# American Institute of Mathematical Sciences

• Previous Article
Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation
• DCDS-B Home
• This Issue
• Next Article
Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two
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  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
##### References:

show all references

##### References:
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 $h$.
 [1] Carmen G. Higuera-Chan, Héctor Jasso-Fuentes, J. Adolfo Minjárez-Sosa. Control systems of interacting objects modeled as a game against nature under a mean field approach. Journal of Dynamics & Games, 2017, 4 (1) : 59-74. doi: 10.3934/jdg.2017004 [2] Shin-Ichiro Ei, Kota Ikeda, Masaharu Nagayama, Akiyasu Tomoeda. Reduced model from a reaction-diffusion system of collective motion of camphor boats. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 847-856. doi: 10.3934/dcdss.2015.8.847 [3] Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 [4] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173 [5] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006 [6] Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013 [7] Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126 [8] Radek Erban, Jan Haskovec. From individual to collective behaviour of coupled velocity jump processes: A locust example. Kinetic & Related Models, 2012, 5 (4) : 817-842. doi: 10.3934/krm.2012.5.817 [9] Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086 [10] Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 189-201. doi: 10.3934/mbe.2014.11.189 [11] Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824 [12] Ken Shirakawa, Hiroshi Watanabe. Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 139-159. doi: 10.3934/dcdss.2014.7.139 [13] Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, Silvia Vilariño. On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories. Journal of Geometric Mechanics, 2011, 3 (1) : 113-137. doi: 10.3934/jgm.2011.3.113 [14] Martin Burger, Alexander Lorz, Marie-Therese Wolfram. Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth. Kinetic & Related Models, 2017, 10 (1) : 117-140. doi: 10.3934/krm.2017005 [15] Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045 [16] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [17] Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039 [18] Edward Allen. Environmental variability and mean-reverting processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2073-2089. doi: 10.3934/dcdsb.2016037 [19] Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217 [20] Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic & Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433

2018 Impact Factor: 1.008