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Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | School of Sciences, Southwest Petroleum University, Chengdu 610500, China |
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &x\in \Omega,\quad t>0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $ |
$ \partial\Omega $ |
$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $ |
$ u_0 $ |
$ v_0 $ |
$ n = 2 $ |
$ n = 3 $ |
$ \chi $ |
$ b\lambda<\mu $ |
$ \chi $ |
$ \xi $ |
$ (u,v,w) $ |
$ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $ |
$ t\rightarrow \infty $ |
$ b\lambda\geq \mu $ |
$ \chi $ |
$ \xi $ |
$ (u,v,w) $ |
$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $ |
$ b\lambda> \mu $ |
$ b\lambda = \mu $ |
References:
[1] |
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. |
[2] |
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. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
T. Cie$\acute{s}$lak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[5] |
E. Espejo, A. Stevens and J. L Velzquez,
Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. |
[7] |
S. Fu and L. Miao,
Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.
doi: 10.1016/j.nonrwa.2019.103079. |
[8] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
|
[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] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
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. |
[13] |
M. Herrero and J. Vel$\acute{a}$zquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[14] |
E. Keller and L. 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. |
[15] |
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. |
[16] |
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. |
[17] |
Y. Li,
Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.
doi: 10.3934/dcdsb.2019066. |
[18] |
X. Li and Y. Wang,
On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.
doi: 10.1016/j.jmaa.2018.10.093. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
K. Lin, C. Mu and H. Zhong,
A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.
doi: 10.1016/j.camwa.2017.10.007. |
[23] |
G. Li, Y. Tao and M. Winkler,
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.
doi: 10.3934/dcdsb.2020102. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
L. Miao, H. Yang and S. Fu,
Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.
doi: 10.1016/j.aml.2020.106639. |
[28] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
|
[29] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[30] |
M. Negreanu and J. Tello,
Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.
doi: 10.1016/j.jmaa.2019.02.007. |
[31] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[32] |
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. |
[33] |
C. Stinner, J. 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. |
[34] |
S. Qiu, C. Mu and X. Tu,
Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.
doi: 10.1016/j.nonrwa.2020.103190. |
[35] |
X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp.
doi: 10.1007/s00033-020-01413-6. |
[36] |
X. Tu, C. Mu and S. Qiu,
Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.
doi: 10.1016/j.na.2020.111923. |
[37] |
J. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[38] |
J. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[39] |
Y. Tao and M. Wang,
Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
[40] |
Y. Tao,
Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.
doi: 10.1016/j.nonrwa.2009.05.005. |
[41] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[42] |
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. |
[43] |
Y. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[44] |
Y. Tao and Z. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[45] |
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. |
[46] |
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. |
[47] |
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. |
[48] |
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. |
[49] |
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. |
[50] |
M. Winkler,
How far can chemotaxis can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
T. Cie$\acute{s}$lak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonliearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[5] |
E. Espejo, A. Stevens and J. L Velzquez,
Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. |
[7] |
S. Fu and L. Miao,
Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. Real World Appl., 54 (2020), 103079.
doi: 10.1016/j.nonrwa.2019.103079. |
[8] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
|
[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] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
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. |
[13] |
M. Herrero and J. Vel$\acute{a}$zquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[14] |
E. Keller and L. 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. |
[15] |
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. |
[16] |
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. |
[17] |
Y. Li,
Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetic, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.
doi: 10.3934/dcdsb.2019066. |
[18] |
X. Li and Y. Wang,
On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.
doi: 10.1016/j.jmaa.2018.10.093. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
K. Lin, C. Mu and H. Zhong,
A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.
doi: 10.1016/j.camwa.2017.10.007. |
[23] |
G. Li, Y. Tao and M. Winkler,
Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383-4396.
doi: 10.3934/dcdsb.2020102. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
L. Miao, H. Yang and S. Fu,
Global boundedness in a two-species predator-prey chemotaxis model, Appl. Math. Lett., 111 (2021), 106639.
doi: 10.1016/j.aml.2020.106639. |
[28] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
|
[29] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[30] |
M. Negreanu and J. Tello,
Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.
doi: 10.1016/j.jmaa.2019.02.007. |
[31] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[32] |
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. |
[33] |
C. Stinner, J. 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. |
[34] |
S. Qiu, C. Mu and X. Tu,
Global dynamics of a two-species chemotaxis-consumption system with signal-dependent motilities, Nonlinear Anal. Real World Appl., 57 (2021), 103190.
doi: 10.1016/j.nonrwa.2020.103190. |
[35] |
X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 18pp.
doi: 10.1007/s00033-020-01413-6. |
[36] |
X. Tu, C. Mu and S. Qiu,
Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.
doi: 10.1016/j.na.2020.111923. |
[37] |
J. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[38] |
J. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonliearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[39] |
Y. Tao and M. Wang,
Global solution for a chemotactic-haptotactic model of cancer invasion, Nonliearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
[40] |
Y. Tao,
Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.
doi: 10.1016/j.nonrwa.2009.05.005. |
[41] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[42] |
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. |
[43] |
Y. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[44] |
Y. Tao and Z. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[45] |
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. |
[46] |
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. |
[47] |
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. |
[48] |
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. |
[49] |
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. |
[50] |
M. Winkler,
How far can chemotaxis can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
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