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Stability and Hopf bifurcation analysis of a delayed phytoplankton-zooplankton model with Allee effect and linear harvesting

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

In this article, a delayed phytoplankton-zooplankton system with Allee effect and linear harvesting is proposed, where phytoplankton species protects themselves from zooplankton by producing toxin and taking shelter. First, the existence and stability of the possible equilibria of system are explored. Next, the existence of Hopf bifurcation is investigated when the system has no time delay. What’s more, the stability of limit cycle is demonstrated by calculating the first Lyapunov number. Then, the condition that Hopf bifurcation occurs is obtained by taking the time delay describing the maturation period of zooplankton species as a bifurcation parameter. Furthermore, based on the normal form theory and the central manifold theorem, we derive the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, by regarding the harvesting effort as control variable and employing the Pontryagin’s Maximum Principle, the optimal harvesting strategy of the system is obtained. Finally, in order to verify the validity of the theoretical results, some numerical simulations are carried out.
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Keywords phytoplankton-zooplankton model; Allee effect; time delay; Hopf bifurcation

Citation: Xinyou Meng, Jie Li. Stability and Hopf bifurcation analysis of a delayed phytoplankton-zooplankton model with Allee effect and linear harvesting. Mathematical Biosciences and Engineering, 2020, 17(3): 1973-2002. doi: 10.3934/mbe.2020105


  • 1. J. Norberg, D. Deangelis, Temperature effects on stocks and stability of a phytoplanktonzooplankton model and the dependence on light and nutrients, Ecol. Model., 95 (1997), 75-86.
  • 2. B. Mukhopadhyay, R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecol. Model., 198 (2006), 163-173.
  • 3. Y. F. Lv, Y. Z. Pei, S. J. Gao, C. G. Li, Harvesting of a phytoplankton-zooplankton model, Nonlinear Anal.: Real World Appl., 11 (2010), 3608-3619.
  • 4. M. Bengfort, U. Feudel, F. M. Hilker, H. Malchow, Plankton blooms and patchiness generated by heterogeneous physical environments, Ecol. Complex., 20 (2014), 185-194.
  • 5. S. Rana, S. Samanta, S. Bhattacharya, K. Alkhaled, A. Goswami, J. Chattopadhyay, The effect of nanoparticles on plankton dynamics: A mathematical model, Biosystems, 127 (2015), 28-41.
  • 6. X. Y. Meng, Y. Q. Wu, Bifurcation and control in a singular phytoplankton-zooplankton-fish model with nonlinear fish harvesting and taxation, Int. J. Bifurcat. Chaos, 28 (2018), 1850042.
  • 7. S. Chakraborty, S. Roy, J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model, Ecol. Model., 213 (2008), 191-201.
  • 8. T. Saha, M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton-zooplankton interactions, Nonlinear Anal.: Real World Appl., 10 (2009), 314-332.
  • 9. M. Banerjee, E. Venturino, A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.
  • 10. M. Javidi, B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system, Ecol. Model., 318 (2015), 8-18.
  • 11. R. J. Han, B. X. Dai, Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton-zooplankton model with nonmonotonic functional response, Int. J. Bifurcat. Chaos, 27 (2017), 1750088.
  • 12. R. J. Han, B. X. Dai, Spatiotemporal pattern formation and selection induced by nonlinear crossdiffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal.: Real World Appl., 45 (2019), 822-853.
  • 13. W. Zheng, J. Sugie, Global asymptotic stability and equiasymptotic stability for a time-varying phytoplankton-zooplankton-fish system, Nonlinear Anal.: Real World Appl., 46 (2019), 116-136.
  • 14. T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Si., 10 (2005), 681-691.
  • 15. L. J. Chen, F. D. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal.: Real World Appl., 11 (2010), 246-252.
  • 16. J. P. Tripathi, S. Abbas, M. Thakur, Dynamical analysis of a prey-predator model with BeddingtonDeAngelis type function response incorporating a prey refuge, Nonlinear Dynam., 80 (2015), 177-196.
  • 17. J. Ghosh, B. Sahoo, S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos Soliton Fract., 96 (2017), 110-119.
  • 18. G. P. Samanta, A. Maiti, M. Das, Stability analysis of a prey-predator fractional order model incorporating prey refuge, Ecol. Genet. Genomi., 7-8 (2018), 33-46.
  • 19. J. Li, Y. Z. Song, H. Wan, H. P. Zhu, Dynamical analysis of a toxin-producing phytoplanktonzooplankton model with refuge, Math. Biosci. Eng., 14 (2017), 529-557.
  • 20. L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends Ecol. Evol., 22 (2007), 185-191.
  • 21. C. Franck, C. B. Tim, G. Bryan, Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405-410.
  • 22. F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008.
  • 23. C. B. Tim, D. Gaynor, G. M. McIlrath, A. Maccoll, R. Kansky, P. Chadwick, et al., Predation, group size and mortality in a cooperative mongoose, Suricata suricatta, J. Anim. Ecol., 68 (1999), 672-683.    
  • 24. M. S. Mooring, T. A. Fitzpatrick, T. T. Nishihira, D. D. Reisig, Vigilance, predation risk, and the Allee effect in desert bighorn sheep, J. Wildlife Manage., 68 (2004), 519-532.
  • 25. D. J. Rinella, M. S. Wipfli, C. A. Stricker, R. A. Heintz, M. J. Rinella, Pacific salmon (Oncorhynchus spp.) runs and consumer fitness: Growth and energy storage in stream-dwelling salmonids increase with salmon spawner density, Can. J. Fish. Aquat. Sci., 69 (2011), 73-84.
  • 26. P. A. Stephens, W. J. Sutherl, R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
  • 27. A. Maiti, P. Sen, D. Manna, G. P. Samanta, A predator-prey system with herd behaviour and strong Allee effect, Nonlinear Dyn. Syst. Theory, 16 (2016), 86-101.
  • 28. H. F. Huo, W. T. Li, J. J. Nieto, Periodic solutions of delayed predator-prey model with the Beddington-DeAngelis functional response, Chaos Soliton Fract., 33 (2007), 505-512.
  • 29. W. Y. Wang, L. J. Pei, Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system, Acta Mech. Sinica, 27 (2011), 285-296.
  • 30. G. D. Zhang, Y. Shen, B. S. Chen, Positive periodic solutions in a non-selective harvesting predator-prey model with multiple delays, J. Math. Anal. Appl., 395 (2012), 298-306.
  • 31. H. Xiang, Y. Y. Wang, H. F. Huo, Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput., 8 (2018), 1535-1554.
  • 32. R. Chinnathambi, F. A. Rihan, Stability of fractional-order prey-predator system with time-delay and Monod-Haldane functional response, Nonlinear Dynam., 92 (2018), 1-12.
  • 33. X. Y. Meng, J. G. Wang, Analysis of a delayed diffusive model with Beddington-DeAngelis functional response, Int. J. Biomath., 12 (2019), 1950047.
  • 34. X. Y. Meng, J. G. Wang, H. F. Huo, Dynamical behaviour of a nutrient-plankton model with Holling type IV, delay, and harvesting, Discrete Dyn. Nat. Soc., 2018 (2018), 9232590.
  • 35. X. Y. Meng, Y. Q. Wu, Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting, Math. Biosci. Eng., 16 (2019), 2668-2696.
  • 36. X. B. Zhang, H. Y. Zhao, Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters, J. Theor. Biol., 363 (2014), 390-403.
  • 37. M. Sen, P. Srinivasu, M. Banerjee, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Appl. Math. Comput., 250 (2015), 19-211.
  • 38. H. Y. Zhao, X. X. Huang, X. B. Zhang, Hopf bifurcation and harvesting control of a bioeconomic plankton model with delay and diffusion terms, Physica A, 421 (2015), 300-315.
  • 39. D. P. Hu, H. J. Cao, Stability and bifurcation analysis in a predator-prey system with MichaelisMenten type predator harvesting, Nonlinear Anal.: Real World Appl., 33 (2017), 58-82.
  • 40. X. Y. Meng, N. N. Qin, H. F. Huo, Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species, J. Biol. Dynam., 12 (2018), 342-374.
  • 41. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.
  • 42. L. Perko, Differential Equations and Dynamical Systems, Springer Science and Business Media, New York, 2013.
  • 43. B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
  • 44. L. S. Pontryagin, Mathematical Theory of Optimal Processes, Routledge, London, 2018.


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