develop an argument that explains which of the three models if representative democracy-participatory, pluralist, or elite-best achieved the founders intent for American democracy in terms of ensuring a stable government run by the people

Representative democracy

This is a system of government where citizens elect public officials to govern on their behalf. This system lead to stability because it allows people to elect people to make decisions on their behalf. Given that there are too many people for everyone to come together to make a decision, direct democracy would cause instability.