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Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity
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 |
$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$ |
$Ω\subset R^n$ |
$n≥2$ |
$χ_{i}$ |
$μ_{i}$ |
$a_{i}$ |
$(i = 1, 2)$ |
$χ_{i}$ |
$μ_{i}$ |
$a_{i}$ |
$(i = 1, 2)$ |
$(u_{0}, w_{0})$ |
$\frac{χ_{i}}{μ_{i}}$ |
$a_{1}, a_{2}∈ (0, 1)$ |
$μ_{1}$ |
$μ_{2}$ |
$(u, v, w, z)$ |
$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$ |
$L^{∞}(Ω)$ |
$t\to ∞$ |
$a_{1}≥1>a_{2}>0$ |
$μ_{2}$ |
$\left(0, 1, 1, 0\right)$ |
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. |
[2] |
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. |
[3] |
J. Cao, W. Wang and H. Yu,
Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392.
doi: 10.1016/j.jmaa.2015.11.058. |
[4] |
E. E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[5] |
A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969. |
[6] |
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. |
[7] |
C. Gai, Q. Wang and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[8] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[9] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[10] |
D. Horstmann,
Generaizing 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. |
[11] |
M. W. Htwe and Y.F Wang,
Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83.
doi: 10.1016/j.crma.2016.10.024. |
[12] |
J. Hu, Q. Wang, J. Yang and L. Zhang,
Globale existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.
doi: 10.3934/krm.2015.8.777. |
[13] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
R. Kowalczyk and Z. Szyma |
[16] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[17] |
Y. Li,
Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304.
doi: 10.1016/j.jmaa.2015.04.052. |
[18] |
K. Lin and C. L. 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. |
[19] |
K. Lin, C. L. Mu and L. C. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[20] |
K. Lin, C. L. Mu and L. C. Wang,
Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[21] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[22] |
P. Liu, J. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[23] |
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. |
[24] |
M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164. Google Scholar |
[25] |
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. |
[26] |
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.
doi: 10.1016/j.jde.2016.05.008. |
[27] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.
|
[33] |
K. Osaki, T. Tsujikawa, T. A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[34] |
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. |
[35] |
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. |
[36] |
Y. S. 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. |
[37] |
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. |
[38] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[39] |
Q. Wang, J. Yang and L. Zhang,
Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.
doi: 10.3934/dcdsb.2017179. |
[40] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[41] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[42] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[43] |
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. |
[44] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[45] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[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 and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[48] |
J. Zheng,
Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480.
|
[49] |
P. Zheng and C. L. 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. |
[50] |
P. Zheng, C. L. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals J. Math. Phys. 58 (2017), 111501, 17pp.
doi: 10.1063/1.5010681. |
[51] |
P. Zheng, C. L. 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. |
[2] |
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. |
[3] |
J. Cao, W. Wang and H. Yu,
Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl., 436 (2016), 382-392.
doi: 10.1016/j.jmaa.2015.11.058. |
[4] |
E. E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[5] |
A. Friedman, Partoal Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que-London, 1969. |
[6] |
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. |
[7] |
C. Gai, Q. Wang and J. Yan,
Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[8] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[9] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[10] |
D. Horstmann,
Generaizing 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. |
[11] |
M. W. Htwe and Y.F Wang,
Boundedness in a full parabolic two-species chemotaxis system, C. R. Acad. Sci. Ser. I., 355 (2017), 80-83.
doi: 10.1016/j.crma.2016.10.024. |
[12] |
J. Hu, Q. Wang, J. Yang and L. Zhang,
Globale existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.
doi: 10.3934/krm.2015.8.777. |
[13] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
R. Kowalczyk and Z. Szyma |
[16] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[17] |
Y. Li,
Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, J. Math. Anal. Appl., 429 (2015), 1291-1304.
doi: 10.1016/j.jmaa.2015.04.052. |
[18] |
K. Lin and C. L. 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. |
[19] |
K. Lin, C. L. Mu and L. C. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[20] |
K. Lin, C. L. Mu and L. C. Wang,
Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[21] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[22] |
P. Liu, J. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[23] |
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. |
[24] |
M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system, AIMS Mathematics, 1 (2016), 156-164. Google Scholar |
[25] |
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. |
[26] |
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.
doi: 10.1016/j.jde.2016.05.008. |
[27] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469.
|
[33] |
K. Osaki, T. Tsujikawa, T. A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Real World Appl., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[34] |
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. |
[35] |
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. |
[36] |
Y. S. 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. |
[37] |
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. |
[38] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[39] |
Q. Wang, J. Yang and L. Zhang,
Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.
doi: 10.3934/dcdsb.2017179. |
[40] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[41] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[42] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[43] |
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. |
[44] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[45] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[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 and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[48] |
J. Zheng,
Boundedness in a two-species quasilinear chemotaxis system with two chemicals, Topol. Methods Nonl. Anal., 49 (2017), 463-480.
|
[49] |
P. Zheng and C. L. 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. |
[50] |
P. Zheng, C. L. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals J. Math. Phys. 58 (2017), 111501, 17pp.
doi: 10.1063/1.5010681. |
[51] |
P. Zheng, C. L. 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 |
[1] |
Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 |
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