  Linear conjugacy offers a new perspective to broaden the scope of stable
biochemical reaction networks to the systems linearly conjugated to the
well-established complex balanced mass action systems ($\ell$cCBMASs). This
paper addresses the challenge posed by time delay, which can disrupt the linear
conjugacy relationship and complicate stability analysis for delayed versions
of $\ell$cCBMASs (D$\ell$cCBMAS). Firstly, we develop Lyapunov functionals
tailored to some D$\ell$cCBMASs by using the persisted parameter relationships
under time delays. Subsequently, we redivide the phase space as several
invariant sets of trajectories and further investigate the existence and
uniqueness of equilibriums in each newly defined invariant set. This enables us
to determine the local asymptotic stability of some D$\ell$cCBMASs within an
updated framework. Furthermore, illustrative examples are provided to
demonstrate the practical implications of our approach.
