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Unformatted text preview: MATHEMATICAL BIOSCIENCES http://www.mbejournal.org/ AND ENGINEERING Volume 4 , Number 1 , January 2007 pp. 1–15 ALTERNATIVE MODELS FOR CYCLIC LEMMING DYNAMICS Hao Wang and Yang Kuang Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287 Abstract. Many natural population growths and interactions are affected by seasonal changes. This suggests that these natural population dynamics should be modeled by nonautonomous differential equations instead of au- tonomous differential equations. Through a series of carefully derived models of the well documented high-amplitude, large-period fluctuations of lemming populations, we argue that when appropriately formulated, autonomous dif- ferential equations may capture much of the desirable rich dynamics such as the existence of a periodic solution with period and amplitude close to that of approximately periodic solutions produced by the more natural but mathemat- ically daunting nonautonomous models. We start this series of models from the Barrow model, a well formulated model for the dynamics of food-lemming interaction at Point Barrow (Alaska, USA) with sufficient experimental data. Our work suggests that autonomous system can indeed be a good approxima- tion to the moss-lemming dynamics at Point Barrow. This together with our bifurcation analysis indicate that neither seasonal factor (expressed by time dependent moss growth rate and lemming death rate in Barrow model), nor the moss growth rate and lemming death rate are the main culprits of the observed multi-year lemming cycles. We suspect that main culprits may in- clude high lemming predation rate, high lemming birth rate and low lemming self-limitation rate. 1. Introduction. Pioneer works on resource-consumer dynamics include the well known works of Lotka (1925) and Volterra (1926) which introduced the classical Lotka-Volterra predator-prey model that arguably forms the foundation of math- ematical ecology. One of the most frequently used resource-consumer models is the Rosenzweig-MacArthur (1963) model, which produces two generic asymptotic behaviors - equilibria and limit cycles. Bazykin (1974) added a self-limitation term to the Rosenzweig-MacArthur model to account for the rather ubiquitous density dependent mortality rate (see also Bazykin et al. 1998). All these models produce oscillatory solutions that seem to mimic the fluctuating populations observed in nature. Indeed, large-scale high-amplitude oscillations in populations of small rodents such as voles and lemmings have been a constant inspiration to numerous influential and thought provoking articles since the pioneering work of Elton (Elton 1924, Hanski et al. 2001). Lemming is a mouse like arctic rodent characterized by a small, short body that is about 13 cm (about 5 in) long, with a very short tail....
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