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Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals
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 |
$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $ |
$ \Omega\subset \mathbb{R}^n $ |
$ n\geq1 $ |
$ \chi_i>0 $ |
$ \mu_i>0 $ |
$ a_i>0 $ |
$ i = 1, 2 $ |
$ q_1\leq a_1; $ |
$ q_2\leq a_2 $ |
$ q_1>a_1 $ |
$ q_2> a_2 $ |
$ (q_1-a_1)(q_2-a_2)<1 $ |
$ q_1: = \frac{\chi_1}{\mu_1} $ |
$ q_2: = \frac{\chi_2}{\mu_2} $ |
$ a_1, a_2\in(0, 1) $ |
$ \mu_1 $ |
$ \mu_2 $ |
$ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $ |
$ t\rightarrow\infty $ |
$ a_1>1>a_2>0 $ |
$ \mu_2 $ |
$ (0, 1, 1, 0) $ |
$ t\rightarrow\infty $ |
$ a_1 = 1>a_2>0 $ |
$ \mu_2 $ |
$ (0, 1, 1, 0) $ |
$ t\rightarrow\infty $ |
References:
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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. |
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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.
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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.
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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. |
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M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. |
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T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
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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.
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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.
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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.
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B. Hu and Y. Tao,
Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.
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Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
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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.
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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. |
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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. |
[15] |
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. |
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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. |
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K. Lin, C. Mu and and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
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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.
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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.
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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. |
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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.
doi: 10.3934/dcdss.2020015. |
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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. |
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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. |
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K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
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K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[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. |
[27] |
H. Qiu and S. Guo,
Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.
doi: 10.3934/dcdsb.2018220. |
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P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007. |
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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. |
[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. |
[31] |
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. |
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J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
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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. |
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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. |
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Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, preprint, arXiv: 1510.07204. |
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L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705, 10 pp.
doi: 10.1016/j.jmaa.2019.123705. |
[37] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[38] |
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. |
[39] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[40] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[41] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[42] |
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. |
[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,
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. |
[45] |
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. |
[46] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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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.
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How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
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On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.
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On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.
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Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.
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Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.
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Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 467 (2018), 751-767.
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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.
|
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] |
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. |
[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. |
[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. |
[5] |
M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. |
[6] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[7] |
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. |
[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.
|
[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. |
[10] |
B. Hu and Y. Tao,
Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.
doi: 10.1016/j.aml.2016.08.003. |
[11] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
K. Lin, C. Mu and and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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.
doi: 10.3934/dcdss.2020015. |
[22] |
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. |
[23] |
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. |
[24] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[25] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[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. |
[27] |
H. Qiu and S. Guo,
Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.
doi: 10.3934/dcdsb.2018220. |
[28] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007. |
[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. |
[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. |
[31] |
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. |
[32] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[33] |
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. |
[34] |
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. |
[35] |
Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, preprint, arXiv: 1510.07204. |
[36] |
L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705, 10 pp.
doi: 10.1016/j.jmaa.2019.123705. |
[37] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[38] |
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. |
[39] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[40] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[41] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[42] |
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. |
[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,
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. |
[45] |
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. |
[46] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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