The Hopf Bifurcation of the Dynamics of Behavior an Ecological Model

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Published: Mar 30, 2026
DOI: https://doi.org/10.37905/jjbm.v7i1.27
Keywords:
Fear effect, Hopf bifurcation, Transcritical, Additional Food

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Dian Savitri
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, Indonesia, Indonesia
https://orcid.org/0000-0002-8064-7191
Abadi
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, Indonesia, Indonesia
https://orcid.org/0000-0002-8475-922X
Riska Wahyu Romadhonia
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, Indonesia, Indonesia
https://orcid.org/0009-0000-7677-6559
Nurul Imamah Ah
Department of Mathematics Education, Universitas Muhammadiyah Jember, Jember, Indonesia, Indonesia
https://orcid.org/0000-0002-9135-1433
An Nisa Salsabila
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, Indonesia, Indonesia
Kurniasih
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, Indonesia, Indonesia

Abstract

We conducted a dynamic analysis of an ecological model that describes the relationship between prey and predator detritivores. Assuming that predators require more food to survive, we applied the Beddington-DeAngelis functional response to examine local stability while taking the fear effect into account. The dynamics of the local stability properties of the equilibrium point ware examined. The two population extinction points, the prey population extinction point, and all population survival were the three points that we were able to determine. The analytical computations were supported by numerical simulations. Some numerical simulations are organized to show the impact of fear effects on prey, additional food on predator and predation using Beddington-DeAngelis on the dynamical behaviors of the model. The first numerical continuation of additional food parameters in the system solution indicated the presence of a Hopf bifurcation at $A = 0.625978$. The greater the supply of additional food, the extinction of the prey that the transcitical bifurcation was found at $A = 8.694143$. The bifurcation indicated that the change remains stable, becoming unstable at the interior equilibrium point and the other system solution. The Hopf bifurcation was also found at $f= 0.081119$ according to the second numerical continuation of the fear effect parameters, and $\beta=1.023308$. In addition, the appearance is that the transcitical bifurcation was found at $f= 2.088891$ and $\beta=3.362426$. We have demonstrated numerically the occurrence of Hopf and transcritical bifurcation driven by those three biological parameters.

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How to Cite
1.
Savitri D, Abadi, Romadhonia R, Ah N, Salsabila AN, Kurniasih D. The Hopf Bifurcation of the Dynamics of Behavior an Ecological Model. Jambura J. Biomath. [Internet]. 2026 Mar. 30 [cited 2026 Apr. 24];7(1):211-23. Available from: https://jjbm.fmipa.ung.ac.id/index.php/ejournal/article/view/27
Issue
Vol. 7 No. 1: March 2026
Section
Ecological Systems and Environmental Modeling
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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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