Isolated d.c.e. degrees and ${\mathrm{\Sigma }}_1$ induction

A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree $\mathbf{d}$ is isolated by a c.e. degree $\mathbf{a}<\mathbf{d}$ if all c.e. degrees that are below $\mathbf{d}$ are also below $\mathbf{a}$; $\mathbf{d}$ is isolated from above by a c.e. degree $\mathbf{a}>\mathbf{d}$ if all c.e. degrees that are above $\mathbf{d}$ are also above $\mathbf{a}$. In this paper, we study the inductive strength of both isolated and upper isolated d.c.e. degrees from the point of view of reverse recursion theory. We show that (1) $P^{-}+B{\mathrm{\Sigma }}_1+\text{Exp}⊢I{\mathrm{\Sigma }}_1\leftrightarrow$ There is an isolated proper d.c.e. degree below ${\mathbf{0}}^{\mathrm{\prime }}$; (2) $P^{-}+B{\mathrm{\Sigma }}_1+\text{Exp}⊢I{\mathrm{\Sigma }}_1\leftrightarrow$ There is an upper isolated proper d.c.e. degree below ${\mathbf{0}}^{\mathrm{\prime }}$.