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Abbott said he had offered additional surveillance resources to bolster the two Australian Orion aircraft already searching for the plane during a recent phone call with Prime Minister Najib Razak.
"He asked that Australia take responsibility for the search in the southern vector, which the Malaysian authorities now think was one possible flight path for this ill-fated aircraft," Abbott told lawmakers in parliament.
"I agreed that we would do so."
The search for flight MH370 is focusing on a wide strip of territory either side of two arcs formed by satellite plots of the aircraft's last known possible position.
The southern Indian Ocean is one of the most remote places in the world and also one of the deepest, posing potentially enormous challenges for the international search effort.
The northern vector of the search stretches through Thailand and China and bends towards India, Pakistan and then Central Asia, over some of the world's most strongly guarded defences.
Abbott said Australian Defence Force chief David Hurley was in discussion with his Malaysian counterparts about how best to deploy the additional resources, without providing further details, in the search for the Boeing 777-200ER.
On Sunday, Australia shifted one of its two Orion aircraft searching for the missing plane further south in the Indian Ocean, at Malaysia's request. The aircraft is now searching the ocean to the north and west of the remote Cocos Islands. The second Orion is continuing to search west of Malaysia.
Australia has a military over-the-horizon radar (OTHR) network, which allows it to observe all air and sea activity north and northwest of Australia for up to 3000km.
However, the Jindalee Operational Radar Network (JORN), which has radar capability extending into the Indian Ocean, does not operate on a 24-hour basis, according to Royal Australian Air Force (RAAF) documents.
JORN is used primarily to provide defence surveillance of Australia's northern approaches but does not continually "sweep" an area like conventional radars. Instead, it "dwells" on a selected area.
Australia's Defence Department did not reply to repeated requests for information on JORN, including details on its operations and whether the network had detected Flight MH370.
Australia's civil aviation radar extends a maximum of just 200 nautical miles (410km) off the coast, a civil aviation authority source said, and is used only for monitoring scheduled aircraft on approach into the country and subsequent landings.
Asked before his parliamentary statement whether any Australian agencies had found information to suggest the plane might have come close to Australia, Abbott said: "I don't have any information to that effect."
"But all of our agencies that could possibly help in this area are scouring their data to see if there's anything that they can add to the understanding of this mystery."