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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 |
$ \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) $ |
$ \ast $ |
$ (u,v,w) $ |
$ (u_\ast,v_\ast,w_\ast) $ |
$ \ast $ |
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] |
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. 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. |
[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. 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. |
[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. 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] |
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. 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. |
[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. 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. |
[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. 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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