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