I don't think the invitations are working properly. I recently just used dryad invite and so far the wild pets have been sporadically. When I used invite last time, there was wild pets appearing in den at ever cycle. Here is record of what is happening on my Map:

12 PM EDT ~ Last wild pet is caught and no more wild pet on map

5 PM EDT ~ Dryad stays at Inn

8 PM EDT ~ No creatures

12 AM EDT ~ Unicorn appears in Magic Glade

4 AM EDT ~ Unicorn appears in Heart of Forest and Puma appears in Big Tree

8 AM EDT ~ No wild pet appears on map

12 AM EDT ~ Unicorn appears in Magic Glade, Unicorn appears in Heart of Forest, and Dragon appears in Cave.