Is it permitted to trap an animal who cause pain but not death on Shabbat?
If the trapping is done on a deorayta level, how can this permitted if its not life threatening?
Can one kill this pain causing animal via “dorso Lefi tumo” (inconspicuously)?
Trapping and killing animals for the sole purpose of preventeing pain (and not to get benefit from the animal) is a “melacha she’eina tzreecha l’gufah”, which is prohibited only rabbinically.
Concerning trapping, there is often an additional point of leniency. Trapping creatures belonging to a species which, as a rule, is not hunted for use, is also only rabbinically prohibited (Shabbos 106b-107b).
Therefore, if one has reason to be afraid that an animal is about to bite him, it may be removed by hand (though this involves trapping) to avoid pain (Tosafos; Shulchan Aruch, Orach Chayim 316:9).
It is more stringent to kill creatures than to trap them, and therefore one cannot permit killing unless there is a strong fear of more significant pain (Shulchan Aruch 316:9-10; Mishna Berurah 316:46).Thus, mosquitoes, which usually cause discomfort but not significant pain, should not be killed on Shabbos.
The concept of “lefi tumo” applies when an animal whose bite causes significant pain, but which is not dangerous, is not chasing after a person. Under these circumstances, it is permitted to kill the animal by stepping on it as part of one’s ordinary walking, “lefi tumo.”