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Global stability in a multi-dimensional predator-prey system with prey-taxis
Department of Mathematics, South China University of Technology, Guangzhou 510641, China |
$ \begin{eqnarray*} { \label{1.1}} \left\{ \begin{array}{llll} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\gamma uv-\rho u, \\ v_{t} = \Delta v-\xi uv+\mu v(1-v), \ \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset\mathbb{R}^{n} $ |
$ (n = 2, 3) $ |
$ \chi $ |
$ \gamma $ |
$ \rho $ |
$ \xi $ |
$ \mu $ |
$ (0, 1) $ |
$ \rho>\gamma $ |
$ (\frac{\mu(\gamma-\rho)}{\xi\rho}, \frac{\rho}{\gamma}) $ |
$ \gamma>\rho $ |
$ L^{\infty}(\Omega) $ |
$ t\rightarrow\infty $ |
References:
[1] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Text in Math. Teubner, Stuttgart, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[3] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete. Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[4] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[5] |
T. Hillen and K. J. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[6] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci, 24 (1997), 633-683.
|
[7] |
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. |
[8] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[9] |
J. Jiang, H. Wu and S. Zheng,
Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differential Equations, 12 (2001), 159-177.
doi: 10.1016/j.jde.2018.01.004. |
[10] |
H.-Y Jin and Z. A. Wang,
Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[11] |
H.-Y Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., (2019). |
[12] |
A. Jüngel, C. Kuehn and L. Trussardi,
A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.
doi: 10.1017/S0956792516000346. |
[13] |
A. Jüngel, Diffusive and Nondiffusive Population Models. Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010,397–425.
doi: 10.1007/978-0-8176-4946-3_15. |
[14] |
P. Kareiva and G. Odell,
Swarms of predators exhibit 'preytaxis' if individual predators use arearestricted search, Amer. Nat., 130 (1987), 233-270.
|
[15] |
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. |
[16] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. Nral'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23, Providence, RI, 1968. |
[17] |
J. M. Lee, T. Hillen and M. A. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[18] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint, 2013.
doi: 10.1016/j.matpur.2013.01.020. |
[19] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[20] |
Y. Tao and M. Winkler,
Global smooth solvability of a parabolic-elliptic nutrient taxis system in domain of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.
doi: 10.1016/j.jde.2019.01.014. |
[21] |
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. |
[22] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemootaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[23] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusivle attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[24] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[25] |
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. |
[26] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations, 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[27] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
show all references
References:
[1] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Text in Math. Teubner, Stuttgart, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[3] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete. Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[4] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[5] |
T. Hillen and K. J. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[6] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci, 24 (1997), 633-683.
|
[7] |
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. |
[8] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[9] |
J. Jiang, H. Wu and S. Zheng,
Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differential Equations, 12 (2001), 159-177.
doi: 10.1016/j.jde.2018.01.004. |
[10] |
H.-Y Jin and Z. A. Wang,
Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[11] |
H.-Y Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., (2019). |
[12] |
A. Jüngel, C. Kuehn and L. Trussardi,
A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.
doi: 10.1017/S0956792516000346. |
[13] |
A. Jüngel, Diffusive and Nondiffusive Population Models. Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010,397–425.
doi: 10.1007/978-0-8176-4946-3_15. |
[14] |
P. Kareiva and G. Odell,
Swarms of predators exhibit 'preytaxis' if individual predators use arearestricted search, Amer. Nat., 130 (1987), 233-270.
|
[15] |
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. |
[16] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. Nral'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23, Providence, RI, 1968. |
[17] |
J. M. Lee, T. Hillen and M. A. Lewis,
Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.
doi: 10.1080/17513750802716112. |
[18] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint, 2013.
doi: 10.1016/j.matpur.2013.01.020. |
[19] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[20] |
Y. Tao and M. Winkler,
Global smooth solvability of a parabolic-elliptic nutrient taxis system in domain of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.
doi: 10.1016/j.jde.2019.01.014. |
[21] |
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. |
[22] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemootaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[23] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusivle attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[24] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[25] |
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. |
[26] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations, 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[27] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
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