Research article Special Issues

Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

  • Received: 09 February 2019 Accepted: 15 April 2019 Published: 28 April 2019
  • In this paper, a delayed phytoplankton-zooplankton system with the coefficient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.

    Citation: Zhichao Jiang, Xiaohua Bi, Tongqian Zhang, B.G. Sampath Aruna Pradeep. Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3807-3829. doi: 10.3934/mbe.2019188

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  • In this paper, a delayed phytoplankton-zooplankton system with the coefficient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.




    [1] S. Jang, J. Baglama and L. Wu, Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton, Appl. Math. Comput., 227 (2014), 717–740.
    [2] F. Rao, The complex dynamics of a stochastic toxic-phytoplankton-zooplankton model, Adv. Difference. Equ., 2014 (2014), 22.
    [3] A. Sharma, A. Kumar Sharma and K. Agnihotri, Analysis of a toxin producing phytoplanktonzooplankton interaction with Holling IV type scheme and time delay, Nonlinear Dynam., 81 (2015), 13–25.
    [4] B. Ghanbari and J. Gómez-Aguilar, Modeling the dynamics of nutrient-phytoplankton-zooplankton system with variable-order fractional derivatives, Chaos Solitons Fractals, 116 (2018), 114–120.
    [5] T. Liao, H. Yu and M. Zhao, Dynamics of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response, Adv. Difference. Equ., 2017 (2017), 5.
    [6] J. Li, Y. Song and H. Wan, Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge, Math. Biosci. Eng., 14 (2017), 529–557.
    [7] Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay, Chaos Solitons Fractals, 104 (2017), 693–704.
    [8] T. Zhang, X. Liu, X. Meng, et al., Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl., 75 (2018), 4490–4504.
    [9] T. Zhang, Y. Xing, H. Zang, et al., Spatio-temporal patterns in a predator-prey model with hyperbolic mortality, Nonlinear Dynam., 78 (2014), 265–277.
    [10] X. Yu, S. Yuan and T. Zhang, The effects of toxin producing phytoplankton and environmental fluctuations on the planktonic blooms, Nonlinear Dynam., 91 (2018), 1653–1668.
    [11] Y. Zhao, S. Yuan and T. Zhang, Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 266– 276.
    [12] Y. Zhao, S. Yuan and T. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 131–142.
    [13] Z. Jiang, W. Zhang, J. Zhang, et al., Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and Holling III functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850162.
    [14] J. Chattopadhayay, R. Sarkar and S. Mandal, Toxcin-producing plankton may act as a biological control for planktonic blooms-field study and mathematical modeling, J. Theoret. Biol., 215 (2002), 333–344.
    [15] J. Dhar, A. Sharma and S. Tegar, The role of delay in digestion of plankton by fish population: A fishery model, J. Nonlinear Sci. Appl., 1 (2008), 13–19.
    [16] J. Chattopadhyay, R. Sarkar and A. El Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Math. Appl. Med. Biol., 19 (2002), 137– 161.
    [17] T. Zhang, W. Ma and X. Meng, Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input, Adv. Difference. Equ., 2017 (2017), 115.
    [18] Y. Tang and L. Zhou, Great time delay in a system with material cycling and delayed biomass growth, IMA J. Appl. Math., 70 (2005), 191–200.
    [19] Y. Tang and L. Zhou, Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay, J. Math. Anal. Appl., 334 (2007), 1290–1307.
    [20] T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplanktonzooplankton interactions, Nonlinear Anal. Real World Appl., 10 (2009), 314–332.
    [21] M. Rehim and M. Imran, Dynamical analysis of a delay model of phytoplankton-zooplankton interaction, Appl. Math. Model., 36 (2012), 638–647.
    [22] Y. Wang, W. Jiang and H. Wang, Stability and global Hopf bifurcation in toxic phytoplanktonzooplankton model with delay and selective harvesting, Nonlinear Dynam., 73 (2013), 881–896.
    [23] Z. Jiang, W. Ma and D. Li, Dynamical behavior of a delay differential equation system on toxin producing phytoplankton and zooplankton interaction, Japan J. Indust. Appl. Math., 31 (2014), 583–609.
    [24] X. Fan, Y. Song and W. Zhao, Modeling cell-to-cell spread of hiv-1 with nonlocal infections, Complexity, 2018 (2018), 2139290.
    [25] M. Chi and W. Zhao, Dynamical analysis of two-microorganism and single nutrient stochastic chemostat model with monod-haldane response Function, Complexity, 2019 (2019), 8719067.
    [26] N. Gao, Y. Song, X. Wang, et al., Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates, Adv. Difference. Equ., 2019 (2019), 41.
    [27] J. Ivlev, Experimental ecology of the feeding of fishes, Yale University Press, New Haven, 1961.
    [28] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
    [29] Z. Wang, X. Wang, Y. Li, et al., Stability and Hopf bifurcation of fractional-order complexvalued single neuron model with time delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750209.
    [30] L. Li, Z. Wang, Y. Li, et al., Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Appl. Math. Comput, 330 (2018), 152–169.
    [31] Z. Jiang and L. Wang, Global Hopf bifurcation for a predator-prey system with three delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750108.
    [32] Y. Dai, Y. Jia, H. Zhao, et al., Global Hopf bifurcation for three-species ratio-dependent predatorprey system with two delays, Adv. Difference. Equ., 2016 (2016), 13.
    [33] J.Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 35 (1998), 4799–4838.
    [34] J. Hale and S. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
    [35] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities (Theory and Application): Ordinary Differential Equations, Academic Press, New York, 1969.
    [36] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144–1165.
    [37] Y. Qu, J. Wei and S. Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays, Phys. D, 23 (2010), 2011–2024.
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