In order for any of us to asses your situation, we need more information. Do you live in an urban or a rural setting? There are two distinctions in bobcat behavior, based on the population density of humans. Bobcats will hunt day, night, and any time inbetween.
OP, has he attacked and killed multiple birds at once, or is he picking them off one by one?
Bobcats have HUGE territories in rural settings, and they have multiple dens they use throughout. They may frequent one den for a few days, then move on to the next throughout the year making them very hard to trap or even spot.
My second question to you is where are you located? Laws for bobcats vary from state to state, and your next available course of action will be determined by your local laws.
How often is the cat attacking? How many chickens do you have, how long has the bobcat been in the area?