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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  July 2021 Early access  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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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