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 {u(t) = d(1)Delta u - del center dot(u chi(1)(w)del w) + mu(1)u(1 - u - a(1)v) in Omega x (0, infinity), v(t) = d(2)Delta u - del center dot(u chi(2)(w)del w) + mu(2)u(1 - a(2)u - v) in Omega x (0, infinity), w(t) = d(3)Delta u + alpha u + beta v -gamma w in Omega x (0, infinity), where Omega is a bounded domain in Rn with smooth boundary partial derivative Omega, n >= 2; chi(i) are functions satisfying some conditions. About this problem, Bai-Winkler [1] first obtained asymptotic stability in (1) under some conditions in the case that a(1), a(2) is an element of (0, 1). Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that a(1), a(2) is an element of (0, 1).