TNR only works in a closed-system. The problem with all the TNR propaganda being spewed all over the net by ignorant cat-lickers and cat-hoarders, is that they don't realize they must sterilize well over 80% of them to be the least bit effective. And that percentage MUST BE sterilized within one breeding cycle of that animal. In the case of cats that's under 4 months. In order to somewhat replicate a "closed system" condition, you must also sterilize as many as occupy an area a little larger than any non-sterile cat can migrate to find another non-sterile cat. In the case of cats, they can easily travel 10 to 20 or more miles to find another to mate with within 4 months. (Some ambitious cats can travel hundreds of miles in that time-frame.) Add-up how many cats are in a 10 to 20 mile radius in your area, then sterilize 80%+ of them in under 4 months. At the cost of (average, across the board in the USA, adding in man-hours, vetting, supplies, and transport, etc.) $140 per cat.

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