Stability analysis in a two-dimensional life energy system model with delay

Bifurcation phenomena can occur in parameter dependent systems. When the parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcations and the parameter values are called bifurcation values. The type of bifurcation that associates equilibria with periodic solution is called Hopf bifurcation. The study of Hopf bifurcation includes determining the bifurcation value, the direction of bifurcation and the stability of the bifurcating periodic solutions. It is a attractive branch in applied mathematics. In this paper, a two-dimensional life energy system model with delay is considered. By analyzing the distribution of the roots of the characteristic equation, the bifurcation set in the parameter plane is drawn. From ecological point of view, the existence of Hopf bifurcation expresses periodic oscillatory behavior of the life energy system.