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Abstract
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.
Mathematics Subject Classification: 92D40, 34K20.
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