One example of character displacement is that when two species of Galapagos finches, Geospiza fuliginosa and Geospiza fortis, occur on the same island, the difference between their beak sizes is accentuated. When either species occurs without the other, their beak sizes occur in a more intermediate range.
In the example of finches, the character displacement that arises occurs because of competition for seeds. Both species are seed eaters, and the size of their bills are optimized to take advantage of their particular environments. Most islands have a range of edible seed sizes, so an intermediate size of beak is preferred when either species is alone. When they occur together, however, they compete with each other heavily for medium-sized seeds. This reduces the effective abundance of that resource for both species.
Each species varies in the average size of their beaks, despite their tendencies when isolated. Thus, when they occur together, the relative abundance of seeds closer to the extremes in size increases, since they do not compete for them. Only the species that has a larger average beak size tends to eat larger seeds, while only the species that has a smaller average beak size tends to eat smaller seeds. This creates an evolutionary pressure on both populations that tends to enhance the differences in beak size over time.