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Competitive exclusion for a two-species chemotaxis system with two chemicals

Research paper by Qingshan Zhang

Indexed on: 29 Apr '18Published on: 15 Apr '18Published in: Applied Mathematics Letters



Abstract

Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Qingshan Zhang In this paper we consider the following competitive two-species chemotaxis system with two chemicals u t = Δ u − χ 1 ∇ ⋅ ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , x ∈ Ω , t > 0 , 0 = Δ v − v + w , x ∈ Ω , t > 0 , w t = Δ w − χ 2 ∇ ⋅ ( w ∇ z ) + μ 2 w ( 1 − a 2 u − w ) , x ∈ Ω , t > 0 , 0 = Δ z − z + u , x ∈ Ω , t > 0 in a smooth bounded domain Ω ⊂ R n with n ≥ 1 , where χ i ≥ 0 , a i ≥ 0 and μ i > 0 ( i = 1 , 2 ) . For the case a 1 > 1 > a 2 ≥ 0 , it will be proved that if χ 1 χ 2 < μ 1 μ 2 , χ 1 ≤ a 1 μ 1 and χ 2 < μ 2 , then the initial–boundary value problem with homogeneous Neumann boundary condition admits a unique global bounded solution and ( u , v , w , z ) → ( 0 , 1 , 1 , 0 ) uniformly on Ω ̄ as t → ∞ .