Bifurcation Analysis of a Class of Food Chain Model with Two Time Delays

This paper investigates the Hopf bifurcation of a three-dimensional food chain model with two timedelays, focusing on the synergistic effect of time delays in energy transfer between different trophic levels on the stability of the system. By analyzing the distribution of the roots of the characteristic equation, the stability conditions of the internal equilibrium point and the criterion for the existence of the Hopf bifurcation are established. Using the paradigm theory and the central manifold theorem, explicit formulas for determining the bifurcation direction and the stability of the bifurcation periodic solution are obtained. Numerical simulations verify the theoretical results. This study shows that increasing the time delay will lead to the instability of the food chain model through Hopf bifurcation and produce limit cycle oscillations. This work simulates the asymmetric propagation mode of population fluctuations observed in natural ecosystems, providing a theoretical basis for analyzing the coevolution of complex food webs.