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January  2020, 25(1): 191-221. doi: 10.3934/dcdsb.2019178

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Liangchen Wang 1,, , Jing Zhang 1, , Chunlai Mu 2, and  Xuegang Hu 1,

1. 

Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Liangchen Wang

Received  December 2018 Revised  March 2019 Published  July 2019

This paper deals with the two-species chemotaxis system with two chemicals
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\nabla\cdot(u\chi_1(v)\nabla v)+\mu_1 u(1-u-a_1w),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v-\alpha v+f_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = d_3\Delta w-\nabla\cdot(w\chi_2(z)\nabla z)+\mu_2 w(1-w-a_2u),\quad &x\in \Omega,\quad t>0,\\ z_t = d_4\Delta z-\beta z+f_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*} $
under homogeneous Neumann boundary conditions in a bounded domain
$ \Omega\subset \mathbb{R}^n $
 (
$ n\geq1 $
), where the parameters
$ d_1,d_2,d_3,d_4>0 $
,
$ \mu_1,\mu_2>0 $
,
$ a_1,a_2>0 $
 and
$ \alpha, \beta>0 $
. The chemotactic function
$ \chi_i $
 (
$ i = 1,2 $
) and the signal production function
$ f_i $
 (
$ i = 1,2 $
) are smooth. If
$ n = 2 $
, it is shown that this system possesses a unique global bounded classical solution provided that
$ |\chi'_i| $
 (
$ i = 1,2 $
) are bounded. If
$ n\leq3 $
, this system possesses a unique global bounded classical solution provided that
$ \mu_i $
 (
$ i = 1,2 $
) are sufficiently large. Specifically, we first obtain an explicit formula
$ \mu_{i0}>0 $
 such that this system has no blow-up whenever
$ \mu_i>\mu_{i0} $
.
Moreover, by constructing suitable energy functions, it is shown that:
$ \bullet $
 If
$ a_1,a_2\in(0,1) $
 and
$ \mu_1 $
 and
$ \mu_2 $
 are sufficiently large, then any global bounded solution exponentially converges to
$\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$
$ f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$
 as
$ t\rightarrow\infty $
;
$ \bullet $
 If
$ a_1>1>a_2>0 $
 and
$ \mu_2 $
 is sufficiently large, then any global bounded solution exponentially converges to
$ (0,f_1(1)/\alpha,1,0) $
 as
$ t\rightarrow\infty $
;
$ \bullet $
 If
$ a_1 = 1>a_2>0 $
 and
$ \mu_2 $
 is sufficiently large, then any global bounded solution algebraically converges to
$ (0,f_1(1)/\alpha,1,0) $
 as
$ t\rightarrow\infty $
.
Keywords: Two-species chemotaxis, boundedness, exponential convergence, algebraical converge, Lotka-Volterra competition.
Mathematics Subject Classification: Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.
Citation: Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178
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. doi: 10.1512/iumj.2016.65.5776. Google Scholar

[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. doi: 10.1142/S021820251550044X. Google Scholar

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T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061. Google Scholar

[5]

M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001. Google Scholar

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T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar

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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. doi: 10.1007/s00332-010-9082-x. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

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J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. Google Scholar

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D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp. doi: 10.1007/s00033-017-0800-1. Google Scholar

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K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar

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K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. doi: 10.3934/dcdsb.2017094. Google Scholar

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K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429. Google Scholar

[16]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[17]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249. doi: 10.1002/mma.4607. Google Scholar

[18]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar

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M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. doi: 10.3934/dcdsb.2017097. Google Scholar

[20]

M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar

[21]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar

[22]

E. Nakaguchi and K. Osaki, $L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66. doi: 10.1619/fesi.59.51. Google Scholar

[23]

E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044. Google Scholar

[24]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar

[25]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009. Google Scholar

[26]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8. Google Scholar

[27]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar

[28]

H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar

[29]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[30]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar

[31]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar

[32]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014. Google Scholar

[33]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165. Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar

[35]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413. Google Scholar

[36]

X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636. doi: 10.3934/dcds.2018156. Google Scholar

[37]

L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401. doi: 10.1016/j.jde.2017.11.019. Google Scholar

[38]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. Google Scholar

[40]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar

[41]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: 10.1017/S0956792501004843. Google Scholar

[42]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032. Google Scholar

[43]

L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729. doi: 10.3934/dcdsb.2017132. Google Scholar

[44]

H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514. doi: 10.1088/1361-6544/aa96c9. Google Scholar

[45]

P. Zheng and C. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. doi: 10.1007/s10440-016-0083-0. Google Scholar

[46]

Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9pp. doi: 10.1063/1.5011725. Google Scholar

[47]

Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32. doi: 10.1016/j.aml.2018.03.012. Google Scholar

[48]

Q. Zhang and Y. Li, Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 467 (2018), 751-767. doi: 10.1016/j.jmaa.2018.07.037. Google Scholar

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P. Zheng, C. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558. Google Scholar

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. doi: 10.1512/iumj.2016.65.5776. Google Scholar

[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. doi: 10.1142/S021820251550044X. Google Scholar

[3]

T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar

[4]

T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061. Google Scholar

[5]

M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001. Google Scholar

[6] M. Eisenbach, Chemotaxis, (Imperial College Press, London, 2004. Google Scholar
[7]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar

[9]

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. doi: 10.1007/s00332-010-9082-x. Google Scholar

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[11]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016. Google Scholar

[12]

D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp. doi: 10.1007/s00033-017-0800-1. Google Scholar

[13]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar

[14]

K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. doi: 10.3934/dcdsb.2017094. Google Scholar

[15]

K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429. Google Scholar

[16]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[17]

M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249. doi: 10.1002/mma.4607. Google Scholar

[18]

M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar

[19]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. doi: 10.3934/dcdsb.2017097. Google Scholar

[20]

M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar

[21]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar

[22]

E. Nakaguchi and K. Osaki, $L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66. doi: 10.1619/fesi.59.51. Google Scholar

[23]

E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044. Google Scholar

[24]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar

[25]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009. Google Scholar

[26]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8. Google Scholar

[27]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar

[28]

H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar

[29]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[30]

C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar

[31]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar

[32]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014. Google Scholar

[33]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165. Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar

[35]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413. Google Scholar

[36]

X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636. doi: 10.3934/dcds.2018156. Google Scholar

[37]

L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401. doi: 10.1016/j.jde.2017.11.019. Google Scholar

[38]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. Google Scholar

[40]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023. Google Scholar

[41]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: 10.1017/S0956792501004843. Google Scholar

[42]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032. Google Scholar

[43]

L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729. doi: 10.3934/dcdsb.2017132. Google Scholar

[44]

H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514. doi: 10.1088/1361-6544/aa96c9. Google Scholar

[45]

P. Zheng and C. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. doi: 10.1007/s10440-016-0083-0. Google Scholar

[46]

Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9pp. doi: 10.1063/1.5011725. Google Scholar

[47]

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