  In this work, the $\beta$-stable region for Z $\geq$ 90 is proposed. The
calculated $\beta$-stable nuclei in the $\beta$-stable region are in good
agreement with the ones obtained by M\"{o}ller \emph{et al}.. The half-lives of
the nuclei close to the $\beta$-stable region are calculated and the
competition between $\alpha$-decay and $\beta$-decay is systematically
investigated. The calculated half-lives and the suggested decay modes are well
in line with the experimental results. The predictions for half-lives and decay
modes of the nuclei with Z = 107$-$110 are presented.
