Pattern formation in fish schooling

Fish schooling is known as one of typical self-organization observed in biological systems and has attracted interests of scientists and engineers in wide fields from biologists to computer engineers. The authors of this paper have introduced in \cite{Uchitane2012} a stochastic ODE model to describe the process of schooling, and have utilized in \cite{Nguyen2014} the model to develop quantitative arguments on fish schooling. In this paper, we want to study our stochastic ODE model from the viewpoint of pattern formation. For this purpose, we set a global obstacle in the space where fish move and investigate obstacle avoidance patterns performed by the fish school. We shall then find that there are clear four avoidance patterns, i.e., Rebound, Pullback, Pass and Reunion, and Separation, and that the emerging pattern changes depending on parameter tuning.