American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise
  • DCDS-B Home
  • This Issue
  • Next Article
    Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China
doi: 10.3934/dcdsb.2021306
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Aichao Liu 1,2, , Binxiang Dai 1,, and  Yuming Chen 3,,

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 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.

Keywords: Boundedness, attraction-repulsion, chemotaxis, two species and chemicals.
Mathematics Subject Classification: Primary: 35A01, 35Q92, 92C17.
Citation: Aichao Liu, Binxiang Dai, Yuming Chen. Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021306
References:
[1]

S. Agmon, A. 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.  Google Scholar

[2]

S. Agmon, A. 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.  Google Scholar

[3]

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

[4]

P. Biler, E. 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.  Google Scholar

[5]

P. Biler, W. 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.  Google Scholar

[6]

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

[7]

K. Fujie, A. Ito, M. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

M. E. Hibbing, C. Fuqua, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Li, C. Mu, K. 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.  Google Scholar

[15]

J. Li, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Luca, A. Chavez-Ross, L. 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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  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]

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

[35]

M. Tian, X. 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.  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]

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.  Google Scholar

[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.  Google Scholar

[39]

L. Wang, J. Zhang, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

H. Yu, Q. 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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

J. Zhao, C. Mu, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

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

show all references

References:
[1]

S. Agmon, A. 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.  Google Scholar

[2]

S. Agmon, A. 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.  Google Scholar

[3]

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

[4]

P. Biler, E. 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.  Google Scholar

[5]

P. Biler, W. 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.  Google Scholar

[6]

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

[7]

K. Fujie, A. Ito, M. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

M. E. Hibbing, C. Fuqua, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Li, C. Mu, K. 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.  Google Scholar

[15]

J. Li, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Luca, A. Chavez-Ross, L. 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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  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]

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

[35]

M. Tian, X. 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.  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]

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.  Google Scholar

[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.  Google Scholar

[39]

L. Wang, J. Zhang, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

H. Yu, Q. 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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

J. Zhao, C. Mu, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

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

[1]

Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197
[2]

Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031
[3]

Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[4]

Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132
[5]

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
[6]

Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[7]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[8]

Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4585-4601. doi: 10.3934/dcdsb.2020114
[9]

Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165
[10]

Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260
[11]

Xu Pan, Liangchen Wang. On a quasilinear fully parabolic two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 361-391. doi: 10.3934/dcdsb.2021047
[12]

Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072
[13]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
[14]

Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027
[15]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[16]

Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
[17]

Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218
[18]

Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131
[19]

Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a two-species kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359-380. doi: 10.3934/krm.2015.8.359
[20]

Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (48)
  • HTML views (56)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy