“I think people are actually frightened,” said Maria Williams, a professor who chairs the University of Alaska’s Faculty Alliance. “I’m frightened because I feel that what is happening is a drastic reshaping of the state of Alaska.”

The showdown in the Legislature this week comes at a time of economic trouble in the state. While much of the United States has benefited from robust economic growth in recent years, Alaska’s fortunes have been largely tied to those of the state’s declining oil and gas industry. Falling oil revenues have brought on a persistent recession that has forced the state to confront lingering questions about how to best revive its economy.

Faced with looming deficits, political leaders avoided imposing a state sales tax or personal income tax, and chose to reduce payouts from the oil dividend fund instead. But the reductions were unpopular with some voters, and Mr. Dunleavy won election last year promising not only to restore the full dividend payments for the future but to fight for catch-up payments to make up for past reductions.

Leaders of the University of Alaska system, which serves more than 26,000 students from Juneau to Fairbanks, expect the governor’s budget cut to result in the shuttering of some satellite campuses, the elimination of hundreds of staff and faculty positions and an unprecedented reduction in the number of students the system is able to serve.

Mr. Dunleavy, a former teacher who got a master’s in education from the University of Alaska system, said his cuts to the state budget, including those for the university system, were necessary to lay a better foundation for private-sector job growth. But Jim Johnsen, the president of the university, said a strong university system was necessary to develop innovators and training for a work force that is increasingly dependent on postsecondary education.