Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop. (September 2020) - Tu, Xinyu; Mu, Chunlai; Qiu, Shuyan
Abstract: In the present study, the initial–boundary value problem for two-species chemotaxis-competition system with two signals ∂ t u 1 = Δ u 1 − χ 11 ∇ ⋅ ( u 1 ∇ v 1 ) − χ 12 ∇ ⋅ ( u 1 ∇ v 2 ) + μ 1 u 1 ( 1 − u 1 − a 1 u 2 ), x ∈ Ω, t > 0, ∂ t u 2 = Δ u 2 − χ 21 ∇ ⋅ ( u 2 ∇ v 1 ) − χ 22 ∇ ⋅ ( u 2 ∇ v 2 ) + μ 2 u 2 ( 1 − u 2 − a 2 u 1 ), x ∈ Ω, t > 0, ∂ t v 1 = Δ v 1 − λ 1 v 1 + α 11 u 1 + α 12 u 2, x ∈ Ω, t > 0, ∂ t v 2 = Δ v 2 − λ 2 v 2 + α 21 u 1 + α 22 u 2, x ∈ Ω, t > 0, is considered under the homogeneous Neumann boundary condition, where Ω ⊂ R n ( n ≤ 2 ) is a bounded domain with smooth boundary ∂ Ω, χ i j > 0, μ i > 0, a i > 0, α i j > 0, λ i > 0 ( i, j = 1, 2 ) . For any sufficiently regular nonnegative initial data, we investigate the global boundedness and regularity of the classical solution. Furthermore, by constructing some appropriate functionals, we establish that when a 1, a 2 ∈ ( 0, 1 ), if μ 1, μ 2 are sufficiently large, then the solution ( u 1, u 2, v 1, v 2 ) converges to 1 − a 1 1 − a 1 a 2, 1 − a 2 1 − a 1 a 2, α 11 ( 1 − a 1 ) + α 12 ( 1 − a 2 ) λ 1 ( 1 − a 1 a 2 ), α 21 ( 1 − a 1 ) + α 22 ( 1 − a 2 ) λ 2 ( 1 − a 1 a 2 ) exponentially as t → ∞ ; when a 1 ≥ 1 > a 2 > 0, if μ 2 is sufficiently large, then the solution ( u 1, u 2, v 1, v 2 ) converges to ( 0, 1, α 12 λ 1, α 22 λ 2 ) with exponential decay when a 1 > 1, and with algebraic decay when a 1 = 1 .