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Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models

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  • Wu, Shi-Liang
  • Li, Wan-Tong

Abstract

This paper deals with the global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts in a class of reaction-diffusion systems. The known results do not apply in solving these problems because the reaction terms do not satisfy the required monotone condition. To overcome the difficulty, a weak monotone condition is proposed for the reaction terms, which is called interval monotone condition. Under such a weak monotone condition, the existence and comparison theorem of solutions is first established for reaction-diffusion systems on R by appealing to the theory of abstract differential equations. The global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts are then proved by the elementary super- and sub-solution comparison and squeezing methods for nonlinear evolution equations. Finally, these abstract results are applied to a two species competition-diffusion model and a system modeling man–environment–man epidemics.

Suggested Citation

  • Wu, Shi-Liang & Li, Wan-Tong, 2009. "Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1229-1239.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:3:p:1229-1239
    DOI: 10.1016/j.chaos.2007.08.075
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    References listed on IDEAS

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