  We study the competition between induced symmetry breaking (ISB),
  Cooper pairing (superconductivity) and chiral condensation at finite
non-asymptotic density. Using a quantum mechanical model studied recently by
Ilieva and Thirring, we analyze the expectation value of the fermion number
density $\la a^*a\ra$ in the presence of chemical potential and discuss
relevance of the ISB phenomenon in the model. By Bogoliubov transformation, we
obtain the ground state energy and find that a phase with both superconducting
and mean field gap is a true ground state. We also study the effective
potentials of a two-dimensional field theory model for competitions between
$\la\bar\psi\psi\ra$, $\la\psi\psi\ra$ and $\la\psi^\dagger\psi\ra$. We find
that in the regime $\mu <\mu_c$, where $\mu_c$ is found to be around $0.4 m_F$
with $m_F$ being the renormalizaton point of chiral condensate, we have a
chiral symmetry-breaking phase with almost zero fermion number density and in
the case of $\mu >\mu_c$ the system sits in the mixed phase characterized by
  $\la \psi^\dagger\psi\ra\neq 0$ and $\la \psi\psi\ra\neq 0$ as long as the
renormalization point, $M_0$, corresponding to a superconducting gap is not
zero, $M_0\neq 0$.
