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Rapid fluctuation for topological dynamical systems
Yu HUANG, Yi ZHOU
Front Math Chin. 2009, 4 (3): 483-494.
https://doi.org/10.1007/s11464-009-0030-8
In this paper, we introduce a new notion called rapid ?uctuation to characterize the complexity of a general topological dynamical system. As a continuation of the former work [Huang, Chen, Ma, J. Math. Anal. Appl., 2006, 323: 228-252], here we prove that a Lipschitz dynamical system de?ned on a compact metric space has a rapid ?uctuation if it has either a quasi shift invariant set or a topological horseshoe. As an application, the rapid ?uctuation of a discrete predator-prey model is considered.
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Spreading and generalized propagating speeds of discrete KPP models in time varying environments
Wenxian SHEN
Front Math Chin. 2009, 4 (3): 523-562.
https://doi.org/10.1007/s11464-009-0032-6
The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.
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