Boundedness in prey-taxis system with rotational flux terms

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October 2020, 19(10): 4839-4851. doi: 10.3934/cpaa.2020214

Boundedness in prey-taxis system with rotational flux terms

Hengling Wang and Yuxiang Li ^,

School of Mathematics, Southeast University, Nanjing 210096, P.R. China

^* Corresponding author

Received October 2019 Revised May 2020 Published July 2020

Fund Project: The authors are supported in part by the National Natural Science Foundation of China (No.11671079, 11701290, 11601127 and 11171063), the Natural Science Foundation of Jiangsu Province(No.BK20170896), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No.KYCX19_0054)

This paper investigates prey-taxis system with rotational flux terms

$ \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*} $

under no-flux boundary conditions in a bounded domain

$ \Omega\subset\mathbb{R}^n\; (n\geq1) $

with smooth boundary. Here the matrix-valued function

$ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{n\times n}) $

fulfills

$ |S(x,u,v)|\leq\frac{S_0(v)}{(1+u)^\theta}(\theta\geq0) $

for all

$ (x,u,v)\in\bar{\Omega}\times[0,\infty)^2 $

with some nondecreasing function

$ S_0 $

. It is proved that for nonnegative initial data

$ u_0\in C^0(\overline{\Omega}) $

and

$ v_0\in W^{1,q}(\Omega) $

with some

$ q>\max\{n,2\} $

, if one of the following assumptions holds: (i)

$ n = 1 $

, (ii)

$ n\geq2, \theta = 0 $

and

$ S_0(m)m<\frac{2}{\sqrt{3n(11n+2)}} $

, (iii)

$ \theta>0 $

, then the model possesses a global classical solution that is uniformly bounded. Where

$ m: = \max\{\|v_0\|_{L^\infty(\Omega)}, \frac{1}{\alpha}\} $

.

Keywords: Chemotaxis, prey taxis, rotational flux, global existence.

Mathematics Subject Classification: 35B45, 35K55, 35K65, 92C17.

Citation: Hengling Wang, Yuxiang Li. Boundedness in prey-taxis system with rotational flux terms. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4839-4851. doi: 10.3934/cpaa.2020214

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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