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Boundedness in prey-taxis system with rotational flux terms
School of Mathematics, Southeast University, Nanjing 210096, P.R. China |
$ \begin{equation*} \begin{cases} u_t = \Delta u-\nabla\cdot(uS(x,u,v)\nabla v)+uv-\rho u,& x\in\Omega,\ t>0,\\ v_t = \Delta v-\xi uv+\mu v(1-\alpha v),&x\in\Omega,\ t>0, \end{cases} \end{equation*} $ |
$ \Omega\subset\mathbb{R}^n\; (n\geq1) $ |
$ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{n\times n}) $ |
$ |S(x,u,v)|\leq\frac{S_0(v)}{(1+u)^\theta}(\theta\geq0) $ |
$ (x,u,v)\in\bar{\Omega}\times[0,\infty)^2 $ |
$ S_0 $ |
$ u_0\in C^0(\overline{\Omega}) $ |
$ v_0\in W^{1,q}(\Omega) $ |
$ q>\max\{n,2\} $ |
$ n = 1 $ |
$ n\geq2, \theta = 0 $ |
$ S_0(m)m<\frac{2}{\sqrt{3n(11n+2)}} $ |
$ \theta>0 $ |
$ m: = \max\{\|v_0\|_{L^\infty(\Omega)}, \frac{1}{\alpha}\} $ |
References:
[1] |
B. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
[2] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[3] |
C. Jin, Y. Wang and J. Yin, Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion, preprint, arXiv: 1804.03964.
doi: 10.1016/j.jde.2018.02.031. |
[4] |
H. Y. Jin and Z. A. Wang,
Global stability of prey-taxis systems, J. Differ. Equ., 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[5] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.
|
[6] |
J. M. Lee, T. Hillen and M. A. Lewis,
Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.
doi: 10.1007/s11538-007-9271-4. |
[7] |
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. |
[8] |
T. Li, A. Suen, M. Winkler and C. Xue,
Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Meth. Appl. Sci., 25 (2015), 721-746.
doi: 10.1142/S0218202515500177. |
[9] |
W. W. Murdoch, J. Chesson and P. L. Chesson,
Biological control in theory and practice, Amer. Nat., 125 (1985), 344-366.
|
[10] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[11] |
N. Sapoukhina, Y. Tyutyunov and R. Arditi,
The role of prey taxis in biological control: A spatial theoretical model, Amer. Nat., 162 (2003), 61-76.
|
[12] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[13] |
J. Wang and M. Wang, Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis, Z. Angew. Math. Phys., 69 (2018), Art. 63, 24 pp.
doi: 10.1007/s00033-018-0960-7. |
[14] |
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. |
[15] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[16] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differ. Equ., 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[17] |
M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, Int. Math. Res. Not., 2019. |
[18] |
Q. Zhang,
Boundedness in chemotaxis systems with rotational flux terms, Math. Nachr., 289 (2016), 2323-2334.
doi: 10.1002/mana.201500325. |
show all references
References:
[1] |
B. Ainseba, M. Bendahmane and A. Noussair,
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.
doi: 10.1016/j.nonrwa.2007.06.017. |
[2] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[3] |
C. Jin, Y. Wang and J. Yin, Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion, preprint, arXiv: 1804.03964.
doi: 10.1016/j.jde.2018.02.031. |
[4] |
H. Y. Jin and Z. A. Wang,
Global stability of prey-taxis systems, J. Differ. Equ., 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[5] |
P. Kareiva and G. Odell,
Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.
|
[6] |
J. M. Lee, T. Hillen and M. A. Lewis,
Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.
doi: 10.1007/s11538-007-9271-4. |
[7] |
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. |
[8] |
T. Li, A. Suen, M. Winkler and C. Xue,
Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Meth. Appl. Sci., 25 (2015), 721-746.
doi: 10.1142/S0218202515500177. |
[9] |
W. W. Murdoch, J. Chesson and P. L. Chesson,
Biological control in theory and practice, Amer. Nat., 125 (1985), 344-366.
|
[10] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[11] |
N. Sapoukhina, Y. Tyutyunov and R. Arditi,
The role of prey taxis in biological control: A spatial theoretical model, Amer. Nat., 162 (2003), 61-76.
|
[12] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[13] |
J. Wang and M. Wang, Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis, Z. Angew. Math. Phys., 69 (2018), Art. 63, 24 pp.
doi: 10.1007/s00033-018-0960-7. |
[14] |
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. |
[15] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[16] |
M. Winkler,
Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differ. Equ., 263 (2017), 4826-4869.
doi: 10.1016/j.jde.2017.06.002. |
[17] |
M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, Int. Math. Res. Not., 2019. |
[18] |
Q. Zhang,
Boundedness in chemotaxis systems with rotational flux terms, Math. Nachr., 289 (2016), 2323-2334.
doi: 10.1002/mana.201500325. |
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