Instead of making birds run after you and get killed, why not put them to a use.
As for now birds are not worth that much and does not serve a purpose. Making birds produce something, such as eggs that can be used in cooking or eaten raw, would give them a purpose and also increese their value.
Eggs could for example be cooked in the deep pot for boiled eggs.
As for the model one already exists "models/props_phx/misc/egg.mdl" reducing the amount of work needed to implement it.