Hi Shaz,
I would expect you to get a letter 'inviting' you to apply for PIP several months before your DLA claim is due to end. As long as you conform to the given time scales, your DLA will continue while the PIP claim is being decided.
You can initiate the PIP claim yourself at any stage by phoning DWP - but you probably won't gain much by doing so now. It might be better to use the time gathering evidence in support of your claim if you can.
Nothing on this board constitutes legal advice - always consult a professional about specific problems