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February  2022, 42(2): 759-779. doi: 10.3934/dcds.2021136

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 Published  February 2022 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 and Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136
References:
[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]

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.

[3]

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.

[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.

[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.

[6]

M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, preprint, arXiv: 2004.04515v2.

[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.

[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.

[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.

[10]

P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

Y. Tao and M. Winkler, Existence theory and qualitative analysis of a fully cross-diffusive predator-prey system, preprint, arXiv: 2004.00529v1.

[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.

[20]

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.

[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.

[22]

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.

show all references

References:
[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]

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.

[3]

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.

[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.

[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.

[6]

M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, preprint, arXiv: 2004.04515v2.

[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.

[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.

[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.

[10]

P. Kareiva and G. Odell, Swarms of predators exhibit "preytaxis" if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

Y. Tao and M. Winkler, Existence theory and qualitative analysis of a fully cross-diffusive predator-prey system, preprint, arXiv: 2004.00529v1.

[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.

[20]

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.

[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.

[22]

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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