  Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M \\ 0 & B
\\\end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. We first construct a
semi-complete duality pair $\mathcal{D}_{T}$ of $T$-modules using duality pairs
in $A$-Mod and $B$-Mod respectively. Then we characterize when a left
$T$-module is Gorenstein $D_{T}$-projective, Gorenstein $D_{T}$-injective or
Gorenstein $D_{T}$-flat. These three class of $T$-modules will induce model
structures on $T$-Mod. Finally we show that the homotopy category of each of
model structures above admits a recollement relative to corresponding stable
categories. Our results give new characterizations to earlier results in this
direction.
