Flocking birds ? ? ? In our study of the dynamics of collective behavior, we are often intrigued by the process that turns individual actions and local interactions into large-scale, global patterns. This happens in many ways in human crowds, bird flocks, fish schools, bacteria swarms, etc. A simple and powerful illustration is Conway’s game of life. Here is a very simple model for bird flocks that assumes away many features but suffices to illustrate the point for a homework problem. Suppose there is a two-dimensional plane with N points moving in it. Each point has neighbors, which are the points within a circle of radius r meters. All the points move with the same constant speed, say, 1 unit, but along different directions. At each timeslot, each point’s direction is updated to be the average of its current direction and all the directions of its neighbors. We can think of a graph in which each node is a point, and each link is a neighbor relationship. But this graph evolves over time as the points’ positions change. Randomly place 100 points in a 10 × 10 units square, and initialize their directions randomly. You should try different values of r. Simulate the above model over time, and describe what happens to the directions of the points. (For more detail of this model, see T. Vicsek, A. Czirok, E. Ben Jacob, I. Cohen, and O. Schochet, “Novel type of phase transitions in a system of self-driven particles,” Physics Review Letters, vol. 75, pp. 1226-1229, 1995. A comprehensive survey of animal behavior can be found in I. D. Couzin and J. Krause, “Selforganization and collective behavior in vertebrates,” Advances in the Study of Behavior, vol. 32, pp. 1–75, 2003.)
Flocking birds ? ? ?
In our study of the dynamics of collective behavior, we are often intrigued by the process that turns individual actions and local interactions into large-scale, global patterns. This happens in many ways in human crowds, bird flocks, fish schools, bacteria swarms, etc. A simple and powerful illustration is Conway’s game of life.
Here is a very simple model for bird flocks that assumes away many features but suffices to illustrate the point for a homework problem. Suppose there is a two-dimensional plane with N points moving in it. Each point has neighbors, which are the points within a circle of radius r meters. All the points move with the same constant speed, say, 1 unit, but along different directions. At each timeslot, each point’s direction is updated to be the average of its current direction and all the directions of its neighbors. We can think of a graph in which each node is a point, and each link is a neighbor relationship. But this graph evolves over time as the points’ positions change.
Randomly place 100 points in a 10 × 10 units square, and initialize their directions randomly. You should try different values of r. Simulate the above model over time, and describe what happens to the directions of the points.
(For more detail of this model, see T. Vicsek, A. Czirok, E. Ben Jacob, I. Cohen, and O. Schochet, “Novel type of phase transitions in a system of self-driven particles,” Physics Review Letters, vol. 75, pp. 1226-1229, 1995. A comprehensive survey of animal behavior can be found in I. D. Couzin and J. Krause, “Selforganization and collective behavior in vertebrates,” Advances in the Study of Behavior, vol. 32, pp. 1–75, 2003.)
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