# American Institute of Mathematical Sciences

August  2017, 22(6): 2301-2319. doi: 10.3934/dcdsb.2017097

## Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  June 2016 Revised  November 2016 Published  March 2017

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.
Citation: Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. [3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [4] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. [5] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. [7] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. [8] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [9] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, 1968. [10] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. [11] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. [12] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. [13] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. [14] Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.

show all references

##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583. [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. [3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [4] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. [5] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. [7] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. [8] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [9] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, 1968. [10] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. [11] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. [12] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. [13] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. [14] Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
 [1] Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071 [2] Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147 [3] De Tang. Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4913-4928. doi: 10.3934/dcdsb.2019037 [4] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [5] Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064 [6] Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055 [7] Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 [8] Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142 [9] Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290 [10] Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 [11] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [12] S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173 [13] Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771 [14] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 [15] Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650 [16] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 [17] Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 [18] Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953 [19] Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011 [20] Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

2020 Impact Factor: 1.327