The basic idea is contained in the old nursery rhyme:

Jack Sprat could eat no fat,

His wife could eat no lean;

And so between them both,

They licked the platter clean.

The two birds of a mated pair, limited to a single area, could be expected to have different food preferences or adaptations for getting it. And we find that there are cases of this. The most striking is that of the huia from New Zealand, of which I've written in a Chicago Museum bulletin. Both sexes have similar food preferences, especially wood-inhabiting insects, but they get them in different ways. The male has a short, straight, stout bill for digging out the wood-boring grubs, woodpecker fashion; the female has a much longer, slender, and curved bill for probing into holes for them, creeper fashion. The female may get grubs in wood too hard for the male to chisel. They supplement each other.

DIET VARIATION BY SEX It is possible that further study may show more sexual differences to have a feeding advantage; the larger size of female hawks fitting them to take larger prey; the smaller size of certain female songbirds fitting them for smaller prey, the smaller bills of female hornbills, the straight bill of the male western grebe, and the upturned bill of the female. Perhaps all are of advantage to the species in giving each sex slightly different advantages in getting food.

Selection could have its effect in the populations with most sexual difference in feeding habits being most successful in raising and leaving progeny. Thus, slowly, differences between the sexes would accumulate. However, it must be kept in mind that this sort of evolution would be limited. The drifting apart of the sexes would be checked by the necessity for their coming together periodically for at least a short period, at nesting time.

WATER IN THE DESERT [Ref]