Alternative models for cyclic lemming dynamics

Many natural population growths and interactions are affected by seasonal changes, suggesting that these natural population dynamics should be modeled by nonautonomous differential equations instead of autonomous 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 differential 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 mathematically 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 an autonomous system can indeed be a good approximation to the moss-lemming dynamics at Point Barrow. This, together with our bifurcation analysis, indicates that neither seasonal factors (expressed by time-dependent moss growth rate and lemming death rate in the 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 the main culprits may include high lemming predation rate, high lemming birth rate, and low lemming self-limitation rate.