DCDS-B
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity
Masaaki Mizukami
Discrete & Continuous Dynamical Systems - B 2017, 22(6): 2301-2319 doi: 10.3934/dcdsb.2017097
This paper deals with the two-species chemotaxis-competition system
$\left\{ {\begin{array}{*{20}{l}}{{u_t} = {d_1}\Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u - {a_1}v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{v_t} = {d_2}\Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - {a_2}u - v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{w_t} = {d_3}\Delta w + h(u,v,w)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\end{array}} \right.$
where
$\Omega$
is a bounded domain in
$\mathbb{R}^n$
with smooth boundary
$\partial \Omega$
,
$n\in \mathbb{N}$
;
$h$
,
$\chi_i$
are functions satisfying some conditions. In the case that
$\chi_i(w)=\chi_i$
, Bai–Winkler [1] proved asymptotic behavior of solutions to the above system under some conditions which roughly mean largeness of
$\mu_1, \mu_2$
. The main purpose of this paper is to extend the previous method for obtaining asymptotic stability. As a result, the present paper improves the conditions assumed in [1], i.e., the ranges of
$\mu_1, \mu_2$
are extended.
keywords: Chemotaxis Lotka–Volterra competition global existence asymptotic behavior stability

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