April  2021, 41(4): 1681-1705. doi: 10.3934/dcds.2020337

Global stability in a multi-dimensional predator-prey system with prey-taxis

Department of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author: shuxuelidandan@163.com

Received  March 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

Fund Project: The first author is supported by NSF grant 12001201

This paper studies the predator-prey systems with prey-taxis
$ \begin{eqnarray*} { \label{1.1}} \left\{ \begin{array}{llll} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\gamma uv-\rho u, \\ v_{t} = \Delta v-\xi uv+\mu v(1-v), \ \end{array} \right. \end{eqnarray*} $
in a bounded domain
$ \Omega\subset\mathbb{R}^{n} $
$ (n = 2, 3) $
with Neumann boundary conditions, where the parameters
$ \chi $
,
$ \gamma $
,
$ \rho $
,
$ \xi $
and
$ \mu $
are positive. It is shown that the two-dimensional system possesses a unique global-bounded classical solution. Furthermore, we use some higher-order estimates to obtain the classical solutions with uniform-in-time bounded for suitably small initial data. Finally, we establish that the solution stabilizes towards the prey-only steady state
$ (0, 1) $
if
$ \rho>\gamma $
and towards the co-existence steady state
$ (\frac{\mu(\gamma-\rho)}{\xi\rho}, \frac{\rho}{\gamma}) $
if
$ \gamma>\rho $
under some conditions in the norm of
$ L^{\infty}(\Omega) $
as
$ t\rightarrow\infty $
.
Citation: Dan Li. Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1681-1705. doi: 10.3934/dcds.2020337
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Text in Math. Teubner, Stuttgart, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[3]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete. Contin. Dyn. Syst., 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701.

[4]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[5]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci, 24 (1997), 633-683. 

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[9]

J. JiangH. Wu and S. Zheng, Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differential Equations, 12 (2001), 159-177.  doi: 10.1016/j.jde.2018.01.004.

[10]

H.-Y Jin and Z. A. Wang, Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[11]

H.-Y Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., (2019).

[12]

A. JüngelC. Kuehn and L. Trussardi, A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.  doi: 10.1017/S0956792516000346.

[13]

A. Jüngel, Diffusive and Nondiffusive Population Models. Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.

[14]

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

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. Nral'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23, Providence, RI, 1968.

[17]

J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

[18]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint, 2013. doi: 10.1016/j.matpur.2013.01.020.

[19]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.

[20]

Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domain of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.

[21]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[22]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemootaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[23]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusivle attractant, J. Differential Equations, 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[24]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[25]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[26]

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.

[27]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Text in Math. Teubner, Stuttgart, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[3]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete. Contin. Dyn. Syst., 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701.

[4]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[5]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci, 24 (1997), 633-683. 

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[9]

J. JiangH. Wu and S. Zheng, Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant, J. Differential Equations, 12 (2001), 159-177.  doi: 10.1016/j.jde.2018.01.004.

[10]

H.-Y Jin and Z. A. Wang, Global stability of prey-taxis system, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[11]

H.-Y Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., (2019).

[12]

A. JüngelC. Kuehn and L. Trussardi, A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.  doi: 10.1017/S0956792516000346.

[13]

A. Jüngel, Diffusive and Nondiffusive Population Models. Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.

[14]

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

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. Nral'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23, Providence, RI, 1968.

[17]

J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

[18]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint, 2013. doi: 10.1016/j.matpur.2013.01.020.

[19]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.

[20]

Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domain of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.

[21]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[22]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemootaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[23]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusivle attractant, J. Differential Equations, 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[24]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[25]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[26]

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.

[27]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

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