Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction-Diffusion Systems

The distribution of the roots of a second order transcendental polynomial is analyzed, and it is used for solving the purely imaginary eigenvalue of a transcendental characteristic equation with two transcendental terms. The results are applied to the stability and associated Hopf bifurcation of a constant equilibrium of a general reaction-diffusion system or a system of ordinary differential equations with delay effects. Examples from biochemical reaction and predator-prey models are analyzed using the new techniques.