Spoilers ahead, regarding a death battle idea

If Lucy from Elfen Lied and Airi Yūnami from Qualidea Code had a Death Battle, what should be the the cause of their fight?

One idea I've thought of is Lucy killing a human child Airi cared about in front of her.

Don't forget that they have similar abilities.

If she has to, Airi may resort to using her true form.