Search Advanced

Mathematical Biosciences and Engineering

2016, Volume 13, Issue 4: 857-885. doi: 10.3934/mbe.2016021
Previous Article Next Article

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

  • 1.
    School of Mathematics and Statistics, Southwest University, Chongqing, 400715
  • Received: 01 August 2015 Accepted: 29 June 2018 Published: 01 May 2016

  • MSC : Primary: 35J55, 35K57; Secondary: 92C15, 92C40.

  • This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

    Citation: Jun Zhou. Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack[J]. Mathematical Biosciences and Engineering, 2016, 13(4): 857-885. doi: 10.3934/mbe.2016021

    Related Papers:

    [1] Dongfu Tong, Yongli Cai, Bingxian Wang, Weiming Wang . Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model. Mathematical Biosciences and Engineering, 2019, 16(5): 3988-4006. doi: 10.3934/mbe.2019197
    [2] Mingzhu Qu, Chunrui Zhang, Xingjian Wang . Analysis of dynamic properties on forest restoration-population pressure model. Mathematical Biosciences and Engineering, 2020, 17(4): 3567-3581. doi: 10.3934/mbe.2020201
    [3] Guangxun Sun, Binxiang Dai . Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting. Mathematical Biosciences and Engineering, 2020, 17(4): 3520-3552. doi: 10.3934/mbe.2020199
    [4] Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang . Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247
    [5] Tingting Ma, Xinzhu Meng . Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism. Mathematical Biosciences and Engineering, 2022, 19(6): 6040-6071. doi: 10.3934/mbe.2022282
    [6] A. Aldurayhim, A. Elsonbaty, A. A. Elsadany . Dynamics of diffusive modified Previte-Hoffman food web model. Mathematical Biosciences and Engineering, 2020, 17(4): 4225-4256. doi: 10.3934/mbe.2020234
    [7] Fang Liu, Yanfei Du . Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator. Mathematical Biosciences and Engineering, 2023, 20(11): 19372-19400. doi: 10.3934/mbe.2023857
    [8] Wanxiao Xu, Ping Jiang, Hongying Shu, Shanshan Tong . Modeling the fear effect in the predator-prey dynamics with an age structure in the predators. Mathematical Biosciences and Engineering, 2023, 20(7): 12625-12648. doi: 10.3934/mbe.2023562
    [9] Yuhong Huo, Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty, Renji Han . Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor. Mathematical Biosciences and Engineering, 2023, 20(10): 18820-18860. doi: 10.3934/mbe.2023834
    [10] Zuolin Shen, Junjie Wei . Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences and Engineering, 2018, 15(3): 693-715. doi: 10.3934/mbe.2018031
  • This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.


    [1] J. Math. Anal. Appl., 376 (2011), 551-564.
    [2] John Wiley & Sons, 2003.
    [3] Phy. Rev. E Stat. Nonlinear & Soft Matter Physics, 85 (2012), 489-500.
    [4] 2009.
    [5] Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507-516.
    [6] Nonlinearity, 10 (1997), 523-563.
    [7] J. Math. Anal. Appl., 366 (2010), 473-485.
    [8] Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413-438.
    [9] In Modern Aspects of the Theory of Partial Differential Equations, volume 216 of Oper. Theory Adv. Appl., pages 153-166. Birkhäuser/Springer Basel AG, Basel, 2011.
    [10] Commun. Contemp. Math., 12 (2010), 661-679.
    [11] Nonlinearity, 21 (2008), 2331-2345.
    [12] Springer Verlag, 2012.
    [13] SIAM J. Appl. Math., 69 (2008), 251-272.
    [14] Appl. Math. Lett., 12 (1999), 59-65.
    [15] CUP Archive, 1981.
    [16] J. Math. Biol., 49 (2004), 358-390.
    [17] J. Dynam. Differential Equations, 16 (2004), 297-320.
    [18] Roc. Mount.J. Math., 43 (2013), 1637-1674.
    [19] Phys. D, 214 (2006), 63-77.
    [20] Amer. Natu., 168 (2006), 36-47.
    [21] IMA J. Numer. Anal., 12 (1992), 405-428.
    [22] Journal of Differential Equations, 131 (1996), 79-131.
    [23] Math. Compu. in Simulation, 40 (1996), 371-396.
    [24] Nonlinear Anal. Real World Appl., 5 (2004), 105-121.
    [25] Handbook of Differential Equations Stationary Partial Differential Equations, 1 (2004), 157-233.
    [26] Trans. Amer. Math. Soc., 357 (2005), 3953-3969.
    [27] J. Math. Anal. Appl., 309 (2005), 151-166.
    [28] Math. Comput. Modelling, 44 (2006), 945-951.
    [29] J. Differential Equations, 241 (2007), 386-398.
    [30] Appl. Math. Lett., 22 (2009), 569-573.
    [31] Nonlinear Anal., 72 (2010), 2337-2345.
    [32] Phys. D, 226 (2007), 129-135.
    [33] European Journal of Biochemistry, 4 (1968), 79-86.
    [34] J. Theoret. Biol., 81 (1979), 389-400.
    [35] J. Differential Equations, 246 (2009), 2788-2812.
    [36] J. Differential Equations, 72 (1988), 1-27.
    [37] Nature Rev. Molecular Cell Bio., 2 (2001), 908-916.
    [38] Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72.
    [39] J. Differential Equations, 190 (2003), 600-620.
    [40] Nonlinearity, 21 (2008), 1471-1488.
    [41] Applied Mathematics Letters, 21 (2008), 1215-1220.
    [42] J. Differential Equations, 251 (2011), 1276-1304.
    [43] Stud. Appl. Math., 109 (2002), 229-264.
    [44] Phys. D, 148 (2001), 20-48.
    [45] J. Math. Biol., 57 (2008), 53-89.
    [46] J. Math. Biol., 64 (2012), 211-254.
    [47] Springer, 1990.
    [48] WSEAS Transac. Math, 10 (2011), 201-209.
    [49] Nonlinear Anal. Real World Appl., 13 (2012), 1961-1977.
    [50] Nonlinear Anal.: Real World Applications, 9 (2008), 1038-1051.
    [51] J. Differential Equations, 246 (2009), 1944-1977.
    [52] Appl. Math. Lett., 22 (2009), 52-55.
    [53] Dyn. Partial Differ. Equ., 4 (2007), 167-196.
    [54] Commun. Pure Appl. Anal., 10 (2011), 1415-1445.
    [55] Nonlinear Anal., 74 (2011), 1969-1986.
    [56] Math. Methods Appl. Sci., 35 (2012), 398-416.
    [57] J. Dynam. Differential Equations, 24 (2012), 495-520.
    [58] J. Math. Anal. Appl., 366 (2010), 679-693.
    [59] Dyn. Partial Differ. Equ., 8 (2011), 363-384.

  • This article has been cited by:

    1. Ying Sun, Jinliang Wang, You Li, Nan Jiang, Juandi Xia, Turing pattern selection for a plant–wrack model with cross-diffusion, 2023, 32, 1674-1056, 090203, 10.1088/1674-1056/acac13
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(2473) PDF downloads(541) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog