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

Shi-Liang Wu and Wan-Tong Li

Chaos, Solitons & Fractals, 2009, vol. 40, issue 3, 1229-1239

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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:40:y:2009:i:3:p:1229-1239

DOI: 10.1016/j.chaos.2007.08.075

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