Practice: Collective Motion
Module 2: Synchronization and Collective Behaviour
This is an ungraded practice page. Choose one option and work on it after the lab. It uses the code you wrote in Session 14 and the models of Session 13.
Option A: Couzin model
Theory: The Couzin Model.
- Implement the Couzin rules and motion in two dimensions.
- Animate the individuals moving. Reuse the structure of Animation.
- Implement the group polarization \(p_{\text{group}}\) and the angular momentum \(m_{\text{group}}\) defined in Couzin et al. (2002). Compare \(p_{\text{group}}\) with the Vicsek order parameter.
- Explore the widths \(\Delta r_o\) and \(\Delta r_a\) and find the four collective behaviours described by Couzin et al. (2002): swarm, torus, dynamic parallel group and highly parallel group.
- Add sliders to change parameters in real time.
- Test collective memory: slowly increase \(\Delta r_o\) and then decrease it again, and check whether the group state follows the same path both ways.
- Implement the model in 3D, as in the original paper.
Option B: Vicsek model with predator avoidance
Guide: Predator Avoidance.
The predator can be a fixed point, a random walker, or your mouse cursor. Particles inside the predator radius turn directly away from it, as in the Couzin repulsion rule.
- Implement the full Vicsek animation from Animation.
- Add the predator avoidance rule and draw the predator.
- Explore the parameter space. How does the predator affect the order parameter? Is there a regime where the group splits and later merges again? Can you find a regime where the particles escape the predator?
Exit question
In the Vicsek model you double both \(N\) and \(L^2\), keeping the density and the noise fixed. Which features of the order parameter curve \(\varphi(\eta)\) should stay the same, and which should change?
The density \(\rho = N/L^2\) is unchanged, so in the ordered phase \(\varphi\) should be about the same. At high noise, \(\varphi\) is set by the finite-size floor of order \(1/\sqrt{N}\), which decreases by a factor \(\sqrt{2}\). The transition region should become sharper. This is the pattern in Figure 2(a) of Vicsek et al. (1995).