  This is a continuation of our previous paper arXiv:1904.07924, which is
devoted to the construction of integrable semi-discretizations of the
Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system; in this
series of papers, we refer to a discretization of one of the two spatial
variables as a semi-discretization. In this paper, we construct an integrable
semi-discrete Davey-Stewartson system, which is essentially different from the
semi-discrete Davey-Stewartson system proposed in the previous paper
arXiv:1904.07924. We first obtain integrable semi-discretizations of the two
elementary flows that compose the Davey-Stewartson system by constructing their
Lax-pair representations and show that these two elementary flows commute as in
the continuous case. Then, we consider a linear combination of the two
elementary flows to obtain a new integrable semi-discretization of the
Davey-Stewartson system. Using a linear transformation of the continuous
independent variables, one of the two elementary Davey-Stewartson flows can be
identified with an integrable semi-discretization of the $(2+1)$-dimensional
Yajima-Oikawa system proposed in
https://link.aps.org/doi/10.1103/PhysRevE.91.062902 .
