The process that produced the difference in the two types of beaks in finches found on the Galapagos Islands is evolution by natural selection.
It may be that the two types of finches had access to different types of foods on their respective islands. In order to continually survive, each finch must find a way to adapt its beak to the available food type. The beneficial trait is thus selected for in the finch's population.
Hence, with time, the two finches become different in the shape of their beaks due to the forces of natural selection as a result of adapting to the kind of food available.
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