How to give a biological interpretation to this phase portrait?
I don't think you missed anything important! 

You could investigate the cyclic behaviour around the equilibrium. For example, looking at the variable aphid population size can start above its equilibrium point, then overshoot it and overshoot it again to finally reach the equilibrium point. If you zoom close to the equilibrium point, you might see long cyclic behaviour before ever reaching the equilibrium.

You sure could do all kind of further analyses such as a stability analysis or searching for cyclical equilibrium but it is easy to see with this simple model that nothing very complicated will arise. You could also investigate for what parameter values this classic pattern breaks up if it does. 

Another option is to make the model more complex by introducing a carrying capacity (and then investigate under what aphid growth rate the behaviour becomes chaotic), or population structure, or whatever.

You might learn much from it but you might still want to have a look at the post What prevents predator overpopulation?.

I highly recommend the book A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Otto and Troy. It will show the typical kind of analyses that can be done for this kind of systems.