Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Volterra competition system
By Xiongxiong Bao, Zhi-Cheng Wang. 2013
107 citations recorded by the source corpus.
Abstract This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system { u t = d 1 ( t ) u x x + u ( r 1 ( t ) − a 1 ( t ) u − b 1 ( t ) v ) , v t = d 2 ( t ) v x x + v ( r 2 ( t ) − a 2 ( t ) u − b 2 ( t ) v ) , x ∈ R , t > 0 , where d i ( t ) , r i ( t ) , a i ( t ) , b i ( t ) ∈ C θ 2 ( R ) are T -periodic functions, 0 θ 1 , i = 1 , 2 . Under certain conditions, the system admits two stable semi-trivial periodic solutions ( p ( t ) , 0 ) and ( 0 , q ( t ) ) and a unique coexistence periodic solution ( u 0 ( t ) , v 0 ( t ) ) , which is unstable and satisfies 0 u 0 ( t ) p ( t ) and 0 v 0 ( t ) q ( t ) for t ∈ [ 0 , T ] . In this paper we prove that the system admits a time periodic traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x + c t , t ) , V ( x + c t , t ) ) connecting two periodic solutions ( p ( t ) , 0 ) and ( 0 , q ( t ) ) as x → ± ∞ , where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution ( U ( x + c t , t ) , V ( x + c t , t ) ) is asymptotically stable and unique modulo translation for front-like initial values.
Subfields: Applied Mathematics, Geometry and Topology, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/be3fa40e3c8f2e40b96809f29f17d49cd683f7c7
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In Applied Mathematics
In Geometry and Topology
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Citations in the corpus
- Traveling waves and spreading speeds for time-space periodic monotone systems (2015, unknown): “, [1, 2, 10, 26, 46, 47, 45] and references therein.”
- Spreading speeds for a two-species competition-diffusion system (2018, background): “As the overall behaviour of solutions from [3] only depends on the means and extrema of the parameters, our results would not be drastically modified ; we thus decided not to take this phenomenon into account.”