If you manipulate habitats to make them unsuitable for a species, these unsuitable habitats will not be reinvaded, even when surrounding populations are healthy. For this reason, exclusion or habitat modification always works better in the long run than does local extermination. Pigeons roost and nest in cubby holes, under bridges and in the eaves and ledges of buildings.
Pigeons are capable of traveling long distances on a daily basis for food or water.