We introduce a novel distributed source seeking framework, DIAS, designed for multi-robot systems in scenarios where the number of sources is unknown and potentially exceeds the number of robots. Traditional robotic source seeking methods typically focused on directing each robot to a specific strong source and may fall short in comprehensively identifying all potential sources. DIAS addresses this gap by introducing a hybrid controller that identifies the presence of sources and then alternates between exploration for data gathering and exploitation for guiding robots to identified sources. It further enhances search efficiency by dividing the environment into Voronoi cells and approximating source density functions based on Gaussian process regression. Additionally, DIAS can be integrated with existing source seeking algorithms. We compare DIAS with existing algorithms, including DOSS and GMES in simulated gas leakage scenarios where the number of sources outnumbers or is equal to the number of robots. The numerical results show that DIAS outperforms the baseline methods in both the efficiency of source identification by the robots and the accuracy of the estimated environmental density function.