I'm no expert ok, but I find these fascinating so I'd like to try: I'd first solve for the equilibrium points in terms of a and b:
Fixed points are created whenever the equilibrium points are real so the root objects completely control the creation and destruction of fixed points. Immediately we see the creation of and whenever becomes non-negative. I would suspect through further analysis, i.e., linearizing, and calculating eigenvalues, we could determine that these two fixed points are indeed a saddle and node hence the saddle-node bifurcation. Probably other ways too though. In the first plot, I depicted the flow in phase space about these two points (red) when a=4 and b=1 (green section in second plot). The top fixed point looks to be a saddle (points move to then away from it). The bottom point looks a source (node). Points move away from it. That would need to be verified algebraically though.
The remaining four have real y-component whenever the quantity or the region above the line . These have real x component whenever or are non-negative. These regions are depicted in the second plot below with the number of fixed points indicated in each.
Again, further analysis could determine the types of remaining fixed points as well as further types of bifurations such as Hopf bifurcations. This is just a start.