for details see Supplementary material Appendix 1 Table A3). We obtained estimated posterior marginal distributions of the parameters K, g, b, s P and s Y. Simulation of time series We found that using the diffusion approximation for calculating the quasi-stationary distribution was very unreliable for this data, probably due to large process error. Thus, following Kendall (009) we estimated the quasistationary distribution by a large number of simulated time series based on the estimated parameters. Sets of estimated parameters that contained biological unrealistic values indicating negative strength of density dependence (g 0) or positive density dependence (b 0) were excluded from the calculations. For each of the observed time series we simulated time series with a length of 1000 time steps from each of the posterior distribution sets of the estimated parameters, using the population model given by the mean and variance in Eq. 8a and 8b: X t X t 1 1 r b exp(x t 1 ) 1 s P 1 e t, (10) where e t ~ N(0, s P ). For each population of a species in a study plot we then calculated the median value as well as the 95% credible intervals of the parameters K, g, b, s P and s Y of the simulated time series. We excluded simulated time series with K 3 and simulated time series in which density dependence was not significant (b not significantly larger than 0) to ensure stationarity of the population fluctuations. The stationary variance s N was then calculated as the variance of the simulated time series. Analysis To analyse the generality of Taylor s temporal power law the log(s N ) log(k ) regression was calculated for each species separately. Interspecific differences in slopes were investigated using a model selection approach (Burnham and Anderson 00). Linear regression models for s N were fitted with species and K as explanatory variables and models were compared using Akaike s information criteria corrected for small sample sizes (AICc) (Burnham and Anderson 00). Since we investigated interspecific comparisons the statistical relationships may be affected by phylogenetic non-independence (Harvey and Pagel 1991). Although we included species as a main effect in the model selection we also conducted separate phylogenetic analyses in the species specific slopes and intercepts of the regressions of g, s N and s P on log(k ). We estimated the lambda statistic of Pagel (Pagel 1999, Freckleton et al. 00) using the phylogeny of British Birds proposed by Thomas (008) and found this to be not statistically significantly different from zero, but quite considerably different from 1. This indicates there is no phylogenetic signal in these relationships. To evaluate the effects of ecological and life history parameters on the relationship between s N and K we fitted linear regression models for all combinations of variables and compared them using AICc. Summing the Akaike weights for each model in which that variable is present can be used to evaluate the relative importance of variables (Burnham and Anderson 00). To examine how the stationary variance, the strength of density dependence and the process variance, respectively, differed with K we fitted linear regression models with the logarithm of K and species as explanatory variables. We performed model selection analysis using AICc. We obtain estimates of s P and K for all fitted time series. A rough estimate of s e and s d can be obtained by utilizing the relationship in Eq. 7. Assuming that s d and s e do not vary among localities for a given species and inserting the estimates of K for N in the relationship given above, we can solve for s d and s e using numerical optimization. More specifically, for each species we maximized the I likelihood N(log( σ P ()) i log(( σ e) ( σ d) / Ki ( )), ( σ )) i 1 with respect to parameters log(s e ), log(s d ), log(s ) and estimates of s P (i) and K(i) for I different localities. N denotes the normal distribution. Results After accounting for observation error, the logarithm of the stationary variance log(s N ) was positively correlated with the logarithm of the median carrying capacity (K ) in all species (Table 1), with slopes significantly larger than 0 in 33 of the 40 species. For most of the significant linear regressions the slopes had values between 1 and, only 3 species of 33 had slopes that were marginally smaller or larger than this. In general, the slopes were mainly located in the interval between 1 and (Fig. a). A model selection approach revealed that the logarithm of the variance of the stationary distribution was best explained by a model containing both species and log(k ) without any interaction effect (Table ). To evaluate whether the differences among species in the relationship between log(s N ) and log(k ) could be explained by species characteristics we again performed a model selection analysis (Table 3). In all models for which ΔAICc, log(k), and the species characteristics clutch size and adult survival rate in addition to the interaction between adult survival and log(k ) were included. From the negative effect of adult survival rate on log(s N ) and the relationship between log(s N ) and log(k ) it follows that increased adult survival decreased the slope of the relationship between the variance and the carrying capacity. An increase in clutch size decreased log(s N ). These variables also had high ( 0.9) summed Akaike weights. This means that species-specific differences in the relationship between log(s N ) and log(k ) were influenced by interspecific differences especially in the adult survival rate. We then estimated how the different components which may influence s N scaled with changes in log(k ). Since changes in the deterministic components can affect the power law an important assumption for this model was that the strength of the density dependence, g, was independent of log(k ). We found that the strength of density dependence significantly decreased with the logarithm of the 111

6 Table 1. The coefficient of Taylor s power law for each species. The significance codes of the slopes are given as: ** 0.01 and * The number of time series used for estimating the Taylor s power law exponents for each species is given together with the mean observed population size. Species Time series mean N (observed) Intercept Slope b t Blackbird ** Blackcap ** Blue tit ** Bullfinch Carrion crow ** Chaffinch ** Chiffchaff ** Coal tit ** Collared dove Dunnock ** Garden warbler * Goldcrest Goldfinch ** Great tit ** Greenfinch ** Greypartridge * Jackdaw * Linnet ** Long-tailed tit * Magpie ** Mallard ** Mistle thrush Moorhen ** Nuthatch Pheasant ** Redstart Reed bunting * Robin ** Skylark ** Song thrush ** carrying capacity in only 5 out of 40 species. A model selection analysis revealed a intraspecific effect of the strength of density dependence and log(k ) but indicated no effect of species differences on this relationship (Table ). We explored the relationship between the process variance and K since it is known that demographic variance should decrease with increasing population size and s P is combined of demographic stochasticity and environmental stochasticity (Eq. 7). Interspecific variation of log(s P ) was best explained by a model which included species and log(k ) but with no interaction effect between species and log(k ). For each species, the logarithm of the process variance (s P ) in general decreased with increasing log(k ). Figure b shows that the slopes of this relationship were mainly negative, and all slopes of significant relationships had values 0. This decrease in process variance with increasing K reflects a decreasing contribution from demographic variance as population size increases, and allowed for a rough estimate of the demographic variance for each species (Eq. 7). As expected (Introduction, Fig. 1), the slope of the temporal Taylor s power law was negatively related (correlation coefficient 0.66, p 0.001, n 40) to the estimated demographic component, s d K, of the process variance. Furthermore, a similar negative relationship (correlation coefficient 0.65, p 0.001, n 40) was found between the estimate of s d and the slope, suggesting that interspecific variation in slopes was caused by interspecific variation in s d rather than by interspecific variation in mean K. Discussion In this study, we have shown that the log stationary variance in the population fluctuations of British birds scale with log mean population size with a slope of approximately b t of 1.5 (Table 1, Fig. a, see also Maurer and Taper 00) which is within the theoretical predicted limits of 1 and. This provides support for the presence of Taylor s power law for temporal variability in population fluctuations. Similar patterns have also been revealed in analyses of population fluctuation of North American birds (Maurer and Taper 00). The process variance which include joint effects of environmental and density dependent demographic stochasticity decreased with increasing carrying capacity (Fig. b). This pattern was expected due to reduced influence of demographic stochasticity at larger population sizes. As predicted from the theoretical model given in the Introduction, we found that species specific slopes of the Taylor s power law decreased with increasing species specific estimates of demographic variance (Fig. 3). In addition, some of the interspecific differences in the relationship between the logarithm of stationary variance and the logarithm of the carrying capacity were explained by variation in life history characteristics (Table 3). The original formulation of Taylor s power laws was based on statistical analyses without any foundation in basic population theory. It soon became clear that different processes could strongly affect the relationship between the variance and mean population size (Taylor 1977, 1980, Taylor and Woiwod 198). An important advance in our understanding of these relationships was provided by Anderson et al. (198), who showed that the magnitude of demographic stochasticity could strongly influence the mean variance relationship. Thus, no complex behavioural mechanism (Taylor 1977) was necessary to invoke to produce these general relationships. Here we extended these analyses to show that demographic and environmental stochasticity contribute differently to the slope of the mean variance relationship (Fig. 1). If the population size is so large that demographic stochasticity can be ignored we would obtain a slope of. In contrast, at smaller that demographic stochasticity can be ignored we would obtain a slope of. A slope of 1 indicates a stable environment with only demographic stochasticity (Desharnais et al. 006). In addition to explaining intraspecific variation in the slope of Taylor s power law, demographic stochasticity can also cause interspecific variation in the slope provided that population sizes are so small that demographic stochasticity contribute to the process variance. For a given population size, a species with a high demographic variance is expected to have a more shallow slope than a species with a smaller demographic variance (Fig. 1). Our results indicate that the slopes of the relationship between log(s N ) and log(k ) of British birds in general were 11

7 Figure. The distribution of the slopes of the linear regression for the stationary variance (s N ), the process variance (s P ) and the strength of density dependence (g) and the carrying capacity (K ) on a logarithmic scale between species. (a) s N. The stipulated lines denote the limits for slope of the relationship between s N and K predicted by theory. (b) s P, (c) g. For all figures the vertical line represents the median of the slope values. between 1 and (Fig. a). We propose that this is caused by the joint influence of demographic and environmental stochasticity on the population dynamics of these species. Firstly, the process variance decreased with log(k ) (Fig. b, see also Sæther et al. 010). The theoretical explorations summarized in Fig. 1 suggest that slope b t should change with increasing population size, reflecting the decreased influence of demographic stochasticity on the annual changes in population size. However, in our data set a second-order polynomial did not in general provide a better fit than a simple linear model (only in one species were there any evidence of curvation). This suggests that mean population sizes in the present study were not sufficiently large to reduce the influence of demographic stochasticity. Infact, the largest observed mean population size was 19. for the species wren (Troglodytes troglodytes). Although not able to directly estimate s d for the populations we were able to estimate interspecific estimates of demographic variance based on theoretical knowledge of how demographic stochasticity contribute to the overall process variance (Eq. 7). We admit these rough estimates are not of similar quality as estimated based on individual fitness 113

8 Table. Model selection analysis of the relationship between the logarithm of respectively the stationary variance (s N ), the process variance (s P ) and the strength of the density dependence (g), and the logarithm of the carrying capacity (K). AICc and AICc weight gives respectively the change in the corrected Akaike s information criteria value and the weight in favour of the model. X indicates that the variable was included in the model. Species log(k) Species log(k) AICc AICc weight X X X X X X X s P X X X X X X X g X X X X X X X s N contributions. In contrast to the intraspecific relationship, we found that the slope of Taylor s power law was related to demographic variance when considering among species variation. Species specific estimates of demographic variance were significantly negatively correlated with species specific slopes of Taylor s power law (Fig. 3). In this study, interspecific variation in mean N(4 19) is not large and variation in demographic stochasticity may then cause interspecific variation in slopes among species (Fig. 1). Elsewhere it has been shown that the interspecific variation of the mean process variance of British birds was related to the overall rarity of the species in the UK (Sæther et al. 010). We would therefore expect that the slopes of relationship between log(s N ) and log(k ) should be related to the total abundance of the species in UK because more abundant species should be less influenced by demographic stochasticity and hence have slopes closer to (Fig. 1) than rare species. However, no such relationship was present (correlation coefficient 0.16, p 0.33, n 40). Accordingly, the total abundance of the species was included only in a few of the models explaining the interspecific variation in slopes of the regression between log(s N ) and log(k ) (Table 3). We found that estimated demographic variance, s d, and the contribution of demographic variance, s d /K had a very similar effect on estimated slopes. This suggests that interspecific variation in slopes was caused by interspecific variation in s d rather than by interspecific variation in mean K, and this may explain why we found no relationship between total UK abundance of species and estimated slopes. However, not only stochastic effects but also changes in the deterministic component of the population dynamics will affect the relationship between the variance and the mean (Anderson et al. 198, Ballantyne 005). No relationship was found between the strength of density regulation and log(k ) (Fig. c). This indicates that changes in the effects of the strength of density dependence with increasing population size do not affect the slopes of the relation between log(s N ) and log(k ) in this data set. Taylor s temporal power law has been explored empirically in birds before, with some indications of slope values Table 3. The 0 best supported models from a model selection for the relationship between the logarithm of the stationary variance (s N ) and the logarithm of the carrying capacity (K) and chosen ecological factors. CS is clutch size, Adsurv is adult survival, Migr is migration, Weight is weight in gram and Est is the total estimated population size in the UK. AICc is the corrected Akaike s information criteria, AICc and AICc weight gives respectively the change in value and the weight infavour of the model. X indicates that the variable was included in the model. At the bottom of the table is the total sum of the AIC weight of all the models in the model selection in which the given predictor variable occurs. log(k) CS Adsurv Migr log (Weight) log (Est) log(k) CS log(k) Adsurv log(k) Migr log(k) log (Weight) log(k) log (Est) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Summed AICc weight for predictor variables ΔAICc AICc weight 114

9 Figure 3. Relation between the slope of the temporal Taylor s power law and the estimated interspecific demographic variance (s d ). outside the range of 1 and (Taylor 1980, Taylor and Woiwod 198, Hanski and Tiainen 1989, Maurer and Taper 00). Taylor and Woiwod (198) found slopes in British birds below 1, however this may be caused by sampling error (Taylor and Woiwod 198). A study of bird species from the North American Breeding Bird Survey with stricter inclusion criteria (longer time series, assessment of error), and thus more comparable to this study, found slope values between 1 and (Maurer and Taper 00). Our model which incorporated both deterministic and stochastic components generated slopevalues between 1 and, and we propose that the deviation from found here is caused by demographic stochasticity (Fig. 3). However, we cannot completely rule out other factors. For instance Kilpatrick and Ives (003) have suggested deviations from as a result of community-level interactions including direct or indirect negative competition. In this study we found large interspecific variation in the slopes of the regression between the logarithm of stationary variance and the logarithm of the carrying capacity (Fig. a). Although our estimates are not directly comparable because of stronger restrictions on inclusion of time series (stationarity of the population fluctuations, minimum length of census period), this supports previous analyses covering a wide range of taxa that mean variance slopes tend to differ among species (Taylor 1980, Hanski 198, Taylor and Woiwod 198, Hanski and Tiainen 1989, Hanski and Woiwod 1993, Curnutt et al. 1996). Our model selection analyses revealed that interspecific differences in the slope of the regression of log(s N ) on log(k ) was related to species-specific life history characteristics (Table 3). In particular, adult survival rate seemed to be a key character because it was included in all the best models, explaining interspecific differences in log(s N ) as well as how log(s N ) depends on log(k ). In general, the value of the slope decreased with increasing survival. Accordingly, previous analyses showed that high process variances s P was associated with high mortality rates (and large clutch sizes) (Sæther et al. 010) and Hanski and Tiainen (1989) found survival to be related to the slope of the temporal power law. Moreover, interspecific differences in demographic stochasticity can be explained by life history variation (Sæther et al. 004). This suggests that smaller population variability in species with high survival rates may be an effect of reduced demographic stochasticity in species towards the slow end of the slow-fast continuum of life history variation. To conclude, this study confirms that there exist a power law relationship between the temporal variation and mean population size at a logarithmic scale in British birds. Differences in the strength of density dependence could however not explain differences among species in the relationship between the variance and mean population size. We show that the interspecific differences in these relationships (Fig. a) can mainly be explained by effects of life history variation on interspecific variation in demographic variance. Such general relationships are important because they provide ecologists of some basic levels for the amount of population variability expected given the knowledge of some basic properties. This can be helpful in identifying populations that show deviating population dynamics as well as in developing coarse models for future population projections. 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