Stability and Bifurcation Analysis of a Fishery Model with Allee Effects

Makwata Harun; Lawi George; Akinyi Colleta; Adu Wasike

doi:doi:10.11648/j.mma.20190401.11

| Peer-Reviewed

Stability and Bifurcation Analysis of a Fishery Model with Allee Effects

Makwata Harun, Lawi George, Akinyi Colleta, Adu Wasike

Published in Mathematical Modelling and Applications (Volume 4, Issue 1)

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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.

Cite This Article

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
SP  - 1
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  -

Copy | Download

Author Information

Makwata Harun

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

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

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

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

Download PDF

Sections

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
SP  - 1
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  -

Copy | Download