Explain why family rivalry hanged to power struggles

Upon Miranda's request, Prospero begins to tell her of his history, and how they came upon the island; Miranda was very young when she left the island, and cannot remember anyone but her father, not even her dead mother. Prospero tells her how his kingdom was usurped by his brother Antonio, while Prospero was distracted by his studies, and how the king of Naples supported Antonio's rule. Antonio then cast Prospero and Miranda out of Milan, and ordered both of them killed; however, Prospero tells his daughter how the good councilor Gonzalo arranged for them not to be killed, which led to their landing on the island.