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doi: 10.3934/dcdsb.2022031
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## Global generalized solutions to a three species predator-prey model with prey-taxis

 a. School of Science, Qingdao University of Technology, Qingdao, 266033, China b. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China c. School of Science, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author: Ruijing Li

Received  October 2021 Revised  January 2022 Early access February 2022

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

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.
Citation: Xin Wang, Ruijing Li, Yu Shi. Global generalized solutions to a three species predator-prey model with prey-taxis. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022031
##### References:
 [1] B. E. Ainseba, M. 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. Bellomo, A. Bellouquid, Y. 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. Cai, C. Zhao, W. 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. Chen, T. 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. Hsu, S. 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. Kratina, R. M. Lecraw, T. 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. Wang, M. 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. Wang, M. 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. Wu, J. 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. Wu, J. 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. Zhou, W. 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.

show all references

##### References:
 [1] B. E. Ainseba, M. 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. Bellomo, A. Bellouquid, Y. 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. Cai, C. Zhao, W. 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. Chen, T. 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. Hsu, S. 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. Kratina, R. M. Lecraw, T. 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. Wang, M. 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. Wang, M. 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. Wu, J. 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. Wu, J. 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. Zhou, W. 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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