Why don't leaders in places like Libya, Syria, Yemen, and so on step down from office in the face of widespread revolts and protests? Why do they hold on to power, even if this means shooting and killing their own citizens?
There is no silver bullet answer to those questions, but there are several factors in play here, ranging from psychological factors such as groupthink, messianic views, and misperceptions to the simple calculation that they can outlast their opponents.
First, let us talk about the psychological factor. Almost every dictator always has a messianic view of their role in their own societies and in the world more generally. Indeed, they seem to believe that they are the only persons capable of saving the state, ending existing political discord, and producing lasting peace and harmony. As a result, they believe that there's nobody after them who will keep the country running together and in good shape. This was evident in the gallery of dictators, from Adolf Hitler, Stalin, and Mao to Kim Il Sung and Qaddafi.
Messianic views, however, do not explain everything. Dictators have resigned and relinquished political power, even though they believed that they were essential to the welfare of their state. Sukarno and Suharto of Indonesia, Hosni Mubarak, and Pinochet are examples of dictators with messianic values that resigned in the end. Even one could argue that Hitler resigned by committing suicide.
What is important here is their perception of vulnerability, which, in turn, is heavily influenced by the "groupthink" around the leaders. Leaders don't operate alone. They invariably create a circle of trusted subordinates who will help these leaders maintain their grip on power. These subordinates in turn rely on the leaders as a source of legitimacy. This creates a dependency relationship between the leader and the system and fosters the need to maintain the survival of both the leader and the system.
Leaders usually fall when either the system collapses (e.g. Hitler) or the system (which includes a host of institutions and groups) has outgrown the leader and believes that the leader is no longer indispensable and can survive better without the leader itself (e.g. Sukarno, Suharto, Pinochet and Mubarak).
Let us look at three current dictators in the Middle East: Saleh (Yemen), Assad (Syria), and Qaddafi (Libya).
In Saleh's case, his leadership is indispensable to his clique (which include friends and family members). Yemeni politics in essence is comprised of a loose coalition of Arab tribes, and Saleh is able to stay on top based on his ability to play each tribe against the others. He blundered in the earlier days of protests, when he cracked down on protesters, creating unity among his political opponents. He then tried to divide the opposition by first declaring that he would resign and later by bringing the Saudis to mediate. Still, his inconsistent patterns of dealing with the protesters gave him credibility problems and in fact his opposition only became emboldened and tried to push for more. Yet, the fragmented nature of the opposition, combined by the indispensability of Saleh to his clique gives Saleh some hope that he might be able to survive the current political crisis, thus explaining his obstinacy.
In the case of Asaad, for him and his associates, it is a question of life and death: they came from the Allawites, the minority tribe that comprises only 10% of the Syrian population, making their survival at stake should they get toppled from power. As a result, Assad chose to crack down on his opponents, grant some carrots of amnesty in order to prevent international intervention, and attempt to divide the opposition.
In Qaddafi's case, the fragmented nature of the state--there is simply no organization at all--actually empowers Qaddafi, making it difficult to dislodge him from power. But at the same time, both the government and the opposition are fragmented, which had led to a stalemate. Qaddafi himself seems to be confident that without the NATO's meddling, he can rely on his mercenaries to get rid of the fragmented, unorganized and poorly trained and equipped opposition. And with both the U.S. and the Europeans unwilling to get dirty and bogged down in Libya, the longer he holds on to power, the higher his chance of survival as sooner or later war fatigue will set in.
Thus, all three dictators hold to power due to a simple logic, that their chance of survival is still high, considering the fragmented opposition and the unwillingness of international community to get their hands dirty.