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July  2018, 38(7): 3617-3636. doi: 10.3934/dcds.2018156

## Global dynamics in a two-species chemotaxis-competition system with two signals

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Depart. of Appl. Math., Chongqing Univ. of Posts and Telecommun., Chongqing 400065, China 3 College of Economic Math., Southwestern Univ. of Finance and Economics, Chengdu 611130, China

* Corresponding author: Xinyu Tu

Received  November 2017 Revised  February 2018 Published  April 2018

Fund Project: The first author is partially supported by the China Scholarship Council (201706050065). The second author is partially supported by National Natural Science Foundation of China (Grant Nos: 11771062, 11371384, 11571062), the Basic and Advanced Research Project of CQCSTC (Grant No: cstc2015jcyjBX0007). Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042). The fourth author is partially supported by Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006).

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type
 $\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$
with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain
 $Ω\subset R^n$
,
 $n≥2$
, 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
 $\frac{χ_{i}}{μ_{i}}$
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
 $\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$
in the norm of
 $L^{∞}(Ω)$
as
 $t\to ∞$
; If
 $a_{1}≥1>a_{2}>0$
and
 $μ_{2}$
is sufficiently large, the solution of the system converges to the constant stationary solution
 $\left(0, 1, 1, 0\right)$
as time tends to infinity, and the convergence rates can be calculated accurately.
Citation: Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156
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