American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence
  • DCDS Home
  • This Issue
  • Next Article
    Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem
July  2021, 41(7): 3031-3043. doi: 10.3934/dcds.2020396

Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source

Lu Xu 1,2, , Chunlai Mu 1, and  Qiao Xin 2,,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2. 

College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

* Corresponding author: Qiao Xin

Received  June 2020 Revised  October 2020 Published  December 2020

Fund Project: The second author is supported by NSFC [No. 11771062, 11971082], Fundamental Research Funds for the Central Universities [No. 2019CDJCYJ001], and Chongqing Key Laboratory of Analytic Mathematics and Applications. The third author is supported by the Youth Doctor Science and Technology Talent Training Project of Xinjiang Uygur Autonomous Region [No. 2017Q087]

This paper deals with the global boundedness of solutions to the forager-exploiter model with logistic sources
$ \begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta u- \nabla\cdot(u\nabla w) + \mu_1 (u-u^m), &x \in \Omega, t>0,\\ v_t = \Delta v - \nabla\cdot(v\nabla u) + \mu_2 ( v-v^l), &x\in \Omega, t>0,\\ w_t = \Delta w - \lambda(u+v)w - \mu w + r(x,t), & x\in \Omega, t>0, \end{array} \right. \end{equation*} $
under homogeneous Neumann boundary conditions in a smoothly bounded domain
$ \Omega \subset R^2 $
, where the constants
$ \mu $
,
$ \mu_1 $
,
$ \mu_2 $
,
$ \lambda $
,
$ m $
 and
$ l $
 are positive. We prove that the corresponding initial-boundary value problem possesses a global classical solution that is uniformly bounded under conditions
$ 2\leq m < 3 $
,
$ l \geq 3 $
,
$ r(x,t) \in C^1(\overline{\Omega}\times[0,\infty))\cup L^{\infty}(\Omega\times(0,\infty)) $
 and the smooth nonnegative initial functions, which improves the results obtained by Wang and Wang (MMMAS 2020).
Keywords: Forager-exploiter model, chemotaxis, boundedness, logistic source.
Mathematics Subject Classification: 35A01, 35K55.
Citation: Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3031-3043. doi: 10.3934/dcds.2020396
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, in: Teubner-Texte Math., 133 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

T. Black, Global generalized solutions to a forager-exploiter model with superlinear degradation and theri eventual regularity properties, Math. Models Methods Appl. Sci., 30 (2020), 1075-1117.  doi: 10.1142/S0218202520400072.  Google Scholar

[3]

X. Cao, Global radial renormalized solution to a producer-scrounger model with singular sensitivities, Math. Models Methods Appl. Sci., 6 (2020), 1119-1165.  doi: 10.1142/S0218202520400084.  Google Scholar

[4]

H. Chen, J.-M. Li and K. Wang, On the vanishing viscosity limit of a chemotaxis model, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1963-1987.  doi: 10.3934/dcds.2020101.  Google Scholar

[5]

A. Friedman, Partial Different Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar

[6]

Y. Giga and H. Sohr, Abstrat $L^p$ estimates for the Cauchy problem with aaplications to the Navier-Sotkes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[7]

B. Hu and Y. Tao, To the exclusion of blow-up in three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.  Google Scholar

[8]

C. Jin, Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3547-3566.  doi: 10.3934/dcds.2018150.  Google Scholar

[9]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.  Google Scholar

[10]

J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 6099-6121.  doi: 10.3934/dcds.2017262.  Google Scholar

[11]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.  Google Scholar

[12]

L. Meng, J. Yuan and X. Zheng, Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 3413-3441.  doi: 10.3934/dcds.2019141.  Google Scholar

[13]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[14]

C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal.: Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022.  Google Scholar

[15]

N. Tania, B. Vanderlei, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. USA, 109 (2012), 11228-11233.   Google Scholar

[16]

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.  Google Scholar

[17]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of larege-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[18]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[19]

Y. Tao and M. Winkler, Large time behavior in a forager-exploiter model with differnet taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.  Google Scholar

[20]

J. Wang and M. Wang, Global bounded solution of the higher-dimensional forager-exploixer modle with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323. doi: 10.1142/S0218202520500232.  Google Scholar

[21]

H. Wang and Y. Li, Boundedness in prey-taxis system with rotational flux terms, Commun. pur Appl.Anal, 19 (2020), 4839-4851.  doi: 10.3934/cpaa.2020214.  Google Scholar

[22]

M. Winkler, Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.  doi: 10.1142/S021820251950012X.  Google Scholar

[23]

M. Winkler, Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2950.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, in: Teubner-Texte Math., 133 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

T. Black, Global generalized solutions to a forager-exploiter model with superlinear degradation and theri eventual regularity properties, Math. Models Methods Appl. Sci., 30 (2020), 1075-1117.  doi: 10.1142/S0218202520400072.  Google Scholar

[3]

X. Cao, Global radial renormalized solution to a producer-scrounger model with singular sensitivities, Math. Models Methods Appl. Sci., 6 (2020), 1119-1165.  doi: 10.1142/S0218202520400084.  Google Scholar

[4]

H. Chen, J.-M. Li and K. Wang, On the vanishing viscosity limit of a chemotaxis model, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1963-1987.  doi: 10.3934/dcds.2020101.  Google Scholar

[5]

A. Friedman, Partial Different Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar

[6]

Y. Giga and H. Sohr, Abstrat $L^p$ estimates for the Cauchy problem with aaplications to the Navier-Sotkes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[7]

B. Hu and Y. Tao, To the exclusion of blow-up in three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.  Google Scholar

[8]

C. Jin, Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3547-3566.  doi: 10.3934/dcds.2018150.  Google Scholar

[9]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.  Google Scholar

[10]

J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 6099-6121.  doi: 10.3934/dcds.2017262.  Google Scholar

[11]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.  Google Scholar

[12]

L. Meng, J. Yuan and X. Zheng, Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 3413-3441.  doi: 10.3934/dcds.2019141.  Google Scholar

[13]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[14]

C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal.: Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022.  Google Scholar

[15]

N. Tania, B. Vanderlei, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. USA, 109 (2012), 11228-11233.   Google Scholar

[16]

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.  Google Scholar

[17]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of larege-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[18]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[19]

Y. Tao and M. Winkler, Large time behavior in a forager-exploiter model with differnet taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.  Google Scholar

[20]

J. Wang and M. Wang, Global bounded solution of the higher-dimensional forager-exploixer modle with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323. doi: 10.1142/S0218202520500232.  Google Scholar

[21]

H. Wang and Y. Li, Boundedness in prey-taxis system with rotational flux terms, Commun. pur Appl.Anal, 19 (2020), 4839-4851.  doi: 10.3934/cpaa.2020214.  Google Scholar

[22]

M. Winkler, Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.  doi: 10.1142/S021820251950012X.  Google Scholar

[23]

M. Winkler, Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2950.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[1]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[2]

Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[3]

Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
[4]

Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
[5]

Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[6]

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
[7]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
[8]

Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094
[9]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150
[10]

Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[11]

Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020334
[12]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[13]

Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
[14]

Wenji Zhang, Pengcheng Niu. Asymptotics in a two-species chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4281-4298. doi: 10.3934/dcdsb.2020288
[15]

Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147
[16]

Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 293-319. doi: 10.3934/dcdss.2020017
[17]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268
[18]

Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260
[19]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[20]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (47)
  • HTML views (150)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences