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Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary
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 |
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\nabla\cdot(u\chi_1(v)\nabla v)+\mu_1 u(1-u-a_1w),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v-\alpha v+f_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = d_3\Delta w-\nabla\cdot(w\chi_2(z)\nabla z)+\mu_2 w(1-w-a_2u),\quad &x\in \Omega,\quad t>0,\\ z_t = d_4\Delta z-\beta z+f_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset \mathbb{R}^n $ |
$ n\geq1 $ |
$ d_1,d_2,d_3,d_4>0 $ |
$ \mu_1,\mu_2>0 $ |
$ a_1,a_2>0 $ |
$ \alpha, \beta>0 $ |
$ \chi_i $ |
$ i = 1,2 $ |
$ f_i $ |
$ i = 1,2 $ |
$ n = 2 $ |
$ |\chi'_i| $ |
$ i = 1,2 $ |
$ n\leq3 $ |
$ \mu_i $ |
$ i = 1,2 $ |
$ \mu_{i0}>0 $ |
$ \mu_i>\mu_{i0} $ |
$ \bullet $ |
$ a_1,a_2\in(0,1) $ |
$ \mu_1 $ |
$ \mu_2 $ |
$\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$ |
$ f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$ |
$ t\rightarrow\infty $ |
$ \bullet $ |
$ a_1>1>a_2>0 $ |
$ \mu_2 $ |
$ (0,f_1(1)/\alpha,1,0) $ |
$ t\rightarrow\infty $ |
$ \bullet $ |
$ a_1 = 1>a_2>0 $ |
$ \mu_2 $ |
$ (0,f_1(1)/\alpha,1,0) $ |
$ t\rightarrow\infty $ |
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. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[6] | M. Eisenbach, Chemotaxis, (Imperial College Press, London, 2004. Google Scholar |
[7] |
T. Hillen and K. Painter,
A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[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] |
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. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp.
doi: 10.1007/s00033-017-0800-1. |
[13] |
K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[14] |
K. Lin and C. Mu,
Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.
doi: 10.3934/dcdsb.2017094. |
[15] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[16] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[17] |
M. Mizukami,
Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.
doi: 10.1002/mma.4607. |
[18] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[19] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[20] |
M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar |
[21] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[22] |
E. Nakaguchi and K. Osaki,
$L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[23] |
E. Nakaguchi and K. Osaki,
Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.
doi: 10.1016/j.na.2010.08.044. |
[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] |
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. |
[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] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[28] |
H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar |
[29] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[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 Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[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] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[35] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[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. |
[37] |
L. Wang, C. Mu, X. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[38] |
M. Winkler,
Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[39] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[40] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[41] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
[42] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[43] |
L. Xie and Y. Wang,
Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.
doi: 10.3934/dcdsb.2017132. |
[44] |
H. Yu, W. Wang and S. Zheng,
Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.
doi: 10.1088/1361-6544/aa96c9. |
[45] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[46] |
Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9pp.
doi: 10.1063/1.5011725. |
[47] |
Q. Zhang,
Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.
doi: 10.1016/j.aml.2018.03.012. |
[48] |
Q. Zhang and Y. Li,
Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 467 (2018), 751-767.
doi: 10.1016/j.jmaa.2018.07.037. |
[49] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558.
|
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. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[6] | M. Eisenbach, Chemotaxis, (Imperial College Press, London, 2004. Google Scholar |
[7] |
T. Hillen and K. Painter,
A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[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] |
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. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp.
doi: 10.1007/s00033-017-0800-1. |
[13] |
K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[14] |
K. Lin and C. Mu,
Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.
doi: 10.3934/dcdsb.2017094. |
[15] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[16] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[17] |
M. Mizukami,
Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.
doi: 10.1002/mma.4607. |
[18] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[19] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[20] |
M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar |
[21] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[22] |
E. Nakaguchi and K. Osaki,
$L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[23] |
E. Nakaguchi and K. Osaki,
Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.
doi: 10.1016/j.na.2010.08.044. |
[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] |
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. |
[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] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[28] |
H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar |
[29] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[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 Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[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] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[35] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[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. |
[37] |
L. Wang, C. Mu, X. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[38] |
M. Winkler,
Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[39] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[40] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[41] |
G. Wolansky,
Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
[42] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[43] |
L. Xie and Y. Wang,
Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.
doi: 10.3934/dcdsb.2017132. |
[44] |
H. Yu, W. Wang and S. Zheng,
Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.
doi: 10.1088/1361-6544/aa96c9. |
[45] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[46] |
Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9pp.
doi: 10.1063/1.5011725. |
[47] |
Q. Zhang,
Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.
doi: 10.1016/j.aml.2018.03.012. |
[48] |
Q. Zhang and Y. Li,
Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 467 (2018), 751-767.
doi: 10.1016/j.jmaa.2018.07.037. |
[49] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558.
|
[1] |
Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137 |
[2] |
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