Along controversy over the issue of density-dependent versus

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1 Seasonality, density dependence, and population cycles in Hokkaido voles Nils Chr. Stenseth, Hildegunn Viljugrein, Takashi Saitoh, Thomas F. Hansen, Marte O. Kittilsen, Erik Bølviken, and Fredrik Glöckner Centre for Ecological and Evolutionary Synthesis, Department of Biology, University of Oslo, P.O. Box 1031 Blindern, N-0316 Oslo, Norway; Field Science Center, Hokkaido University, North 11, West 10, Sapporo , Japan; Department of Biological Science, Florida State University, Conradi Building, Tallahassee, FL 32306; and Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway Edited by Thomas W. Schoener, University of California, Davis, CA, and approved July 11, 2003 (received for review September 2, 2002) Voles and lemmings show extensive variation in population dynamics regulated across and within species. In an attempt to develop and test generic hypotheses explaining these differences, we studied 84 populations of the gray-sided vole (Clethrionomys rufocanus) in Hokkaido, Japan. We show that these populations are limited by a combination of density-independent factors (such as climate) and density-dependent processes (such as specialist predators). We show that density-dependent regulation primarily occurs in winter months, so that populations experiencing longer winters tend to have a stronger delayed density-dependence and, as a result, exhibit regular density cycles. Altogether, we demonstrate that seasonality plays a key role in determining whether a vole population is cyclic or not. Clethrionomys rufocanus seasonal and annual density dependence state-space modeling sampling variance Along controversy over the issue of density-dependent versus -independent population regulation has led to the conclusion that both factors are important for understanding population fluctuations (1 8). It is, however, less clear how such density-dependent and -independent factors interact with each other in shaping the dynamic pattern of populations across a larger part of the species range. In an attempt to disentangle these issues, we analyze a set of 30-year seasonal (spring and fall) time series from 84 populations of the gray-sided vole [Clethrionomys rufocanus (Sundevall, 1846)] from Hokkaido, Japan (Fig. 1A) (9, 10). To investigate the role of seasonality in the generation of population cycles, we decompose the annual (fall-to-fall) density dependence, as well as the densityindependent stochasticity into their seasonal components. The added detail provided by pinpointing the seasonal arena of population regulation (see supporting information on the PNAS web site, provides us with a better basis for suggesting and evaluating hypotheses about the biological mechanisms that cause density dependence, stochasticity, and population fluctuations. A perennial problem in the study of population dynamics has been the relative lack of extensive and accurate data. In this study we attempt to reach more accurate conclusions than are usually possible by two means. First, we use comparative time-series data from a large number of very similar populations. Second, we address the very substantial problem of biased and imprecise measures of population density through use of a state-space modeling approach (11 13), where time-series observations are related to unobserved states of the real population through a probabilistic observation model accounting for sampling variation. Our study confirms and extends an earlier study of ours (14); whereas the earlier study used only fall data (and as a result could cover the entire island of Hokkaido), the present study used both fall and spring data. The greater detail of the data used in this paper makes a much more detailed analysis of the seasonal structure possible; a picture we assume, based on earlier studies (14), applies to the entire island of Hokkaido. The Study System, Its Seasonal Structure, the Sample Sites, and the Data Hokkaido is the northernmost island ( N, E) of Japan and covers 78,073 km 2. It neighbors the Asian continent and is surrounded by the Sea of Okhotsk, the Pacific Ocean, and the Sea of Japan. A southern warm current prevails in the Sea of Japan along the western shore of Hokkaido, whereas northern cold currents prevail in the Pacific Ocean along the northern and eastern shores. Mountain ridges run north south through the middle of the island; in the southwestern part there is another mountain ridge. Except for the temperate deciduous forests of the southwestern peninsula, most of the natural forests in Hokkaido are regarded as the transition between the temperate and the subarctic zones (15). The dominant tree genera are Abies, Acer, Betula, Picea, and Quercus. Hokkaido represents the easternmost part of the gray-sided voles distribution, which extends as far as Fennoscandia in the west (16, 17). The lengths of the seasons vary from one year to the next. Information on this variability is not available. A related measure, an index of vegetation growth called the warmth index (WI) (see supporting information) is, however, available for a large number of meteorological sites, but not for all trapping sites used in our study. This index is used as a basis for developing a measure of relative length of the winter ( w ), being 1 (relative length of the summer) (for details, see supporting information). The gray-sided vole (C. rufocanus) represents a pest on plantations of larch (Larix leptolepis) and todo-fir (Abies sachalinensis). Since 1954, the Forestry Agency of the Japanese Government has carried out censuses of vole populations for management purposes in forests all over Hokkaido. The forests managed by the Forestry Agency cover 28,400 km 2 [21,500 km 2 natural forests and 6,900 km 2 planted forests (1992 figures)]. These forests were in 1992 managed by 76 district offices, which were further divided into several ranger offices, giving a total of 433 ranger offices. The individual ranger office, which also represented our basic unit of analysis, carried out the censuses; personnel at the ranger offices were regularly trained, including in species identification, for performing the censuses (16). Rodent censuses were carried out twice a year [spring (May or June) and fall (September or October)] on a 0.5-hectare ( m) plot [the latter being used for the annual analysis (14, 17, 19)]. At each plot, 50 snap traps were set in m grids for five or three nights. For each site, one to three separate grids were monitored in fixed preselected natural forests. The census grids were occasionally relocated within the local area. Altogether, 84 time series for both spring and fall covering 30 years ( ) in the central and northernmost part of Hokkaido (the Asahikawa Regional Office, Forestry Agency of the Japa- This paper was submitted directly (Track II) to the PNAS office. To whom correspondence should be addressed. n.c.stenseth@bio.uio.no by The National Academy of Sciences of the USA PNAS September 30, 2003 vol. 100 no cgi doi pnas

2 Fig. 1. The study area (A) and the dynamics of the gray-sided voles (B and C). (A) Hokkaido is the northernmost island ( N, E) of Japan and covers 78,073 km 2. Average relative length of winter is (SD 0.009) in the western populations of Group 1 (red), (SD 0.010) in the eastern populations of Group 2 (blue), and (SD 0.014) in the southern and mountainous populations of Group 5 (yellow). (B) Population dynamics are investigated by autoregressive coefficients. Below the semicircle the dynamics are cyclic with shorter periods the stronger the density dependence (28). An observed annual scatter-plot of the autoregressive coefficients for 84 Hokkaido time series is shown. To the right of the triangle, the annual density dependencies ( i ) are plotted against the relative length of the winter. Ordinary regressions are shown as lines. (C) Predicted change for an increase of 0.1 in w from the estimated position in the parameter space, using the estimated density dependencies from the pooled analyses (Table 1; see text). For each area (the northern and southern part of Groups 1, 2, and 5), seasonal estimates are divided by average season length to estimate per-time-unit seasonal density dependencies. The direct and delayed annual density dependencies are generally predicted to become stronger the longer the winter. To the right of the triangle, the weighted regressions of annual density dependencies on relative length of winter are given; the weights [based on (WI ŴI) 2 ; see text] for Group 1, Group 2, and the more mountainous Group 5 are 0.96, 0.85, and 0.19, respectively. ECOLOGY nese Government) are used; this region is the one for which cycles occur (10) and for which we have sufficient data. We have grouped these populations according to topographic characteristics in our previous studies (19). Trapping efforts were represented by multiplying the number of traps by the number of nights and the number of census plots (grids). For instance, when three-night censuses were carried out at three grids in a ranger office, the trapping efforts by the ranger office at the census was 450 ( ). The spring-to-fall season (summer) corresponds to the period of population growth (because of reproduction), whereas the fall-to-spring season (winter) corresponds to the period of population decline [winter reproduction is negligible (refs. 17 and 20 25; see supporting information)]. Reliable data are available on food habits of voles (16, 20). C. rufocanus of Hokkaido prefer green plants as food (20). Leaves of bamboo grasses (Sasa senanesis, Sasa kurilensis, and Sasa nipponica) constitute a large part of the winter diet, and their shoots occupy a considerable part of the summer diet (although summer diet is much more varied than winter diet). Thus, bamboo leaves influence survival during the winter and bamboo shoots influence reproduction during the summer (most likely primarily during the spring because other plants do not grow as well at this time of the year). Bamboo grasses may be particularly important during winter when other plants wither, even though their leaves do seem to be nutritionally poor during the winter. Even though there are many predators such as snakes, birds, and mammals in Hokkaido (21), their activity under the snow cover is often restricted. An exception is the least weasel Mustela nivalis, which is a highly efficient predator under the snow cover. Stenseth et al. PNAS September 30, 2003 vol. 100 no

3 The gray-sided vole is the most common rodent species in Hokkaido (20, 25). Three other microtine and murine rodent species are recorded as an integral part of the census: Clethrionomys rutilus (Pallas, 1779), Apodemus speciosus (Temminck, 1844), and Apodemus argenteus (Temminck, 1844). In addition, Clethrionomys rex Imaizumi, 1971, Apodemus peninsulae (Thomas, 1907), and shrews (Sorex spp.) are occasionally caught. Personnel of the Forestry Agency identified the specimens. Clethrionomys is easily distinguished from other genera. Distinguishing C. rufocanus from its congeners may, however, be difficult, although the abundance of C. rufocanus and C. rutilus was separately reported. The numbers for the other Clethrionomys species is, however, very low in Hokkaido (20, 26), reducing the impact of any misclassification. The Ecological Model Let spring and fall abundances in year t (log-transformed and centered around their means) be denoted x t and y t, respectively. The seasonal models for the net winter population growth rate, R wt ( x t y t 1 ), and the net summer population growth rate, R st ( y t x t ), are given as (27) R wt x t y t 1 a w1 y t 1 a w2 x t 1 a w3 y t 2 a w4 x t 2 wt R st y t x t a s1 x t a s2 y t 1 a s3 x t 1 a s4 y t 2 st, [1a] [1b] where wt is the time-independent process noise during the winter [ wt N(0, w 2 )] and st is the corresponding summer component [ st N(0, s 2 )]. The parameters a w and a s define the seasonal density-dependent structure for the winter and summer growth rates, respectively. The annual net growth rate is obtained by rewriting Eqs. 1 and 2 as a model in y only (27): R t y t y t 1 a w1 a s1 a w2 a s2 a w1 a s1 y t 1 a w3 a s3 a w4 a s4 a w1 a s3 a s1 a w3 a w2 a s2 y t 2 a w3 a s3 a w2 a s4 a s2 a w4 y t 3 a w4 a s4 y t 4 t, [2] where t is the annual noise term (see supporting information). The two last autoregressive terms in Eq. 2 are typically negligible (as is also the case in this study), rendering a second-order process an appropriate approximation (6, 8, 25): R t y t y t 1 1 y t 1 2 y t 2 t, [3] where t is the annual noise term [ t N(0, 2 )] and the parameters 1 and 2 define the direct and delayed annual density dependencies; for this system to correspond to a stationary process, the two autoregressive parameters (1 1 ) and 2 must fall within the triangle defined by ( 2, 1), (0, 1), and (2, 1) (28). The link between the annual and the seasonal parameters are given as 1 a w1 a s1 a w2 a s2 a w1 a s1 and 2 a w3 a s3 a w4 a a4 a w1 a s3 a s1 a w3 a w2 a s2. Assuming a white-noise term (i.e., no delayed noise in the noise process of Eq. 3), 2, the annual noise variance may be expressed by 2 given as a function f( s 2, w 2 ) (see supporting information). Linking the Data to the Ecological Model Sampling variance is known to bias the estimates of, and invalidate tests for, density dependence (5, 29); the degree of direct density dependence is typically overestimated when ignoring sampling variance. Despite this, sampling variance is generally ignored (25) in empirical studies, partly because of availability of data and methodological shortcomings. Sampling Fig. 2. Kernel density estimates of the posterior distribution of parameters for the seasonal processes (Eqs. 1a and 1b) are shown for two populations. (A) Takinoshita (44 11 N, E). (B) Mitsumata (44 04 N, E). Posterior distributions of parameters from the corresponding observation model (see supporting information) are not shown. variance may be accounted for by adopting a state-space modeling approach where time-series observations are related to unobserved states of the real population through a stochastic observation model (11 13). Let the number of voles caught during the spring and fall session at a given site in year t be given by z s,t and z f,t, respectively. Similarly, let the trapping effort at a given site in the spring and the fall of year t be given by T s,t and T f,t, respectively. Our core data consist of the total number of gray-sided voles caught (z s,t and z f,t ) and the corresponding trapping effort (T s,t and T f,t ). The expected number of voles caught during a trapping session will be related to the abundance of voles at the site (y t and x t, log-abundances scaled around their respective mean values, y and x ) and on the trapping effort. As a first approximation, the ecological process model as described by Eqs. 1a and 1b or Eq. 3 is assumed to be linked to the observed number of voles caught through a Poisson model. For the fall samples this will then be given as: P(z f,t z y t ) exp( f,t ) z f,t /z!, where the Poisson mean (assumed to be proportional to the trapping effort and the population abundance) is defined as f,t qt f,t exp( y y f,t ). The trapping effort T f,t is a known parameter. The parameter represents the mean of the logtransformed true population abundances. Because of the nature of the data, the average level of the true population ( ) is confounded by the trappability (i.e., the average probability for any present individual to be trapped) in the given environment q. However, although cannot be directly estimated, we can estimate the product of the average abundance level and the trappability, defined as exp( ) q exp( ). This yields the following reparameterization (for the fall; and equivalently for the spring): f,t T f,t exp( f y f,t ). By using the software package BUGS (Bayesian inference using Gibbs sampling; ref. 30), a Bayesian framework was adopted to estimate parameters in the state-space model. The resulting posterior distributions of parameters from the seasonal model are shown for two typical populations (Fig. 2). Only vague (i.e., essentially flat) prior distributions were used. For further details, see supporting information. Posterior means are used as parameter estimates and used in further analyses. Seasonal parameter cgi doi pnas Stenseth et al.

4 Table 1. Strength of density-dependent processes as estimated by the mean posterior distribution obtained from pooled seasonal (Eqs. 1a and 1b) and annual (Eq. 3) analyses (SD of posterior distribution in parentheses) Dynamics Group 1 (south; n 15) Group 1 (north; n 15) Group 2 (south; n 14) Group 2 (north; n 16) Group 5 (south; n 12) Group 5 (north; n 12) Winter a w (0.08) 0.99 (0.09) 0.98 (0.08) 0.84 (0.08) 0.72 (0.10) 0.82 (0.08) a w (0.10) 0.25 (0.09) 0.26 (0.11) 0.09 (0.09) 0.07 (0.12) 0.03 (0.13) a w (0.08) 0.09 (0.08) 0.22 (0.07) 0.23 (0.08) 0.09 (0.09) 0.08 (0.08) a w (0.09) 0.05 (0.08) 0.16 (0.09) 0.07 (0.10) 0.32 (0.11) 0.27 (0.11) Summer a s (0.06) 0.47 (0.05) 0.25 (0.07) 0.25 (0.06) 0.24 (0.07) 0.21 (0.08) a s (0.08) 0.20 (0.07) 0.07 (0.09) 0.21 (0.07) 0.16 (0.10) 0.15 (0.08) a s (0.08) 0.22 (0.06) 0.27 (0.10) 0.03 (0.08) 0.30 (0.10) 0.07 (0.11) a s (0.05) 0.15 (0.06) 0.04 (0.06) 0.14 (0.05) 0.04 (0.06) 0.08 (0.06) Annual Fall-to-fall (0.06) 0.86 (0.06) 1.00 (0.06) 1.10 (0.06) 0.83 (0.07) 1.03 (0.07) (0.06) 0.07 (0.06) 0.22 (0.06) 0.27 (0.06) 0.28 (0.06) 0.11 (0.07) Spring-to-spring (0.07) 0.75 (0.07) 0.74 (0.08) 0.85 (0.07) 0.88 (0.07) 0.84 (0.09) (0.07) 0.09 (0.07) 0.29 (0.07) 0.14 (0.07) 0.42 (0.08) 0.28 (0.09) A common dynamic structure is assumed within each of six areas (the northern and southern part of Groups 1, 2 and 5; see Fig. 1A). The state-space model is fitted simultaneously to each site within an area, restricting the parameters of density dependence to be common for each of these n sites, allowing the process noise to vary. (Estimates in bold indicate strong evidence for the corresponding density-dependent process. Group 1 tends to exhibit noncyclic dynamics, whereas populations in the other groups are cyclic.) estimates being judged as statistically appropriate were obtained from only 74 sites (see supporting information). All 84 time series are, however, used for obtaining the main results summarized in Table 1, as well as those depicted in Figs. 1 B and C. Notice that the annual density dependencies given by Eq. 3 are both estimated to be stronger when sampling error is not accounted for (Fig. 3A). Results The obtained parameter estimates (Table 1 and Figs. 1B and 2) suggest that a fourth-order process (with the existence of some long lags) underlies the dynamics in at least one of the seasons (see Eqs. 1 and 2). The direct fall-to-spring effect (a w1 ) is clearly stronger than the direct spring-to-fall effect (a s1 ), suggesting that winter processes are more important than summer processes in determining the annual direct density dependence. These, as well as fall-to-fall effects (a s2 ) and spring-to-spring effects (a w2 ) all contribute to the annual direct density dependence (see Eq. 2 and supporting information), a result consistent with earlier findings both for Hokkaido (see refs. 25, 31, and 32) and the same species in northern Finland (refs. 27 and 33; also see refs ). As seen from Fig. 3B, we are able to predict both the direct and delayed annual density dependencies from the seasonal components. Assuming the model given by Eq. 2, the seasonal components explain 70% and 79% of the (deterministic) among-population variability in the direct ( 1 ) and delayed ( 2 ) annual density dependencies, respectively. A major part of the among-populations variation in the level of densityindependent stochasticity (i.e., among-population variation in ) is, however, accounted for by variation in the level of summer stochasticity (i.e., among-population variation in s ; Fig. 4). Variation in degree of summer stochasticity accounts for 32.0% of the total variation in annual stochasticity, whereas the winter stochasticity accounts for 11.7%. If summer stochasticity already is accounted for, the winter stochasticity only accounts for 1.2% of the remaining variation. In the multiple regression of annual stochasticity on the summer and winter stochasticities, the effect of the winter stochasticity is not statistically significantly different from zero. It should finally be noticed that the fall-to-fall annual densitydependent structure corresponds to the spring-to-spring densitydependent structure (see supporting information). This close correspondence clearly strengthens the validity of the obtained results. Discussion The Seasonal Structure of the Population Dynamics. The summer is the main reproductive season for the gray-sided vole; essentially no winter reproduction is observed (see supporting information and refs. 16 and 20). Hence, strong, direct density dependence during the winter must necessarily involve winter survival. Fig. 3. Decomposing the annual density-dependent structure from the seasonal density-dependent structure. Autoregressive coefficients estimated from the annual ecological model (Eq. 3) incorporating an observation model (see text) are given by 1 (Left) and 2 (Right). (A) The relation between AR(2) estimates as found by ordinary autoregressive analyses ignoring sampling variance and the corresponding estimates incorporating sampling variance [Left, (AR) 2 vs. 1 ; Right, (AR) 2 vs. 2 ]. As expected, the direct density dependence is overestimated when sampling variance is ignored [on average overestimated with 0.23 (SD 0.14) compared with the state-space model]. As can be seen, this is also the case for the delayed density dependence [on average overestimated with 0.16 (SD 0.15) compared with the state-space model]. (B) The relation between 1, the first-order coefficients calculated on the basis of seasonal analyses incorporating observation models, and the directly estimated 1 (R , F 1, , P 0.001); Right depicts the equivalent relationship for the second-order coefficients 2 (as given in the text) and 2 (R , F 1, , P 0.001). The direct density dependence calculated on the basis of seasonal analyses is on average underestimated with 0.15 (SD 0.32) compared with 1. The delayed density dependence calculated on the basis of seasonal analyses is on average overestimated with 0.13 (SD 0.23) compared with 2. ECOLOGY Stenseth et al. PNAS September 30, 2003 vol. 100 no

5 Fig. 4. Relating the annual-noise process to its seasonal components. (A) The annual-noise SD ( ) as estimated from Eq. 3 plotted against the annual noise ( ) calculated on the basis of seasonal analyses (see supporting information). (B) The annual-noise SD ( ) as a function of its main seasonal noise component, the summer SD ( s ). Mechanistically, such density dependence may be generated through a functional response of predators or through the limited amount of food (such as bamboo; Sasa spp.) produced during the preceding summer. On the other hand, the strong influence of summer stochasticity (Fig. 4) may stem from a variety of summer events (such as plant production and generalist predation), all of which may be sensitive to erratic temporal and spatial variation in climatic conditions. Notice, however, that when interpreting the statistical parameter a w1, given a mechanistic process model, this will typically involve direct density dependence of the vole during the winter, in addition to other processes from other seasons. Hence, biological interpretation of these parameters is difficult in the absence of a detailed mechanistic process model. The same applies to the other seasonal statistical parameters. If we are to explain the observed phenomenological lagstructure, involving four lags (Table 1), in terms of a process model that only allow interactions within a season, we would have to postulate four underlying variables. One of these must be the density of voles. The character of the remaining three variables is unclear. We suggest three possibilities: (i) voles, winter food (bamboo grass), and a specialist predator (the least weasel, M. nivalis) with two age classes (the two age classes of the predator are reasonable on this time scale; weasels require one year to mature); (ii) voles, bamboo grass, specialist predator, and additional members of the small-mammal community that either compete directly with the gray-sided vole or induce apparent competition (39) through some common predator; and, finally, (iii) an interacting web consisting of voles, food plants (e.g., bamboo grass), weasels, and generalist predators. A range of other hypotheses (involving, e.g., intrinsic properties of the vole populations) are certainly possible; however, the three suggested hypotheses are more plausible based on general natural-history insights (see ref. 40 for details). We know, for instance, that the vole specialist (41 43) least weasel is found all across the island (41, 44) and that it may have a particularly strong effect during the winter, because it is able to hunt in the subnivean space where the voles are protected from other predators. Bamboo grass is common all across Hokkaido (45); their new shoots in the spring are known to affect reproduction during the summer, and leaves of bamboo-grass (laid down by the snow) are known to affect survival during the winter. It is difficult to explore these alternatives within a time-series setting, because the number of possible interactions in a four-variable model leads to a large number of parameters. Thus, to further test such process hypotheses, one would need to search for more direct evidence of interacting variables (or perform appropriate experiments; see ref. 40 for further discussion). The striking difference between population regulation during summer and winter suggests that seasonality may hold a clue to understanding geographic differences in population dynamics. Notice, for instance, that the western populations (Group 1; see Fig. 1A) tend to fall in the upper right part of the triangle of Fig. 1B (corresponding to stable populations), whereas eastern and the mountainous populations tend to fall further down toward the central part of the triangle (corresponding to periodically fluctuating populations). Because of warm currents along the western coast of Hokkaido, the Group 1 populations are exposed to the longer summers (17). On the other hand, colder currents along the eastern coast expose the Group 2 populations to longer winters (17). The more mountainous Group 5 populations are also exposed to long winters (17). On this basis, it is reasonable to hypothesize that the observed systematic differences in the population dynamics of these groups are influenced by differences in season length, a view that is substantiated by observing the relation between the annual density dependencies and the length of the winter, particularly so when using the pooled estimates (Fig. 1). What we in effect might see is a detailed example of more cyclic dynamics being observed toward the north (18), where the summer season is fairly short. Incorporating the Length of the Seasons into the Population-Dynamics Model. To further investigate the hypothesis that season length is essential for determining the dynamics, we reparameterized the seasonal model to reveal the relative length of the winter as a parameter w. This allows separation of the effects of season length per se from the effect of density dependence per time within the season. A seasonal model will then be given as x t y t 1 a* w1 y t 1 a* w2 x t 1 a* w3 y t 2 a* w4 x t 2 w [4a] y t x t a* s1 x t a* s2 y t 1 a* s3 x t 1 a* s4 y t 2 1 w, [4b] where seasonal density dependencies a wi and a si in Eqs. 1a and 1b now are given as a* wi w and a* si (1 w ), respectively. The annual direct ( 1 ) and delayed ( 2 ) density dependencies expressed in terms of seasonal components are and 1 a* w1 a* s1 1 w a* w2 w a* s2 1 w a* w1 a* s1 w 1 w 2 a* w3 w a* s3 1 w a* w4 w a* s4 1 w a* w1 a* s3 w 1 w a* s1 a* w3 w 1 w a* w2 a* s2 w 1 w. The linear effects of winter length on 1 and 2 are then and 1 w a* w1 a* w2 a* s1 a* s2 a* w1 a* s1 1 2 w 2 w a* w3 a* w4 a* s3 a* s4 a* w1 a* s3 a* s1 a* w3 a* w2 a* s2 1 2 w. Given that density dependencies are stronger (i.e., more negative) during the winter than the summer, and that the winter is Our estimates for the relative length of the winter ( w) provide the best prediction [(WI ŴI) 2 ] for Groups 1 and 2. To observe that Group 5 (the more mountainous group) is more difficult to estimate seems reasonable because topographic effects in mountainous regions (north/south sloping) are likely to be more pronounced cgi doi pnas Stenseth et al.

6 the longer season, 1 / w and 2 / w will quite often be negative. Using the pooled estimates reported in Table 1, the annual direct and delayed density dependencies typically become smaller (i.e., the strength increases) as winter becomes longer (i.e., d 1 /d w and d 2 /d w both become negative) (Fig. 1C). On this basis, we predict that noncyclic dynamics are more likely in areas with short winters and that periodic cycles are more likely in areas with long winters, just as seen empirically (ref. 9; also see Fig. 1B). Although the density-dependent structure is best explained by whether the population belongs to Group 1, 2, or 5 (Fig. 1B), the length of winter ( w ) seems to be a good predictor of the density-dependent structure. This dependency of the relative winter length can, however, only be substantiated when using the (better) pooled estimates reported in Table 1, not when using the estimates for the individual populations. Conclusion Altogether we have shown that winter length has a clear effect on the population dynamics of Hokkaido voles. Essentially, increasing winter length may be seen as a bifurcation parameter shifting the dynamics from an intrinsically stable regime with irregular fluctuations (generated by density-independent mechanisms) to larger-amplitude, periodic cycles influenced by density-dependent mechanisms. As the required delayed density-dependent structure is widespread in voles (8), such a seasonbased bifurcation process may apply quite generally. The underlying mechanisms generating this density-dependent structure are the subject of contention, and may differ from one system to another; however, the relative length of the seasons will be a generic bifurcation parameter irrespective of the underlying mechanism. Hence, we suggest that seasonality represent a key for unlocking the mystery of population cycles. We thank Kung-Sik Chan for many very helpful discussions during the formulation of the model and the linking of the seasonal and annual processes. We also thank Torbjørn Ergon, Tero Klemola and Ole Chr. Lingjærde. Comments by Tim Coulson and two additional anonymous reviewers were greatly appreciated. We are indebted to the Forestry Agency of Japanese Government for providing the data analyzed in this paper. We acknowledge the support we, over the years, have received from Japan Society for the Promotion of Science, the Forestry and Forest Products Research Institute of Japan, the Norwegian Research Council (NFR/NT), the Center for Advanced Studies of the Norwegian Academy of Science and Letters, and the University of Oslo. This study was also supported in part by Grant-in-Aid from the Ministry of Education, Science, and Culture of Japan (to T.S.). 1. Nicholson, A. J. (1933) J. Anim. Ecol., Suppl. 2, Andrewartha, H. G. & Birch, L. C. 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7 Supporting Text The Bayesian Approach. A graphical representation of the quantities in the annual and seasonal model, the corresponding observation models, and their relationships are given in Fig. 6. A complete Bayesian model consists of the joint prior distribution for all unobservables, here (for the annual model) α 1, α 2, σ, λ f, and γ f and the unknown states y 1, y 2,..., y N, and the joint distribution of the observables, here the abundance data (z f,1, z f,2,..., z f,n ) and the trapping effort (T f,1, T f,2,..., T f,n ). Bayesian inference is then based on the posterior distribution of the unobservables given the data. By conditioning on the data, the posterior distribution (i.e., the conditional probability of the unobservable quantities of interest, given the observed data) is calculated [by successive application of Bayes theorem (1)]. The posterior distribution can usually not be obtained analytically but can be computed by using Markov Chain Monte Carlo methods, for instance, Gibbs sampling (2). Bayesian inference is readily applicable through BUGS (Bayesian inference using Gibbs sampling; To fully specify our model, a prior distribution has to be defined for all unobservable parameters that are not directly conditioned on other parameters or observed data (for the annual model these are: α 1, α 2, σ, and γ f ). Vague (i.e., essentially flat) prior distributions were used [α 1 ~ N(0,100), α 2 ~ N(0,100), γ f ~ N(0,100), and 1/σ 2 ~ Gamma(0.001, 0.001)], meaning that the prior tells us little about the parameters relative to what is learned from the sample. For both the annual and the bivariate seasonal model we performed iterations of the Gibbs sampler, using WINBUGS 0.6, after a burn-in of 10,000 iterations. This was done for each of the 84 locations for which both spring and fall data were available. Convergence. Autocorrelations within the Markov chains seemed to be reasonably low [indicating that the mixing of the Gibbs sampler is not too slow (3)]. Convergence was also assessed through the runs of multiple chains and Gelman Rubin convergence statistics (4). The annual model converged for all 84 sites. For the seasonal model, data from 74 of the total of 84 sites were judged to converge appropriately. The seasonal model failed

8 to converge for 10 sites (assf6, assf7, assf9, assf10, aassf28, assf37, assf59, assf68, assf74, and assf78) because of excessive auto- and cross-correlations for some of the Markov chains. Only the remaining 74 of 84 sites were used in analyses of the seasonal parameters. Including all seasonal estimates in the analyses will, we are convinced, not change the conclusions of the paper; indeed, including only 74 of the 84 is a conservative restriction of our total data set. Notice, furthermore, that data from all 84 sites are used in our main results summarized in Table 1 and depicted in Fig. 1 B and C. The Parameter Estimates (BUGS estimates). Parameter estimates together with credible intervals (the Bayesian equivalent to confidence intervals) are given in Tables 3 and 4 for the annual and seasonal models, respectively (parameters of the corresponding observation model are not shown). The Parameter Estimates (Standard Autoregressive Modeling Ignoring Sample Variance). Parameter estimates for an order-two annual model are given in Table 5. Deducing the Annual Variance from the Seasonal Variances. We have the following model: R wt = x t y t-1 = a w1 y t-1 + a w2 x t-1 + a w3 y t-2 + a w4 x t-2 + ε wt R st = y t x t = a s1 x t + a s2 y t-1 + a s3 x t-1 + a s4 y t-2 + ε st, [S1] [S2] where ε wt is a process noise during the winter (being normally distributed with mean 0 and variance σ 2 w ) and ε st is a process noise during the summer (being normally distributed with mean 0 and variance σ 2 s ). The parameters a w and a s define the seasonal density-dependent structure. Together Eqs. S1 and S2 define the annual net growth rate, defined for instance as the fall-to-fall-growth net rate, R t = y t y t-1. We can now rewrite Eqs. S1 and S2 as a model in y only: R t = y t y t-1

9 = (a w1 + a s1 + a w2 + a s2 + a w1 a s1 )y t-1 + (a w3 + a s3 + a w4 + a s4 + a w1 a s3 + a s1 a w3 a w2 a s2 )y t-2 + (a w3 a s3 a w2 a s4 a s2 a w4 )y t-3 a w4 a s4 y t-4 + η t,. [S3] where η t = ε st + (1 + a s1 )ε wt a w2 ε st-1 + a s3 ε wt-1 a w4 ε st-2 is no longer a white noise process (even though its components are). The covariance structure of η t, assuming no covariation between summer and winter noise, is defined by (5) c 0 = σ s 2 (1 + a w2 2 + a w4 2 ) + σ w 2 [(1 + a s1 ) 2 + a s3 2 ] c 1 = a w2 σ s 2 + (1 + a s1 )a s3 σ w 2 + a w2 a w4 σ s 2 c 2 = a w4 σ s 2, [S4] [S5] [S6] where, c i denotes the covariance of η t components having a time-lag difference of i. More appropriately, the expression for η t may be written as η t = η t + β 1 η t-1 + β 2 η t-2 where η t is normally distributed with N(0, σ 2 η ). The two expressions for η t should have an equal covariance structure, and as a result, σ 2 η may be expressed, implicitly and assuming no covariation between the variance during the winter and summer, as a function f(σ 2 s, σ 2 w ) given as follows: σ η 2 + σ η 2 c 1 2 /(σ η 2 + c 2 ) 2 + c 2 2 /σ η 2 c 0 = 0 [S7] The solutions for each of the time series are given in Table 6. This has several solutions; whenever there is ambiguity with respect to solutions, we give all solutions in Table 6. Additional Methodological Aspects. Simulated data were used to investigate the bias observed in the one-to-one line of Fig. 3B. Data were simulated from the seasonal bivariate model (Eqs. S1 and S2) by using the parameter estimates obtained from previous fitting of this model to data from site assf12 or assf36 (two typical populations of group 2). The simulated data represent true abundance values and therefore the sampling process was ignored. Each of 30 data sets (15 data sets were simulated for each

10 of the two sets with estimates) was then fitted to the seasonal bivariate model (Eqs. S1 and S2) and the annual second-order model (Eq. 3) by using BUGS. The annual densitydependent coefficients predicted from the seasonal model were compared with the corresponding coefficients directly estimated from the annual model (Fig. 7). As can be seen, there are deviations from the one-to-one expectations. A similar pattern was found when using an annual model of order four instead of order two when estimating the first and second coefficients (although the bias for the first coefficient no longer became significant; results not shown). Our results suggest that the similar bias observed when using real data (Fig. 3B) was not due to methodological aspects, such as state-space modeling (the incorporation of a sampling process), ignoring the predicted order four in the annual model or the noise term in the bivariate seasonal model not being white. Estimating the Relative Length of Winter. Earlier, Stenseth et al. (6) used a proxy for the length of the breeding season, τ, assumed to be directly related to the warmth index, WI, given as Σ(T 5), where the sum is taken over months for which the average monthly temperature, T, is equal to or above 5 C (7, 8). By so doing, Stenseth et al. demonstrated that τ is closely related to the geographic scores. This relation is further improved when we assume that the winter length (τ w ; τ w = 1 τ) is related to the WI through a logit function: τ w = blogit(cwi). WI may furthermore be approximated (and estimated) by using geographic coordinates [i.e., position in the south north and west east direction (denoted g sn and g we, respectively), and altitude above sea level (denoted H)]. (The latter approximation is necessary, because the meteorological stations where the WI is measured do not correspond to the sampling stations for the voles. However, all sampling stations have geographical coordinates and altitude.) The geographic coordinates g we and g sn are ordinary longitude and latitude, respectively, given in minutes [defining ' E and 41 00' N as the origin (0,0)]. However, because 1 min of latitude does not correspond to the same geographic distance as 1 min of longitude, the values of g we have to be standardized. In Hokkaido, 60 min of longitude corresponds to km, whereas 60 min of latitude is km. Thus, the original values of g we are multiplied by When regressing WI on geographic coordinates and altitude (h = log e (H + 1)), using Akaike s Information Criterion (AIC) as a selection criterion, we

11 obtain the following model: ŴI = g we g sn h g we g sn g sn h. We assume that the winter length (τ w ) is related to the warmth index through the relation τ w = blogit(cwi), where the parameters b and c are determined by assuming the relative length of the winter to be 7/12 where the WI is the lowest and 4/12 where the WI is the highest. Furthermore, we assume that WI is appropriately modeled by using geographic coordinates [i.e., position in the south north and west east direction (denoted g sn and g we respectively), and altitude above sea level (denoted H)]. The relationship between WI and g sn, g we, and h [where h = log e (H)] is determined by linear regression; the best model obtained, using AIC (9), starting with the most general model WI = d 0 + d 1 g we + d 2 g sn + d 3 h + d 4 g we g sn + d 5 g sn h + d 6 g we h. On this basis, we arrive at the following expression for the relative length of winter: τ w = blogit(p 0 + p 1 g we + p 2 g sn + p 3 h + p 4 g we g sn + p 5 g sn h) + error, where p i = cd i. We obtained the following model τ w = 3.63logit( g we g sn h g we g sn g sn h) (see ref. 10 for further information). We have performed robustness testing by varying the assumed length of the winters (7/12 and 4/12, respectively); as can be seen, our results are robust against this variation. To estimate the values of b and c in τ w = blogit(cŵi), we assume that the longest summer is 8 months and is found at sea level in the southwestern part of Hokkaido, and that the shortest one is 5 months and found in the northeastern part of Hokkaido (10-12). Two meteorological stations are selected in each part of the island (Fig. 8), all having WI values relatively well predicted by the model described above [for selected stations: (residual) 2 < 12; overall mean (residual) 2 = 24]. Combining both selected stations in the southwest with both stations in the northeast produces four sets of equations: τ wswi = blogit(cwi SWi ) and τ wnei = blogit(cwi NEi ), where i = 1, 2. The values of b and c are then determined by using the function uniroot in S-PLUS 2000 (13) (we obtain the estimates b = 3.63 and c = 0.04). The model for ŴI is subsequently entered into the resulting expression, producing the following model of the relative length of winter τ w :

12 τ w = 3.63logit( g we g sn h g we g sn g sn h) + error Fig. 9A shows the curvilinear relation between the predicted τ w and the predicted ŴI. Fig. 9B shows the relation between the predicted τ w and the predicted ŴI, using observed WI values to calculate τ w. Fig. 10 summarizes sensitivity analyses varying the maximum and minimum winter lengths. The panel shown in the main paper is highlighted in the frame. As can be seen, the overall pattern is rather robust to deviations form our assumed maximum and minimum length of the winter. Estimating the Annual Model on the Basis of the Spring-to-Spring Dynamics. Fig. 11 show the pattern depicted in Fig. 1C (also see Table 1). As can be seen, the overall same pattern emerges suggesting that our results in that respect are robust. 1. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (1995) Bayesian Data Analysis (Chapman & Hall, London). 2. Gilks, W. R., Richardson, S. & Spiegelhalter, D. J. (1996) Markov Chain Monte Carlo in Practice (Chapman & Hall, London). 3. Raftery, A. L. & Lewis, S. (1992) in Bayesian Statistics 4, eds. Bernardo, J. M., Berger, J. O., Dawid, A. P. & Smith, A. F. M. (Oxford Univ. Press, Oxford), pp Spiegelhalter, D. J., Thomas, A., Best, N. & Gilks, W. R. (1999) WINBUGS 1.2 User Manual (MRC Biostatistics Unit, Institute of Public Health, Cambridge, U.K.). 5. Priestley, M. B. (1981) Spectral Analysis and Time Series (Academic, London). 6. Stenseth, N. C., Bjørnstad, O. N. & Saitoh, T. (1998) Res. Pop. Ecol. 40, Kira, T. (1949) Seibutsu Kagaku (Biological Science) 1, 193 (Japanese). 8. Kira, T. (1971) Nature from the Ecological Viewpoint (Kawaide-shobo, Tokyo) (Japanese). 9. Burnham, K. P. & Anderson, D. R. (1998) Model Selection and Inference: A Practical Information-Theoretic Approach (Springer, New York).

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