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

The first author is supported by NSF grant 12001201
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  • 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 $.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • [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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