Ah yes, the great ASI cat saga.
Back in rum, bum and baccy days, some of pusser's wooden was canoes visited the place. The ships' rats thought it was Christmas and legged it out of said ships and into the local area as fast as their legs would carry them. The then bred like wildfire and became a distinct pest....
So some bright spark thought "Hmm, what animal catches rats?" Presumably the idea of hordes of Jack Russells rushing about was too expensive, because they introduced a few well-practised rat-catching moggies to the island. Which caught the odd rat - and, in the well-known feline life cycle (eat/sleep/shag/eat/sleep/shag/die) also bred like crazy.
But they'd also discovered that pusser's wily rats were too much like hard work and the indigenous bird population, never having seen a cat in their lives before, were a much easier target....
So now the feral monsters eat birds, baby turtles......but only the stupidest descendants of the ex-naval rats get caught.