Bifurcation with regard to combined interaction parameter in a life energy system dynamic model of two components with multiple delays (Q430206)

The authors consider a life energy system dynamic model of the form NEWLINE\[NEWLINE\begin{cases} \dot{x}_1(t)=-a_1x_1^2(t)+bx_1(t)+[c_{12}-d_{12}x_1(t)]x_2(t-\tau_1),\\ \dot{x}_2(t)=-a_2x_2^2(t)+bx_2(t)+[c_{21}-d_{21}x_2(t)]x_1(t-\tau_2).\end{cases}\tag{1}NEWLINE\]NEWLINE By using the characteristic equation approach developed in [\textit{S. Guo, Y. Chen} and \textit{J. Wu}, J. Differ. Equations 244, No. 2, 444--486 (2008; Zbl 1136.34058)], they obtain the asymptotic stability of the zero solution of (1) under certain technical conditions. Furthermore, they discuss the Hopf bifurcation from \((0,0)\) with \(\eta=\sqrt{|c_{21}c_{12}|}\) as bifurcation parameter, including the bifurcation direction and stability of the bifurcating periodic solutions. The methods are the classical normal form theory and the center manifold theorem.