How to protect public api

I have a asp.net web API, which I retieve the information while building up the page with AJAX's.

Recently I notices that some bot are leeching my API, I do not mean that they DDoS but they're just going in to the site and collecting data from the API every night and post it in thier own websites.

I wounder how can I protect myself, I mean, I know that as long as it's public someone even can go into the webpage and scrap it, but how can I make a honey tarap in order to catch them.

So the easier thing that was to use KEYS, but I'm using proxy cache to and my server is not strong enough to handle all the actual requests. So I was thinking to block in the server whoever that don't "referrer" for my site and make a "check cookie", so everytime the proxy won't see the referrer it willp pass therequest tot the server and then it will check with the cookie and in case of bot it will Email me.

So, my question is, do I think proper? Am I wrong? Maybe it's not worth the effort or maybe there is already a pattern for this?

Thank you for your time.