Local Stability, the Existence of Chaotic Behavior and Bifurcations in an Open-access Fishery Model

Mirela Garić-Demirović; Mustafa Kulenović; Zehra Nurkanović

doi:10.5644/SJM.21.02.09

Authors

Mirela Garić-Demirović University of Tuzla, Departmen of Mathematics, U. Vejzagića 4, Tuzla, Bosnia and Herzegovina https://orcid.org/0000-0001-9284-1891
Mustafa Kulenović University of Rhode Island, Department of Mathematics https://orcid.org/0000-0002-2936-7098
Zehra Nurkanović University of Tuzla, Departmen of Mathematics, U. Vejzagića 4, Tuzla, Bosnia and Herzegovina https://orcid.org/0000-0002-0917-8433

DOI:

https://doi.org/10.5644/SJM.21.02.09

Keywords:

Area preserving map, difference equations, fixed point, Neimark-Sacker bifurcation, KAM theory

Abstract

In this paper, we investigate an open-access fishery model which is used to examine the dynamics of the resource and industry and to explain the current economic status of the anchovy fishery. We consider the local character of the interior and boundary equilibrium points. Also, we show that the considered system of difference equations exhibits Neimark-Sacker bifurcation under certain conditions. The existence of the repelling curve and invariant curve is demonstrated. We show that in a certain parameter region the corresponding map of the considered system is an area-preserving map, so the positive equilibrium point in that case is stable. Also, we produce numerical simulations to support our findings.

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