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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 & Applied Analysis, 2020, 19 (10) : 4839-4851. doi: 10.3934/cpaa.2020214
References:
[1]

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[17]

M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, Int. Math. Res. Not., 2019. Google Scholar

[18]

Q. Zhang, Boundedness in chemotaxis systems with rotational flux terms, Math. Nachr., 289 (2016), 2323-2334.  doi: 10.1002/mana.201500325.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. W. Murdoch, J. Chesson and P. L. Chesson, Biological control in theory and practice, Amer. Nat., 125 (1985), 344-366.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems?, Int. Math. Res. Not., 2019. Google Scholar

[18]

Q. Zhang, Boundedness in chemotaxis systems with rotational flux terms, Math. Nachr., 289 (2016), 2323-2334.  doi: 10.1002/mana.201500325.  Google Scholar

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