Size and Scaling of Predator-Prey Dynamics to appear in Ecology Letters, 2006 Joshua S. Weitz 1, ∗ and Simon A. Levin 1, † 1 Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544 (Dated: April 28, 2006) We propose a scaled version of the Rosenzweig-MacArthur model using both Type I and Type II functional responses that incorporates the size-dependence of interaction rates. Our aim is to link the energetic needs of organisms with the dynamics of interacting populations, for which survival is a result of a game-theoretic struggle for existence. We solve the scaled model of predator-prey dynamics and predict population level characteristics such as the scaling of coexistence size ranges and the optimal predator-prey size ratio. For a broad class of such models, the optimal predator- prey size ratio given available prey of a fixed size is constant. We also demonstrate how scaling predictions of prey density differ under resource limitation vs. predator draw-down. Finally, we show how evolution of predator size can destabilize population dynamics, compare scaling of predator- prey cycles to previous work, as well as discuss possible extensions of the model to multispecies communities. I. INTRODUCTION Body size is a major factor constraining the struc- ture and functioning of organisms, and in particular their bioenergetic needs. The scaling of many organismal prop- erties can be shown empirically (and in some cases justi- fied theoretically) to obey simple allometric scaling rela- tionships of the form y = am b where y is an organismal property, m is organismal mass, a is a prefactor and b is a scaling exponent. Such scaling laws imply that some of the seemingly endless complex- ity in the biological world can be reduced when viewing systems in light of their inherent scale. In recent years, theories of optimal resource supply net- works (1–3) have reinvigorated and extended formative work on the relevance of scaling in biology (4–6) to a much broader and more ambitious goal. Briefly, there is currently an important debate unfolding on whether scaling laws can be extended beyond the individual to the population and even ecosystem level to develop a synthetic view of ecological structure based on under- lying energetic requirements (7–10). Population lev- el characteristics such as food web structure (11, 12), productivity (13), elemental ratios (14), the spread of pathogens (15), and home-range area (16), are all prop- erties that depend on the size of constituent individuals. In that sense, the ambitious goal of scaling up the theory of bioenergetics seems likely to contribute significantly to our understanding of what constrains the range of expect- ed ecological communities and ecological functions. However, there is a significant distinction between theories of optimal behavior/design versus those that * Electronic address: jsweitz@princeton.edu; URL: http://www. eeb.princeton.edu/~jsweitz † Electronic address: slevin@eno.princeton.edu describe dynamics whose interacting agents have possibly competing goals. The success of size-dependent strate- gies of foraging, predation, resource competition, repro- duction, etc. all depend on the strategies used by other organisms often in conflict and/or competition with one another. If we are to reconcile these views, the size and scaling of individuals must be incorporated into a game- theoretic, evolutionary framework (17–19). The basic rationale behind such an approach is that the fitness of a given type must be considered in light of other types and their strategies. Only then may we assess how the bioenergetic scaling of distinct individuals and popula- tions interact to form persistent, structured communities. The Rosenzweig-MacArthur model of predator-prey dynamics and the related Lotka-Volterra model of com- petition for a common resource are the basis for a mathe- matical description of interacting communities (20). We focus on predator-prey dynamics as a means of elucidat- ing how conclusions reached on the basis of individual optimality may differ from those obtained when multiple types are placed in competition. In so doing, we show how body size is central to the steady state behaviors that arise as a consequence of ecological interactions, a perspective first formalized in Yodzis & Innes (21). To frame the problem, consider the population dynamics of prey and predators with densities N and P respectively: dN dt = rN (1 − N/K) − g(N )P, (1) dP dt = ǫg(N )P − dP, (2) where r is the population growth rate of prey, K is stand- ing density of prey, ǫ is the conversion ratio of prey eaten to predator reproduction, d is the predator mor- tality rate, and g(N ) is the functional response of the predator. The body size-dependence of parameters in the Rosenzweig-MacArthur equation are discussed extensive- ly in the literature, e.g. (1, 4, 5, 21–23). What has not been accomplished, thus far, is to integrate these param- eterizations into a self-consistent dynamic model of eco- Typeset by REVT E X