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.{u1t=Δu1−χ1∇⋅(u1∇w)+μ1u1(1−∑i=1na1iui),x∈Ω,t>0,u2t=Δu2−χ2∇⋅(u2∇w)+μ2u2(1−∑i=1na2iui),x∈Ω,t>0,⋯⋯unt=Δun−χn∇⋅(un∇w)+μnun(1−∑i=1naniui),x∈Ω,t>0,−Δw+λw=∑i=1nui,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN(N≥1)Ω⊂RN(N≥1) with smooth boundary. We prove that if μiμi, χiχi and the following matrixA=(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann) satisfy certain properties, then all solutions of this system will stabilize towards a positive equilibrium {ui⁎}i=1,…,n which is globally asymptotically stable.