Katie the dog and Sarah Darby entered Prospect Park in Brooklyn beneath a full hunter’s moon. Ms. Darby turned on Katie’s red LED collar, unclipped her leash and Katie sprinted off, looking over her shoulder to make sure her mistress was following.

Katie disappeared around a bend in the paved path that skirts the Long Meadow. When Ms. Darby caught up, Katie was circling a trash can. “Ready, Katie?” She tilted back the trash can. A plump rat darted out.

A blurred chase lasted only a second. Katie snatched the squeaking rat in her jaws, whipped her head back and forth, and snapped its neck. She dropped the limp rat, and Ms. Darby used a folded paper towel to pick it up by the tail and drop it in the trash can, among banana peels and muffin liners.

It was Katie’s 363rd kill of the year.

Her 364th came nine minutes later. “Watch out,” Ms. Darby called quietly to a woman in a hooded sweatshirt who was jogging by the furry corpse, more than a foot long from nose to tail. “Oh, my god,” the runner said as she hurried off.

Katie weighs 12 ½ pounds and is about 4 years old. She is 28 percent Chihuahua, 16 percent pug, 16 percent rat terrier and 40 percent mutt. Ms. Darby, 50, an educational consultant by day, adopted Katie from a shelter in Texas, near the Mexican border, in 2022. Not too long after, Ms. Darby said, “I was walking her in a playground that had tons and tons of rats and she sort of revealed to me her interest in doing this.”