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Global existence and stability in a two-species chemotaxis system
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China |
$\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), & x\in \Omega ,t>0, & \\ \ \ {{v}_{t}}=\Delta v-v+h(w), & x\in \Omega ,t>0, & \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), & x\in \Omega ,t>0, & \\ \ \ {{z}_{t}}=\Delta z-z+h(u),& x\in \Omega ,t>0, & \\\end{array} \right.$ |
$Ω\subset\mathbb{R}^{n}$ |
$(u_{*}, v_{*}, w_{*}, z_{*})$ |
$0 < a_{1}, a_{2} < 1$ |
$a_{1}>1>a_{2}>0$ |
$a_{1}, a_{2}>1$ |
$(u_{*}, v_{*}, w_{*}, z_{*})$ |
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,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[3] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters, 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[4] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang,
Stability of spiky solution of Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134.
doi: 10.1016/j.jde.2014.06.008. |
[5] |
A.-K. Drangeid,
The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 13 (1989), 1091-1113.
doi: 10.1016/0362-546X(89)90097-7. |
[6] |
S. Guo,
Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477.
doi: 10.1016/j.nonrwa.2018.01.011. |
[7] |
S. Guo and S. Yan,
Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.
doi: 10.1016/j.jde.2015.09.031. |
[8] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[9] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber Deutsch. Math.-Verein., 106 (2004), 51-69.
|
[10] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[13] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of 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 and Continuous Dynamical Systems - Series B, 20 (2017), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[15] |
D. Li and S. Guo,
Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247.
|
[16] |
D. Li and S. Guo,
Stability and Hopf bifurcation in a reaction-diffusion model with chemotaxis and nonlocal delay effect, International Journal of Bifurcation and Chaos, 28 (2018), 1850046.
doi: 10.1142/S0218127418500463. |
[17] |
P. L. Lions,
Résolution de problemes elliptiques quasilinéaires, Archive for Rational Mechanics and Analysis, 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[18] |
D. Liu and Y. Tao,
Global boundedness in a fully parabolic attractionrepulsion chemotaxis model, Mathematical Methods in the Applied Sciences, 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[19] |
A. Lunardi,
Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis: Theory, Methods & Applications, 9 (1985), 563-586.
doi: 10.1016/0362-546X(85)90041-0. |
[20] |
J. D. Murray, Mathematical Biology. Ⅱ Spatial Models and Biomedical Applications {Interdisciplinary Applied Mathematics V. 18}. Springer-Verlag, New York, 2003. |
[21] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete and Continuous Dynamical Systems - Series B, 18 (2014), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[22] |
E. Nakaguchi and K. Osaki,
Lp-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[23] |
E. Nakaguchi and K. Osaki, et al., Global existence of solutions to an n-dimensional parabolicparabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka
Journal of Mathematics, 55 (2018), 51-70. |
[24] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[25] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[26] |
C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded
and exterior domains. applications, In Nonlinear Analysis, Function Spaces and Applications
Vol. 4, Springer, 119 (1990), 180-223. |
[27] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[28] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[29] |
Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional
Keller-Segel-Navier-Stokes system, Zeitschrift für angewandte Mathematik und Physik, 67
(2016), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[30] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[31] |
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, Journal of Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[32] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[33] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[34] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis & Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[35] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[36] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation,
Zeitschrift für angewandte Mathematik und Physik, 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[37] |
S. Yan and S. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect,
International Journal of Bifurcation and Chaos 27 (2017), 1750105, 24pp.
doi: 10.1142/S021812741750105X. |
[38] |
P. Zheng, C. Mu, R. Willie and X. Hu,
Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Computers & Mathematics with Applications, 75 (2018), 1667-1675.
doi: 10.1016/j.camwa.2017.11.032. |
[39] |
R. Zou and S. Guo,
Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049, 22pp.
doi: 10.1142/S0218127417500493. |
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,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[3] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters, 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[4] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang,
Stability of spiky solution of Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134.
doi: 10.1016/j.jde.2014.06.008. |
[5] |
A.-K. Drangeid,
The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 13 (1989), 1091-1113.
doi: 10.1016/0362-546X(89)90097-7. |
[6] |
S. Guo,
Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477.
doi: 10.1016/j.nonrwa.2018.01.011. |
[7] |
S. Guo and S. Yan,
Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.
doi: 10.1016/j.jde.2015.09.031. |
[8] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[9] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber Deutsch. Math.-Verein., 106 (2004), 51-69.
|
[10] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[13] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of 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 and Continuous Dynamical Systems - Series B, 20 (2017), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[15] |
D. Li and S. Guo,
Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247.
|
[16] |
D. Li and S. Guo,
Stability and Hopf bifurcation in a reaction-diffusion model with chemotaxis and nonlocal delay effect, International Journal of Bifurcation and Chaos, 28 (2018), 1850046.
doi: 10.1142/S0218127418500463. |
[17] |
P. L. Lions,
Résolution de problemes elliptiques quasilinéaires, Archive for Rational Mechanics and Analysis, 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[18] |
D. Liu and Y. Tao,
Global boundedness in a fully parabolic attractionrepulsion chemotaxis model, Mathematical Methods in the Applied Sciences, 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[19] |
A. Lunardi,
Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis: Theory, Methods & Applications, 9 (1985), 563-586.
doi: 10.1016/0362-546X(85)90041-0. |
[20] |
J. D. Murray, Mathematical Biology. Ⅱ Spatial Models and Biomedical Applications {Interdisciplinary Applied Mathematics V. 18}. Springer-Verlag, New York, 2003. |
[21] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete and Continuous Dynamical Systems - Series B, 18 (2014), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[22] |
E. Nakaguchi and K. Osaki,
Lp-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[23] |
E. Nakaguchi and K. Osaki, et al., Global existence of solutions to an n-dimensional parabolicparabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka
Journal of Mathematics, 55 (2018), 51-70. |
[24] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis: Theory, Methods & Applications, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[25] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[26] |
C. G. Simader, The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded
and exterior domains. applications, In Nonlinear Analysis, Function Spaces and Applications
Vol. 4, Springer, 119 (1990), 180-223. |
[27] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[28] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[29] |
Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional
Keller-Segel-Navier-Stokes system, Zeitschrift für angewandte Mathematik und Physik, 67
(2016), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[30] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[31] |
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, Journal of Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[32] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[33] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[34] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis & Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[35] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[36] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation,
Zeitschrift für angewandte Mathematik und Physik, 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[37] |
S. Yan and S. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect,
International Journal of Bifurcation and Chaos 27 (2017), 1750105, 24pp.
doi: 10.1142/S021812741750105X. |
[38] |
P. Zheng, C. Mu, R. Willie and X. Hu,
Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Computers & Mathematics with Applications, 75 (2018), 1667-1675.
doi: 10.1016/j.camwa.2017.11.032. |
[39] |
R. Zou and S. Guo,
Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049, 22pp.
doi: 10.1142/S0218127417500493. |

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