814 Ecology, 82(3), 2001, pp. 814���830 q 2001 by the Ecological Society of America THE GEOMETRY OF A POPULATION CYCLE: A MECHANISTIC MODEL OF SNOWSHOE HARE DEMOGRAPHY AARON A. KING1,3 AND WILLIAM M. SCHAFFER1,2 1Program in Applied Mathematics, University of Arizona, Tucson, Arizona 85721 USA 2Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721 USA Abstract. The phenomenology and causes of snowshoe hare cycles are addressed via construction of a three-trophic-level population dynamics model in which hare populations are limited by the availability of winter browse from below and by predation from above. In the absence of predators, the model predicts annual oscillations, the magnitude of which depends on habitat quality. With predators in the system, a wide range of additional dy- namics are possible: multi-annual cycles of various periods, quasiperiodicity, and chaos. Parameterizing the model from the literature leads to the conclusion that the model is compatible with the principal features of the cycle in nature: its regularity, mean period, and the observed range of peak-to-trough amplitudes. The model also points to circum- stances that can result in the cycle���s abolition as observed, for example, at the southern edge of the hare���s range. The model predicts that the increase phase of the cycle is brought to a halt by food limitation, while the decline from peak numbers is a consequence of predation. This is consistent with factorial field experiments in which hare populations were given supplemental food and partial surcease from predators. The results of the ex- periments themselves are also reproducible by the model. Analysis of the model was carried out using a recently developed method in which the original dynamical system is reformulated as a perturbation of a Hamiltonian limit wherein exist infinite numbers of periodic, quasiperiodic, and chaotic motions. The periodic orbits are continued numerically into non-Hamiltonian regions of parameter space corresponding to the situation in nature. This procedure allows one to obtain an overall understanding of the geometry of parametric dependencies. The present study represents the first formulation of a full three-trophic-level snowshoe hare model and the first time any model of the cycle has been parameterized entirely using independently measured quantities. Key words: applied bifurcation theory boreal forest ecosystem food-chain dynamics mathe- matical model multi-annual population cycle snowshoe hare (Lepus americanus) ten-year cycle. INTRODUCTION The snowshoe hare cycle As any ecology student can readily relate, cyclic fluc- tuations in population density of the ������snowshoe,������ or ������varying,������ hare (Lepus americanus) and its mamma- lian predators in Canada���s north woods have been the subject of continuing inquiry and debate since the days of Elton and his collaborators (1924, 1927). The im- portance of this phenomenon, arguably ecology���s most celebrated oscillation, is accentuated both by its mag- nitude and inherent romance: since colonial times at a minimum, hares, lynx, martens, and other fur-bearing creatures of the spruce���fir forest have fluctuated in abundance, attaining maximal densities every 8���11 yr (Fig. 1). By ecological, if not celestial, standards, it is an extraordinarily precise metronome which, over the years, has attracted the attention of ecologists (Elton and Nicholson 1942a, b, MacLulich 1957), field bi- Manuscript received 18 June 1999 revised 3 February 2000 accepted 20 February 2000. 3 Present address: Department of Environmental Science and Policy, University of California, Davis, California 95616 USA. E-mail: aking@ucdavis.edu ologists (Wolff 1980, Keith 1990, Boutin et al. 1995, Poole 1995, Slough and Mowat 1996, and references therein), statisticians (Bulmer 1974, Finerty 1980, Roy- ama 1992, Stenseth et al. 1997), and theorists (Leigh 1968, Fox and Bryant 1984, Trostel et al. 1987, Ak- cakaya 1992, Blasius et al. 1999). The most important features of these oscillations are as follows (Norrdahl 1995): 1. Regularity.���Although the cycles are by no means perfectly periodic, hare population peaks and those of their predators succeed one another at fairly regular intervals. The most famous evidence of this regularity, as seen in the lynx fur harvest, was gathered by Elton and Nicholson (1942b). The lynx is strongly, if not obligately, dependent on the hare (Brand et al. 1976), and changes in its abundance are generally be- lieved to reflect changes in the availability of its pre- ferred food species. As is well known, Elton and Nich- olson (1942b) found that most peaks in most localities occurred at intervals of 8���11 yr. Not surprisingly, there is a strong and statistically significant peak in the power spectrum at a frequency of 0.1 yr21 (Finerty 1980). Reviews of more recent fur statistics point to a similar conclusion, suggesting that the cycle continues to this

March 2001 815 GEOMETRY OF A POPULATION CYCLE FIG. 1. Cycles in abundance of Canadian lynx and snow- shoe hare as revealed in the fur harvest records of the Hudson���s Bay Company. (a) Lynx fur harvest from the MacKenzie River District (after Elton and Nicholson 1942b). (b) Snowshoe hare harvest from the regions near Hudson���s Bay (after MacLulich 1957). day across much of North America (Stenseth et al. 1998b). 2. Period.���Historical interest in the sunspot cycle notwithstanding, the 8���11-yr period appears unrelated to any known meteorological or other extrinsic oscil- lation (Finerty 1980). In addition, it differs dramati- cally from the 3���5-yr oscillation of lemmings in the arctic tundra and voles in the subarctic taiga. 3. Amplitude.���The magnitude of fluctuations has long elicited comment. Recent population studies (Keith and Windberg 1978, Keith 1990, Boutin et al. 1995) suggest peak to trough ratios ranging from 13 to 50. Fur returns evidence much larger amplitudes���two to three orders of magnitude (Elton and Nicholson 1942b, MacLulich 1957)���but these almost surely exaggerate the variance in actual animal numbers (e.g., Finerty 1980). 4. Distribution.���The 10-yr cycle is essentially a northern phenomenon. Southern populations of snow- shoe hare manifest annual oscillations, but not the mul- ti-annual variations observed farther north (Keith 1990). Of interest, but uncertain import, is the fact that hare populations in Wisconsin formerly exhibited mul- ti-annual cycles but, since 1950, these have abated (Green and Evans 1940, Keith 1990). 5. Synchrony.���Over wide geographic areas, popu- lation cycles are roughly synchronous (Elton and Nich- olson 1942b, Bulmer 1974, Smith 1983, Sinclair et al. 1993, Ranta et al. 1997). As noted above, ������Wildlife���s Ten-Year Cycle������ (Keith 1963) belongs to the boreal forest, a community which, to a first approximation, is organized into three trophic levels (Fig. 2) and in which the hare occupies a central, if unenviable, place. Active throughout the year, hares grow fat on an abundance of herbaceous forage when the days are long and subsist barely, or not at all, on the terminal twigs of woody plants in winter (Pease et al. 1979, Vaughan and Keith 1981). During the latter period, hares constitute the bulk of the prey biomass available to predators, their primacy in this regard re- flecting both absolute abundance and the fact that other prey species hibernate (Boutin et al. 1995). Hypotheses As reviewed, for example, by Lack (1954) and Fi- nerty (1980), many explanations have been proposed to account for recurrent fluctuations in animal popu- lations. In the case of the 10-yr cycle, the current pre- vailing opinion is that exploitative interactions (pre- dation, herbivory) are at the heart of the matter (Nor- rdahl 1995, Korpima ��ki and Krebs 1996). There are three principal variations on this theme: 1. Hare���vegetation interaction.���Lack (1954) in- terpreted the 10-yr cycle as resulting from cyclic de- clines in the hare ������due to food shortage (with perhaps secondary disease)������ from which follows the subse- quent decimation of both the hares��� predators and the latter���s alternative prey that suffer disproportionately when hare populations crash. In the same vein, Fox and Bryant (1984) propose that the cycle results from the interaction between hares and their winter food sup- ply. According to their scenario, hare populations crash because of a dearth of palatable winter forage, and the cycle���s period is determined by the recovery time of the browse. Fluctuating predator populations, by this account, are consequential rather than causal. 2. Hare���predator interaction.���This hypothesis (Trostel et al. 1987, Korpima ��ki and Krebs 1996) views the oscillations as resulting from exploitation of the hares by a ������complex������ of predator species. The latter include the Canadian lynx (Lynx canadensis), coyote (Canis latrans), Great Horned Owl (Bubo virginianus), Goshawk (Accipiter gentilis), and Red-tailed Hawk (Buteo jamaicencis), among others. 3. Three trophic level hypothesis.���Hypotheses 1 and 2 conjecture mechanisms involving two trophic levels. By way of contrast, Keith and his colleagues have promoted the opinion that the cycle results from interactions among species distributed across three lev- els (Keith 1974). It was hypothesized that hare popu- lations are food limited and that starvation due to over- browsing at high densities initiates the decline from peak numbers. The downward turn is then accentuated by predation with the consequence that hare popula- tions remain at low density even after the browse has recovered, i.e., until the predators, in their turn, decline. In recent years, Keith and his associates (Keith et al. 1984, Keith 1990) have ascribed a greater importance to predation at all phases of the cycle while continuing

816 AARON A. KING AND WILLIAM M. SCHAFFER Ecology, Vol. 82, No. 3 FIG. 2. Food web of the Canadian boreal forest (after Stenseth et al. 1997). There are essentially three trophic levels. A wide range of forbs and grasses support a community of herbivores during the summer season. During the winter, some of the herbivores hibernate, while others depend upon woody browse. There are a wide variety of predators which differ in degree of dependence on hares. to stress the indirect effects (reduced recruitment and increased vulnerability to predators) of resource limi- tation when hares are in greatest abundance. In light of recent experiments, Krebs et al. (1995) have also argued for the hypothesis that food shortage and pre- dation together are causally involved in hare population regulation (see also Stenseth 1995). Nonlinear time se- ries analysis of fur returns (Stenseth et al. 1997) points to a similar conclusion: that whereas fluctuations in lynx abundance can be accounted for by a two-variable model, variations in hare density require three vari- ables. Stenseth et al. (1997) interpreted this as an in- dication that lynx population dynamics depend only on the numbers of hares, whereas hare dynamics are af- fected by both the density of predators and the abun- dance of winter food supplies. Our own perspective on the 10-yr cycle is that eco- logical dynamics are per force nonlinear, and the fun- damental lesson of nonlinear dynamics is that quali- tative differences in behavior can result from quanti- tative differences in parameter values. In the face of such parametric sensitivity, the articulation of alter- native verbal hypotheses and their evaluation by strong inference���the conventional biological approach��� may be doomed from the outset. This is one reason for the fact that hypotheses which are sharp in a linear world can be blunt in a nonlinear world. Consider, for example, the determination that mortality in one pop- ulation results primarily from factor A and in another from factor B. Such an observation can, of course, be indicative of fundamental differences in ecology. But it can also reflect nothing more substantial than min- imal variations in factors such as rates of resource re- newal or predator recruitment. In the latter case, the essential equivalence of the two situations might easily be missed. An alternative approach is to embrace the possibility of sensitivity to parameter values. In the case of pop- ulation dynamics, this necessitates the formulation of mathematical models encapsulating the essential mech- anisms and the independent estimation of their param- eters. These preliminaries accomplished, one can then determine whether or not a particular model reproduces both the essential phenomenology and, in cases where they are available, the results of experimental manip- ulations. Such an approach is necessarily iterative, and, indeed, one can go round many times. On the other hand, it is, we submit, the only means of dealing ef- fectively with nonlinear dynamical phenomena, be they physical, chemical, or ecological. This approach is hardly novel. To the contrary, it has been the basis of the scientific method since Newton. Field studies and experiments There are two primary sources of data on the cycle. Fur returns, dating to the 1700s, provide an extended,