Hopf bifurcation in an autonomous system with logistic growth and Holling type II functional response
DOI:
https://doi.org/10.17268/sel.mat.2023.02.16Keywords:
Dynamical system, Hopf bifurcation, Hartman-Grobman theorem, phase portrait, Holling type II functional responseAbstract
The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis,
values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated
to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation.
The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.
References
Navas J. Modelos matemáticos en biología[Internet]. Departamento de Matemáticas, Universidad de Jaén; 2009. Disponible en http://matema.ujaen.es/jnavas/web_modelos/index.htm
Puchuri Medina, L. Limit Cycles in Predator-Prey Models. Selecciones Matemáticas, 2017; 4(01), 70-81. https://doi.org/10.17268/sel.mat.2017.01.08
Yi F, Wei J, Shi J. Bifurcation and Spatiotemporal Patterns in a Homogeneous Diffusive Predator-Prey System. J. of Differential Equations. 2009; 246(5):1944-1977.
Gálvez García M. Estudio y comparación de diversos modelos de depredador-presa. Trabajo de fin de grado de Matemáticas. Universidad de Sevilla; 2018.
Kuznetsov Y. Elements of Applied Bifurcation Theory. 2da ed. Springer; 1998.
Perko L. Differential Equations and Dynamical Systems. 3rd ed. Estados Unidos: Springer-Verlag; 2001.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
