Skip to main content
SpringerLink
Log in

Persistence and Stability of a Seasonally Perturbed Three Species Stochastic Model of Salmonoid Aquaculture

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

Dynamics of a seasonally perturbed stochastic three species (phytoplankton-zooplankton-fish) system with harvesting is investigated in this article. The effect of periodic variations is considered on three different parameters of the system, growth rate of prey and mortality rates of middle and top predator. These seasonally varying parameters are considered to be in the different phases. Uniform boundededness of the system is proved. Existence of unique positive global solution of the system is established. It is observed that the system is strongly persistent in mean under certain parametric conditions. Gaussian white noise term is introduced into the system to represent the effect of random harvesting of fish population through poaching, uncontrolled recreational fishing or escape into wild habitat during storm. It is observed that the system is stable in mean square when the intensity of noise is small.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Rinaldi, S., Muratori, S., Kuznetsov, Y.A.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull. Math. Biol. 55, 15–35 (1993)

    Article  Google Scholar 

  2. Yu, H.G., Zhong, S.M., Agarwal, R.P., Sen, S.K.: Effect of seasonality on the dynamical behavior of an ecological system with impulsive control strategy. J. Franklin Inst. 348, 652–670 (2011)

    Article  MathSciNet  Google Scholar 

  3. Popova, E.E., Fasham, M.J.R., Osipov, A.V., Ryabchenko, V.A.: Chaotic behaviour of an ocean ecosystem model under seasonal external forcing. J. Plankton Res. 19, 1495–1515 (1997)

    Article  Google Scholar 

  4. Dai, C.J., Zhao, M., Chen, L.S.: Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances. Math. Comp. Simul. 84, 83–97 (2012)

    Article  MathSciNet  Google Scholar 

  5. Hastings, A., Powell, T.: Chaos in three-species food chain. Ecology 72, 896–903 (1991)

    Article  Google Scholar 

  6. Malchow, H., Petrovskii, S., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology. Chapman and Hall/CRC, New York (2008)

    MATH  Google Scholar 

  7. Mukherjee, D.: Stability analysis of a stochastic model for prey-predator system with disease in the Prey. Nonlinear Anal. Model. Control. 8(2), 83–92 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Das, P., Mukandavire, Z., Chiyaka, C., Sen, A., Mukherjee, D.: Bifurcation and chaos in S-I-S epidemic model. Differ. Equ. Dyn. Syst. 17(4), 393–417 (2009)

    Article  MathSciNet  Google Scholar 

  9. Das, P., Mukherjee, D., Sarkar, A.K.: Study of an SI epidemic model with nonlinear incidence rate: discrete and stochastic version. Appl. Math. Comput. 218(6), 2509–2515 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Mandal, P.S., Banerjee, M.: Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model. Phys. A. 391, 1216–1233 (2012)

    Article  Google Scholar 

  11. Bandyopadhyay, M., Chattopadhyay, J.: Ratio-dependent predatorprey model: effect of environmental fluctuation and stability. Nonlinearity 18, 913–936 (2005)

    Article  MathSciNet  Google Scholar 

  12. Bahar, A., Mao, X.: Stochastic delay Lotka-Volterra model. J. Math. Anal. Appl. 292, 364–380 (2004)

    Article  MathSciNet  Google Scholar 

  13. Liu, M., Wang, K., Wu, Q.: Survival analysis of stochastic competitive models in a polluted envioronment and stochastic competitive exclusion principle. Bull. Math. Biol. (2010). doi:10.1007/s11538-010-9569-5

    Article  MathSciNet  Google Scholar 

  14. Ton, T.V., Yagi, A.: Dynamics of a stochastic predator-prey model with the Beddington-De Angelis functional response. Commun. Stochast. Anal. 5(2), 371–386 (2011)

    MATH  Google Scholar 

  15. Mandal, P.S., Banerjee, M.: Deterministic and stochastic dynamics of a competitive phytoplankton model with allelopathy. Differ. Equ. Dyn. Syst. 21(4), 341–372 (2013)

    Article  MathSciNet  Google Scholar 

  16. Saha, T., Banerjee, M.: Effect of small time delay in a predator-prey model within random environment. Differ. Equ. Dyn. Syst. 16(3), 225–250 (2008)

    Article  MathSciNet  Google Scholar 

  17. Jumarie, G.: Stochastics of order n in biological system: application to population dynamics, thermodynamics, nonequilibrium phase and complexity. J. Biol. Syst. 11(2), 113–137 (2003)

    Article  Google Scholar 

  18. Sokol, W., Howell, J.A.: Kinetics of phenol oxidation by washed cell. Biot. Bioe. 23, 2039–2049 (1981)

    Article  Google Scholar 

  19. Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)

    Article  Google Scholar 

  20. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982)

    MATH  Google Scholar 

  21. Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stochast. Process. Appl. 97, 95–110 (2002)

    Article  MathSciNet  Google Scholar 

  22. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)

    Article  MathSciNet  Google Scholar 

  23. Kolmanovskii, V.B., Shaikhet, L.E.: Construction of Lyapunov functionals for stochastic hereditary systems. a survey of some recent results. Math. Comput. Model. 36, 691–716 (2002)

    Article  MathSciNet  Google Scholar 

  24. Friedman, A.: Stochastic Differential Equations and their Applications. Academic Press, New York (1976)

    MATH  Google Scholar 

  25. Mao, X.: Stochastic Differential Equations and Applications. Horwood, New York (1997)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the anonymous reviewers for their helpful comments and suggestions to improve the paper. We pay homage to Late Prof. A. B. Roy, Department of Mathematics, Jadavpur University, Kolkata, for his kind help and suggestions given before his demise in writing this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    Partha Ghosh & Pritha Das

  2. Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata, 700063, India

    Debasis Mukherjee

Authors
  1. Partha Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pritha Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Debasis Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pritha Das.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghosh, P., Das, P. & Mukherjee, D. Persistence and Stability of a Seasonally Perturbed Three Species Stochastic Model of Salmonoid Aquaculture. Differ Equ Dyn Syst 27, 449–465 (2019). https://doi.org/10.1007/s12591-016-0283-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-016-0283-0

Keywords

Search

Navigation