Mathematical Modelling and Applications

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Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form

Received: 01 May 2020    Accepted: 18 June 2020    Published: 04 August 2020
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Abstract

In this paper, a three species eco system, involving three pairs is considered modeled to examine the stability. Among the three species, one plays dual roles which are a host and an enemy with Monod response. In the first place model assumptions and formulation was carried out for investigations. The biological feasibility of the system is checked. That is positivity and boundedness of the model is verified. It is shown that biologically valid. The dynamical behavior of the proposed model system was analyzed qualitatively. The dynamical analysis includes the determination of all possible equilibrium points and their stability properties. All the equilibrium states are identified and the local asymptotic stability of some of the equilibrium states is examined by considering the set criteria. It is observed that among the states, the state in which the Prey and its Host species are exist is stable and the state where the Predator/Ammensal species is washed out is asymptotically stable. The global stability of the co-existence of the species was investigated by constructing a suitable Lyapunov function. To support our analytical studies, some numerical simulations was performed susing some mathematical software and the results were forwarded in the last section.

DOI 10.11648/j.mma.20200503.14
Published in Mathematical Modelling and Applications (Volume 5, Issue 3, September 2020)
Page(s) 156-166
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), 2024. Published by Science Publishing Group

Keywords

Prey, Predator, Ammensal, Commensal, Host, Continuous Time, Stability, Numerical Simulation

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

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

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  • APA Style

    Geremew Kenassa Edessa, Purnachandra Rao Koya. (2020). Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form. Mathematical Modelling and Applications, 5(3), 156-166. https://doi.org/10.11648/j.mma.20200503.14

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    Geremew Kenassa Edessa; Purnachandra Rao Koya. Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form. Math. Model. Appl. 2020, 5(3), 156-166. doi: 10.11648/j.mma.20200503.14

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    AMA Style

    Geremew Kenassa Edessa, Purnachandra Rao Koya. Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form. Math Model Appl. 2020;5(3):156-166. doi: 10.11648/j.mma.20200503.14

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  • @article{10.11648/j.mma.20200503.14,
      author = {Geremew Kenassa Edessa and Purnachandra Rao Koya},
      title = {Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form},
      journal = {Mathematical Modelling and Applications},
      volume = {5},
      number = {3},
      pages = {156-166},
      doi = {10.11648/j.mma.20200503.14},
      url = {https://doi.org/10.11648/j.mma.20200503.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.mma.20200503.14},
      abstract = {In this paper, a three species eco system, involving three pairs is considered modeled to examine the stability. Among the three species, one plays dual roles which are a host and an enemy with Monod response. In the first place model assumptions and formulation was carried out for investigations. The biological feasibility of the system is checked. That is positivity and boundedness of the model is verified. It is shown that biologically valid. The dynamical behavior of the proposed model system was analyzed qualitatively. The dynamical analysis includes the determination of all possible equilibrium points and their stability properties. All the equilibrium states are identified and the local asymptotic stability of some of the equilibrium states is examined by considering the set criteria. It is observed that among the states, the state in which the Prey and its Host species are exist is stable and the state where the Predator/Ammensal species is washed out is asymptotically stable. The global stability of the co-existence of the species was investigated by constructing a suitable Lyapunov function. To support our analytical studies, some numerical simulations was performed susing some mathematical software and the results were forwarded in the last section.},
     year = {2020}
    }
    

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    T1  - Modeling and Stability Analysis of a Three Species Ecosystem with the Third Species Response to the First Species in Sigmoid Functional Response Form
    AU  - Geremew Kenassa Edessa
    AU  - Purnachandra Rao Koya
    Y1  - 2020/08/04
    PY  - 2020
    N1  - https://doi.org/10.11648/j.mma.20200503.14
    DO  - 10.11648/j.mma.20200503.14
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
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    EP  - 166
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20200503.14
    AB  - In this paper, a three species eco system, involving three pairs is considered modeled to examine the stability. Among the three species, one plays dual roles which are a host and an enemy with Monod response. In the first place model assumptions and formulation was carried out for investigations. The biological feasibility of the system is checked. That is positivity and boundedness of the model is verified. It is shown that biologically valid. The dynamical behavior of the proposed model system was analyzed qualitatively. The dynamical analysis includes the determination of all possible equilibrium points and their stability properties. All the equilibrium states are identified and the local asymptotic stability of some of the equilibrium states is examined by considering the set criteria. It is observed that among the states, the state in which the Prey and its Host species are exist is stable and the state where the Predator/Ammensal species is washed out is asymptotically stable. The global stability of the co-existence of the species was investigated by constructing a suitable Lyapunov function. To support our analytical studies, some numerical simulations was performed susing some mathematical software and the results were forwarded in the last section.
    VL  - 5
    IS  - 3
    ER  - 

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