Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

This paper deals with the two-species chemotaxis-competition system. About the problem, Bai--Winkler first obtained asymptotic stability under some conditions. Recently, the conditions assumed in the previous work were improved; however, there is a gap between the conditions assumed in the previous works. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior.