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February  2019, 24(2): 851-879. doi: 10.3934/dcdsb.2018210

## Mean field model for collective motion bistability

 1 Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France 2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA 3 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

* Corresponding author

Received  October 2017 Published  February 2019 Early access  June 2018

We consider the Czirók 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 collectivemotion. 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.

Citation: Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210
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The real (a) and imaginary (b) parts of the complex growth rate of the first mode (the most unstable mode) as a function of $\sigma$ for $h = 5$ (solid black), $h = 6$ (dashed red), $h = 8$ (magenta dotted), and $h = 10$ (blue dot-dashed). Here $\phi$ is given by (46) and $L = 10$. $G$ is given by (47) and it derives from the double-well potential plotted in picture c. We address the linear stability of the order state $\rho_{\xi_e}$. One can see that the threshold value for the noise level $\sigma$ to ensure linear stability is $1.8$ for $h = 5$, $0.85$ for $h = 6$, $1.4$ for $h = 8$, and $2.2$ for $h = 10$. The most stable situation is the one corresponding to $h = 6$.
The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ at each time step $t_n$ for $h = 2$ (a-b) and $h = 6$ (c-d). The dashed lines in the velocity plots are the solutions of $G(\xi) = \xi$. The dashed lines in the discrepancy plots stand for the value (53) corresponding to a uniform sampling. Here $\Delta t = 0.1$, $N = 500$, and $\sigma = 2$. One can see that the spatial distribution is uniform and the velocity average is 0 when $h = 2$ and $- \xi_e$ when $h = 6$.
The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ for $\sigma = 0.5$ (a-b), $\sigma = 1$ (c-d), $\sigma = 1.5$ (e-f). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, and $h = 6$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform when $\sigma = 0.5$ while the average velocity is $\xi_e$ and the spatial distribution is uniform when $\sigma = 1$ or $1.5$.
The empirical position distribution smoothed by kernel density estimation (a and c) and the first $200$ trajectories (b and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 0.5$, and $h = 6$. One can see that the spatial distribution is not uniform and a cluster is moving with an apparent velocity of $3.6$.
The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ for $\sigma = 0.5$ (a-b), $\sigma = 1$ (c-d), $\sigma = 1.5$ (e-f). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean 0 and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, and $h = 6$. One can see that the average velocity is not $\pm \xi_e$ and the spatial distribution is not uniform when $\sigma = 0.5$ or $1$ while the average velocity is $\xi_e$ and the spatial distribution is uniform when $\sigma = 1.5$.
The empirical average velocity $\bar{u}^n$ (a), the square centered $L^2$-discrepancy $CL_2^2(n)$ (b), and the empirical position distribution smoothed by kernel density estimation (c and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 1$, and $h = 10$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform.
The empirical average velocity $\bar{u}^n$ (a), the square centered $L^2$-discrepancy $CL_2^2(n)$ (b), and the empirical position distribution smoothed by kernel density estimation (c and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 1$, and $h = 5$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform.
The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $N = 80$ (a), $N = 100$ (b), $N = 120$ (c), and $N = 140$ (d). Here $\Delta t = 0.1$, $h = 6$ and $\sigma = 5$. The frequencies of the transitions between the two stable order states decay when $N$ increases. The system has less transitions with a higher number of agents.
The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $\sigma = 4$ (a), $\sigma = 4.5$ (b), $\sigma = 5$ (c), and $\sigma = 5.5$ (d). Here $\Delta t = 0.1$, $N = 100$, and $h = 6$. The frequencies of the transitions between the two stable order states increase when $\sigma$ increases. The system has more transitions with a higher $\sigma$.
The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $h = 5$ (a), $h = 5.5$ (b), $h = 6$ (c), and $h = 6.5$ (d). Here $\Delta t = 0.1$, $N = 100$, and $\sigma = 5$. The frequencies of the transitions between the two stable order states decay with $h$.
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