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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 |
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.
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. |
[2] |
N. Bellomo, A. Bellouquid, Y. 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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Eftimie, G. 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. |
[8] |
G. Furioli, A. Pulvirenti, E. 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. |
[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. |
[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.
|
[11] |
W. Hoffman, D. Heinemann and J. A. Wiens,
The ecology of seabird feeding flocks in Alaska, The Auk, 98 (1981), 437-456.
|
[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. |
[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. |
[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. |
[15] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[16] |
K. Lin, C. 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. |
[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. |
[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. |
[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. |
[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.
|
[21] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[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. |
[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. |
[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. |
[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. |
[26] |
C. Stinner, J. 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. |
[27] |
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. 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. |
[28] |
N. Sfakianakis, N. Kolbe, N. 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.
|
[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. |
[30] |
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. U.S.A., 109 (2012), 11228-11233.
doi: 10.1073/pnas.1201739109. |
[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. |
[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. |
[33] |
L. Wang, J. Zhang, C. 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.
|
[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. |
[35] |
M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Mathématique, 141 (2020), 583-624.
|
[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. |
[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. |
[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. |
[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. |
[40] |
M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint. |
[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.
|
[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. |
[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. |
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. |
[2] |
N. Bellomo, A. Bellouquid, Y. 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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Eftimie, G. 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. |
[8] |
G. Furioli, A. Pulvirenti, E. 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. |
[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. |
[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.
|
[11] |
W. Hoffman, D. Heinemann and J. A. Wiens,
The ecology of seabird feeding flocks in Alaska, The Auk, 98 (1981), 437-456.
|
[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. |
[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. |
[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. |
[15] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[16] |
K. Lin, C. 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. |
[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. |
[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. |
[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. |
[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.
|
[21] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[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. |
[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. |
[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. |
[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. |
[26] |
C. Stinner, J. 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. |
[27] |
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. 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. |
[28] |
N. Sfakianakis, N. Kolbe, N. 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.
|
[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. |
[30] |
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. U.S.A., 109 (2012), 11228-11233.
doi: 10.1073/pnas.1201739109. |
[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. |
[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. |
[33] |
L. Wang, J. Zhang, C. 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.
|
[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. |
[35] |
M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Mathématique, 141 (2020), 583-624.
|
[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. |
[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. |
[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. |
[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. |
[40] |
M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint. |
[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.
|
[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. |
[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. |
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