JOHANNESBURG — South African officials say game rangers killed three poachers in the past week in an escalation of armed incursions into South Africa’s flagship wildlife reserve, where hundreds of rhinos have been killed for their horns this year.

Paul Daphne, a spokesman for South Africa’s national parks service, said Saturday that poaching in Kruger National Park had traditionally increased in November and December before tapering off around the year-end holiday season.

“There is definitely an increase compared to previous months in the number of poachers coming into the Kruger park,” Mr. Daphne said. “It’s not unusual for this time of year.”

Many poachers cross into Kruger from neighboring Mozambique. Demand for rhino horn is surging in parts of Asia where it is seen as a status symbol and a cure for illness despite a lack of evidence that it can heal. The horn is made of keratin, a protein also found in human fingernails.