Global generalized solutions to the forager-exploiter model with logistic growth

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September 2022, 27(9): 5255-5282. doi: 10.3934/dcdsb.2021273

Global generalized solutions to the forager-exploiter model with logistic growth

Qian Zhao ^1,2, and Bin Liu ^1,2,,

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 Published September 2022 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.

Keywords: Chemotaxis, forager-exploiter model, global existence, generalized solution.

Mathematics Subject Classification: Primary: 35D30; Secondary: 35A01, 35Q92.

Citation: Qian Zhao, Bin Liu. Global generalized solutions to the forager-exploiter model with logistic growth. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5255-5282. 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.
[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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