Science, Vol. 287, Feb. 4, 2000
B.E. Sæther, ^{1*} J. Tufto, ^{2} S. Engen, ^{2} K. Jerstad, ^{3} O. W. Røstad, ^{4} J. E. Skåtan ^{5}
Predicting the effects of an expected climatic change requires estimates and modeling of stochastic factors as well as densitydependent^{ }effects in the population dynamics. In a population of a small^{ }songbird, the dipper (Cinclus cinclus), environmental stochasticity^{ }and density dependence both influenced the population growth rate.^{ }About half of the environmental variance was explained by variation^{ }in mean winter temperature. Including these results in a stochastic^{ }model shows that an expected change in climate will strongly affect^{ }the dynamics of the population, leading to a nonlinear increase^{ }in the carrying capacity and in the expected mean population size.^{ }
1 Department of Zoology,
^{2} Department of Mathematical
Sciences, Norwegian University of Science and Technology, N7491 Trondheim,
Norway.
^{3} Aurebekk, N4500 Mandal, Norway.
^{4}
Department of Biology and Nature Conservation, Agricultural University of
Norway, Post Office Box 5014, N1432 Ås, Norway.
^{5} N4592
Birkeland, Norway.
^{*} To whom correspondence should be addressed.
Email: BerntErik.Sather@chembio.ntnu.no
A central question in ecology for decades has been how to quantify the relative importance of stochastic and densitydependent^{ }factors for fluctuations in population size (1). This^{ }question has received increased attention because of the need^{ }to predict the biological consequences of climate change (2).^{ }To answer this question, we must obtain separate estimates of^{ }different forms of stochasticity, such as demographic and environmental^{ }variances (3) and the strength of density dependence,^{ }and use these estimates to model the impact of a climate change^{ }on the population fluctuations. Several studies have predicted^{ }changes in species ranges, demographic rates, or average population^{ }sizes in response to climate change (4). However, missing^{ }are quantitative analyses that explicitly link climate change^{ }and population fluctuations in a mechanistic population model.^{ }Here we provide such a link and obtain predictions of markedly^{ }altered population dynamics for a songbird, mediated primarily^{ }through winter temperature.
Modeling the dynamics of populations in a stochastic environment involves estimating the separate effects of density regulation^{ }and stochastic factors. The variance of the change in population^{ }size can be split into the demographic and environmental variances^{ }(5). The demographic variance is caused by stochastic^{ }variation among individuals in their contribution to the next^{ }generation, whereas the environmental variance is due to stochasticity^{ }similarly affecting a certain group of individuals at a certain^{ }time (3). Several studies (6) have now shown^{ }that knowledge of demographic as well as environmental stochasticity^{ }is important for understanding temporal fluctuations in population^{ }size. Thus, quantifying the effects and predicting the consequences^{ }of an expected climate change, which possibly involves changes^{ }in both the mean and the variance of several climatic variables,^{ }will require estimates of how these changes will affect the behavior^{ }of the population dynamic processes. The dipper (Cinclus cinclus),^{ }a 50 to 60g passerine species widely distributed in aquatic^{ }habitats close to running water all over the Palearctic region^{ }(7), is suitable for studying those questions because^{ }at northern latitudes the amount of ice strongly affects which^{ }areas have available winter feeding habitats. Thus, this relation^{ }provides a possible link between population dynamics and climate.^{ }
We studied a population of dippers in southern Norway (8), where a large proportion of all individuals was colorringed^{ }for individual recognition (9) over a 20year period^{ }(197897). Large fluctuations in population size occurred during^{ }the study period (Fig. 1A), from a minimum of 27 pairs^{ }in 1982 to a maximum of 117 pairs in 1993. The recruitment rate^{ }of the population, R_{t} = X_{t plus;1}X_{t}," src="/content/vol287/issue5454/fulltext/854/img001.gif" where X_{t}^{ }is the size of the breeding population in year t, also showed^{ }large annual variation (Fig. 1B). The recruitment rate^{ }decreased with increasing population size (r = 0.49, n = 18,^{ }P < 0.05) (Fig. 1B). Accounting for annual variation^{ }in the number of immigrants M_{t}+1 (10), no significant^{ }densitydependent decrease was found in the net reproductive rate^{ }NR_{t} = X_{t & plus;1}M_{t+1}X_{t}," src="/content/vol287/issue5454/fulltext/854/img002.gif" (r = 0.27,^{ }n = 18, P 0.1) (Fig. 1B). Thus, the decrease in recruitment^{ }rate was mainly due to a reduction of the immigration rate IR_{t}^{ }= M_{t+1}X_{t}" src="/content/vol287/issue5454/fulltext/854/img003.gif" with increasing population size^{ }(r = 0.56, n = 18, P < 0.05) (Fig. 1C). However, no^{ }significant density dependence in the absolute number of immigrating^{ }females was found (r = 0.27, n = 18, P 0.1).
/cgi/content/full/287/5454/854/F1
/cgi/content/full/287/5454/854/F1Fig. 1. (A) The annual variation in population size (X_{t}). (B) The recruitment rate R_{t} (solid circles) and the net recruitment rate NR_{t} (open circles) and (C) the immigration rate IR_{t}, in relation to population size (X_{t}). [View Larger Version of this Image (18K GIF file)]
To estimate parameters, we modeled the dynamics of the population by assuming that the change in the logarithm of the net^{ }recruitment rate (log NR_{t}) was normally distributed
Dgr;ln NR_{t} = lnX_{t+1}−M_{t+1}X_{t}∼" src="/content/vol287/issue5454/fulltext/854/img004.gif" 
r−&agr;X,+&bgr;C_{t},&sfgr;′_{e}^{2} + &sfgr;_{d}^{2}X_{t}" src="/content/vol287/issue5454/fulltext/854/img005.gif" 
(1) 
where C_{t} is the climatic variable (11, 12), _{e}^{2} is the residual environmental variance not accounted for by the^{ }variation in C_{t}, _{d}^{2} is the demographic variance estimated from individualbased data^{ }(13), is the strength of density regulation, and^{ }denotes the strength of dependency on the climatic variable.^{ }Observations of mean winter temperatures C_{t} were centered such^{ }that the expectation E(C_{t}) = 0 for the whole period with available^{ }climate data (38 years). It follows that the environmental variance^{ }is about
e^{2} = &sfgr;′_{e}^{2}+var(&bgr;C_{t}) = &sfgr;′_{e}^{2} + &bgr;^{2}var(C_{t})" src="/content/vol287/issue5454/fulltext/854/img006.gif" 
(2) 
The number of immigrants was correlated with the winter temperature (r = 0.62, n = 18, P = 0.003) (Fig. 2A). To incorporate^{ }this in the model used for estimation and prediction, we assumed^{ }that M_{t} was Poissondistributed with parameter _{t}, with^{ }each _{t} being independently lognormally distributed with^{ }expectation depending on mean winter temperature by letting each^{ }
log(&lgr;_{t}) ∼ N(&mgr;_{0}+&mgr;_{1}C_{t},&sfgr;_{&lgr;}^{2})" src="/content/vol287/issue5454/fulltext/854/img007.gif" 
(3) 
where '^{2} is the variance in the log of the immigration rate, µ_{0} is the mean log immigration rate at C_{t} = 0 and^{ }µ_{1} measures the dependence of the immigration rate on temperature.^{ }
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/cgi/content/full/287/5454/854/F2Fig. 2. The number of immigrants plotted against mean winter temperature (A) and the relative change in population size in relation to mean winter temperatures that were centered (B) and the NAO index (C). [View Larger Version of this Image (13K GIF file)]
To facilitate modeling of the effects of a climatic change, we used mean winter (January to March) temperature (°C) and total^{ }winter precipitation as C_{t} in Eqs. 1 and 3.^{ }These variables were chosen because they are commonly used when^{ }developing climatic scenarios (2). We estimated the^{ }posterior distribution (quantifying degree of belief in alternative^{ }parameter values conditional on the data) for parameters in the^{ }model (Eqs. 1 and 3) by Markov Chain Monte Carlo^{ }methods (14). These analyses showed that variation^{ }in the logarithm of the net reproductive rate was influenced by^{ }population density and by climate (Table 1). Low recruitment^{ }occurred in years with high population densities. Furthermore,^{ }fewer individuals for a given population size were recruited after^{ }cold winters (Fig. 2B). This may be because mean winter^{ }temperature was closely correlated with the annual variation in^{ }the number of days with ice cover in the study area (r = 0.83,^{ }n = 38, P < 0.001). However, the recruitment rate was not significantly^{ }correlated with winter precipitation (r = 0.13, n = 12, P 0.1).^{ }On the basis of the annual variation in mean winter temperature^{ }C_{t} over the past 40 years (SD = 2.09°C) and the estimate of the^{ }parameters substituted into Eq. 2, we find that about^{ }half of the total environmental variance was explained by variation^{ }in winter temperature (Table 1).
The slope (µ_{1}) in the^{ }regression of log(_{t}) on C_{t} was of similar magnitude^{ }but somewhat smaller than the slope () in the regression of the^{ }log net recruitment rate on C_{t} (Table 1), indicating^{ }that cold winters have similar effects on the survival of both^{ }dispersing and nondispersing individuals.
Table 1. The estimates of the parameters describing the population dynamics (log NR_{t}, see Eq. 1) of the dipper Cinclus cinclus in southern Norway and posteriori probabilities (degree of belief conditional on the observed data) of some hypotheses.  
 
Parameter 
Estimate (mean ± SD) 
Posteriori probabilities 
 
Population growth rate (r) 
0.086 ± 0.186 
P(r 0) = 0.31 
Density regulation () 
0.0042 ± 0.0014 
P( 0) = 0.998 
Effects of winter temperature () 
0.15 ± 0.03 
P( 0) = 0.9999 
Total environmental variance (_{e}^{2}) 
0.21 ± 0.06 

Environmental variance from winter 

temperature (_{ce}^{2}) 
0.10 ± 0.05 

Residual environmental variance ('_{e}^{2}) 
0.11 ± 0.04 

Demographic variance (_{d}^{2}) 
0.268 ± 0.018 

Expected number of immigrants at c = 0 
50.4 ± 4.4 

at c = 2.5 
67.7 ± 7.5 

Slope in Poisson regression for M_{t} (µ_{1}) 
0.11 ± 0.03 
P(µ_{1} 0) = 0.999 
The deterministic carrying capacity K was found numerically (for each realization from the posterior) by setting X_{t} and X_{t}+1^{ }equal to K and M_{t} equal to its expectation and then solving for^{ }K.
In the Northern Hemisphere, climatic conditions during winter are often determined by annual variation in largescale fluctuations^{ }in atmospheric mass between the subtropic and the subpolar North^{ }Atlantic regions, the North Atlantic Oscillation (NAO). The NAO^{ }index is an integrated measure that influences a large number^{ }of climatic variables that affect the winter weather over large^{ }areas of the Northern Hemisphere (15). For instance,^{ }a high positive NAO index is, in general, associated with relatively^{ }warm winters with much precipitation in northern Atlantic coastal^{ }Europe (16). Variation in the NAO index is closely^{ }correlated with global fluctuations in temperature (17).^{ }Because the net recruitment rate is positively correlated with^{ }the NAO index (Fig. 2C), such largescale climatic changes^{ }are likely to affect the dynamics of the local dipper population.^{ }
The Intergovernmental Panel on Climate Change (IPCC) has developed different scenarios of future greenhouse gas and aerosol^{ }precursor emissions on the basis of socioeconomic assumptions^{ }for the period 19902100 (2). These predicted emissions^{ }were then used to project atmospheric concentrations of greenhouse^{ }gases and aerosols and their effects on natural radiation processes.^{ }When these effects are entered into climatic models, some scenarios^{ }suggest an increase in mean winter temperature of 2° to 3°C in^{ }the region in which our study population is located (2),^{ }although the quantitative effects of the anthropogenic influences^{ }on the atmosphere must be considered uncertain (18).^{ }
To examine the effects of the fluctuations in climate and a change^{ }in longterm mean on the dipper population dynamic, we model the^{ }climatic process C_{t} as a firstorder autoregressive process (19)^{ }
t+1−c∼N[a(C_{t}−c),&sfgr;_{c}^{2}]" src="/content/vol287/issue5454/fulltext/854/img008.gif" 
(4) 
where c and _{c}^{2} are the mean and variance, respectively, and a determines the return time in the process. According^{ }to the scenarios from IPCC for the region (2), we model^{ }the change in the longterm mean winter temperature by setting^{ }c = 2.5 in Eq. 4.
We find that an increase in mean winter^{ }temperature is likely to strongly affect the population dynamics^{ }of the dipper in the study area. The expected carrying capacity^{ }(K) increased by 58% (Fig. 3A), from K = 115 (SD = 10)^{ }for c = 0 to K = 182 (SD = 22) for c = 2.5. Similarly, such an^{ }increase in mean temperature increased the expectation of the^{ }stationary distribution of the population size X_{t} from 117 to^{ }184 (Fig. 3B). When we run the model, assuming no relation^{ }between climate and immigration rate, the effect of a change from^{ }c = 0 to c = 2.5 on K was still large [K = 167 (SD = 20) for c^{ }= 2.5]. Thus, the major effect of a change in winter climate on^{ }the dipper population will occur through an influence on the local^{ }dynamics.
/cgi/content/full/287/5454/854/F3
/cgi/content/full/287/5454/854/F3Fig. 3. (A) The posterior distribution of the deterministic carrying capacity (K) and (B) the stationary distribution of the population size (X_{t}) before (solid lines) and after (dashed lines) an increase in winter temperature of 2.5°C. (C) The change in the carrying capacity (K) as a function of a change in mean winter temperature (°C). [View Larger Version of this Image (17K GIF file)]
To quantitatively examine how the magnitude of a change in mean winter temperature affected the population dynamics of the^{ }dipper, we computed the change in carrying capacity K (evaluated^{ }at the estimated mean of all model parameters) over a range of^{ }2.5° to 2.5°C for the mean value of c in Eq. 4. Our^{ }analysis suggests a nonlinear relation between the carrying capacity^{ }K and change in mean temperature c (Fig. 3C). Thus, for^{ }c = 2.5, K will decrease to about 45 pairs, whereas a similar^{ }increase in the temperature will increase K considerably more,^{ }to about 178 pairs.
Several studies have documented effects of changes in climate on several demographic characteristics of birds and mammals.^{ }For instance, many temperate bird species lay eggs earlier in^{ }the year, probably because of warmer springs (20, 21).^{ }Similarly, the spawning dates of amphibians such as Rana spp.^{ }were positively correlated with the NAO index (20).^{ }Our study complements those results by showing that climatic changes^{ }may also strongly affect the dynamical characteristics of a population^{ }(Figs. 2 and 3). These large effects on the^{ }population dynamics of relative small changes in climate (Fig. 3)^{ }were due to an effect on the local dynamics as well as higher^{ }immigration rate in mild winters. These interactions between processes^{ }operating at different geographical scales suggest that climate^{ }changes may have major consequences for the pattern of fluctuations^{ }in bird populations over space and time (22).^{ }
27 August 1999; accepted 7 December 1999