Availability of Food and the Population Dynamics of Arvicoline Rodents

Ecology, 82(6), 2001, pp. 1521±1534 ᭧ 2001 by the Ecological Society of America

AVAILABILITY OF FOOD AND THE POPULATION DYNAMICS OF ARVICOLINE RODENTS

PETER TURCHIN1,3 AND GEORGE O. BATZLI2 1Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs, Connecticut 06269-3043 USA 2Department of Ecology, Ethology and Evolution, University of Illinois, Champaign, Illinois 61820 USA

Abstract. Availability of food may play a number of different dynamical roles in rodent±vegetation systems. Consideration of a suite of rodent±vegetation models, ranging from very simple ones to a model of medium complexity tailored to a speci®c system (brown lemmings at Point Barrow, Alaska, USA), suggested several general principles. If vegetation grows logistically following an herbivory event (a standard assumption of previously advanced models for herbivore±plant interactions), then almost any biologically reasonable combinations of parameters characterizing rodent±vegetation systems would result in population cycles. We argue, however, that the assumption of logistic growth of the food supply may be appropriate for only a few species, such as moss-eating lemmings. The dynamics of food supply for many arvicoline (microtine) rodents may be better described by a ``linear initial regrowth'' model, which exhibits globally stable dynamics. If this is so, quantitative interactions with food supply are unlikely to explain multiannual population cycles for most boreal or temperate voles. The role of food in population dynamics, however, is not limited to its potential to generate cycles. A tritrophic model including vegetation, rodents, and their specialist predators suggests that food limitation may provide direct density dependence needed for sustained oscillations in this system (which is usually modeled by a phenomenological logistic term in the prey equation). We relate the general theory that we developed to one speci®c system where we have enough data to arrive at reasonable estimates for most of the parametersÐbrown lemmings at Point Barrow. The Barrow model exhibits oscillations of the approximately correct period and amplitude, thus giving some theoretical support to the food hypothesis. Nevertheless, we suggest that this result should be treated cautiously because key events explaining the population cycle in the model occur during winter, but winter biology of lemmings is still poorly understood. Key words: consumer±resource interactions; herbivore±plant interactions, mathematical models; lemmings; logistic growth vs. linear growth, dynamical results; modeling population dynamics; population cycles in arvicoline (microtine) rodents; rodents, arvicoline; small-mammal population dynamics and food supply; tritrophic interactions; voles.

INTRODUCTION Henttonen et al. 1987, Klemola et al. 2000), but theoretical ecologists largely neglected this issue. Unlike Consumer±resource systems have an inherent ten- predator±prey theory in general, and its application to dency to cycle (Lotka 1925, Volterra 1926, May 1981), vole populations in particular (see Hanski et al. 2001), and thus it is natural to look to predators and food resources of rodents for an explanation of their pop- models for population interactions between arvicoline ulation cycles. Initially, however, the search for mech- rodents and their food supply have not been a focus of anisms underlying rodent oscillations turned to other intensive theoretical development (for two exceptions factors than the availability of food. Three decades see Stenseth et al. [1977] and Oksanen [1990]). passed after Elton's (1924) pioneering paper before The general question addressed in this paper is: What Lack (1954) and Pitelka (1957a) proposed the expla- quantitative characteristics of rodents and their food nation of vole and lemming cycles based on periodic plants (for example, the dynamics of plant growth and overexploitation of their food supply followed by pop- rodent consumption) are likely to result in vegetation± ulation crashes (additionally, changes in food quality herbivore cycles? To make this question more precise, could explain rodent cycles; see review by Batzli consider the possible roles that vegetation can play in [1992]). Empirical ecologists have subsequently sub- rodent dynamics. First (and of most interest to the topic jected this suggestion to several experimental tests of our paper), the availability of high-quality food (e.g., Cole and Batzli 1978, Lindroth and Batzli 1986, plants can be a slow dynamic variable, so that food shortages persist for a substantial period of time (e.g., more than one herbivore generation) after the herbivore Manuscript received 2 October 1998; revised and accepted 11 May 2000. population peak. General theory of resource±consumer 3 E-mail: [email protected] dynamics (Lotka 1925, Volterra 1926) suggest that this 1521 1522 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6 should result in second-order population cycles (``sec- [1981] varied both the plant standing crop and growth ond-order dynamical processes'' are those that are rate simultaneously, as one parameter). Furthermore, it characterized by delayed density dependence). is dif®cult to relate measurements of primary produc- However, the in¯uences of food supply are not lim- tivity in the empirical literature to these two theoretical ited to their potential to generate cycles. For instances, quantities. Authors often report the amount of regrowth food may be a fast dynamic variable, imposing direct of vegetation that has occurred some time after her- (immediate) density dependence when population den- bivory, but rarely report the short-term rates of re- sity reaches the level at which the amount of food per growth. Yet, when developing a mechanistic, quanti- individual is insuf®cient for maximum survival and tative model of vegetation±rodent interaction, we must reproduction. In this case, there are no delayed effects explicitly distinguish between the growth rate when of food shortage persisting after population density de- vegetation is sparse, and the ``carrying capacity'' (max- clines to a low value, and food will play a stabilizing imum biomass) of the vegetation. role in rodent dynamics. In other words, such vege- In addition to these quantitative features of produc- tation±herbivore systems will behave as ``®rst-order'' tivity, there is an important, but little-appreciated, qual- processes (Royama 1992). In addition to being a dy- itative distinction, which affects how we model veg- namic variable (fast or slow), availability of food, or etation dynamics. This distinction is illustrated by the some other aspect of it, may play the role of a param- two extreme functional forms that can be used in the eter that changes the dynamic properties of the sys- vegetation equation. One is the logistic growth equa- temÐfor example, a bifurcation from stable to cyclic tion, dynamics. Finally, food may have no in¯uence on ro- dV K Ϫ V dent dynamics (a null factor) because other density- ϭ uV (1) dependent factors stop rodent population growth before dt K signi®cant depletion of the food supply occurs. where V is the vegetation biomass (per unit of area), Our speci®c goals are, therefore, to develop theo- u is the (per capita) rate of plant growth at V near zero, retical expectations regarding the circumstances under and K is the maximum biomass approached in the ab- which the different roles that food may play in rodent sence of herbivory (``carrying capacity''). The other dynamics, as delineated above, are expressed. In par- alternative is ticular, we are interested in determining under what dV K Ϫ V conditions theoretical vegetation±herbivore systems ϭ U (2) will behave as ®rst- vs. second-order dynamical pro- dt K cesses. We begin by surveying and, where necessary where U is the initial regrowth rate (at V near 0). This developing a suite of simple models for dynamics of equation has been previously used in models of nutrient rodent±vegetation systems. Using these models, we in- dynamics in a chemostat (see, for example, Edelstein- vestigate how rodent±vegetation dynamics are affected Keshet 1988:121), as well as in theoretical treatments by such qualitative features as the dynamics of vege- of species competing for ``abiotically'' growing re- tation regrowth after herbivory, by quantitative varia- sources (e.g., MacArthur 1972, Schoener 1976, Abrams tion in primary productivity, and by adding the third 1977, Gurney and Nisbet 1998). trophic level (specialist predators). Next, we relate the These two modelsÐEqs. (1 and 2)Ðmake very dif- general theory to one speci®c system, brown lemmings ferent assumptions about the dynamics of plant growth, (Lemmus sibiricus) near Barrow, Alaska, USA, for and have very different dynamic consequences. The which we have enough data to arrive at reasonable logistic implies that when vegetation biomass V is near estimates for most of the model parameters. Finally, 0, its growth rate is an accelerating function of V that we discuss the implications of our theoretical results reaches its maximum at K/2, and then slows to 0 as V for small-rodent cycles in general. approaches K (Fig. 1). The logic underlying the logistic model is that the more plant biomass is present, the GENERAL MODELS AND THEIR DYNAMICS more solar energy it can ®x, and the faster it will grow Background: how should vegetation dynamics (until it starts approaching the limit K). By contrast, be modeled? Eq. 2 implies no acceleration period; instead, when V is low it increases linearly at the maximum rate U, and Perhaps the most important factor in¯uencing food gradually slows to 0 as V approaches K (Fig. 1). Thus, supply for herbivorous rodents is primary productivity V refers not to the total plant biomass, but only to the of the plant community. Productivity has two very dif- part easily accessible to herbivores (e.g., the above- ferent quantitative aspects: it determines the maximum ground partsÐstems and leavesÐon which rodents biomass achieved in the absence of herbivory and the feed). The model assumes that initial regrowth is fueled rate at which plant biomass is replenished after being by energy stored in the roots and rhizomes, and there depleted by herbivores. These two aspects are not al- is no period of accelerating growth. We will call this ways modeled separately in the theoretical literature model the ``linear initial regrowth'' equation (or the (for example, the in¯uential paper by Oksanen et al. ``regrowth equation,'' for short). June 2001 RODENT±VEGETATION INTERACTIONS 1523

(Table 1: Model I), one of the commonly used approaches to modeling resource±consumer interactions in theoretical ecology (e.g., May 1981). This model makes the following assumptions: (1) vegetation biomass grows logistically, with maximum per capita rate of increase u, and carrying capacity K; (2) the functional response of herbivores to plant density is of Type II, with maximum consumption per herbivore A, and the half-saturation plant density B; (3) Reproduction by herbivores is directly proportional to biomass consumed, with the constant of proportionality R; and (4) Herbivores have a constant death rate. The speci®c form in Table 1 is a slight variation on the standard model, since the parameter G is placed FIG. 1. Temporal dynamics of vegetation obeying re- inside the parentheses. The biological interpretation of growth (solid curve) and logistic (broken curve) equations; G is the threshold consumption rate of plant biomass K is the ``carrying capacity''Ðthe maximum biomass approached in the absence of herbivory. (per herbivore) at which herbivore birth rate is just balanced by their death rate. This parameterization was chosen because it is more straightforward to estimate Both models are oversimpli®cations of reality, and G using bioenergetic arguments, instead of attempting it is best to think of them as ideal cases, rather than to estimate consumer death rate. Furthermore, we can representing the growth dynamics of actual plants. For estimate R by assuming that at maximum consumption example, mosses may be better described by the lo- rate (when V ϭ K), the per capita rate of herbivore gistic because nearly all of their living biomass is ac- population growth is rmax (Table 2). cessible to herbivores, while many perennial grami- This basic model is capable of two nontrivial kinds noids (grasses and sedges), in which at least 80% of of dynamic behaviors: a stable equilibrium and a limit biomass is underground (Wielgolaski 1975), may be cycle. Cycles occur for parameter values satisfying the better described by the regrowth equation. However, inequality B/K Ͻ (A Ϫ G)/(A ϩ G). A reference set of even graminoid dynamics may be better described by parameters for Model I (median values in Table 2) can the logistic if there is extensive damage to their root be extracted from the ones we used for the Barrow systems resulting from a herbivore outbreak. For ex- model (see below). Thus, u and G are the averages of ample, at high population densities during the spring the summer and winter values of these parameters in thaw brown lemmings grub for rhizomes (Pitelka Table 3, weighted by the lengths of summer and winter 1957b), and root voles eat rhizomes of graminoids dur- periods. A and B are unchanged, and K is the biomass ing winter (Tast 1974). of mosses (we interpret V as the moss biomass, since Historically, theoretical population ecologists inter- we have assumed logistic growth of the food supply). ested in the dynamics of herbivory have represented Although useful to provide estimates of reasonable pa- plant dynamics with the logistic model (e.g., Caughley rameters for the general model, these assumptions are and Lawton 1981, Crawley 1983). By contrast, both not meant to represent Barrow or any other location. empirical and theoretical ecosystem ecologists inter- Because B/K ϭ 0.035 is much less than (A Ϫ G)/(A ested in vegetation dynamics developed models that ϩ G) ϭ 0.625, the median parameter values place the imply the regrowth equation (see Parton et al. 1993: model deep in the limit-cycle region. Numerical so- Eq. 14, AÊ gren and Bosatta 1996: Eqs. 8.1 and 9.5). For lutions indicate that Model I exhibits cycles of high example, in the Century model of grassland primary amplitude and long period (Table 1, Fig. 2a). To ex- productivity, the aboveground-production rate in the amine other effects of parameter values on the behavior beginning of the season is not affected by accumulating of this and subsequent models, we ran simulations us- biomass, and foliage initially grows linearly (Parton et ing combinations of values within the ranges shown in al. 1993). As the season progresses, growth slows down Table 2. In general, decreasing the B/K ratio destabi- and eventually stops as a result of several processes lizes dynamics (via the well-known mechanism of the (e.g., increased shading, shoot death, and depletion of ``paradox of enrichment''). In addition, the model be- nutrients in the soil). Clearly, the implied growth of comes more prone to cycle when G is decreased in aboveground biomass is of the regrowth type, with K relation to A. Parameter u does not affect qualitative a phenomenological parameter re¯ecting the combined stability. However, low values of u result in long cycles action of several mechanistic processes. with high amplitude, while high values produce shorter, Model IÐRosenzweig-MacArthur model less-extreme oscillations in herbivore density. In sum- (logistic vegetation) mary, Model I generates second-order oscillations for Our starting point for exploring rodent±vegetation practically all biologically plausible values of its pa- dynamics is the Rosenzweig-MacArthur (1963) model rameters. 1524 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6

TABLE 1. Models for vegetation±herbivore (lemming) population interaction. Population dynamics are described for models when using the median parameters shown in Tables 2 and 3.

Dynamics Quantitative characteristics Equations² Type (median parameters)³ Variables Model I, Rosenzweig-MacArthur model (logistic vegetation) dV V AVH Cycles Period: 16 yr V ϭ vegetation biomass ϭ uV 1 ϪϪ Ampl.: Ͻ0.001±800 ind./ha dt΂΃ K V ϩ B H ϭ herbivore density dH AV ϭ RH Ϫ G dt΂΃ V ϩ B Model II, Bazykin model (rodent self-limitation) dV V AVH Cycles Period: 12 yr same variables as above ϭ uV 1 ϪϪ Ampl.: Ͻ0.001±500 ind./ha dt΂΃ K V ϩ B dH AV ϭ RH Ϫ G Ϫ EH 2 dt΂΃ V ϩ B Model III, Variable-territory model dV V AVH Stable Eq. density: 70 ind./ha same variables as above ϭ uV 1 ϪϪ dt΂΃ K V ϩ B dH AV eH 2 ϭ RH Ϫ G Ϫ dt΂΃ V ϩ BV Model IV, Regrowth±herbivory model dV V AVH Stable Eq. density: 420 ind./ha same variables as above ϭ U 1 ϪϪ dt΂΃ K V ϩ B dH AV ϭ RH Ϫ G dt΂΃ V ϩ B Model V, Regrowth±herbivory model with seasonality dV V AVH Cycles ``Period'': 2 yr same variables as above, plus ϭ U(␶)1ϪϪ Ampl.: 1±1400 ind./ha ␶ϭseason (0 Յ ␶Ͻ1) dt΂΃ K V ϩ B dH AV ϭ RH Ϫ G dt΂΃ V ϩ B Model VI, Herbivore±logistic/regrowth vegetation model dV V AVH Stable Eq. density: 40 ind./ha Two kinds of vegetation: U 1 ϭ ϪϪ V ϭ ``logistic'' vegetation dt΂΃ KV V ϩ M ϩ B M ϭ ``regrowth'' vegetation dM M AMH ϭ uM 1 ϪϪ H ϭ herbivore density dt΂΃ KM V ϩ M ϩ B dH A(V ϩ M) ϭ RH Ϫ G dt[] V ϩ M ϩ B Model VII, Tritrophic model dV V AVH Cycles Periodmed : 5 yr V ϭ vegetation biomass ϭ U 1 ϪϪ H ϭ herbivore density dt΂΃ K V ϩ B Ampl.med : 0.2±900 ind./ha P ϭ specialist predator density dH AV CHP ϭ RH Ϫ G Ϫ dt΂΃ V ϩ BHϩ D dP P ϭ sP 1 Ϫ Q dt΂΃ H June 2001 RODENT±VEGETATION INTERACTIONS 1525

TABLE 1. Continued.

Dynamics Quantitative characteristics Equations² Type (median parameters)³ Variables Model VIII, the Barrow model dV V AVH Cycles Period: 5.5 yr V ϭ vascular biomass ϭ U(␶)1ϪϪ Ampl.: 1±390 ind./ha M ϭ moss biomass dt΂΃ KV V ϩ␣M ϩ B H ϭ lemming density dM M ␣AMH ϭ u(␶)M 1 ϪϪ ␶ϭseason (0 Յ ␶Ͻ1) dt΂΃ KM V ϩ␣M ϩ B dH A(V ϩ␣M) ϭ RH Ϫ G(␶) dt[] V ϩ␣M ϩ B ² A ϭ maximum rate of vegetation consumption by an herbivore; B ϭ herbivore half-saturation constant; C ϭ maximum killing rate by predators; D ϭ predator half-saturation constant; E ϭ herbivore density-dependence parameter; e ϭ vegetation/ herbivore ratio at equilibrium; G ϭ herbivore consumption rate at zero population growth; H ϭ herbivore density; K ϭ vegetation carrying capacity (maximum biomass per unit area); KH, KV, KM ϭ carrying capacities of the herbivore population, vascular plants (regrowth vegetation), and mosses (logistic vegetation), respectively; M ϭ moss biomass density (or logistically growing vegetation biomass density); P ϭ specialist predator density; Q ϭ herbivore/predator ratio at equilibrium; R ϭ conversion rate of vegetation biomass into herbivore biomass; s ϭ maximum per capita rate of predator population growth; U ϭ maximum rate of regrowth; V ϭ vegetation biomass; ␣ϭdiscounting parameter for relative consumption of mosses (compared to vascular plants); ␮ϭmaximum per capita rate of growth (logistic vegetation); and ␶ϭseason (dummy variable for seasonality). ³ Quantitative dynamics are obtained for median parameter. `Àmpl.'' ϭ amplitude, i.e., min/max range. `Àmpl.'' is used when dynamics are oscillatory; thus there is a number for the minimum and another for the maximum densities. `Èq. density'' ϭ equilibrium density; this is used when there is stable equilibrium, thus only a single number.

Model IIÐBazykin model (rodent self-limitation) The only mechanism by which herbivore density is regulated in the Rosenzweig-MacArthur model is food limitation. However, rodents may have additional, intraspeci®c mechanisms of population regulation. Add- TABLE 2. Parameter values for generic Models I±VII in Table 1. ing such a self-limitation term leads to the model investigated by Bazykin (1974) (see Table 1). The Ba- Median zykin model is inherently more stable than Model I. Parameter² value Range³ For it to cycle, not only must the condition for Model Generic models I be satis®ed, but also intraspeci®c density dependence, u 2yrϪ1 1±10 E, must be relatively weak: K 2000 kg/ha 500±5000 KM 2000 kg/ha n.a. 4RA(AK Ϫ AB Ϫ GK Ϫ BG) K 300 kg/ha n.a. E Ͻ . (3) V u(K B)(K B)2 U 10 000 kg´haϪ1ýrϪ1 1000±10 000 Ϫ ϩ Ϫ1 Ϫ1 A 15 kgýr índ. 10±20 Assuming the value of E from Table 2 (which implies B 70 kg/ha 50±200 G 0.6A 0.4±0.8 A that the population of herbivores will equilibrate at KH Ϫ1 rmax 6yr n.a. ϭ 1000 individuals/ha if food is provided ad libitum; Ϫ1 R rmax[AK/(K ϩ B) Ϫ G] n.a. KH 1000 ind./ha 100±1000 Ϫ1 ABLE E rmaxKH n.a. T 3. Parameter values for Barrow (Alaska, USA), Mod- q 10 kg/ind. 1±10 el VIII in Table 1. e rmax q n.a. Specialist predator§ Parameter Median value Range² Ϫ1 Ϫ1 Ϫ1 Ϫ1 C 600 ind.ýr ´predator n.a. Us 10 000 kg´ha ýr 5000±20 000 Ϫ1 D 6 ind./ha n.a. us 12 yr 6±24 Ϫ1 s 1.25 yr n.a. KV 1000 kg/ha 100±2000 Q 40 ind./predator n.a. KM 2000 kg/ha 1000±4000 A 15 kgýrϪ1índ.Ϫ1 10±20 ² KH ϭ carrying capacity of the herbivore population; q ϭ B 70 kg/ha 35±140 vegetation/herbivore ratio at equilibrium; rmax ϭ maximum R 10.7 AϪ1 n.a. per capita rate of herbivore population growth. For all other G 0.44A n.a. variables, see Table 1: footnote ². s Gw 0.63A n.a. ³ n.a. ϭ not applicable; we did not investigate a range of ␣ 0.5 0.1±1 values for this parameter. § Model VII in Table 1. ² n.a. ϭ not applicable. 1526 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6

for the relationship between KH and E, see Table 2) and median values of other parameters, the Bazykin model predicts essentially the same dynamics as Model I, although the cycle period is somewhat shorter, and the peak density is somewhat lower (Table 1, Fig. 2b). Note that although the limit imposed by intraspeci®c competition is 1000 ind./ha, the population density reaches at most (and only for brief periods of time) one half of that density. This observation re¯ects the fact that peak density is jointly governed by food limitation and

intraspeci®c competition. Decreasing KH to ϳ130 ind./ ha results in a stable equilibrium, with herbivore density of 25±30 ind./ha (again, the equilibrium is much

lower than KH, re¯ecting joint regulation by food and intraspeci®c competition). For intraspeci®c regulation to become the only determinant of the equilibrium den-

sity, KH must be reduced to ϳ70 ind./ha. Below this

level, the herbivore equilibrium is very close to KH (less than 10% difference), and food in Model II becomes a ``null factor''Ðfood has neither a dynamic effect on herbivore density, nor does it affect the parameters gov- erning the rate of herbivore population change.

Model IIIÐVariable-territory model The Bazykin model offers one simple way to model herbivore self-limitation. Although a thorough investigation of different functional forms of density dependence in the consumer and how they might affect consumer±resource dynamics is beyond the scope of this paper, we think it would be useful to consider at least one alternative to the Bazykin model. Note that the Bazykin model assumes that the carrying capacity,

KH, is a ®xed number of herbivores per unit of area. An alternative, and equally simple assumption is to

make KH directly proportional to food availability: KH ϭ V/q (Leslie 1948). Here V is vegetation biomass density, as before, and q is a proportionality constant.

Rewriting the Bazykin model in terms of rmax and KH, and then substituting V/q, we obtain the following equation for consumer rate of change: dH AV r qH2 ϭ RH Ϫ G Ϫ max : (4) dt΂΃ V ϩ BV We will call this modi®cation of the Bazykin model a ``variable-territory model,'' because it can be derived by assuming that the territory size changes in response to food availability (P. Turchin, unpublished manuscript). Note that Model III is closely related to the FIG. 2. Representative dynamics of herbivores and veg- Leslie-May predator±prey model (Leslie 1948, May etation for selected generic models (Table 1) using median 1981) that has served as the basis of theoretical in- parameter values shown in Table 2: (a) Model I, (b) Model vestigation into vole±weasel interactions (see Hanski II, (c) Model V, and (d) Model VII. Thick lines represent et al. 2001; see also Model VII). Indeed, if we ap- herbivores; thin lines represent vegetation. Note that density is plotted using a logarithmic scale; the units are those that proximate the term R(AV/(VϩB) Ϫ G) in Eq. 4 with are appropriate for each variable (e.g., rodents are individuals rmax (this is a reasonable approximation as long as V per hectare; vegetation is measured in kilograms per hectare). k B, but it breaks down for small V), then we obtain the following predation equation of the Leslie-May form: June 2001 RODENT±VEGETATION INTERACTIONS 1527

dH r qH2 H (Maynard Smith 1974; see McNair [1986] for the dis- ϭ rHϪϭmax rH1 Ϫ q . (5) dtmax V max ΂΃ V tinction between ``constant-number'' and ``constant- proportion'' refuges, and the potentially contrasting ef- In other words, the Leslie-May model can be viewed fects of these refuge types on stability). as an approximation of the variable-territory variant of the Bazykin model. Model VÐRegrowth with seasonality model The value of q ϭ 10 kg/ind. is a reasonable guess In this model, the initial regrowth rate, U, is a func- for a vole such as Clethrionomys rufocanus. The dy- tion of season, ␶ (0 Յ ␶Ͻ1). During summer, which namics of Model III for this q are stable (Table 2). is assumed to be 1/6th of a year (2 mo), U(␶) ϭ Us.In Reducing q by a factor of 10 (which implies KH ϭ 1000 winter (the rest of the year), there is no vegetation ind./ha, a value that we used in Model II above), results growth, so U(␶) ϭ 0. in mild oscillations with a period of 2±3 yr and am- Adding a non-growing season in Model IV intro- plitude 15±40 ind./ha. Reducing q even further we duces a period when vegetation can be consumed, but eventually recover the dynamics of Model I. This nu- cannot grow back. As a result of this time lag for re- merical result con®rms the intuition that dividing food growth, the dynamics of Model V for median parameter among competitively superior individuals should exert values are characterized by a 2-yr cycle, rather than a strongly stabilizing effect on dynamics. Furthermore, stability as in Model IV (Fig. 2). For somewhat dif- numerical explorations suggest that for high q the ap- ferent parameter values the model can exhibit chaotic proach to the equilibrium is exponential, thus preclud- oscillations with an average period of 3±4 yr and high ing the possibility of pseudoperiodic oscillations sus- amplitude. This model appears to be very similar (both tained by environmental noise. In summary, as q is in structure and in the resulting dynamics) to the model increased from zero, the dynamics of Model III shift investigated by Oksanen (1990). from second-order oscillations to ®rst-order stability. Although Model V's dynamics super®cially resemble the oscillations obtained in Models I or II, the dy- Model IVÐLinear initial regrowth model namic mechanism is quite different. Model V is char- Replacing the logistic growth in the vegetation equa- acterized by ®rst-order dynamics, whereas Models I tion of Model I with the regrowth term has a profound and II are second-order processes. In practice, this effect on model dynamics (Table 1). Model IV is glob- means that dynamics of Models I and II will be char- ally stable for all values of its parameters. The key acterized by delayed density dependence, while output difference between this model and Model I is that lo- of Model V should show no evidence of delayed den- gistic growth has an inherent lag time built in itÐthe sity dependence. Thus, the fundamental difference be- more vegetation is depleted by herbivory, the longer it tween logistic and regrowth vegetation models persists takes to grow back. Thus, logistically growing vege- whether we are dealing with seasonal or non-seasonal tation consumed down to 0.01% of its maximum stand- environments, and this difference can be detected by ing crop will take a much longer time to grow back an appropriate statistical test for delayed density de- compared to vegetation decreased to 1% of K. By con- pendence (e.g., Turchin and Ellner 2000). The stability trast, regrowing vegetation in Model IV will need es- of Model V is strongly affected by seasonality (as the sentially the same time to get back to K whether it length of the nongrowing season is decreased to zero starts from 1%, 0.01%, or even 0% of K. Peak lemming we recover Model IV, and thus stability). One other densities often result in depressing vegetation levels to parameter has strong effects on stability in this model: less than 10% of K (see Discussion, below). Thus, the K. The greater the maximum standing biomass, the difference in after-peak periods of low food between more violently population density ¯uctuates. regrowth-type and logistic vegetation may be substantial. Model VIÐBoth regrowth and logistic vegetation For biologically plausible range of U values in Tables Suppose that the total maximum standing crop, K, 2 and 3, regrowth-type vegetation will essentially come is divided between two types of vegetation, plants with back within one growing season. As a result, it acts as regrowth and plants with logistic growth functions, so a fast dynamical variable, which explains why Model that K ϭ KV ϩ KM. Clearly, Model VI will cycle when

IV behaves as a ®rst-order dynamical system. all vegetation has a logistic growth function (KM ϭ K), Another way to think about Model IV is to consider and be stable when all vegetation has a regrowth func- it a model of consumer±resource interaction in which tion (KV ϭ K). For KM ϭ 2000 and for KV ϭ 300 resources possess an absolute refuge. In this interpre- (weighted average of summer and winter graminoid tation, the belowground plant biomass, inaccessible to standing crop biomass at Barrow) logistic vegetation herbivores, is a refuge, and there is movement of bio- declines to zero, while rodent and regrowth vegetation mass from the refuge to the vulnerable, aboveground settle on a stable equilibrium. Cycles occur for KV Յ biomass, represented by V. An absolute, or ``constant- 100, because regrowth vegetation cannot sustain the number,'' refuge should exert a powerful stabilizing herbivore population by itself (since KV is close to the in¯uence on dynamics in consumer±resource models half-saturation constant B) and, therefore, logistic veg- 1528 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6 etation does not go extinct, which allows cycles to by seasonal die-off in late August±September, before occur. it is preserved by being frozen. During winter, therefore, lemmings switch to a much higher utilization of Model VIIÐThree-trophic-level model mosses (see Batzli 1993: Table 1). The seasonal dietary The ®nal general model that we consider consists of shift from monocots to green mosses appears to be three trophic levels: vegetation±rodent herbivore±spe- paralleled by increased ability of brown lemmings to cialist predator. As our intent is to compare the dy- utilize mosses. Experimental feeding of brown lem- namics of this model to the model developed by Hanski mings with green mosses showed that during winter and co-authors for the vole±weasel interaction (Hanski months animals can survive for long periods on this et al. 1993, Hanski and KorpimaÈki 1995, Turchin and kind of food, while during summer they usually die Hanski 1997; see Hanski et al. 2001), we assume that within 2±3 d (Chernyavsky et al. 1981). vegetation dynamics are of regrowth type (since Mi- The starting point for developing equations describ- crotus voles eat primarily vascular plants). We also will ing lemming dynamics at Barrow (Alaska, USA) is assume the same functional form and parameter values provided by Model VI. Let V(t) and M(t) be the edible for predators used by Turchin and Hanski (1997). In biomass (in kilograms of dry mass per hectare) of vas- particular, the model assumes that predator population cular plants and mosses, respectively. We model dy- growth is logistic, with the ``carrying capacity'' pro- namics of monocot shoot biomass with a modi®cation portional to prey density and Q being the constant of of the regrowth equation that takes into account their proportionality (this is the Leslie-May model to which seasonal dynamics: growth in summer (two months we referred when discussing Model III). Perhaps not from melt-off in mid-June to ®rst heavy frosts in mid- surprisingly, the dynamics of Model VII, especially the August), rapid die-off of 90% of biomass during the cycle period (5 yr), are similar to those exhibited by transition period between summer and winter, and no the vole±weasel model. Thus, it appears that the her- change under snow during the winter months, except bivore±vegetation interaction in our tritrophic model for consumption by lemmings. Moss dynamics are plays a similar dynamical role to that of the phenom- modeled as a seasonally modi®ed logistic equation with enological logistic term in the prey equation of the growth in summer and no growth in winter. We assume vole±weasel model. no direct competition between mosses and monocots, so that in the absence of herbivory, both resources LEMMING±VEGETATION DYNAMICS AT BARROW, would increase to their respective ``carrying capacity'' ALASKA, USA (maximum standing crop), KM and KV, respectively. It is clear from this survey of general models that Note that the carrying capacity of these two types of (1) the dynamic interaction between plants and her- vegetation have somewhat different biological inter- bivores can, in theory, produce population cycles; and pretations. The carrying capacity of graminoids re¯ects (2) whether any particular system would oscillate de- the maximum biomass reached within one growing sea- pends very much on the details of functional forms and son. By contrast, the carrying capacity of moss is speci®c parameter values. This observation raises an achieved when the rate at which green biomass is added important theoretical question: Can any speci®c veg- is balanced by the rate of moss dieback (green moss etation±rodent system exhibit population oscillations turning into ``peat''). for biologically reasonable functional forms and pa- Because our model is not intended as an accurate, rameter values? That is the question we address in this detailed representation of population dynamics of lem- section. Our goal is to develop an empirically based mings at Barrow, but only as an exploration of the model for the interaction between the brown lemming, likely dynamical effects of the trophic link between Lemmus sibiricus (ϭtrimucronatus), and its food sup- lemmings and vegetation, we assume that lemming ply. We chose a lemming for this investigation because populations are regulated solely by food availability. unlike many vole species, whose food supply is better Let H(t) be the density of lemmings (individuals per modeled by a regrowth equation, both brown and Nor- hectare). Lemming consumption of vegetation is mod- wegian lemmings are known to rely on mosses (es- eled as a Type II functional response, and the lemming pecially during winter), whose dynamics we expect to equation is in the Rosenzweig-MacArthur (1963) form be well approximated by the logistic equation. (that is, the amount of food consumed directly affects the growth/decline rate of H). Because winter condi- Description of model tions impose much greater energy demands on lem- We begin by making the important distinction be- mings, the parameter G is assumed to change with sea- tween the two kinds of vegetation that together com- son, in turn affecting the maximum rate of lemming prise the food supply of the brown lemming: mosses population growth when food is abundant, rmax. The and shoots of vascular plants (primarily grasses and maximum rate of increase is, therefore, high in summer sedges). Graminoids have higher nutritional value and (assumed to be ϳ6yrϪ1), while during the rest of the provide the bulk of summer food for lemmings (Batzli year (the freeze-up period, period under snow, and the Ϫ1 1993). Monocot biomass, however, is greatly reduced melt-off period) the average rmax is lower at ϳ4yr June 2001 RODENT±VEGETATION INTERACTIONS 1529

(see Turchin and Ostfeld [1997] for a recent discussion of population growth rates characterizing arvicoline rodents). Finally, we explore lemming preference for vascular plants by including a discounting factor ␣, de®ned as follows: given equal biomass per hectare of graminoids and mosses, for each unit of graminoids, lemmings will consume ␣ units of mosses (thus, ␣Ͻ1). The equations resulting from these assumptions are shown in Table 1, Model VIII. The variable ␶ indicates season (0 Յ ␶Ͻ1), with ␶ϭ0 corresponding to the fall (transition between summer and winter). Seasonal dynamics are included in growth rates U(␶) and u(␶) and the consumption rate G(␶), which are the following functions of time: 5 Summer ( ⁄ Յ ␶Ͻ1): U(␶) ϭ Us, u(␶) ϭ us, G(␶) ϭ Gs. 5 FIG. 3. Maximal foraging rates of brown lemmings in Winter (0 Յ ␶ Յ ⁄): U(␶) ϭ 0, u(␶) ϭ 0, G(␶) ϭ Gw. In addition, at ␶ϭ0 (transition between summer and relation to availability of monocotyledons in arctic tundra (data from Batzli et al. [1981]). winter), V(t) is reduced by 90%.

Estimates of parameters (in the previous paragraph) into milligrams per minute, 1. Maximum rate of vegetation consumption, A.Ð we adjust the estimate of B by calculating the value on Based on the information on energy requirements of the x-axis corresponding to A/2 using the estimated brown lemmings (Collier et al. 1975), we calculate that curve in Fig. 3. This yields an estimate of B ϭ 7 g/m2 in summer a 50-g lemming needs to consume grami- or 70 kg/ha. noids at a rate of at least 12 g dry mass (dm)/d. We 3. Consumption rate at zero population growth, use an average mass of 50 g for lemmings, even though G(␶), and conversion factor, R.ÐWhen vegetation is adults grow to over twice that size, because the struc- not a limiting factor, the model assumes that the lem- ture of reproducing populations becomes skewed to- ming population grows exponentially with rmax ϭ R(A ward juveniles and subadults (Batzli 1975). A repro- Ϫ G(␶)). G(␶) is the rate of food consumption at which ductive female needs 1.8 times as much energy (22 the consumer population is just managing to replace gdm/d), and during winter energy demands increase by itself. It is related to the metabolic requirements of a 40±50%, so that the maximum intake (the amount lemming, and will change with season, since lemmings needed by a reproducing female in winter) is likely to need more energy in winter. We can estimate Gw, Gs, be about 32 gdm/d (12 kg/yr). Actual daily consump- and R from the following three equations: (a) R(A Ϫ tion of graminoids in summer by brown lemmings has Gs) ϭ 6; (b) R(A Ϫ Gw) ϭ 4; (c) Gs/Gw ϭ 0.7. The ®rst Ϫ1 been measured as 13.6 gdm/d for 24-g juveniles and two equations use the information that rmax is 6 yr in 22.9 gdm/d for 65-g nonbreeding adults by Melchior summer and 4 yrϪ1 in winter. The third relationship was (1972), as 12.5 g/d for 35-g nonbreeding subadults by obtained by assuming that G is directly proportional Batzli and Cole (1979), and as 23.1 g/d for nonbreeding to average daily metabolic rate (ADMR) and by solving adult brown lemmings (average weight: 66 g) by Kir- the empirical equation of Collier et al. (1975). Using yuschenko (1985). In light of these results, an estimate an average live body mass of 50 g and average tem- of A of ϳ12 kg/yr seems reasonable. However, this peratures of 5ЊC in summer and Ϫ15ЊC in winter (Batzli may be an underestimate of damage to plants because 1975, Batzli et al. 1980), this equation gives a ratio of lemmings also reduce vegetation by clipping, digging, 0.73 for summer to winter ADMR. Solving the equa- and trampling. Thus, we estimate that A is ϳ15 kg/yr. tions (a)±(c), we calculate that Gs/A ϭ 0.44 and Gw/A 2. Half-saturation constant, B.ÐThis parameter can ϭ 0.63. To check this result, we also calculate the ratio be estimated from the data of Batzli et al. (1981) who of minimum energy requirement to maximum intake, measured maximum rates of consumption by brown A. In summer this ratio is 12 g/d divided by 32 g/d ϭ lemmings (no search time) at various densities of veg- 0.38, and in winter 18 g/d (1.5 ϫ 12 g/d) divided again etation (Fig. 3). The relationship between dry plant by 32 g/d ϭ 0.56 (numerical values are taken from biomass consumed (in milligrams per minute) and Maximum rate of vegetation consumption, A, above). available forage (in grams per square meter) is well These ratios, 0.38 and 0.56, are of the same order of approximated by Type II functional response curve magnitude but somewhat lower than Gs/A ϭ 0.44 and with parameters A ϭ 83 mg/min ϭ 83 g/d (assuming Gw/A ϭ 0.63. The difference, presumably, is a re¯ection 70% of time spent foraging) and B ϭ 23.5 g/m2. How- of the fact that G includes both the minimum energy ever, this is surely an overestimate because lemmings requirement necessary to sustain one lemming, and en- do not forage at maximal rates all the while that they ergy necessary to replace it with another one of the are active. Translating our estimate of A ϭ 15 kg/yr next generation. 1530 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6

4. Maximum rate of moss growth, u.ÐCrude estimates of the parameter u can be obtained by estimating how long vegetation would take to recover from a low value to a value near carrying capacity. For example, if u ϭ 5yrϪ1 then it takes just under 1 yr for vegetation to grow from 10% to 90% of carrying capacity; if u ϭ 2yrϪ1, then it takes ϳ2 yr. The recovery time for mosses in Arctic ecosystems is at least 2±3 yr (Koshkina 1970), which implies u of ϳ2yrϪ1, or less. Observations on mosses at Barrow indicate that green moss at the beginning of the growing season is composed of growth from the previous two years and that the oldest (2-yr- old) tissue senesces and is replaced by new growth during the course of the season (Tieszen et al. 1980). Thus, most of the standing crop of green material rep- resents two years' growth, and we use an estimate of u ϭ 2yrϪ1 as an average over the whole year. Since mosses can grow only during the summer (that equals Ϫ1 1/6 yr), us ϭ 6u ϭ 12 yr .

5. Regrowth rate of graminoids, Us.ÐUsing Eq. 14 in Parton et al. (1993), and assuming that aboveground potential plant production rate is limited in Barrow only Ϫ2 Ϫ1 by temperature, we estimate Us ϭ 100 g´m ´mo ,or about 10 000 kg´haϪ1´yrϪ1.

6. Maximum standing crop, KV, and KM.ÐThe peak biomass of vascular plants (nearly all graminoids) for a mesic meadow at Barrow is about 80±100 g/m2,or 800±1000 kg/ha (Dennis and Tieszen 1972). We will assume KV ϭ 1000 kg/ha. Moss biomass is in the range of 400±800 g/m2, of which about 30% is green and probably edible (Rastorfer et al. 1974). This translates into KM of about 2000 kg/ha. 7. The discounting parameter for the relative consumption rate of mosses, ␣.ÐBecause we do not know how to estimate this parameter (except that it is Ͻ1), we will explore the dynamics of the model for a wide range of values, ␣ϭ0.1±1. FIG. 4. Dynamics of lemming populations and vegetation for the Barrow model (Model VIII in Table 1) using median Dynamics of the model values of parameters shown in Table 3. For the median values of parameters, and assuming that the discount factor ␣ϭ0.5, the model exhibits limit cycles with period of ϳ6 yr and amplitude 1±390 approach to the equilibrium is exponential; thus, for ␣ lemmings/ha (Fig. 4). Essentially the same kind of dy- ϭ 0 the model does not exhibit pseudoperiodic oscil- namics (4±7 yr cycle, and similar amplitude) hold for lations in the presence of noise. In other words, when values of ␣ϭ0.2±1. Shorter cycle periods obtain for ␣ approaches 0 very closely, model dynamics shift smaller values of ␣. For ␣Ͻ0.2, the amplitude of from second to ®rst order. cycles diminishes rapidly, and for ␣ϭ0.1, the long- This result suggests that when ␣ is not very low, term dynamics exhibit only seasonal oscillations. If we model dynamics are dominated by the interaction be- sample population trajectory once a year, it will ap- tween lemmings and mosses. If this is the case, then proach a stable equilibrium. However, the approach to the formula describing the stability of Model I may the equilibrium is oscillatory, and therefore, in the pres- also give the approximate conditions for the stability ence of noise, the dynamics will resemble noisy mul- of the Barrow model. Interpreting K in Model I as ␣KM tiannual (second-order) cycles. Such dynamics have and G as a seasonally weighted average of Gs and Gw been termed ``pseudoperiodic oscillations'' (Poole we obtain the following formula: 1977). As ␣ is decreased toward 0, the model dynamics BAϪ GÅ (when sampled once a year) approach the non-oscil- Ͻ (6) ␣KAϩ GÅ latory dynamics predicted by the model in which only M Å graminoids (regrowth-type vegetation) are present. The where G is an average of GW and GS. This conjectured June 2001 RODENT±VEGETATION INTERACTIONS 1531 formula was investigated numerically by varying each system. The models we developed here provide ex- parameter in the Barrow model one at a time (while amples of all four roles. Thus, food is a null factor in keeping the rest ®xed at their median values). As we Model II with the carrying capacity (®xed number of found above, decreasing ␣ below 0.2 stabilizes the dy- herbivores per unit area) KH (equilibrium density de- namics. Similarly, dynamics are stabilized when B is termined by intraspeci®c interactions) set low enough. increased above 250 kg/ha, KM is decreased below 600 It is a fast dynamic variable leading to stability in Mod- kg/ha, or Gs and Gw are increased by 50%. Changing el IV. Even when seasonality is added to the regrowth A or R does not change the qualitative type of dynamics. model (Model V), the resulting dynamics are ®rst-order All these numerical results con®rm that formula 6 gives oscillationsÐthat is, we should observe only direct, a good approximation for when dynamics will be os- undelayed density dependence when analyzing popu- cillatory vs. stable. lation data on the yearly time scale. In Models I, II, Using our best estimates of parameters, the Barrow and VIII food is a slow dynamic variable resulting in model predicts second-order oscillatory dynamics (ei- second-order oscillations, characterized by delayed ther limit cycles or pseudoperiodic oscillations), but density dependence. Depending on the parameter val- there are reasons to be cautious in interpreting this ues, the dynamics may be either unstable (limit cycles result. First, the model-predicted amplitude (about 400- or chaos) or an oscillatory approach to a stable equi- fold) is probably a bit low, since lemmings at Barrow librium. The difference between these two kinds of can ¯uctuate by a factor of at least 600-fold (Pitelka dynamics is not qualitative, however, because in the 1973: Table 1). Additionally, the period of six years in presence of noise pseudoperiodic oscillations behave the model output is longer than the ``typical'' microtine similarly to a perturbed limit cycle. Finally, in Model 4-yr cycle (but decreasing ␣ to 0.2 results in 4-yr cy- VII, food plays a double role: food availability provides cles). Second, to the best of our knowledge, brown direct density dependence necessary to stabilize the lemmings cannot survive and reproduce on a diet ex- predator±prey cycle (it is a fast dynamic variable), and clusively consisting of mosses. Feeding trials con- certain food properties also determine the stability of ducted in summer indicated poor digestibility and low the predator±prey interaction (it is a parameter). In par- voluntary intake of fresh mosses at Barrow (Batzli and ticular, reducing the maximum food supply, K, lowers Cole 1979). The only samples of winter diets of brown the prey population equilibrium in the absence of pred- lemmings at Barrow indicate substantial amounts of ators. This, in turn, reduces the amplitude and the pe- graminoids in their diets (Batzli and Pitelka 1983), and riod of oscillations, and eventually (around K ϭ 150) the same is true for Wrangel Island (Chernyavsky and stabilizes the system. The same effect was noted by Tkachev 1982). However, we do not have information Hanski et al. (1993) in the context of a ``pure'' pred- on nutrition of lemmings during winters when popu- ator±prey model that had direct density dependence in lation grows to peak densities. In the model, medium the form of the logistic growth term in the prey equation to high lemming densities (more than ϳ30 lemmings/ (see below). ha) quickly exhaust winter supply of graminoids and Although our mathematical and numerical analyses the lemmings have to subsist on a diet of almost 100% of Models I±VIII (Table 1) were necessarily limited by mosses. Furthermore, modifying the model in such a considerations of space, they nevertheless suggest sev- way that lemmings can eat at most 50% of mosses, we eral general principles. First, the dichotomy between ®nd that winter supply of food becomes a severe bot- logistic growth and linear initial regrowth is very im- tleneck for population increase. The dynamics of this portant. It is extremely easy to obtain second-order modi®ed model are characterized by summer increases cycles in the Rosenzweig-MacArthur (1963) model. to 40 lemmings/ha, and winter decreases to Ͻ10 lem- Almost any biologically reasonable combination of pa- mings/ha. In general, the winter supply of graminoids, rameters produces oscillations, except possibly when even when supplemented by an equal amount of mosses maximum biomass, K, is very low (e.g., in an arctic consumed, is suf®cient to produce at most 10 lem- desert). By contrast, the regrowth model cannot pro- mings/ha at the time of snowmelt. In reality, more than duce second-order oscillations (at least, for the pos- an order of magnitude greater lemming densities are tulated functional forms; the model can produce cycles observed at snowmelt during lemming peaks. Clearly, when the functional response of the herbivore is of a our understanding of lemming winter biology has not decreasing kind, see Abrams [1989]). According to our progressed to the point where we can construct a model theoretical results, we should not expect delayed den- for lemming dynamics tailored to the Barrow condi- sity-dependent cycles driven by the interaction with tions (or to any other location) with any degree of food when plants regrow rapidly, which seems likely con®dence. for voles (Microtus) at lower latitudes. Support for this prediction comes from the study by Ostfeld et al. (1993) DISCUSSION who observed a strong immediate, but no delayed (one The theoretical overview in the Introduction sug- year later) effect of high population densities of M. gested that food can play a number of different dy- pennsylvanicus on their food supplies. Thus, our the- namical roles depending on biological properties of the oretical results suggest that the difference in the food 1532 PETER TURCHIN AND GEORGE O. BATZLI Ecology, Vol. 82, No. 6 habits between lemmings and voles should have im- 1981, Moen et al. 1993). As the snow melts, remaining portant consequences for their population dynamics, lemmings often grub for rhizomes, preventing complete echoing the proposal of Oksanen and Oksanen (1992) recovery of graminoids during the following summer. that lemming oscillations are driven by their interaction This mechanism may add delayed density dependence with vegetation, while vole cycles are driven by their in graminoid recovery similar to that for mosses. Thus, interactions with predators. We note, however, that although the results of our models may not apply to ®rst-order oscillations are possible for folivores, if the all arvicolines, it seems clear that herbivore±plant in- growing season is short or maximum vegetation bio- teractions can account for the cyclic dynamics of at mass (K) is high. Furthermore, general insight based least some populations. on the logisitic/regowth dichotomy should be tempered Further development of realistic models for the dy- with the observation that the two alternative plant namics of arvicoline rodents and their interaction with growth functions are rather extreme simpli®cations of food availability will require additional data. In par- the complex reality. Better measurements of vegetation ticular, we need to have better measurement of the func- recovery after a herbivore outbreak are needed to de- tions for regrowth of vegetation and better information termine how closely vegetation dynamics conform to on rodent food habits, food quality and availability, either of the two postulated growth functions. and potential population growth during winter. Final A second principle is that the phenomenological pa- evaluation of the role of food availability in population rameter, vole carrying capacity, in vole±weasel models cycling will depend on innovative experiments that ma- (Hanski et al. 1993, Hanski and KorpimaÈki 1995, Tur- nipulate availability of food and measure associated chin and Hanski 1997) can be explicitly modeled as impact on population dynamics. Such experiments have population regulation based on the limitation by food often been hampered by the mobility of predators that supply, producing dynamics that are broadly similar to respond to local increases in voles when food avail- those previously reported. In these models, though nec- ability is increased (the pantry effect ®rst reported by essary, predation is not suf®cient for cycling to occur. Schultz [1969]), but experiments using large enclosures Some kind of nondelayed density-dependent regulation in the ®eld show promise of sorting out the effects of in the vole populations is required because specialist predation and food supply (Batzli 1992, Krebs et al. predator populations cannot increase rapidly enough to 1995, Klemola et al. 2000). stop the growth of increasing vole populations by them- ACKNOWLEDGMENTS selves. Our results indicate that food limitation can We are grateful to N. C. Stenseth and E. Framstad for provide the required fast dynamic feedback (this is re- extensive discussions of dynamics of lemming±vegetation in- ally an application of the general principle discussed teractions. We also thank P. Abrams, P. HambaÈck, I. Hanski, by Schaffer [1981]). L. Oksanen, and N. C. Stenseth for very helpful comments A third conclusion from our models is that popula- and criticisms of the manuscript. Our collaborative work was made possible by the support of the Norwegian Academy of tions of arvicolines should regularly overconsume their Sciences for the 1996 Oslo Workshop on arvicoline popu- food supply, if herbivore±food interactions produce cy- lation dynamics. P. Turchin and G. O. Batzli were also sup- cling dynamics (see Figs. 2 and 4). However, data based ported by NSF grants 95-09237 and 95-28571, respectively. upon total herbaceous production suggest that Cleth- LITERATURE CITED rionomys and Microtus voles living in temperate hab- Abrams, P. A. 1977. Density-independent mortality and in- itats may consume no more than 5% of available plant terspeci®c competition: a test of Pianka's niche overlap material (Krebs and Myers 1974: Table XIII). The dif- hypothesis. American Naturalist 111:539±552. ®culty with such ®gures is that all herbaceous plant Abrams, P. A. 1989. 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