Doug Corwin knew there was a problem at his family’s commercial duck farm in Long Island when he spotted scores of dead or lethargic birds during a barn inspection in January. Within days, Crescent Duck Farm became a casualty of the global avian flu outbreak, one of many farms around the U.S. that had to cull their entire flock, sending the prices of eggs and other agricultural commodities soaring. Now the more than century-old farm — the last duck farm remaining in a New York region once synonymous with the culinary delicacy — is cautiously rebuilding.