Historical effect in the territoriality of ayu fish.

Ayu fish form algae-feeding territories in a river during a non-breeding (growing) season. We build a cost-benefit theory to describe the breakdown and formation of territory. In the early stage of a growing season, all fish hold territories at low densities. Once all territory sites are occupied, excess fish become floaters. When fish density further increases, a phase transition occurs: all the territories suddenly break down and fish form a school. In contrast, when the fish density is decreased, territories are suddenly formed from the school. Both theory and experiments demonstrate that ayu should exhibit a historical effect: the breakdown and formation processes of territory are largely different. In particular, the theory in formation process predicts a specific fish behavior: an "attempted territory holder" that tries to have a small territory emerges just before the formation of territory.