Self-organizing Dynamic Model of Fish Schooling

Fish schools show a high degree of polarization in the absence of a leader or external stimuli. In this paper the problems of the collective motion of fish schooling are analyzed, and the question of how polarized patterns or structures arise spontaneously is addressed. I attempt to show collective properties on the basis of elemental properties. Individual fish are regarded as gas molecules having locomotion, inbuilt responses with respect to each other and fluctuations of motion. A non-linear Langevin equation describing self-organized formation of fish schools is obtained with practical approximation methods. It is shown that fish schools are governed by specific mathematical relations, i.e. synergetics, which represent the principal feature of order-disorder (polarization-non-polarization) transitions. Systems composed of many fish dramatically change their structure when certain parameters are varied. Moreover, transient behavior and the onset of polarized schooling structure are discussed with reference to the non-linear Langevin equation near the instability point. Experimental data on the transient behavior of the formation of schooling structure are shown for comparison with the theoretical result.