In The Tempest, what is the conflict between Prospero and some of the passengers on the ship caught in the tempest?

In Act I of the play we discover the history of Prospero,
what led him to end up exiled on the island and also how the passengers we meet at the
beginning of the play were involved in his exile. Prospero tells Miranda his story and
how he had been a duke until his brother, Antonio, conspired with Alsonso the King of
Naples to overthrow his power and exile him. Prospero had left his brother in charge
whilst he studied and this gave Antonio the opportunity he was waiting for to become the
new Duke of Milan. Thus it is that Propsero and Miranda, who was three at the time, were
stranded on an old boat. Gonzalo, who was a member of the council of Alonso, was the one
character who showed some compassion and stocked this boat with food and water and also
Prospero's books. Now fate has brought Prospero's old enemies to his island, and the
rest of the play shows how this conflict is resolved.