A new approach toward stabilization in a two-species chemotaxis model with logistic source. (1st February 2018) - Lin, Ke; Mu, Chunlai; Zhong, Hua
Abstract: This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem u t = d 1 Δ u − χ 1 ∇ ⋅ ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ), x ∈ Ω, t > 0, v t = d 2 Δ v − χ 2 ∇ ⋅ ( v ∇ w ) + μ 2 v ( 1 − a 2 u − v ), x ∈ Ω, t > 0, 0 = d 3 Δ w − w + u + v, x ∈ Ω, t > 0, in a bounded domain Ω ⊂ R n, n ≥ 1, with smooth boundary, where d 1, d 2, d 3, χ 1, χ 2, μ 1, μ 2, a 1, a 2 are positive constants. When a 1 ∈ ( 0, 1 ) and a 2 ∈ ( 0, 1 ), it is shown that under some explicit largeness assumptions on the logistic growth coefficients μ 1 and μ 2, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state ( u ∗, v ∗, w ∗ ) of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential. When a 1 ≥ 1 and a 2 ∈ ( 0, 1 ), if μ 2 is suitable large, for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 ≢ 0 and v 0 ≢ 0, the globally bounded solution of above system will stabilize toward ( 0, 1, 1 ) as t → ∞ in algebraic.