  In this paper, we present an algorithm for deriving the normal forms of
Bautin bifurcations in reaction-diffusion systems with time delays and Neumann
boundary conditions. On the center manifold near a Bautin bifurcation, the
first and second Lyapunov coefficients are calculated explicitly, which
completely determine the dynamical behavior near the bifurcation point. As an
example, the Segel-Jackson predator-prey model is studied. Near the Bautin
bifurcation we find the existence of fold bifurcation of periodic orbits, as
well as subcritical and supercritical Hopf bifurcations. Both theoretical and
numerical results indicate that solutions with small (large) initial conditions
converge to stable periodic orbits (diverge to infinity).
