A Stochastic Model for Three Species

  • Authors

    • Y. Suresh Kumar
    • N. Seshagiri Rao
    • B. V AppaRao
    https://doi.org/10.14419/ijet.v7i4.10.21211

    Received date: October 7, 2018

    Accepted date: October 7, 2018

    Published date: October 2, 2018

  • Global stability, Local stability, Lyapunov's technique, Mutual species, predator, Stochastic process.

  • Abstract

    The present work is related to a three species ecosystem including a mutualism interaction between two species and a predator, where the predator is depending on both the mutual species. All three species in this model are considered in limited resources. The sustainability of the system (local stability) is discussed through the perturbed technique at the possible existing each equilibrium points. Using Lyapunov's technique the global stability of the system is also described. Further the nature of the system is observed by introducing the stochastic process to the species and the numerical simulations are studied to know the interaction among the species.

  • References

    1. Afanas’ev V N., Kolmanowski V B., Nosov V R, 1996. Mathemat-ical Theory of Global Systems Design, Kluwer Academic, Dor-drecht.
    2. Cushing J M, 1977. Integro differential equations and delay models in population Dynamics, Lecture Notes in Bio-Mathematics, 20, Springer Verlag, Berlin, Heidelberg, Germany.
    3. Freedman H I, 1934. Stability analysis of Predator -Prey model with mutual Interference and density dependent death rates, Wil-liams and Wilkins, Baltimore.
    4. Gause G F, 1934. The Struggle for Existence. Williams and Wilkins, Baltimore.
    5. Haberman R, 1977. Mathematical Models, Prentice Hall, New Jer-sey, USA.
    6. Kapur J N, 1988. Mathematical Modeling, Wiley-Eastern, New Delhi.
    7. Kapur J N., 1985. Mathematical models in biology and medicine. Affiliated East West Press.
    8. Lotka A J, 1925. Elements of Physical Biology, Williams and Wil-kins, Baltimore.
    9. Meyer W J, 1985. Concepts of Mathematical Modeling, McGraw-Hill.
    10. Mukherjee D, 1988. Uniform persistence in a generalized prey-predator system with parasite infection, Biosystems, 47, 149–155.
    11. Mukherjee D, 2003. Stability Analysis of a Stochastic Model for Prey-Predator System with disease in the prey, Nonlinear Analysis; Modeling and Control,Vol.8, No.2, 83-92.
    12. Paparao AV, Lakshmi Narayana K, Shahnaz Bathul, 2012, A Three Species Ecological Model with a Prey, Predator and a Competitor to both the Prey and Predator International Journal of Mathematics and Scientific Computing, No.1, Vol.2, 70-75.
    13. Paparao A V, Lakshmi Narayana .K, Shahnaz Bathul, 2014, Appli-cation of Homotopy Analysis Method to a Three Species Eco-logical Model, International Journal of Ecological Economics and Statistics, No.1, Vol.32, 83-92.
    14. Paparao AV, Lakshmi Narayana K, 2017. Dynamics of a Prey Predator and Competitor Mode with Time Delay, International Journal of Ecology& Development, Vol.32, 75-86.
    15. Paul Colinvaux. 1986. Ecology, John Wiley and Sons, Inc., New York.
    16. Pielou E C, 1977. Mathematical Ecology, John Wiley and Sons, Inc., New York.
    17. Seshagiri Rao N, Kalyani K & Pattabhi Ramacharyulu N.Ch., 2011. A host-mortal commensal ecosystem with host harvesting at a constant rate, ARPN Journal of Engineering and Applied Sci-ences, Vol.6 (11), 79 - 99.
    18. Seshagiri Rao N, Kalyani K & Acharyulu K.V.L.N., 2014. Thresh-old results for host-Mortal commensal ecosystem with limited re-sources, Global Journal of Pure and Applied Mathematics, Vol.10 (6), 787-791.
    19. Seshagiri Rao N, Kalyani K & Acharyulu K.V.L.N., 2014. Host-Monad ecological model with phase plane analysis, International Journal of Applied Engineering Research, Vol.9 (23), 21605-21610.
    20. Seshagiri Rao N, Acharyulu K.V.L.N. & Kalyani K, 2015. Thresh-old results of a Host-Mortal commensal ecosystem with a constant harvesting of the commensal species, ARPN Journal of Engineering and Applied Sciences, Vol.10 (2), 802-805.
    21. Suresh.Y and Seshagiri Rao.N, 2017. An Ecological Model of Mu-tualism Interaction between Two Species and a Predator, In-ternational Journal of Mathematics and Computation, Vol.28, No.4, PP: 48-57.
    22. Srinivas M N, Shiva Reddy K & Sabarmathi A, 2014. Optimal har-vesting strategy andstochastic analysis for a two species commen-saling system, AIN Shams Engineering Journal, 5, 515–523.
    23. Thompson DW, 1917.On Growth and Form, Cambridge, Cam-bridge University Press, USA.
    24. Volterra V, 1931. Lecons sen Lu theorie mathematique de la luitte pour la vie, Gauthier- Villars, Paris.

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  • How to Cite

    Suresh Kumar, Y., Seshagiri Rao, N., & V AppaRao, B. (2018). A Stochastic Model for Three Species. International Journal of Engineering and Technology, 7(4.10), 497-503. https://doi.org/10.14419/ijet.v7i4.10.21211