American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064
[2]

Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136
[3]

Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120
[4]

Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
[5]

Dan Li. Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1681-1705. doi: 10.3934/dcds.2020337
[6]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[7]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[8]

Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133
[9]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
[10]

Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic & Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51
[11]

Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[12]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[13]

Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239
[14]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[15]

Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262
[16]

Guoqiang Ren, Heping Ma. Global existence in a chemotaxis system with singular sensitivity and signal production. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 343-360. doi: 10.3934/dcdsb.2021045
[17]

Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
[18]

Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017
[19]

Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021
[20]

Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks & Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (213)
  • HTML views (65)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy