Abstract
Fish school is an example of the familiar form of animal social behavior. However, the mechanism of the schooling behavior has not been clear yet. In our earlier papers, a mathematical model has been proposed for describing the behavior of fish school by using observation data from water tank experiments. Each fish regulates his movement, speed and direction, by the information which he receives from other fish around him. The mechanism of this information exchange has been demonstrated by carrying out computer simulation.
In this paper, a special attention is paid to analyzing the model for the schooling behavior of two fish. This model is represented by a system of nonlinear differential equations with five state variables. We investigate the relationship between a stable steady state of the system and a schooling behavior of fish swimming in the two-dimensional unbounded space. A result of analysis shows that the steady states of fish are given as a rectilinear motion and a circular motion. From an investigation of the local stability of the steady states it is found that the fish behavior model constructed by water tank experiment data is classified into four groups. Each group has different stable steady states. The most dominant group of the model has such a stable steady state as two fish swim along a linear trajectory and the leader corresponds to the fish with larger characteristic velocity.
