Pledges to stand down from Hosni Mubarak and Ali Abdullah Saleh of Yemen signal a new trend across the Middle East

President Ali Abdullah Saleh, who famously compares ruling Yemen to "dancing on the heads of snakes", tapped into the restive mood of the Arab world today by announcing that he would not stand for re-election in 2013.

Scepticism may well be in order. Saleh has made similar promises before – and has still held on to power since 1978. And his timing – in advance of a planned Yemeni "day of rage", looked suspicious.

But his pledge, just hours after Egypt's Hosni Mubarak said he would stand down this year, is part of what it is starting to look like a trend across the Middle East.

Stunned by events in Tunisia and Egypt, and with rumblings of serious unrest from Algeria to Jordan, authoritarian leaders are taking pre-emptive action to inoculate themselves against the "contagion" of people power.

All face anger over unemployment, poverty and corruption. Maintaining food and fuel subsidies, raising salaries and shuffling cabinets are useful options. Ending repression and starting meaningful political reform is much harder.

"Just a short time ago people tried to argue that the Tunisian crisis was an isolated case and that it was different from any other Arab country," said Marwan Muasher, a former Jordanian minister who is now with the Carnegie Foundation. "It is now difficult, if not impossible, to make the same argument with Egypt … in turmoil. If the largest Arab country is faced with unrest, people need to draw the right lessons."

The biggest casualty so far of this early "Arab spring" has been the phenomenon of the president-for-life – and the related issue of dynastic republican succession that has so often accompanied it.

Until the Egyptian unrest 82-year-old Mubarak had no designated successor and was seen as still likely to run for a seventh term in September.

Failing that, there was a good chance he would be succeeded by his banker son Gamal, a key figure in the ruling National Democratic Party. Tawrith (succession) has been endlessly debated. No longer. "Where's daddy now?" asked a scornful poster in Tahrir Square.

Direct succession was not on the cards in Tunisia, where the flight of Zine el-Abidine Ben Ali on 14 January electrified the Arab world. But the kleptocratic role of his wife's family and his son in-law Sakher al-Materi were bitter reminders of the untrammelled power of a "semi-mafia" presidency.

Saleh had signalled before that he might not stand again. But even if he did not, his son Ahmed had been groomed to follow him. So by rejecting MPs' accusations of backing "hereditary" rule Saleh made a major concession he must hope, like Mubarak and Egypt's generals, will allow the regime to survive.

But the trend is only partial. Libya's Muammar al-Gaddafi – the veteran of them all, with 42 years in power – remains as vigorous and eccentric as ever. Talk of the succession of his reformist-minded son, Seif al-Islam, has faded recently in the face of resistance by the old guard.Still, another son, Mutasim, is a rising star as his father's national security adviser.

Syria, bastion of Arab nationalism and close ally of Iran that is far from the orbit of US power, is another significant exception. It is 10 years since President Bashar al-Assad succeeded his late father Hafez and, despite slick PR and economic liberalisation, there has been no easing of his grip on a repressive regime that is widely seen as a bulwark against sectarianism.

In the mostly placid, wealthy monarchies of the Gulf, where there is little political life and succession is always hereditary, the odd man out could be the island state of Bahrain. This is where the Sunni King Hamad and his Al-Khalifa dynasty rule over a restive Shia majority angry at discrimination and corruption.

The forthcoming "day of rage", and others planned in the coming days in Syria and Algeria, will be closely watched across a suddenly hopeful and nervous region.