  We study Gorenstein flat objects in the category ${\sf Rep}(Q,R)$ of
representations of a left rooted quiver $Q$ with values in ${\sf Mod}(R)$, the
category of all left $R$-modules, where $R$ is an arbitrary associative ring.
We show that a representation $X$ in ${\sf Rep}(Q,R)$ is Gorenstein flat if and
only if for each vertex $i$ the canonical homomorphism $\varphi_i^X:
\oplus_{a:j\to i}X(j)\to X(i)$ is injective, and the left $R$-modules $X(i)$
and ${\rm Coker}\varphi_i^X$ are Gorenstein flat. As an application of this
result, we show that there is a hereditary abelian model structure on ${\sf
Rep}(Q,R)$ whose cofibrant objects are precisely the Gorenstein flat
representations, fibrant objects are precisely the cotorsion representations,
and trivial objects are precisely the representations with values in the right
orthogonal category of all projectively coresolved Gorenstein flat left
$R$-modules.
