Advanced Search
Article Contents
Article Contents

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

  • * Corresponding author: Jinfeng Wang

    * Corresponding author: Jinfeng Wang 

Partially supported by National NSFC grant 11971135, NSFH grant LH2019A017, LH2020A019 and 2018-KYYWF-0999

Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an $ n $-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.

    Mathematics Subject Classification: Primary: 35K57; Secondary: 35K59.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Change of $ \chi_i $ with non-negative integer $ i $

    Figure 2.  $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ is asymptotically stable for $ \chi = 0 $

    Figure 3.  $ (\tilde{u}_1,\tilde{u}_2,\tilde{v}) $ remains stable for $ \chi = 200<\hat{\chi} $

    Figure 4.  Spatial heterogenous and time-periodic patterns for $ \chi = 308>\hat{\chi} $

  • [1] H. Amann, Dynamic theory of quasilinear parabolic equations Ⅱ.Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 
    [2] H. Amman, 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, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathe. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [4] N. BellomoK. J. PainterY. Tao and M. Winkler, Occurrence vs. absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math, 79 (2019), 1990-2010.  doi: 10.1137/19M1250261.
    [5] E. X. Dejesus and C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equation, Phys. Rev. A, 35 (1987), 5288-5290.  doi: 10.1103/PhysRevA.35.5288.
    [6] A. K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7.
    [7] Y. H. DuP. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173.
    [8] X. X. Guo and J. F. Wang, Dynamics and pattern formations in diffusive predator-prey models with two prey-taxis, Math. Methods Appl. Sci., 42 (2019), 4197-4212.  doi: 10.1002/mma.5639.
    [9] X. He and S. N. 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.
    [10] 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.
    [11] W. J$\ddot{a}$ger 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 prey-taxis if individual predators use area-restricted search, American Naturalist, 130 (1987), 233-270. 
    [14] 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.
    [15] P. LiuJ. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.
    [16] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. 
    [17] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. 
    [18] Y. S. 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.
    [19] Y. S. 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.
    [20] Y. S. Tao and M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.  doi: 10.1142/S021820251950043X.
    [21] J. F. WangS. N. Wu and J. P. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1273-1289.  doi: 10.3934/dcdsb.2020162.
    [22] J. P. Wang and M. X. Wang, Global bounded solution of the higher-dimensional forager-exploiter model with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323.  doi: 10.1142/S0218202520500232.
    [23] K. WangQ. Wang and F. Yu, Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis, Discrete Contin. Dyn. Syst., 37 (2017), 505-543.  doi: 10.3934/dcds.2017021.
    [24] X. L. WangW. D. Wang and G. H. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431-443.  doi: 10.1002/mma.3079.
    [25] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1644-1673.  doi: 10.1002/mana.200810838.
    [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
    [27] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.
    [28] 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.
    [29] 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.
    [30] S. N. WuJ. P. Shi and B. Y. 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.
    [31] 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.
  • 加载中



Article Metrics

HTML views(499) PDF downloads(486) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint