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In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type        ut = ∆u − χ 1 ∇ • (u∇v) + µ 1 u(1 − u − a 1 w), x ∈ Ω, t > 0, 0 = ∆v − v + w, x ∈ Ω, t > 0, wt = ∆w − χ 2 ∇ • (w∇z) + µ 2 w(1 − a 2 u − w), x ∈ Ω, t > 0, 0 = ∆z − z + u, x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain Ω ⊂ R n , n ≥ 2, where χ i , µ i and a i (i = 1, 2) are positive constants. It is shown that for any positive parameters χ i , µ i , a i (i = 1, 2) and any suitably regular initial data (u 0 , w 0), this system possesses a global bounded classical solution provided that χ i µ i are small. Moreover, when a 1 , a 2 ∈ (0, 1) and the parameters µ 1 and µ 2 are sufficiently large, it is proved that the global solution (u, v, w, z) of this system exponentially approaches to the steady state

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