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October  2022, 27(10): 6037-6062. doi: 10.3934/dcdsb.2021306

Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals

1. 

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China

2. 

School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China

3. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada

* Corresponding author: Binxiang Dai and Yuming Chen

Received  May 2021 Revised  November 2021 Published  October 2022 Early access  January 2022

This paper deals with a class of attraction-repulsion chemotaxis systems in a smoothly bounded domain. When the system is parabolic-elliptic-parabolic-elliptic and the domain is $ n $-dimensional, if the repulsion effect is strong enough then the solutions of the system are globally bounded. Meanwhile, when the system is fully parabolic and the domain is either one-dimensional or two-dimensional, the system also possesses a globally bounded classical solution.

Citation: Aichao Liu, Binxiang Dai, Yuming Chen. Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 6037-6062. doi: 10.3934/dcdsb.2021306
References:
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S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Commun. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

N. BellomoA. BellouquidY. 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.

[4]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[5]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[6]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[7]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[9]

Q. GuoZ. Jiang and S. Zheng, Critical mass for an attraction-repulsion chemotaxis system, Appl. Anal., 97 (2018), 2349-2354.  doi: 10.1080/00036811.2017.1366989.

[10]

X. He, M. Tian and S. Zheng, Large time behavior of solutions to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 54 (2020), 103095, 14 pp. doi: 10.1016/j.nonrwa.2020.103095.

[11]

M. E. HibbingC. FuquaM. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nat. Rev. Microbiol., 8 (2010), 15-25.  doi: 10.1038/nrmicro2259.

[12]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Kesser-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.

[13]

H.-Y. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimenssions, Discrete Contin. Dyn. Syst. B., 23 (2018), 3071-3085.  doi: 10.3934/dcdsb.2017197.

[14]

D. LiC. MuK. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, J. Math. Anal. Appl., 448 (2017), 914-936.  doi: 10.1016/j.jmaa.2016.11.036.

[15]

J. LiY. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002.

[16]

J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051.

[17]

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

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.

[19]

Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942.

[20]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.

[21]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.

[22]

N. Mizoguchi and Ph. 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.

[23]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

[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.

[25]

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.

[26]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Q., 10 (2002), 501-543. 

[27]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theoret. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[28]

H. Qiu and S. Guo, Global existence and stablity in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. B., 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.

[29]

C. StinnerJ. 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.

[30]

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.

[31]

Y. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.

[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.

[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.

[34]

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.

[35]

M. TianX. He and S. Zheng, Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 30 (2016), 1-15.  doi: 10.1016/j.nonrwa.2015.11.004.

[36]

X. TuC. MuP. 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.

[37]

G. Viglialoro, Explicit lower bound of blow-up time for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 479 (2019), 1069-1077.  doi: 10.1016/j.jmaa.2019.06.067.

[38]

L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.

[39]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stablization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.

[40]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[41]

P. Xu and S. Zheng, Global boundedness in an attraction-repulsion chemotaxis system with logistic source, Appl. Math. Lett., 83 (2018), 1-6.  doi: 10.1016/j.aml.2018.03.007.

[42]

H. YuQ. Guo and S. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342.  doi: 10.1016/j.nonrwa.2016.09.007.

[43]

H. YuW. 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.

[44]

Y. Zeng, Existence of global bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal., 161 (2017), 182-197.  doi: 10.1016/j.na.2017.06.003.

[45]

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.

[46]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

[47]

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, 9 pp. doi: 10.1063/1.5011725.

[48]

J. ZhaoC. MuD. Zhou and K. Lin, A parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source, J. Math. Anal. Appl., 455 (2017), 650-679.  doi: 10.1016/j.jmaa.2017.05.068.

[49]

J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/TMNA.2016.082.

[50]

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.

[51]

P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys., 58 (2017), 111501, 17 pp. doi: 10.1063/1.5010681.

[52]

P. ZhengC. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Differential Integral Equations, 31 (2018), 547-558. 

show all references

References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Commun. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

N. BellomoA. BellouquidY. 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.

[4]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[5]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[6]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[7]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[9]

Q. GuoZ. Jiang and S. Zheng, Critical mass for an attraction-repulsion chemotaxis system, Appl. Anal., 97 (2018), 2349-2354.  doi: 10.1080/00036811.2017.1366989.

[10]

X. He, M. Tian and S. Zheng, Large time behavior of solutions to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 54 (2020), 103095, 14 pp. doi: 10.1016/j.nonrwa.2020.103095.

[11]

M. E. HibbingC. FuquaM. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nat. Rev. Microbiol., 8 (2010), 15-25.  doi: 10.1038/nrmicro2259.

[12]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Kesser-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.

[13]

H.-Y. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimenssions, Discrete Contin. Dyn. Syst. B., 23 (2018), 3071-3085.  doi: 10.3934/dcdsb.2017197.

[14]

D. LiC. MuK. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, J. Math. Anal. Appl., 448 (2017), 914-936.  doi: 10.1016/j.jmaa.2016.11.036.

[15]

J. LiY. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002.

[16]

J. Li and Y. Wang, Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.  doi: 10.1016/j.jmaa.2018.07.051.

[17]

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

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.

[19]

Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942.

[20]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.

[21]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.

[22]

N. Mizoguchi and Ph. 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.

[23]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.

[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.

[25]

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.

[26]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Q., 10 (2002), 501-543. 

[27]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theoret. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[28]

H. Qiu and S. Guo, Global existence and stablity in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. B., 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.

[29]

C. StinnerJ. 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.

[30]

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.

[31]

Y. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.

[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.

[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.

[34]

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.

[35]

M. TianX. He and S. Zheng, Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 30 (2016), 1-15.  doi: 10.1016/j.nonrwa.2015.11.004.

[36]

X. TuC. MuP. 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.

[37]

G. Viglialoro, Explicit lower bound of blow-up time for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 479 (2019), 1069-1077.  doi: 10.1016/j.jmaa.2019.06.067.

[38]

L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.

[39]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stablization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.

[40]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[41]

P. Xu and S. Zheng, Global boundedness in an attraction-repulsion chemotaxis system with logistic source, Appl. Math. Lett., 83 (2018), 1-6.  doi: 10.1016/j.aml.2018.03.007.

[42]

H. YuQ. Guo and S. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342.  doi: 10.1016/j.nonrwa.2016.09.007.

[43]

H. YuW. 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.

[44]

Y. Zeng, Existence of global bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal., 161 (2017), 182-197.  doi: 10.1016/j.na.2017.06.003.

[45]

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.

[46]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

[47]

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, 9 pp. doi: 10.1063/1.5011725.

[48]

J. ZhaoC. MuD. Zhou and K. Lin, A parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source, J. Math. Anal. Appl., 455 (2017), 650-679.  doi: 10.1016/j.jmaa.2017.05.068.

[49]

J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/TMNA.2016.082.

[50]

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.

[51]

P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys., 58 (2017), 111501, 17 pp. doi: 10.1063/1.5010681.

[52]

P. ZhengC. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Differential Integral Equations, 31 (2018), 547-558. 

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