Dual interaction producing both territorial and schooling behavior in fish

Fish schools are frequently observed without a leader and an explicit condition forming school. Several models of schooling have been proposed focusing on the influence of neighbors, and introducing probability distributions, while these models are based on the separation of schooling and territorial behavior. We frequently consider the duality of aggregation of animals, in which behavioral patterns involve both attraction and repulsion, antagonistic with each other. The idea of probability does not explain this duality that can provide both schooling and territorial behavior. From these biological facts, we have constructed a behavior model in which the influence of neighbors is formulated by the interface between the states of neighbors and a map of changes in these states. This interface uses a self-similar nowhere differentiable transition map which is temporally constructed, encompassing a crucial duality of repulsive and attractive forces hidden in the interaction among fishes. We tested it with computer simulations against the biological reality of schooling and territorial behavior. Due to the influence neighbors can have on duality, the same model can show both schooling behavior with a high degree of polarization and territorial behavior.