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Stability analysis of coexistence of three species prey–predator model

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Abstract

In this paper, we have proposed a prey–predator model for the study of dynamical behaviors of three species such as toxin-producing Phytoplankton, Zooplankton and Fish in a fishery system. The stability condition, existence condition of equilibrium and bifurcation have also been established. In this paper, Holling type II functional response function has been considered to analysis of the proposed model. All equilibriums of the proposed system are determined, and the behavior of the system is also investigated near the positive equilibrium point. At the end, local stability of the system is analyzed by numerical illustrations.

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References

  1. Abdllaoui, A.E., Chattopadhyay, J., Arino, O.: Comparisons, by models, of some basic mechanisms acting on the dynamics of the zooplankton toxic phytoplankton systems. Math. Mod. Meth. Appl. S. 12(10), 1421–1451 (2002)

    Article  MATH  Google Scholar 

  2. Chattrjee, A., Pal, S., Chattrjee, S.: Bottom up and top down effect on toxin producing phytoplankton and its consequences on the formation of plankton bloom. Appl. Math. Comput. 218, 3387–3398 (2011)

    Article  MathSciNet  Google Scholar 

  3. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1990)

    MATH  Google Scholar 

  4. Chakraborty, K., Jana, S., Kar, T.K.: Effort dynamics of a delay-induced prey–predator system with reserve. Nonlinear Dyn. 70, 1805–1829 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chattopadhayay, J., Sarkar, R.R., Mondal, S.: Toxin producing plankton may act as a biological control for planktonic bloom—field study and mathematical modelling. J. Theor. Biol. 215, 333–344 (2002)

    Article  Google Scholar 

  6. Chakraborty, K., Jana, S., Kar, T.K.: Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting. Appl. Math. Comput. 218, 9271–9290 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Duinker, J., Wefer, G.: Das \(CO_2\) Und Die Rolle Des Ozeans. Naturwissenschahtn 81, 237–242 (1994)

    Article  Google Scholar 

  8. Faria, T., Magalhaes, L.T.: Normal forms for retarded functional differential equations with parameters and Applications to Hopf bifurcations. J. Differ. Equa. 122, 181–200 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gakkhar, S., Negi, K.: A mathematical model for viral infection in toxin producing phytoplankton and zooplankton system. Appl. Math. Comput. 179, 301–313 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hugo, A., Massawe, E.S., Makinde, O.D.: An eco-epidemiological mathematical model with treatment and disease infection in both prey and predator population. J. Ecol. Nat. Environ. 4(10), 266–279 (2012)

    Google Scholar 

  11. Kar, T.K., Chakraborty, K.: Effort dynamics in a prey–predator model with harvesting. Int. J. Inf. Syst. Sci. 6(3), 318–332 (2010)

    MATH  MathSciNet  Google Scholar 

  12. Liu, W.M.: Criterion of Hopf bifurcation without using eigenvalues. J. Math. Anal. Appl. 182, 250 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ma, Z., Wang, Y., Jiang, G.: Bifurcation analysis of a linear Hamiltonian system with two kinds of impulsive control. Nonlinear Dyn. 70, 2367–2374 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Makinde, O.D.: Solving ratio-dependent predator–prey system with constant effort harvesting using Adomain decomposition method. Appl. Math. Comput. 187, 17–22 (2007)

    Article  MathSciNet  Google Scholar 

  15. Odum, E.P.: Fundamentals of Ecology. Saunders, Philadelphia (1971)

    Google Scholar 

  16. Pradhan, T., Chaudhuri, K.S.: A dynamical reaction model of two species fishery with taxation as a control instrument: a capital theoretic analysis. Ecol. Model. 121, 1–16 (1999)

  17. Pei, Y., Chen, L., Zhang, Q., Li, C.: Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control. J. Theor. Biol. 235, 495–503 (2005)

    Article  MathSciNet  Google Scholar 

  18. Pei, Y., Zeng, G., Chen, L.: Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control. Nonlinear Dyn. 52, 71–81 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Panja, P., Mondal, S.K.: A mathematical study on the spread of Cholera. South Asian J. Math. 4(2), 69–84 (2014)

    Google Scholar 

  20. Qu, Y., Wei, J.: Bifurcation analysis in a time-delay model for preypredator growth with stage-structure. Nonlinear Dyn. 49, 285–294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Samayda, T.: What is a bloom? A commentary. Limnol Oceonogr. 42, 1132–1136 (1997)

    Article  Google Scholar 

  22. Sarkar, R.R., Chattopadhyay, J.: The role of environmental stochasticity in a toxic phytoplankton-non-toxic phytoplankton zooplankton system. Environmetrics 14, 775–792 (2003)

    Article  Google Scholar 

  23. Saha, T., Bandyopadhyay, M.: Dynamical analysis of toxin producing phytoplankton–zooplankton interactions. Nonlinear Anal. Real World Appl. 10, 314–332 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Vankatsubramaninan, V., Schattler, H., Zaborszky, J.: Local bifurcation and feasibility regions in differential–algebraic systems. IEEE. Trans. Autom. Control 40(12), 1992–2013 (1995)

    Article  Google Scholar 

  25. Wang, J., Jiang, W.: Bifurcation and chaos of a delayed predator–prey model with dormancy of predators. Nonlinear Dyn. 69, 1541–1558 (2012)

    Article  MATH  Google Scholar 

  26. Xiao, D., Li, W., Han, W.: Dynamics in a ratio dependent predator prey model with predator harvesting. J. Math. Anal. Appl. 324(1), 4–29 (2006)

    Article  MathSciNet  Google Scholar 

  27. Yunfei, L., Yongzhen, P., Shujing, G., Changguo, L.: Harvesting of a phytoplankton–zooplankton model. Nonlinear Anal. Real World Appl. 11, 3608–3619 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721 102, WB, India

    Prabir Panja & Shyamal Kumar Mondal

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  1. Prabir Panja
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  2. Shyamal Kumar Mondal
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Correspondence to Prabir Panja.

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Panja, P., Mondal, S.K. Stability analysis of coexistence of three species prey–predator model. Nonlinear Dyn 81, 373–382 (2015). https://doi.org/10.1007/s11071-015-1997-1

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  • DOI: https://doi.org/10.1007/s11071-015-1997-1

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