On his ranch southwest of Johannesburg, John Hume chokes up when he talks about the first of his rhinos to be poached. “We tame our rhinos, we feed them, we want to manage them,” he says, fighting back tears. “I make it easy for the poachers to kill my rhinos.”

Mr. Hume, a 73-year-old retired developer of timeshare resorts, spends more than $200,000 a month protecting his rhinos with a veritable private army, some of whom buzz over his 16,000-acre spread near Klerksdorp, South Africa, by helicopter. Still, he has lost 13 rhinos...