Hopf bifurcation in an autonomous system with logistic growth and Holling type II functional response


  • Danny Estefany Paz Vidal Universidad del Cauca, Colombia.
  • Joan Esteban Salazar Gordillo Universidad del Cauca, Colombia.




Dynamical system, Hopf bifurcation, Hartman-Grobman theorem, phase portrait, Holling type II functional response


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.

Author Biography

Joan Esteban Salazar Gordillo, Universidad del Cauca, Colombia.

Universidad del Cauca , Colombia


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.



How to Cite

Paz Vidal, D. E., & Salazar Gordillo, J. E. (2023). Hopf bifurcation in an autonomous system with logistic growth and Holling type II functional response. Selecciones Matemáticas, 10(02), 444 - 461. https://doi.org/10.17268/sel.mat.2023.02.16



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