You do not own such a thing because Ludwig doesn't need food! Ghostdogs do not have a need for sustenance of any kind.
That aside, your inventory is quite barren. All that you currently have with you is your BOW and your humble supply of ARROWS.
You never planned on staying out for too long, so you didn't take much with you.
Ludwig: Hear the girl, run after the voice. Then Five: Run after Ludwig. AND THEN Nino: Run after five.
...But they were already gone.
How could you be so careless? How could you lose your precious companion?
Oh, sweet precious Ludwig Grimm von Phantomhound is probably wandering all alone right now! You can't even begin to imagine how scared and lonely he must be!!!
Just how did this whole debacle happen?
...Come on, you already know how. It's obvious, really.
You zoned out like you always do!
You were just going for a short walk to see the ECLIPSE. A rare moment in which the CRESCENT is visible through the BARRIER.
It was supposed to be a quick detour before you and Ludwig go back to the town. But you were so enamored by the ECLIPSE, you never noticed when he left your side.
However, you are not to blame.
The Nightlights are quite captivating.
Mysterious archer girl: Use the secret super sensitive grim dog harmonica to call back Ludwig!
You just saw her inventory! She hasn't got anything like that.
These names are getting kind of redundant. Just because you have a bow doesn't mean your name is Archer Girl! Or that your name would be Blond Girl because your hair happens to be blond!!
You do have an actual name. It's a very good one, too.
And that name is...
It's a nice name.
Now that formalities and introductions are out of the way, you survey your current location.
You were looking for Ludwig, your precious companion, and you ended up here. The ECLIPSE ended long ago, too. You need to return home to your town soon! Staying alone in the FOREST OF ILLUSIONS is a bad idea.
But before that, most importantly, you have to find Ludwig!! The animal will not be happy if it's not with you.
And if he's not happy, then you will not be happy either.
Now, where to look?
I spy another Homestuck reference
tl;dr: Unfortunately, Purple Archer doesn't have any food to lure Ludwig out, and thinks back to how she lost him: she was too captivated by the moon and night sky and didn't notice he left. Purple Archer introduces herself as Aria Elwyn. Aria resolves to find Ludwig fast.