Stabilization in an n-species chemotaxis system with a logistic source.

In this paper we consider the following system involving more than two competitive populations of biological species all of which are attracted by the same chemoattractant. { u 1 t = Δ u 1 − χ 1 ∇ ⋅ ( u 1 ∇ w ) + μ 1 u 1 ( 1 − ∑ i = 1 n a 1 i u i ) , x ∈ Ω , t > 0 , u 2 t = Δ u 2 − χ 2 ∇ ⋅ ( u 2 ∇ w ) + μ 2 u 2 ( 1 − ∑ i = 1 n a 2 i u i ) , x ∈ Ω , t > 0 , ⋯ ⋯ u n t = Δ u n − χ n ∇ ⋅ ( u n ∇ w ) + μ n u n ( 1 − ∑ i = 1 n a n i u i ) , x ∈ Ω , t > 0 , − Δ w + λ w = ∑ i = 1 n u i , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R N ( N ≥ 1 ) with smooth boundary. We prove that if μ i , χ i and the following matrix A = ( a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a n 1 a n 2 ⋯ a n n ) satisfy certain properties, then all solutions of this system will stabilize towards a positive equilibrium { u i ⁎ } i = 1 , … , n which is globally asymptotically stable. [ABSTRACT FROM AUTHOR]