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This paper is concerned with the two‐species chemotaxis‐competition systemu  t   =   d  1   Δ  u  −   χ  1   ∇  ·  (  u  ∇  w  )  +   μ  1   u  (  1  −  u  −   a  1   v  )    in    Ω  ×  (  0  ,  ∞  )  ,v  t   =   d  2   Δ  v  −   χ  2   ∇  ·  (  v  ∇  w  )  +   μ  2   v  (  1  −   a  2   u  −  v  )    in    Ω  ×  (  0  ,  ∞  )  ,0  =   d  3   Δ  w  +  α  u  +  β  v  −  γ  w    in    Ω  ×  (  0  ,  ∞  )  ,where Ω is a bounded domain in    R  n    with smooth boundary  ∂ Ω,  n ≥2;  χ   i   and  μ   i   are constants satisfying some conditions. The above system was studied in the cases that  a  1 , a  2 ∈(0,1) and  a  1 >1> a  2 , and it was proved that global existence and asymptotic stability hold whenχ  iμ  iare small. However, the conditions in the above 2 cases strongly depend on  a  1 , a  2 , and have not been obtained in the case that  a  1 , a  2 ≥1. Moreover, convergence rates in the cases that  a  1 , a  2 ∈(0,1) and  a  1 >1> a  2  have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all  a  1 , a  2 >0 which covers the case that  a  1 , a  2 ≥1, and lead to convergence rates for solutions of the above system in the cases that  a  1 , a  2 ∈(0,1) and  a  1 ≥1> a  2 .

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type