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Global generalized solutions to a three species predator-prey model with prey-taxis

  • * Corresponding author: Ruijing Li

    * Corresponding author: Ruijing Li 

This work was supported by the National Natural Science Foundation of China (Grant No. 11901112)

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  • In this paper, we study the following three species predator-prey model with prey-taxis:

    $ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+r_1u(1-u-kv-b_1w), &\quad x\in \Omega, t>0, \\ v_t = d_2\Delta v+r_2v(1-hu-v-b_2w), &\quad x\in \Omega, t>0, \\ w_t = d_3\Delta w-\chi_2\nabla\cdot(w\nabla u)-\chi_3\nabla\cdot(w\nabla v)\\ \ \ \ \ \ \ \ +r_3w(-1+au+av-w), &\quad x\in \Omega, t>0. \end{array}\right. $

    We prove that if (1.7) and (1.6) hold, the model ($ \ast $) admits at least one global generalized solution in any dimension.

    Mathematics Subject Classification: Primary: 35A01, 35K57; Secondary: 35Q92, 92C17.


    \begin{equation} \\ \end{equation}
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  • [1] B. E. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal., Real World Appl., 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.
    [2] N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [3] T. Black, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.
    [4] Y. CaiC. ZhaoW. Wang and J. Wang, Dynamics of a Leslie-Gower predator-prey model with additive Allee effect, Appl. Math. Model., 39 (2015), 2092-2106.  doi: 10.1016/j.apm.2014.09.038.
    [5] Y. ChenT. Giletti and J. Guo, Persistence of preys in a diffusive three species predator-prey system with a pair of strong-weak competing preys, J. Differential Equations, 281 (2021), 341-378.  doi: 10.1016/j.jde.2021.02.013.
    [6] F. Dai and B. Liu, Global solution for a general cross-diffusion two-competitive-predator and one-prey system with predator-taxis, Commun. Nonlinear Sci. Numer. Simulat., 89 (2020), 105336.  doi: 10.1016/j.cnsns.2020.105336.
    [7] Z. Feng and M. Zhang, Boundedness and large time behavior of solutions to a prey-taxis system accounting in liquid surrounding, Nonlinear Anal., Real World Appl., 57 (2021), 103197.  doi: 10.1016/j.nonrwa.2020.103197.
    [8] H. I. Freedman, Deterministic mathematical models in population ecology, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1980. doi: 10.2307/2530090.
    [9] M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, SIAM J. Math. Anal., 52 (2020), 5865-5891.  doi: 10.1137/20M1344536.
    [10] X. He and S. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.
    [11] S. B. HsuS. Ruan and T. H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, J. Math. Anal. Appl., 426 (2015), 659-687.  doi: 10.1016/j.jmaa.2015.01.035.
    [12] H. Y. Jin and Z. A Wang, Global stability of prey-taxis systems, J. Differ. Equ., 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.
    [13] P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270. 
    [14] P. KratinaR. M. LecrawT. Ingram and B. R. Anholt, Stability and persistence of food webs with omnivory: Is there a general pattern?, Ecosphere, 3 (2012), 1-18.  doi: 10.1890/ES12-00121.1.
    [15] N. Krikorian, The Volterra model for three species predator-prey systems: Boundedness and stability, J. Math. Biol., 7 (1979), 117-132.  doi: 10.1007/BF00276925.
    [16] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equation of Parabolic Type, Amer. Math. Soc. Transl., Amer. Math. Soc., Providence, RI, 23 (1968).
    [17] E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.
    [18] J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, Nonlinear Differ. Equ. Appl., 24 (2017), 49 pp. doi: 10.1007/s00030-017-0472-8.
    [19] W. Lv, Global generalized solutions for a class of chemotaxis-consumption systems with generalized logistic source, J. Differential Equations, 283 (2021), 85-109.  doi: 10.1016/j.jde.2021.02.043.
    [20] T. Namba, K. Tanabe and N. Maeda, Omnivory and stability of food webs, Ecological Complexity, 5 (2008) 73–85. doi: 10.1016/j.ecocom.2008.02.001.
    [21] G. Ren, Boundedness and stabilization in a two-species chemotaxis system with logistic source, Z. Angew. Math. Phys., 71 (2020) 177. doi: 10.1007/s00033-020-01410-9.
    [22] G. Ren, Global solvability in a two-species chemotaxis system with logistic source, J. Math. Phys., 62 (2021), 041504.  doi: 10.1063/5.0040652.
    [23] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal., Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.
    [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 boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source, Commun. Pure Appl. Anal., 19 (2020), 3843-3883.  doi: 10.3934/cpaa.2020170.
    [26] G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differential Equations, 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.
    [27] G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differ. Equ., 269 (2020), 1484-1520.  doi: 10.1016/j.jde.2020.01.008.
    [28] 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.
    [29] G. Ren and B. Liu, Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis, Discrete Contin. Dyn. Syst. Ser. A, 42 (2022), 759-779.  doi: 10.3934/dcds.2021136.
    [30] G. Ren and Y. Shi, Global boundedness and stability of solutions for prey-taxis model with handling and searching predators, Nonlinear Anal., Real World Appl., 60 (2021), 103306.  doi: 10.1016/j.nonrwa.2021.103306.
    [31] Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal., Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.
    [32] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-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.
    [33] Y. Tao and M. Winkler, Global smooth solvability of a parabolic.elliptic nutrient taxis system in domains of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.
    [34] Y. Tao and M. Winkler, 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.  doi: 10.1142/S021820251950043X.
    [35] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and Its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
    [36] J. Wang and M. X. Wang, Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis, Z. Angew. Math. Phys., 69 (2018), 63.  doi: 10.1007/s00033-018-0960-7.
    [37] J. Wang and M. X. Wang, Global solution of a diffusive predator-prey model with prey-taxis, Comput. Math. Appl., 77 (2019), 2676-2694.  doi: 10.1016/j.camwa.2018.12.042.
    [38] Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.
    [39] Y. WangM. Winkler and Z. Xiang, Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation, Adv. Nonlinear Anal., 10 (2021), 707-731.  doi: 10.1515/anona-2020-0158.
    [40] Y. WangM. Winkler and Z. Xiang, Local energy estimates and global solvability in a three dimensional chemotaxis fluid system with prescribed signal on the boundary, Commun. Partial Differ. Equ., 46 (2021), 1058-1091.  doi: 10.1080/03605302.2020.1870236.
    [41] M. Winkler, Global large-data solutions in a chemotaxis-(navier-)stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.
    [42] M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.
    [43] M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations, 263 (2017), 4826-4869.  doi: 10.1016/j.jde.2017.06.002.
    [44] M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Functional Analysis, 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009.
    [45] 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.
    [46] M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$, Adv. Nonlinear Anal., 9 (2020), 526-566.  doi: 10.1515/anona-2020-0013.
    [47] M. Winkler, Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-stokes) systems?, International Mathematics Research Notices, 2021 (2021), 8106-8152.  doi: 10.1093/imrn/rnz056.
    [48] S. N. WuJ. P. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differ. Equ., 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.
    [49] S. N. WuJ. F. Wang and J. P. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.  doi: 10.1142/S0218202518400158.
    [50] T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal., Real World Appl., 39 (2018), 278-299.  doi: 10.1016/j.nonrwa.2017.07.001.
    [51] S. R. ZhouW. T. Li and G. Wang, Persistence and global stability of positive periodic solutions of three species food chains with omnivory, J. Math. Anal. Appl., 324 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.12.021.
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