Yudhoyono went on national television on Wednesday to announce that he was freezing military and intelligence cooperation, including over the issue of asylum seekers, that has long been an irritant in relations.
"Local police advise that a demonstration is planned for outside the Australian Embassy on 21 November 2013," Australia's Department of Foreign Affairs and Trade said in updated travel advisory for Indonesia, the country's No. 2 tourist destination after New Zealand.
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"Australians should monitor local media, avoid protests, maintain high levels of vigilance and security awareness."
Australian Prime Minister Tony Abbott has not confirmed the spying actions or apologized, but on Wednesday expressed regret for the embarrassment the media reports had caused Yudhoyono and has promised a "swift, full and courteous response" to the president's concerns.
Relations hit a nadir in 1999 when Australia sent troops into East Timor after Indonesia's military pulled out following a scorched-earth policy in the one-time Indonesian colony.
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Ties with Jakarta have improved significantly since then but deteriorated since Abbott took office in September, because of the spying reports and tension over how to prevent asylum seekers sailing from Indonesia to Australia.