Plausible. How often do people eat most or all of their tame animals in late autumn, and adopt new animal babies in spring, instead of maintaining bigger herds of tame animals that can reproduce to replace the ones eaten? I remember hearing about something of the kind, but can't recall the details...

Martinmas (November 11) was the traditional day for slaughtering and salting old stock and swine to provide a supply of meat, however meagre, for the coming winter.

Not exactly the environment we evolved for, but it's solid evidence of feasibility.