Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional tipo II de Holling.

Resumo

El sistema autónomo presa-depredador con crecimiento logístico y respuesta funcional tipo II de Holling, que describe la dinámica poblacional de dos especies, presa-depredador, se modela mediante un sistema de ecuaciones diferenciales ordinarias: \begin{ecuación} \label{equivalente:S}      \izquierda\{        \begin{matriz}{ll}         \dot {x_1}=F_1(x_1,x_2)=x_1\left(1-\dfrac{x_1}{k}\right)-\dfrac{cx_1x_2}{1+x_1}\\         \dot{x_2}=F_2(x_1,x_2)=-\delta x_2 + \dfrac{cx_1x_2}{1+x_1}        \end{matriz}      \bien.  \end{ecuación} donde $x_1(t)$ y $x_2(t)$ representan la cantidad poblacional de presas y depredadores en el tiempo $t$, respectivamente, y $ \delta, c, k$ son parámetros positivos. \\   In this study, the equilibrium points of the system (\ref{eq:S}) were identified. Two saddle-node points and one non-trivial $P_3$ equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (\ref{eq:S}), evaluated at $P_3$, to have a pair of purely complex eigenvalues  (necessary condition for the Hopf bifurcation). Through this analysis, values $c_0, \delta_0, k_0$ 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 MATLAB software, which enabled graphical visualization of the obtained results. \\   \textbf{Palabras clave:} Sistema dinámico, bifurcación de Hopf, punto de equilibrio, teorema de Hartman-Grobman, retrato de fase, curva logística, respuesta funcional Holling tipo II.