Statistical Modeling.

Data were log-transformed to stabilize the variance (25). This transformation is biologically suitable due to the multiplicative nature of the population dynamics process involving birth and death processes (26). The log-transformed series {X_t} were scaled to have zero mean and variance equal to one.

In our search for parsimonious models for the time series under study, we start with a general model and simplify it as far as statistically permissible. Throughout we rely on recent developments in time series analysis (refs. 21, 23, 27–32; see references therein for details).

We follow the long standing conjecture (24, 33–37) that the transformed series may adequately be modeled in delay coordinates as a general autoregressive model of order [or dimension; sensu Royama (38)] d; that is, X_t = F_d(X_t−1, X_t−2, . . . , X_t−d) + ɛ_t. We assume that {ɛ_t} is time- and state-independent white noise (30). As suggested by Cheng and Tong (27), the function F_d determining the laws of the population dynamics can be estimated for a given choice of d with minimal assumptions, using nonparametric regression. Provided F_d is continuous, Cheng and Tong (28, 29) showed that the sample size requirement is exponential in d for estimation of the functional form of F_d but is at most quadratic in d for the estimation of the dimension. To find the appropriate dimension for the series (and their underlying process), we employ cross-validation (39–42). That is, for each d considered (d = 1, . . . , 4), we estimate F_d based on all but one data point, which is subsequently predicted. This is repeated for each data point. The sum of squared out-of-sample prediction errors [the cross-validation (CV) value; a rough estimate of the percentage noise is used as a measure of the model suitability. The d corresponding to the autoregressive model minimizing the CV value provide a consistent estimate of d, under minimal assumptions (25–28). To estimate the functions F_d, we employ two different nonparametric regression methods: (i) the Nadaraya–Watson kernel regression (21, 27, 28) and (ii) a locally linear regression (31, 32, 43). We employ a Gaussian product kernel for both. The consensus arising (Table 1) is that the snowshoe hare may parsimoniously be described by a model of dimension three (note that H2 appears cleaner in terms of predictability, based on its CV value, than H1), whereas the series for the lynx is more likely to be governed by a process of dimension two.§ The advantage of employing flexible models for F_d is that conclusions reached are not biased by any parametric prejudices. The disadvantage is that such general models have a high number of degrees of freedom (ref. 44, chapter 3) causing loss in precision (high statistical variance). These models may also be victims of the curse of dimensionality (45). Indeed, the statistical uncertainty associated with any estimate may be paramount when the number of data points is relatively low. The high degree of consistency across the different data sets (Table 1) is, however, encouraging.

To increase precision we investigate the possibilities of imposing some restrictive assumptions. One such simplifying assumption involve assuming that there is no interaction, on the logarithmic scale, between years (44, 46): 1 On the basis of the test for additivity by Chen et al. (23), this restrictive assumption appears permissible for 15 of 16 time series (Table 1); the rejection level of the deviating series (L12) is not far from the nominal 0.05 level. Additivity may, of course, be an artifact of low power due to the small sample size. However, the conclusion is biologically reasonable (see below) and the sample size is, after, all not particularly low. Estimating the optimal dimension by using CV of the additive model (Table 1) yields conclusions consistent with the analyses based on the more general models. Note, specifically that the additive models are usually as good as the more complicated models for prediction (see CV values in Table 1).

Since we have demonstrated approximate additivity, a linear model (on the logarithmic scale) represents the next level of simplification: 2 Two tests of nonlinearity were employed: (i) The nonparametric test by Hjellvik and Tjøstheim (21) and (ii) the Tukey one-degree-of-freedom test by Tsay (22). The null hypothesis of linearity is rejected in 5 of the 14 lynx series and is close to a nominal 0.05 level for one of the hare series (H1). This (31% at a nominal 0.05 level and 50% at a 0.1 level) is more than expected under linearity (assuming the different series to be independent). However, it is less than one may expect under extreme nonlinearity (see ref. 47). The estimates of d based on cross-validation of the linear models are generally higher than that based on the nonlinear models, as expected for nonlinear systems (48). Thus, it appears that the hare series may be represented an additive process on a logarithmic scale of order around three, while the order for the lynx series is around two. The lynx is certainly a nonlinear process (as previously concluded for series L3; see, for example, refs. 24 and 49). The same seems to be the case for the hare. Note, however, that the nonlinear function appear monotonic and not highly curved (Fig. 3).

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Figure 3

The f_i functions (as given by Eq. 1; i = 1, . . . , d) for the hare series (H2) with d = 3 (A) and the lynx series (L10) with d = 2 (B). See the main text for interpretation. All f_i functions have been smoothed by natural cubic splines with 2 degrees of freedom using the backfitting algorithm implemented in the function generalized additive models in s-plus (50). Estimated confidence intervals and residuals are also shown.

Despite significant nonlinearities, the monotoniety of the f_i functions justifies the linear autoregressive model (Eq. 2) as a useful average of the state dependent (functional) autoregressive coefficients of the dynamics, because these may be interpreted as coefficients of statistical density dependence (13, 47, 51).

Both the log-linear and the log-additive model indicate negative direct density dependence for the hare (α₁-1; see refs. 13 and 51) ranging from −0.38 to −0.05 with a mean of −0.23. There is essentially no dependency on the second lag (α₂ ranging from 0.06 to 0.11 with a mean of 0.09), but a strong negative dependency on the third lag is observed (α₃ ranging from −0.52 to −0.24 with a mean of −0.38) (Table 1 and Fig. 3A). The three-dimensional structure of the hare system is consistent with earlier theoretical arguments (17, 52, 53) and with experimental results (4). For the lynx, the direct density dependency is positive (α₁-1 ranging from 0.01 to 0.48 with a mean of 0.26); the lagged dependency is strongly negative (α₂ ranging from −0.84 to −0.27 with a mean of −0.63) (Table 1 and Fig. 3B). The two-dimensional structure for the lynx (L3) has been deduced in several earlier studies including Moran’s (ref. 33; but see §).

All additive skeleton models (with ɛ_t ≡ 0 for all values of t) shown in Table 1 give rise to dampened oscillations (as judged by the Schur–Cohn stability criterion for linear difference equations; refs. 54 and 55). Thus, we do not observe limit cycles as reported by Tong (36) [based on the SETAR(2;2,2) model for the L3 lynx series]. The dampened oscillations will, however, be sustained in the presence of environmental stochasticity. The difference between limit cycles and weakly dampened cycles are, therefore, not so conspicuous in stochastic systems (see, e.g., ref. 47).

Ecological Interpretations.

Recent studies have shown that the hares feed on a variety of food plants and are eaten by an array of predators (Fig. 1A) (56–59). The lynx is a specialist on hares but may utilize other prey species (Fig. 2A) (60, 61). When prey are scarce some predators may act as top predators on the lynx (62).

The present statistical results are consistent with the three-level trophic model for the hare (4, 56, 58). Other models involving spatial structure or age–structure are also possible. Due to recent field experiments and observations (e.g., ref. 4), we emphasis the trophic hypothesis as the more plausible scenario. In contrast to the more complex regulation of the hare, the lynx dynamics are thought to be food-driven (60, 63–65). Thus, the classic view of the lynx–hare cycle as a simple and symmetric predator–prey interaction seems to be an oversimplification (refs. 13 and 64; see also below).

Because several food species are eaten by the hare and the hare itself is eaten by many predators (Fig. 1B), the deduced three-dimensional structure of the time series lead us to hypothesize that the food species act as a group from the hare’s point of view and that the predators act as another group; if not, we might expect higher dimensionality. There is evidence from the Kluane Ecosystem Project that the predator community is strongly affected by intraguild predation, particularly as hares decline in abundance (62). Additional circumstantial evidence for this hypothesized compensation exists: Until about 1980, lynx were extensively trapped in northern Canada, but this predator removal has had no reported gross effect on the hare cycle (C.J.K., unpublished observations). During the 1986–1994 hare cycle, lynx were a major predator at Kluane. During this same period, northern goshawk numbers at Kluane were below normal (66). Finally, on Anticosti Island in the St. Lawrence River, there are no lynx present, yet the hare cycle persists (3). Both red fox and great-horned owls are particularly abundant on Anticosti Island and appear to compensate for the absence of lynx (2, 3). Keith (ref. 3, p. 115) documents cases of the extermination of lynx in southern Canada with continued hare cycles.

The vegetation appears to segregate into palatable and nonpalatable species for the hare (56, 57). Among the palatable species (primarily Betula glandulosa, Salix glauca, and Picea glauca), the hare has a mixed diet (66). Preferred foods are typically reduced during the peak of the hare cycle, whereas less preferred (but still palatable) species remain common (59). The impact of hares on their food plants is transient (58) and the recovery of the preferred food plants is substantial even before the decline is complete. At Kluane essentially no effect on the herbaceous vegetation of excluding hares from a 4-ha plot for 8 years has been found (R. Turkington, personal communication).

Snowshoe hares constitute a major part of the lynx’s diet. Lynx may themselves be eaten by several top predators (Fig. 1A; refs. 60 and 65 and M. O’Donoghue, personal communication). Studies in North America suggest, however, the hare to be the one major factor influencing the population dynamics of the lynx (60, 63, 67–69). The statistical models for the time series support the view of the dynamics of the lynx as a hare–lynx interaction.

Through our analyses we have simplified the bewildering complexity portrayed in Figs. 1A and 2A: If we are to study the dynamics of the lynx, we need to focus on the hare—in addition to studying the lynx. If we are to study the dynamics of the snowshoe hare, we need to focus on the guild of predators as a unit and the vegetation as a unit—in addition to the hare. The analyses have further documented detailed patterns of statistical density-dependence (Figs. 1B and 2B and Table 1). A viable hypothesis for the dynamics should be able to account for these patterns as well as for the dimensionality. In the following section, we suggest two simple mathematical models for trophic interactions within the boreal ecosystem. We investigate which constraints are required on the ecological interactions to generate the observed patterns of statistical density dependence. The detailed dynamic behavior of these models are deemed outside the scope of this report.

A Vegetation–Hare Predator Model.

Let H_t be the abundance of hares at time t, V_t be the abundance of the vegetation at time t, and P_t be the abundance of predators at time t; notice that P_t does not consist of lynx only but the combination of a variety of predators preying upon the hare. The functions F_h, F_v, and F_p describe per capita ecological interactions such that 3a 3b 3c where ɛ_v, ɛ_h, and ɛ_p represent stochastic influences. There is no general agreement about the impact of hares on edible vegetation (see also above): Wolff (56) and Keith (57) have measured stronger impacts on vegetation at the peak of the cycle than measured in the Kluane Ecosystem Project (R. Turkington, unpublished results). However, rapid vegetation recovery after the cyclic peak have been observed in all studies (56–58). The effect of the hare on the vegetation seems by and large negligible, thus allowing us to assume ∂F_v/∂h ≈ 0 (h = ln H). The total predator community is strongly affected by the hare cycle, and the abundance of all the major hare predators—lynx, coyote, great-horned owl, and northern goshawk—follow the hare cycle closely (58, 70), suggesting ∂F_p/∂h > 0 and ∂F_h/∂p < 0 (where p = ln P).

Approximating the F functions in Eq. 3 by the first terms in a Taylor expansion in log-transformed abundances, we may, under minimal additional assumptions, write the log-linear model in delay coordinates (see Eq. 2): 4 where α₁, α₂, and α₃ are statistical parameters that may be written as functionals of the ecological system portrayed by Eq. 3.¶ Interpreting the statistical results on the basis of Eqs. 3 and 4, the observed patterns of direct and delayed density dependence (α₁ > 0, α₂ ≈ 0, and α₃ < 0) may be shown to be highly plausible: given internal regulation through intraguild predation within the predator guild, (∂F_p/∂p ≤ 0), the condition α₃ < 0 is satisfied by assuming fairly weak self-regulation in both the vegetation and the hare population (∂F_v/∂v and ∂F_h/∂h are slightly negative, as documented in refs. 71–74). If ∂F_p/∂p is too negative, the relation α₁ > 0 is violated: our statistical results, thus, introduces restrictions on the strength of the intraguild predator interaction—it must be strong, but not too strong. The relation α₂ ≈ 0 is fulfilled under a wide range of interaction strengths.

A Hare–Lynx Model.

Let H′_t be the abundance of hares (with the possible inclusion of other herbivores preyed upon by the lynx; H′_t may be different from H_t used in the hare model) at time t and let L_t the abundance of lynx at time t. The functions G_h and G_l describe the ecological interactions such that 5a 5b Lynx occupy discrete territories during most of the hare cycle (63, 65, 67, 68) and only at the low phase (when hares are scarce) does the territorial system break down as many lynx go nomadic. A few lynx individuals survive the low phase by diversifying their diet with red squirrels and rodents. Hares represent the main food, however, even in the low phase (refs. 65, 67, and 68; M. O’Donoghue, personal communication; B. Slough, personal communication).

Approximating the G functions in Eq. 5 by the first terms in a Taylor expansion in log-transformed abundances, we may write: l_t = β₁l_t−1 + β₂l_t−2 + ɛ_t, where β₁ and β₂ are statistical parameters that may be written as functionals of the ecological system portrayed by Eq. 5.**

Due to the tight trophic interactions between the lynx and the hare (i.e., ∂G_h/∂l < 0 and ∂G_l/∂h > 0), the empirical patterns of statistical density dependence seen in the parsimonious time series model for the lynx (Table 1; β₁ > 0 and β₂ < 0), are easily fulfilled in the predator–prey system if self-regulation within both the lynx and the hare populations are not too strong (the hare model requiring similar weak self-regulation in the hare).