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Stability and Bifurcation Analysis of a Fishery Model with Allee Effects

Received: 13 March 2019     Accepted: 26 April 2019     Published: 20 May 2019
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Abstract

We study the equilibrium point (n*, E*) of the fishery model with Allee effect in its population growth dynamics. The Allee effect is considered to be induced by the harvesting of the fish stock. The aggregated model is a set of two differential equations with the fish population and harvesting effort as the dependent variables, with the market price having been taken to evolve faster hence the aggregation from a three dimensional system to a two dimensional system. The analysis of the equilibrium point is performed by looking at three cases in which the threshold population is set at three different values; , and . Three different equilibrium solutions are obtained: A stable equilibrium, coexistence of three equilibria points with two being saddles and the other stable and the co-existence of three equilibria points with two being stable and a saddle between them. The equilibrium solutions depicts three kinds of fishery: A fishery with fish population maintained at high levels far from extinction but with little economic activity, a fishery with co-existence of an over-exploited and an under-exploited state, which is a dilemma since neither of the state supports sustainable fish resource exploitation, and a fishery that is well managed with fish population being harvested in a sustainable manner thus a balance between commercial harvesting and species existence.

Published in Mathematical Modelling and Applications (Volume 4, Issue 1)
DOI 10.11648/j.mma.20190401.11
Page(s) 1-9
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), 2019. Published by Science Publishing Group

Keywords

Allee Effect, Fishing Mortality, Equilibrium Solution

References
[1] Auger P., Mchich R., Nadia R., Kooi B. W. (2009), Effects of market price on the dynamics of a spatial model: Over exploited fishery / Traditional fishery. Ecological complexity.
[2] Auger P. Arnold D. (2009), a model of a fishery with fish stock involving delay equations, Phil, Trans Royal Society A 367.
[3] Barbier E. B., Strand L., Sathirathai S. (2002), Do open access conditions affect the valuation of an externality? Estimating the welfare effects of mangrove-fishery linkages, Env.resour.Econ. 21, 343-367.
[4] Bapan Ghosh et. al (2014), Relationships between exploitation, oscillation, MSY and extinction, Mathematical Biosciences 256, 1-9.
[5] Clark C. W. (1990), Mathematical Bioeconomics: The optimal Management of renewable resources, 2nd (Ed), Wiley, New York.
[6] Corrine Wentworth et. al (2011)m Optimum harvesting models for fishery population, Project report, St. Marys’ College of Maryland, U. S. A
[7] IUCN 2019, The IUCN red list of threatened species version 20119-1, http://www.iucnredlist.org. Downloaded on 21 march 2019.
[8] Kar T. K. and Matsuda H. (2007), Sustainable Management of a Fishery with a strong Allee effect, Trends In Applied Sciences Research 2(4), 271-283.
[9] Makwata H. et. al (2018), Stability Bifurcation Analysis of a fishery model with nonlinear variation in Market Price, Applied Mathematical SciencesVol. 12, 2018, no. 7 337-350
[10] Mchich R., Auger P. M., Bravo de la parra R., Raissi N. (2002), Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control, Ecological modeling 158(2), 51-62
[11] Murray J. D (2003), Mathematical biology, Springer Verlag Berlin.
[12] Poggiale J. C., Auger P. (1998), Aggregation and Emergence in Systems of Ordinary Differential Equations, Math. comput. Modelling 27(4), 1-21.
[13] Stephens P. A., Sutherland W. J. and Freckleton R. P. (1999), What is Allee effect? Oikos 87(1), 185-190.
[14] Smith V. L (1969), On Models of commercial Fishing, Political economy 77(2), 181-192.
[15] Schaefer M. B (1957. Some considerations of population dynamics and economics in relation to the management of the commercial marine fisheries. J. Fish Res. Board Canada 14, 669-681.
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  • APA Style

    Makwata Harun, Lawi George, Akinyi Colleta, Adu Wasike. (2019). Stability and Bifurcation Analysis of a Fishery Model with Allee Effects. Mathematical Modelling and Applications, 4(1), 1-9. https://doi.org/10.11648/j.mma.20190401.11

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    ACS Style

    Makwata Harun; Lawi George; Akinyi Colleta; Adu Wasike. Stability and Bifurcation Analysis of a Fishery Model with Allee Effects. Math. Model. Appl. 2019, 4(1), 1-9. doi: 10.11648/j.mma.20190401.11

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    AMA Style

    Makwata Harun, Lawi George, Akinyi Colleta, Adu Wasike. Stability and Bifurcation Analysis of a Fishery Model with Allee Effects. Math Model Appl. 2019;4(1):1-9. doi: 10.11648/j.mma.20190401.11

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  • @article{10.11648/j.mma.20190401.11,
      author = {Makwata Harun and Lawi George and Akinyi Colleta and Adu Wasike},
      title = {Stability and Bifurcation Analysis of a Fishery Model with Allee Effects},
      journal = {Mathematical Modelling and Applications},
      volume = {4},
      number = {1},
      pages = {1-9},
      doi = {10.11648/j.mma.20190401.11},
      url = {https://doi.org/10.11648/j.mma.20190401.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20190401.11},
      abstract = {We study the equilibrium point (n*, E*) of the fishery model with Allee effect in its population growth dynamics. The Allee effect is considered to be induced by the harvesting of the fish stock. The aggregated model is a set of two differential equations with the fish population and harvesting effort as the dependent variables, with the market price having been taken to evolve faster hence the aggregation from a three dimensional system to a two dimensional system. The analysis of the equilibrium point is performed by looking at three cases in which the threshold population is set at three different values; ,  and . Three different equilibrium solutions are obtained: A stable equilibrium, coexistence of three equilibria points with two being saddles and the other stable and the co-existence of three equilibria points with two being stable and a saddle between them. The equilibrium solutions depicts three kinds of fishery: A fishery with fish population maintained at high levels far from extinction but with little economic activity, a fishery with co-existence of an over-exploited and an under-exploited state, which is a dilemma since neither of the state supports sustainable fish resource exploitation, and a fishery that is well managed with fish population being harvested in a sustainable manner thus a balance between commercial harvesting and species existence.},
     year = {2019}
    }
    

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    T1  - Stability and Bifurcation Analysis of a Fishery Model with Allee Effects
    AU  - Makwata Harun
    AU  - Lawi George
    AU  - Akinyi Colleta
    AU  - Adu Wasike
    Y1  - 2019/05/20
    PY  - 2019
    N1  - https://doi.org/10.11648/j.mma.20190401.11
    DO  - 10.11648/j.mma.20190401.11
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
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    EP  - 9
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20190401.11
    AB  - We study the equilibrium point (n*, E*) of the fishery model with Allee effect in its population growth dynamics. The Allee effect is considered to be induced by the harvesting of the fish stock. The aggregated model is a set of two differential equations with the fish population and harvesting effort as the dependent variables, with the market price having been taken to evolve faster hence the aggregation from a three dimensional system to a two dimensional system. The analysis of the equilibrium point is performed by looking at three cases in which the threshold population is set at three different values; ,  and . Three different equilibrium solutions are obtained: A stable equilibrium, coexistence of three equilibria points with two being saddles and the other stable and the co-existence of three equilibria points with two being stable and a saddle between them. The equilibrium solutions depicts three kinds of fishery: A fishery with fish population maintained at high levels far from extinction but with little economic activity, a fishery with co-existence of an over-exploited and an under-exploited state, which is a dilemma since neither of the state supports sustainable fish resource exploitation, and a fishery that is well managed with fish population being harvested in a sustainable manner thus a balance between commercial harvesting and species existence.
    VL  - 4
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

  • Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya

  • Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya

  • Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

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