Spreading and generalized propagating speeds of discrete KPP models in time varying environments

doi:10.1007/s11464-009-0032-6

Front Math Chin

2009, Vol. 4

Issue (3) : 523-562 https://doi.org/10.1007/s11464-009-0032-6

RESEARCH ARTICLE

Spreading and generalized propagating speeds of discrete KPP models in time varying environments

Wenxian SHEN(

)

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

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Abstract

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.

Keywords Discrete KPP (Kolmogorov Petrovsky and Paskunov) model time varying environment front front solution spreading speed interval generalized propagating speed interval traveling wave solution linear determinacy variational principle compact flow recurrent function

Corresponding Author(s): SHEN Wenxian,Email:wenxish@mail.auburn.edu

Issue Date: 05 September 2009

Cite this article:

Wenxian SHEN. Spreading and generalized propagating speeds of discrete KPP models in time varying environments[J]. Front Math Chin, 2009, 4(3): 523-562.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-009-0032-6
https://academic.hep.com.cn/fmc/EN/Y2009/V4/I3/523

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