  A few years ago, Campos investigated the critical phenomena of thick branes
in warped spacetimes [Phys. Rev. Lett. {\bf 88} (2002) 141602]. Inspired by his
work, we consider a toy model of thick branes generated by a real scalar field
with the potential $V(\phi)=a\phi^2-b\phi^4+c\phi^6$, and investigate the
variation of the mass parameter $a$ on the branes as well as the localization
and resonances of fermions. An interesting result is found: there is a critical
value for the mass parameter $a$, and when the critical value of $a$ is reached
the solution of the background scalar field is not unique and has the shape of
a double kink. This happens in both cases with and without gravity. It is also
shown that the numbers of the bound Kaluza-Klein modes of fermions on the
gravity-free brane and the resonant states of fermions on the brane with
gravity increase with the value of $a$.
