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Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting

Received: 25 May 2018     Accepted: 26 June 2018     Published: 31 July 2018
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Abstract

This paper deals with the study of mutuality interactions between two species population with type II functional response and also with the inclusion of harvesting. Harvesting functions are introduced to express the rate of reductions of the species separately. Mathematical model have been constructed and considered for the analysis and results. In this model, the first population species benefited according to type II functional response and the second species benefited from the first according to type I functional response and also harvested proportional to its density. It is shown that the model has positive and bounded solutions. Stability analysis is carried out. The local and global stability of biologically interested equilibrium point are considered and analyzed. Numerical examples supporting theoretical results such as phase plane and simulation study using DSolver are also included. Assumptions and results are presented and discussed lucidly in the text of the paper.

Published in American Journal of Applied Mathematics (Volume 6, Issue 3)
DOI 10.11648/j.ajam.20180603.12
Page(s) 109-116
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Mutualism, Functional Response, Harvesting, Phase Plane Analysis, Positivity and Boundedness

References
[1] L. H. Erbe, V. S. H. Rao and H. I. Freedman, “Three-species food chain models with mutual interference and time delays”, Math Biosci 80 (1986), 57–80.
[2] J. L. Bronstein, (1994). “Our current understands of mutualism”. Quarterly Review of Biology. 69 (1): 31–51.
[3] M. Begon, J. L. Harper, and C. R. Townsend. 1996. “Ecology: individuals, populations, and communities”, Third Edition. Blackwell Science Ltd., Cambridge, Massachusetts, USA.
[4] Morin. PJ, (2011). “Community ecology”, John Wiley and Sons, Hoboken, USA.
[5] Levins, R. 1966. “The strategy of model building in population biology”. American Scientist 54:421–431.
[6] MacArthur, R. H., and R. Levins. 1967. “The limiting similarity, convergence, and divergence of coexisting species”. American Naturalist 101:377–385.
[7] P. F. Verhulst, Recherches mathematiques sur la loi d'accroissement de la population [Mathematical researches into the law of population growth increase], Nouveaux Memoires de VAcademic Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 1—42.
[8] Freedman, H. I., “Deterministic Mathematical Models in Population Ecology”, Marcel Dekker, New York, USA 1980.
[9] D. H. Wright, “A simple, stable model of mutualism incorporating handling time”, The American Naturalist, 134 (1989), 664-667.
[10] J. Ollerton, 2006. “Biological Barter: Interactions of Specialization Compared across Different Mutualisms”. pp. 411–435 in: Waser, N. M. & Ollerton, J. (Eds) Plant-Pollinator Interactions: From Specialization to Generalization. University of Chicago Press.
[11] C. W. Clark, “Bio economic Modeling and Fisheries Management”, Wiley, New York, 1985.
[12] C. W. Clark, “Mathematical Bio economics: The Optimal Management of Renewable Resources”, Wiley, New York, 1990.
[13] D. R. Jana, Agrawal, R. K. Upadhyay and G. P. Samanta, “Ecological dynamics of age selective harvesting of fish population: Maximum sustainable yield and its control strategy”, Chaos, Solitons and Fractals 93 (2016), 111–122.
[14] D. Jana, and G. P. Samanta, “Role of Multiple Delays in ratio-dependent prey-predator system with prey harvesting under stochastic environment”, Neural, Parallel and scientific Computations 22 (2014), 205–222.
[15] T. R. Das, N. Mukherjee and K. S. Chaudhuri, “Harvesting of a prey–predator fishery in the presence of toxicity”, Appl Math Model 33 (2009), 2282–2292.
[16] M. Kot, “Elements of Mathematical Ecology”, Cambridge University Press, Cambridge, 2001.
[17] R. Ouncharoen, Pinjai S, Dumrongpokaphan T, and Lenbury Y (2012). “Global stability analysis of predator-prey model with harvesting and delay”. Thai Journal of Mathematics, 8(3): 589-605.
[18] Rusliza Ahmad (2017). “Global stability of two-species mutualism model with proportional harvesting”. International Journal of Advanced and Applied Sciences, 4(7) 2017, Pages: 74-79.
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  • APA Style

    Solomon Tolcha, Boka Kumsa, Purnachandra Rao Koya. (2018). Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting. American Journal of Applied Mathematics, 6(3), 109-116. https://doi.org/10.11648/j.ajam.20180603.12

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    ACS Style

    Solomon Tolcha; Boka Kumsa; Purnachandra Rao Koya. Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting. Am. J. Appl. Math. 2018, 6(3), 109-116. doi: 10.11648/j.ajam.20180603.12

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    AMA Style

    Solomon Tolcha, Boka Kumsa, Purnachandra Rao Koya. Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting. Am J Appl Math. 2018;6(3):109-116. doi: 10.11648/j.ajam.20180603.12

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  • @article{10.11648/j.ajam.20180603.12,
      author = {Solomon Tolcha and Boka Kumsa and Purnachandra Rao Koya},
      title = {Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting},
      journal = {American Journal of Applied Mathematics},
      volume = {6},
      number = {3},
      pages = {109-116},
      doi = {10.11648/j.ajam.20180603.12},
      url = {https://doi.org/10.11648/j.ajam.20180603.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180603.12},
      abstract = {This paper deals with the study of mutuality interactions between two species population with type II functional response and also with the inclusion of harvesting. Harvesting functions are introduced to express the rate of reductions of the species separately. Mathematical model have been constructed and considered for the analysis and results. In this model, the first population species benefited according to type II functional response and the second species benefited from the first according to type I functional response and also harvested proportional to its density. It is shown that the model has positive and bounded solutions. Stability analysis is carried out. The local and global stability of biologically interested equilibrium point are considered and analyzed. Numerical examples supporting theoretical results such as phase plane and simulation study using DSolver are also included. Assumptions and results are presented and discussed lucidly in the text of the paper.},
     year = {2018}
    }
    

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    T1  - Modeling and Simulation Study of Mutuality Interactions with Type II functional Response and Harvesting
    AU  - Solomon Tolcha
    AU  - Boka Kumsa
    AU  - Purnachandra Rao Koya
    Y1  - 2018/07/31
    PY  - 2018
    N1  - https://doi.org/10.11648/j.ajam.20180603.12
    DO  - 10.11648/j.ajam.20180603.12
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 109
    EP  - 116
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20180603.12
    AB  - This paper deals with the study of mutuality interactions between two species population with type II functional response and also with the inclusion of harvesting. Harvesting functions are introduced to express the rate of reductions of the species separately. Mathematical model have been constructed and considered for the analysis and results. In this model, the first population species benefited according to type II functional response and the second species benefited from the first according to type I functional response and also harvested proportional to its density. It is shown that the model has positive and bounded solutions. Stability analysis is carried out. The local and global stability of biologically interested equilibrium point are considered and analyzed. Numerical examples supporting theoretical results such as phase plane and simulation study using DSolver are also included. Assumptions and results are presented and discussed lucidly in the text of the paper.
    VL  - 6
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

  • Department of Mathematics, Hawassa University, Hawassa, Ethiopia

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