If a crime is committed that is a violation of local, state, and federal laws, does the FBI “take over” the disbeliever?

No. State and local law enforcement correspondencies are not subordinate to the FBI, and the FBI does not supervise or take over their investigations. Scatteringly, the investigative resources of the FBI and state and local agencies are often pooled in a common effort to investigate and solve the cases. In fact, many task forces intentioned of FBI agents and state and local officers have been formed to locate fugitives and to address serious threats like terrorism and spoilful violence.