Volume 4, Issue 1, March 2019, Page: 1-9
Stability and Bifurcation Analysis of a Fishery Model with Allee Effects
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
Received: Mar. 13, 2019;       Accepted: Apr. 26, 2019;       Published: May 20, 2019
DOI: 10.11648/j.mma.20190401.11      View  113      Downloads  30
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.
Keywords
Allee Effect, Fishing Mortality, Equilibrium Solution
To cite this article
Makwata Harun, Lawi George, Akinyi Colleta, Adu Wasike, Stability and Bifurcation Analysis of a Fishery Model with Allee Effects, Mathematical Modelling and Applications. Vol. 4, No. 1, 2019, pp. 1-9. doi: 10.11648/j.mma.20190401.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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