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Availability of Food and the Population Dynamics of Arvicoline Rodents

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Availability of Food and the Population Dynamics of Arvicoline Rodents Ecology, 82(6), 2001, pp. 1521±1534 q 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 pre- viously 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 de- scribed 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 dy- namics, 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; pop- ulation cycles in arvicoline (microtine) rodents; rodents, arvicoline; small-mammal population dy- namics and food supply; tritrophic interactions; voles. INTRODUCTION Henttonen et al. 1987, Klemola et al. 2000), but the- oretical 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 2 V dent dynamics (a null factor) because other density- 5 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 2 V conditions theoretical vegetation±herbivore systems 5 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,
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