doi: 10.3934/dcdsb.2021273
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Global generalized solutions to the forager-exploiter model with logistic growth

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

* Corresponding author: Bin Liu

Received  July 2021 Revised  September 2021 Early access November 2021

Fund Project: This work is supported by National Natural Science Foundation of China grant 11971185

This paper presents the global existence of the generalized solutions for the forager-exploiter model with logistic growth under appropriate regularity assumption on the initial value. This result partially generalizes previously known ones.

Citation: Qian Zhao, Bin Liu. Global generalized solutions to the forager-exploiter model with logistic growth. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021273
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

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N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006.  doi: 10.1142/s0218202511400069.  Google Scholar

[4]

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

[5]

X. Bai and M. Winkler, Equilibrium in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[6]

F. Dai and B. Liu, Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production, J. Differential Equations, 269 (2020), 10839-10918.  doi: 10.1016/j.jde.2020.07.027.  Google Scholar

[7]

R. EftimieG. de Verirs and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.  Google Scholar

[8]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Mod. Meth. Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.  Google Scholar

[9]

S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. RWA., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079.  Google Scholar

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M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[11]

W. HoffmanD. Heinemann and J. A. Wiens, The ecology of seabird feeding flocks in Alaska, The Auk, 98 (1981), 437-456.   Google Scholar

[12]

H. Y. Jin and Z. A. Wang, Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.  Google Scholar

[13]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[14]

K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar

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K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar

[16]

K. LinC. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007.  Google Scholar

[17]

K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with$\setminus$without loop, Calc. Var. Partial Equations, 59 (2020), 35pp. doi: 10.1007/s00526-020-01777-7.  Google Scholar

[18]

Y. Liu, Global existence and boundedness of classical solutions to a forager-exploiter model with volume-filling effects, Nonlinear Anal. Real World Appl., 50 (2019), 519-531.  doi: 10.1016/j.nonrwa.2019.05.015.  Google Scholar

[19]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[20]

T. Nagai T. Senb and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[22]

G. Ren, Boundedness and stabilization in a two-species chemotaxis system with logistic source, Z. Angew. Math. Phys., 71 (2020), 177, 17pp. doi: 10.1007/s00033-020-01410-9.  Google Scholar

[23]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differential Equations, 269 (2020), 1484-1520.  doi: 10.1016/j.jde.2020.01.008.  Google Scholar

[24]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.  Google Scholar

[25]

G. Ren and B. Liu, Global solvability and asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra competitive kinetics, Math. Models Methods Appl. Sci., 31 (2021), 941-978.  doi: 10.1142/S0218202521500238.  Google Scholar

[26]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[27]

M. B. ShortM. R. D'OrsognaV. B. PasourG. E. TitaP. J. BrantinghamA. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.  doi: 10.1142/S0218202508003029.  Google Scholar

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N. SfakianakisN. KolbeN. Hellmann and M. Lukáčová-Medvid'ová, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.   Google Scholar

[29]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[30]

N. TaniaB. VanderleiJ. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 11228-11233.  doi: 10.1073/pnas.1201739109.  Google Scholar

[31]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[32]

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

[33]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.   Google Scholar

[34]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[35]

M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Mathématique, 141 (2020), 583-624.   Google Scholar

[36]

M. Winkler, Boundedness in the high-dimensional parabolic-parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[37]

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

[38]

M. Winkler, Large-data global generalized solution in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3029-3115.  doi: 10.1137/140979708.  Google Scholar

[39]

M. Winkler, Small-mass solution in the two-dimensionsl Keller-Segel system coupled to the Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.  doi: 10.1137/19M1264199.  Google Scholar

[40]

M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint. Google Scholar

[41]

M. Winker, Global boundedness of solutions on the two-dimensional forager-exploiter model with logistic source, Discrete Contin. Dyn. Syst. Ser., 41 (2021), 3031-3043.   Google Scholar

[42]

T. Xiang, Chemotactic aggregation versus logistic damping on boundedness in the 3D minimal Keller-Segel Model, SIAM J. Appl. Math., 78 (2018), 2420-2438.  doi: 10.1137/17M1150475.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Mech., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006.  doi: 10.1142/s0218202511400069.  Google Scholar

[4]

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

[5]

X. Bai and M. Winkler, Equilibrium in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[6]

F. Dai and B. Liu, Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production, J. Differential Equations, 269 (2020), 10839-10918.  doi: 10.1016/j.jde.2020.07.027.  Google Scholar

[7]

R. EftimieG. de Verirs and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.  Google Scholar

[8]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Mod. Meth. Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.  Google Scholar

[9]

S. Fu and L. Miao, Global existence and asymptotic stability in a predator-prey chemotaxis model, Nonlinear Anal. RWA., 54 (2020), 103079.  doi: 10.1016/j.nonrwa.2019.103079.  Google Scholar

[10]

M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[11]

W. HoffmanD. Heinemann and J. A. Wiens, The ecology of seabird feeding flocks in Alaska, The Auk, 98 (1981), 437-456.   Google Scholar

[12]

H. Y. Jin and Z. A. Wang, Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.  Google Scholar

[13]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[14]

K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar

[15]

K. LinC. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar

[16]

K. LinC. Mu and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl., 75 (2018), 837-849.  doi: 10.1016/j.camwa.2017.10.007.  Google Scholar

[17]

K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with$\setminus$without loop, Calc. Var. Partial Equations, 59 (2020), 35pp. doi: 10.1007/s00526-020-01777-7.  Google Scholar

[18]

Y. Liu, Global existence and boundedness of classical solutions to a forager-exploiter model with volume-filling effects, Nonlinear Anal. Real World Appl., 50 (2019), 519-531.  doi: 10.1016/j.nonrwa.2019.05.015.  Google Scholar

[19]

M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar

[20]

T. Nagai T. Senb and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[22]

G. Ren, Boundedness and stabilization in a two-species chemotaxis system with logistic source, Z. Angew. Math. Phys., 71 (2020), 177, 17pp. doi: 10.1007/s00033-020-01410-9.  Google Scholar

[23]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differential Equations, 269 (2020), 1484-1520.  doi: 10.1016/j.jde.2020.01.008.  Google Scholar

[24]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.  Google Scholar

[25]

G. Ren and B. Liu, Global solvability and asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra competitive kinetics, Math. Models Methods Appl. Sci., 31 (2021), 941-978.  doi: 10.1142/S0218202521500238.  Google Scholar

[26]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[27]

M. B. ShortM. R. D'OrsognaV. B. PasourG. E. TitaP. J. BrantinghamA. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.  doi: 10.1142/S0218202508003029.  Google Scholar

[28]

N. SfakianakisN. KolbeN. Hellmann and M. Lukáčová-Medvid'ová, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.   Google Scholar

[29]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[30]

N. TaniaB. VanderleiJ. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 11228-11233.  doi: 10.1073/pnas.1201739109.  Google Scholar

[31]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[32]

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

[33]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.   Google Scholar

[34]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[35]

M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Mathématique, 141 (2020), 583-624.   Google Scholar

[36]

M. Winkler, Boundedness in the high-dimensional parabolic-parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[37]

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

[38]

M. Winkler, Large-data global generalized solution in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3029-3115.  doi: 10.1137/140979708.  Google Scholar

[39]

M. Winkler, Small-mass solution in the two-dimensionsl Keller-Segel system coupled to the Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.  doi: 10.1137/19M1264199.  Google Scholar

[40]

M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint. Google Scholar

[41]

M. Winker, Global boundedness of solutions on the two-dimensional forager-exploiter model with logistic source, Discrete Contin. Dyn. Syst. Ser., 41 (2021), 3031-3043.   Google Scholar

[42]

T. Xiang, Chemotactic aggregation versus logistic damping on boundedness in the 3D minimal Keller-Segel Model, SIAM J. Appl. Math., 78 (2018), 2420-2438.  doi: 10.1137/17M1150475.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Mech., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

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