Stability and Bifurcation Analysis of a Fishery Model with Allee Effects

Makwata Harun

doi: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; <img width="37" height="35" src="http://article.sciencepublishinggroup.com/journal/389/3891035/image001.png" />, <img width="37" height="35" src="http://article.sciencepublishinggroup.com/journal/389/3891035/image002.png" /> and <img width="42" height="36" src="http://article.sciencepublishinggroup.com/journal/389/3891035/image003.png" />. 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.

Highlights

IntroductionThis paper is concerned with the local and bifurcation analysis of the equilibrium point (n∗, E∗) of the dynamical system in nɺ
Equation (7) becomes c(n*) = n4 − n3 − n3 + n2 + n. We investigate this Equilibrium point (n∗, E∗) by having the threshold population T, as a factor of the fish population at a particular time n, at three different values, Laboratory results on specific fish species like tuna have shown that if the stock is harvested to a quarter of the stock available at a time, the population recovers to optimum levels in a duration of five years when harvesting is stopped see for instance
We have considered a fishery model with Allee effects in the population growth

Summary

Introduction

This paper is concerned with the local and bifurcation analysis of the equilibrium point (n∗, E∗) of the dynamical system in nɺ. This equilibrium is of interest because its analysis gives the long - term behavior of the system when harvesting is done it predicts whether the Fish stock recovers or extinct on harvesting guide sustainable harvesting of the fish resource, this is of importance to economists and conservationists. E(n) = 1 (A − c ) qn qn Makwata Harun et al.: Stability and Bifurcation Analysis of a Fishery Model with Allee Effects which yields E1, E2, E3, denoting unstable equilibrium, stable equilibrium and the co-existence of three strictly positive equilibria where two are stable, separated by a saddle. The co-existence of three positive equilibria predicts existence of the fishery in either an over-exploited or in an underexploited state which causes a dilemma since the two states cannot co-exist in the same fishery

Local Stability Analysis

Bifurcation Analysis

Bifurcation

Discussion

Conclusion