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A gradient from stable to cyclic populations of Clethrionomys rufocanus in Hokkaido, Japan
, 1996
"... this paper were collected in forested regions of northern Hokkaido (Fig. 1; 41 ..."
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this paper were collected in forested regions of northern Hokkaido (Fig. 1; 41
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling
, 2007
"... Abstract: This paper is to show that all but one discrete models used for population dynamics in ecology are inherently paradoxical that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to resolve an ongoi ..."
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Abstract: This paper is to show that all but one discrete models used for population dynamics in ecology are inherently paradoxical that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to resolve an ongoing controversy regarding ecological chaos. Another implication of the result is that all continuous dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.
Can noise induce chaos?
, 2003
"... An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended ..."
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An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a longterm average over the deterministic attractor while the SLE is the longterm average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE’s should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that ‘‘chaos’ ’ should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.