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Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator

Received: 1 January 2020     Accepted: 13 March 2020     Published: 31 March 2020
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Abstract

In this paper, the interactions among three species populations are considered. The system includes two mutuality preys and one predator. The second prey is harvested. While dependent on preys, the predator has an alternative food source also. The three species interaction can be described as a food chain in which two preys help each other but the predator attacks both the preys according to type I and II functional responses respectively. These population interactions are modeled mathematically using ordinary differential equations. It is shown that the solution of the model is both positive and bounded. The equilibrium points of the model are found and they are analyzed to identify a threshold that will guarantee the coexistence of the populations. Positive equilibrium points of the system are identified and their local and global stability analysis is carried out. Numerical simulation study of the model is conducted to support the results of the mathematical analysis. It is pointed out that as long as harvesting rate on the prey population is smaller than its intrinsic growth rate the coexistence of the system can be achieve. The results of the analysis and the discussion of the population dynamics is lucidly presented in the text of the paper.

Published in Mathematical Modelling and Applications (Volume 5, Issue 2)
DOI 10.11648/j.mma.20200502.12
Page(s) 55-64
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), 2020. Published by Science Publishing Group

Keywords

Prey – predator, Normalization, Positivity, Boundedness, Harvesting, Functional Response, Stability

References
[1] Ali N. (2016). Stability and bifurcation of a prey predator model with qiwu’s growth rate for prey, International Journal of Mathematics and Computation, 27 (2): 30–39.
[2] Cramer N. and May R. (1972). Inter-specific competition, predation and species diversity: a comment, Journal of Theoretical Biology, 34 (2): 289–293.
[3] Elettre M. (2009). Two-prey one-predator model, Chaos, Solitons & Fractals, 39 (5): 2018–2027.
[4] Neelima Daga, et al. (2014). Stability analysis of a prey-predator model with a reserved area, Advances in Applied Science Research, 5 (3): 293-301.
[5] Nomdedeu M., Willen C., Schieffer A. and Arndt H. (2012). Temperature-dependent ranges of coexistence in a model of a two-prey-one-predator microbial food web, Marine Biology, 159 (11): 2423–2430.
[6] Paine R. (1966). Food web complexity and species diversity, American Naturalist, 100: 65-75.
[7] Parrish J. and Saila S. (1970). Inter-specific competition, predation and species diversity, Journal of Theoretical Biology, 27 (2): 207–220.
[8] Renato C. Coexistence in a One-Predator, Two-Prey System with Indirect Effects, Journal of Applied Mathematics Volume 2013, 1-13.
[9] S. Sharma, G. P. Samanta. (2014). Dynamical Behaviour of a Two Prey and One Predator System, International Journal for Theory, Real World Modelling and Simulations.
[10] S. Sarwardi et al. (2013). Dynamical behaviour of a two-predator model with prey refuge, J Biol Phys 39: 701–722.
[11] Song X. and Li Y. (2007). Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect, Chaos, Solitons & Fractals, 33 (2): 463–478.
[12] Solomon T. et al. (2018). Application of Refuges and Harvesting on Prey-Prey- Predator System with Types of Functional Response.
[13] Tripathi J., Abbas S. and Thakur M. (2014). Local and global stability analysis of a two prey one predator model with help, Communications in Nonlinear Science and Numerical Simulation, 19 (9): 3284–3297.
[14] Xu C., Li P. and Shao Y. (2012). Existence and global attractivity of positive periodic solutions for a Holling II two-prey one-predator system, Advances in Difference Equations 2012 (1): 1–14.
Cite This Article
  • APA Style

    Solomon Tolcha, Boka Kumsa Bole, Purnachandra Rao Koya. (2020). Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator. Mathematical Modelling and Applications, 5(2), 55-64. https://doi.org/10.11648/j.mma.20200502.12

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    ACS Style

    Solomon Tolcha; Boka Kumsa Bole; Purnachandra Rao Koya. Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator. Math. Model. Appl. 2020, 5(2), 55-64. doi: 10.11648/j.mma.20200502.12

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    AMA Style

    Solomon Tolcha, Boka Kumsa Bole, Purnachandra Rao Koya. Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator. Math Model Appl. 2020;5(2):55-64. doi: 10.11648/j.mma.20200502.12

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  • @article{10.11648/j.mma.20200502.12,
      author = {Solomon Tolcha and Boka Kumsa Bole and Purnachandra Rao Koya},
      title = {Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator},
      journal = {Mathematical Modelling and Applications},
      volume = {5},
      number = {2},
      pages = {55-64},
      doi = {10.11648/j.mma.20200502.12},
      url = {https://doi.org/10.11648/j.mma.20200502.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200502.12},
      abstract = {In this paper, the interactions among three species populations are considered. The system includes two mutuality preys and one predator. The second prey is harvested. While dependent on preys, the predator has an alternative food source also. The three species interaction can be described as a food chain in which two preys help each other but the predator attacks both the preys according to type I and II functional responses respectively. These population interactions are modeled mathematically using ordinary differential equations. It is shown that the solution of the model is both positive and bounded. The equilibrium points of the model are found and they are analyzed to identify a threshold that will guarantee the coexistence of the populations. Positive equilibrium points of the system are identified and their local and global stability analysis is carried out. Numerical simulation study of the model is conducted to support the results of the mathematical analysis. It is pointed out that as long as harvesting rate on the prey population is smaller than its intrinsic growth rate the coexistence of the system can be achieve. The results of the analysis and the discussion of the population dynamics is lucidly presented in the text of the paper.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Population Dynamics of Two Mutuality Preys and One Predator with Harvesting of One Prey and Allowing Alternative Food Source to Predator
    AU  - Solomon Tolcha
    AU  - Boka Kumsa Bole
    AU  - Purnachandra Rao Koya
    Y1  - 2020/03/31
    PY  - 2020
    N1  - https://doi.org/10.11648/j.mma.20200502.12
    DO  - 10.11648/j.mma.20200502.12
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
    SP  - 55
    EP  - 64
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20200502.12
    AB  - In this paper, the interactions among three species populations are considered. The system includes two mutuality preys and one predator. The second prey is harvested. While dependent on preys, the predator has an alternative food source also. The three species interaction can be described as a food chain in which two preys help each other but the predator attacks both the preys according to type I and II functional responses respectively. These population interactions are modeled mathematically using ordinary differential equations. It is shown that the solution of the model is both positive and bounded. The equilibrium points of the model are found and they are analyzed to identify a threshold that will guarantee the coexistence of the populations. Positive equilibrium points of the system are identified and their local and global stability analysis is carried out. Numerical simulation study of the model is conducted to support the results of the mathematical analysis. It is pointed out that as long as harvesting rate on the prey population is smaller than its intrinsic growth rate the coexistence of the system can be achieve. The results of the analysis and the discussion of the population dynamics is lucidly presented in the text of the paper.
    VL  - 5
    IS  - 2
    ER  - 

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Author Information
  • Department of Mathematics, Addis Ababa Science and Technology University, Addis Ababa, Ethiopia

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

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