First, you really have to identify the culprit here. Spread out some flour and look at the tracks. Go out at night with a flashlight and take a peek. Look at teeth marks in the lily leaves. It may be squirrels, opossums, or raccoons.

You should experiment with different strategies. Can you remove the trellis, making the site unattractive or unaccessible? Can you cover or fence the lilies? Try leaving a light on all night. Can you plant something at this site that is less attractive?

Trapping and removal is an option, but remember that animals don't live in a location in a vacuum. Remove one, and another may take its place. Manipulating the factors in your control will prove to be the best long-term strategy.

There are additional identification strategies at the link below.

Online Resources:
Identification Techniques
Wildlife Control Technique Information