# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021136
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## Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

*Corresponding author: Bin Liu

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: The first author is supported by NSF grant No. 12001214 and the second author is supported by NSF grant No. 11971185

In this work we consider a two-species predator-prey chemotaxis model
 $\left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+u(a_1-b_{11}u-b_{12}v), &x\in \Omega, t>0, \\[0.2cm] v_t = d_2\Delta v-\chi_2\nabla\cdot(v\nabla u)+v(-a_2+b_{21}u-b_{22}v-b_{23}w), & x\in \Omega, t>0, \\[0.2cm] w_t = d_3\Delta w-\chi_3\nabla\cdot(w\nabla v)+w(-a_3+b_{32}v-b_{33}w), & x\in \Omega, t>0 \\ \end{array}\right.(\ast)$
in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model (
 $\ast$
) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions
 $(u,v,w)$
exponentially converges to constant stable steady state
 $(u_\ast,v_\ast,w_\ast)$
. Inspired by [5], we employ the special structure of (
 $\ast$
) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.
Citation: Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021136
##### 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.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar [6] M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, preprint, arXiv: 2004.04515v2. Google Scholar [7] 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.  Google Scholar [8] 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.  Google Scholar [9] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [10] P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.   Google Scholar [11] 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.  Google Scholar [12] J. Liu and Y. Ma, Asymptotic behavior analysis for a three-species food chain stochastic model with regime switching, Math. Probl. Eng., 2020 (2020), 1-17.  doi: 10.1155/2020/1679018.  Google Scholar [13] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A, 230 (1996), 449-543.  doi: 10.1016/0378-4371(96)00051-9.  Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] Y. Tao and M. Winkler, A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation, J. Functional Analysis, 281 (2021), 109069.  doi: 10.1016/j.jfa.2021.109069.  Google Scholar [18] Y. Tao and M. Winkler, Existence theory and qualitative analysis of a fully cross-diffusive predator-prey system, preprint, arXiv: 2004.00529v1. Google Scholar [19] 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). doi: 10.1007/s00033-018-0960-7.  Google Scholar [20] 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.  Google Scholar [21] 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.  Google Scholar [22] 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.  Google Scholar

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##### 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.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar [6] M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, preprint, arXiv: 2004.04515v2. Google Scholar [7] 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.  Google Scholar [8] 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.  Google Scholar [9] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [10] P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.   Google Scholar [11] 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.  Google Scholar [12] J. Liu and Y. Ma, Asymptotic behavior analysis for a three-species food chain stochastic model with regime switching, Math. Probl. Eng., 2020 (2020), 1-17.  doi: 10.1155/2020/1679018.  Google Scholar [13] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A, 230 (1996), 449-543.  doi: 10.1016/0378-4371(96)00051-9.  Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] Y. Tao and M. Winkler, A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation, J. Functional Analysis, 281 (2021), 109069.  doi: 10.1016/j.jfa.2021.109069.  Google Scholar [18] Y. Tao and M. Winkler, Existence theory and qualitative analysis of a fully cross-diffusive predator-prey system, preprint, arXiv: 2004.00529v1. Google Scholar [19] 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). doi: 10.1007/s00033-018-0960-7.  Google Scholar [20] 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.  Google Scholar [21] 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.  Google Scholar [22] 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.  Google Scholar
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