How to prove universality in complex systems?
In order to show that your computation model is universal, you have to show that you can implement every computable function in your model. You do this by picking some other computation model $M$ and showing how to simulate $M$ in your model. For somewhat more formal definitions, see this answer.

As stated, this sounds difficult, and the first step is to look at various proofs of universality to get some ideas of what is involved. Some examples are listed in the Wikipedia article. It's possible that your model can easily simulate one of the simpler computation models such as counter machines, and this is perhaps the first thing to check.

It is also possible that your model is not Turing-complete but is still quite strong, say PSPACE-complete or EXPTIME-complete.