HASTINGS-ON-HUDSON, N.Y. — This suburban village overlooking the Hudson River is a mere 2 square miles, home to a hip downtown, neighborhoods of neatly kept homes and an ever-growing population of deer that overrun woods, chew through gardens and cause more than a dozen car crashes a year.Grasping for a way to control the deer without hunting the animals, leaders of this village of 7,900 have proposed an ambitious compromise to shoot them up — not with bullets but with birth control.Scientists and humane groups hope the program, which seeks to capture and inject female white-tailed deer with a contraceptive made from pigs’ ovaries, can become a model for other places that are too congested or compassionate to consider killing.“We’re hearing all about `Don’t kill Bambi’ and all the jokes about deer condoms,” Mayor Peter Swiderski said. “People are having their little chuckles. But deer have a pretty big negative effect on the community.”Under the plan, which will begin this winter if approved by the state Department of Environmental Conservation, as many as 90 percent of the does in Hastings will be tranquilized, inoculated with the contraceptive, then tagged and released. The deer population is estimated at up to 120, a density of 60 per square mile. That’s three times the deer density that some studies have tied to a decline in plant and animal species.