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Large time behavior in a predator-prey system with indirect pursuit-evasion interaction

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  • In a bounded domain $ \Omega\subset \mathbb{R}^n $ with smooth boundary, this work considers the indirect pursuit-evasion model

    $ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u -\chi\nabla \cdot(u \nabla w) +u(\lambda -u+a v), \\ v_t = \Delta v +\xi \nabla \cdot(v\nabla z) +v(\mu-v-b u), \\ 0 = \Delta w -w +v, \\ 0 = \Delta z -z +u, \end{array} \right. \end{eqnarray*} $

    with positive parameters $ \chi, \xi, \lambda, \mu $, $ a $ and $ b $.

    It is firstly asserted that when $ n\le 3 $, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if $ b\lambda<\mu $ and under some explicit smallness conditions on $ \chi $ and $ \xi $, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if $ b\lambda>\mu $, however, then under an explicit smallness assumption on $ \chi $ but without any restriction on $ \xi $, any bounded classical solution $ (u, v) $ with $ u\not\equiv 0 $ stabilizes to $ (\lambda, 0) $ as $ t\to \infty $.

    Mathematics Subject Classification: Primary: 5B40, 35B65, 35K57, 35Q92, 92C17.


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  • [1] P. AmorimB. Telch and L. M. Villada, A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing, Math. Biosci. Eng., 16 (2019), 5114-5145.  doi: 10.3934/mbe.2019257.
    [2] H. Bréezis and W. A. Strauss, Semilinear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.
    [3] T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system. Parabolic and Navier-Stokes equations, Part 1, Polish Acad. Sci. Inst. Math., Banach Center Publ., 81 (2008), 105–117. doi: 10.4064/bc81-0-7.
    [4] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.
    [5] A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York-Montreal, Que.-London, 1969.
    [6] T. GoudonB. NkongaM. Rascle and M. Ribot, Self-organized populations interacting under pursuit-evasion dynamics, Phys. D, 304/305 (2015), 1-22.  doi: 10.1016/j.physd.2015.03.012.
    [7] T. Goudon and L. Urrutia, Analysis of kinetic and macroscopic models of pursuit-evasion dynamics, Commun. Math. Sci., 14 (2016), 2253-2286.  doi: 10.4310/CMS.2016.v14.n8.a7.
    [8] 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.
    [9] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. 
    [10] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 
    [11] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.
    [12] H.-Y. Jin and Z.-A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.
    [13] P. Kareiva and G. Odell, Swarms of predators exhibit `preytaxis' if individual predators use area-restricted search, The American Naturalist, 130 (1987), 233-270.  doi: 10.1086/284707.
    [14] 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.
    [15] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.
    [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 Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.
    [18] Y. Tao and M. Winkler, Boundedness and stabilization in a population model with cross-diffusion for one species, Proc. London Math. Soc., 119 (2019), 1598-1632.  doi: 10.1112/plms.12276.
    [19] 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.
    [20] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.
    [21] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.
    [22] Y. Tao and M. Winkler, Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension, J. Differential Equations, 267 (2019), 388-406.  doi: 10.1016/j.jde.2019.01.014.
    [23] M. A. Tsyganov, J. Brindley, A. V. Holden and V. N. Biktashev, Quasi-soliton interaction of pursuit-evasion waves in a predator-prey system, Phys. Rev. Lett., 91 (2003), 218102. doi: 10.1103/PhysRevLett.91.218102.
    [24] M. A. TsyganovI. B. KrestevaA. B. Medvinsky and G. R. Ivanitsky, A novel mode bacterial population wave interaction, Dokl. Akad. Nauk, 333 (1993), 532-536. 
    [25] Y. TyutyunovL. Titova and R. Arditi, A minimal model of pursuit-evasion in a predator-prey system, Math. Model. Nat. Phenom., 2 (2007), 122-134.  doi: 10.1051/mmnp:2008028.
    [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, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.
    [28] S. WuJ. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.
    [29] 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.
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