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

  • * Corresponding author: Bin Liu

    * Corresponding author: Bin Liu

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

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  • 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.

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


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  • [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. 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.
    [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. 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.
    [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.
    [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. HoffmanD. 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. 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.
    [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.
    [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. 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.
    [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.
    [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. 
    [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. 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.
    [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. 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. 
    [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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