  We study the moduli space of rank 2 instanton sheaves on $\p3$ in terms of
representations of a quiver consisting of 3 vertices and 4 arrows between two
pairs of vertices. Aiming at an alternative compactification for the moduli
space of instanton sheaves, we show that for each rank 2 instanton sheaf, there
is a stability parameter $\theta$ for which the corresponding quiver
representation is $\theta$-stable (in the sense of King), and that the space of
stability parameters has a non trivial wall-and-chamber decomposition. Looking
more closely at instantons of low charge, we prove that there are stability
parameters with respect to which every representation corresponding to a rank 2
instanton sheaf of charge 2 is stable, and provide a complete description of
the wall-and-chamber decomposition for representation corresponding to a rank 2
instanton sheaf of charge 1.
