  This paper reports on a theoretical analysis of the rich variety of
spatio-temporal patterns observed recently in inclined layer convection at
medium Prandtl number when varying the inclination angle $\gamma$ and the
Rayleigh number $R$. The present numerical investigation of the inclined layer
convection system is based on the standard Oberbeck-Boussinesq equations. The
patterns are shown to originate from a complicated competition of
buoyancy-driven and shear-flow driven pattern forming mechanisms. The former
are expressed as \rm{longitudinal} convection rolls with their axes oriented
parallel to the incline, the latter as perpendicular \rm{transverse} rolls.
Along with conventional methods to study roll patterns and their stability, we
employ direct numerical simulations in large spatial domains, comparable with
the experimental ones. As a result, we determine the phase diagram of the
characteristic complex 3D convection patterns above onset of convection in the
$\gamma-R$ plane, and find that it compares very well with the experiments. In
particular we demonstrate that interactions of specific Fourier modes,
characterized by a resonant interaction of their wavevectors in the layer
plane, are key to understanding the pattern morphologies.
