  Recently, research on the complex periodic behavior of multi-scale systems
has become increasingly popular. Krupa et al. \cite{krupa2} provided a way to
obtain relaxation oscillations in slow-fast systems through singular Hopf
bifurcations and canard explosion. The authors derived a $O(1)$ expression $A$
for the first Lyapunov coefficient (under the condition $A \neq 0$), and
deduced the bifurcation curves of singular Hopf and canard explosions.
  This paper employs Blow-up technique, normal form theory, and Lyapunov
coefficient formula to present higher-order approximate expressions for the
first Lyapunov coefficient when $A=0$ for slow-fast systems. As an application,
we investigate the bifurcation phenomena of a predator-prey model with Allee
effects. Utilizing the formulas obtained in this paper, we identify both
supercritical and subcritical Hopf bifurcations that may occur simultaneously
in the system. Numerical simulations validate the results. Finally, by normal
form and slow divergence integral theory, we prove the cyclicity of the system
is 1.
