Mean Field Model for Collective Motion Bistability

TL;DR

This paper analyzes a one-dimensional model of locust collective motion, identifying conditions for stable ordered states influenced by noise and interaction strength, supported by analytical and numerical results.

Contribution

It provides a detailed analysis of stationary states and their stability in a collective motion model, including fluctuation and large deviation analysis.

Findings

Stable ordered states depend on noise level and interaction strength.

Transition probabilities between states are characterized by large deviation principles.

Numerical simulations confirm analytical stability conditions.

Abstract

We consider the Czir\'ok model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collective motion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.