As more people come to the planet, expansion will begin and that means we will have multiple Martian communities. Of course, a leader will be established among the original settlers but this will not last forever. Either because the original leader decides to go back to Earth or he or she passes away. This means we will likely have a democratic voting process (considering its popularity on Earth). That person will become the head of the initial community. However, more communities mean more leaders will be needed as one person cannot run things for an entire planet alone. From here, more people will be given political power for their specific community.
Think of it like a Mayoral or Gubernatorial role where that person makes the decisions for their community. Of course, some communities might be broken up based on the jobs they hold. For example, the original settlers will have various roles key to what is needed for scientific experimentation. As time goes on, more people with more skills will be around. We also cannot forget those that might be brought over to specifically work for private space programs. Thus, we might have a community for just scientists and another that is responsible for plants and crops, for instance. Thus, just like on Earth, the leadership of a community will be crucial.