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On a discrete threedimensional LeslieGower competition model
The hipster effect: When anticonformists all look the same
Department of Mathematics and Volen National Center for Complex Systems, Brandeis University, 415 South Street, Waltham, MA 02453, USA 
In such different domains as statistical physics, neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals evolving following (mainstream) or against (hipsters) the majority are ubiquitous. Moreover, in a variety of applications, interactions between agents occur after specific delays that depends on the time needed to transport, transmit or take into account information. This paper focuses on the role of opposition to majority and delays in the emerging dynamics in a population composed of mainstream and anticonformist individuals. To this purpose, we introduce a class of simple statistical system of interacting agents taking into account (ⅰ) the presence of mainstream and anticonformist individuals and (ⅱ) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Another modality synchronizing hipsters are asymmetric interactions, particularly when the crossinteraction between hipsters and mainstreams aree prominent, i.e. when hipsters radically oppose to mainstream and mainstreams wish to follow the majority, even when led by hipsters. We demonstrate this phenomenon analytically using bifurcation theory and reduction to normal form. We find that, in the case of asymmetric interactions, the level of randomness in the decisions themselves also leads to synchronization of the hipsters. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.
References:
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L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush University of California Press, BerkeleyLos Angeles, Calif. 1964. 
[2] 
N. Brunel and V. Hakim, Fast global oscillations in networks of integrateandfire neurons with low firing rates, Neural Computation, 11 (1999), 16211671. doi: 10.1162/089976699300016179. 
[3] 
J. Carr, Applications of Centre Manifold Theory, SpringerVerlag, 1981. 
[4] 
D. Challet, M. Marsili and Y.Ch. Zhang, Modeling market mechanism with minority game, Physica A: Statistical Mechanics and its Applications, 276 (2000), 284315. doi: 10.1016/S03784371(99)00446X. 
[5] 
D. Challet, M. Marsili and Y.Ch. Zhang, Minority Games: Interacting Agents in Financial Markets, OUP Catalogue, 2013. 
[6] 
A. Clauset, C. R. Shalizi and M. E. J. Newman, Powerlaw distributions in empirical data, SIAM review, 51 (2009), 661703. doi: 10.1137/070710111. 
[7] 
F. Collet, M. Formentin and D. Tovazzi, Rhythmic behavior in a twopopulation meanfield ising model, Physical Review E, 94 (2016), 042139, 7pp. doi: 10.1103/PhysRevE.94.042139. 
[8] 
A. Crisanti and H. Sompolinsky, Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model, Physical Review A, 36 (1987), 49224939. doi: 10.1103/PhysRevA.36.4922. 
[9] 
A. Crisanti and H. Sompolinsky, Dynamics of spin systems with randomly asymmetric bounds: Ising spins and glauber dynamics, Phys. Review A, 37 (1987), 48654874. doi: 10.1103/PhysRevA.37.4865. 
[10] 
P. Dai Pra, M. Fischer and D. Regoli, A curieweiss model with dissipation, Journal of Statistical Physics, 152 (2013), 3753. doi: 10.1007/s1095501307562. 
[11] 
P. Dai Pra, E. Sartori and M. Tolotti, Rhytmic behavior in large scale systems: a model related to meanfield games, preprint, http://www.math.unipd.it/ daipra/draftGamePaolo.pdf, 2017. 
[12] 
P. D. Pra, E. Sartori and M. Tolotti, Climb on the bandwagon: Consensus and periodicity in a lifetime utility model with strategic interactions, arXiv: 1804.07469, 2018. 
[13] 
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom, A. M. Riet and B. Sautois, Matcont and cl Matcont: Continuation Toolboxes in Matlab, Universiteit Gent, Belgium and Utrecht University, The Netherlands, 2006. 
[14] 
A. Dhooge, W. Govaerts and Y. A. Kuznetsov, Matcont: A matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 141164. doi: 10.1145/779359.779362. 
[15] 
B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14., Siam, 2002. doi: 10.1137/1.9780898718195. 
[16] 
T. Faria, On a planar system modelling a neuro network with memory, Journal of Differential Equations, 168 (2000), 129149. doi: 10.1006/jdeq.2000.3881. 
[17] 
S. Galam, Sociophysics: A Physicist's Modeling of PsychoPolitical Phenomena, Springer, 2012. doi: 10.1007/9781461420323. 
[18] 
F. Giannakopoulos and A. Zapp, Local and global hopf bifurcation in a scalar delay differential equation, Journal of Mathematical Analysis and Applications, 237 (1999), 425450. doi: 10.1006/jmaa.1999.6431. 
[19] 
J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, volume 42 of Applied mathematical sciences, Springer, 1983. doi: 10.1007/9781461211402. 
[20] 
J. Haidt, Disagreeing virtuously, Disagreeing Virtuously, page 138, 2017. 
[21] 
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Verlag, 1993. doi: 10.1007/9781461243427. 
[22] 
G. Hermann and J. Touboul, Heterogeneous connections induce oscillations in largescale networks, Physical Review Letters, 109 (2012), 018702. doi: 10.1103/PhysRevLett.109.018702. 
[23] 
A. Jackson and D. Ladley, Market ecologies: The effect of information on the interaction and profitability of technical trading strategies, International Review of Financial Analysis, 47 (2016), 270280. doi: 10.1016/j.irfa.2016.02.007. 
[24] 
J. S. Juul and M. A. Porter, Hipsters on networks: How a small group of individuals can lead to an antiestablishment majority, arXiv: 1707.07187, 2017. 
[25] 
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112., SpringerVerlag, New York, 1995. doi: 10.1007/9781475724219. 
[26]  
[27] 
C. Quiñinao and J. Touboul, Limits and dynamics of randomly connected neuronal networks, Acta Applicandae Mathematicae, 136 (2015), 167192. doi: 10.1007/s1044001499455. 
[28] 
A. Roxin, N. Brunel and D. Hansel, Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks, Physical Review Letters, 94 (2005), 238103. doi: 10.1103/PhysRevLett.94.238103. 
[29] 
M. Scheutzow, Noise can create periodic behavior and stabilize nonlinear diffusions, Stochastic Processes and Their Applications, 20 (1985), 323331. doi: 10.1016/03044149(85)902194. 
[30] 
M. Scheutzow, Some examples of nonlinear diffusion processes having a timeperiodic law, The Annals of Probability, 13 (1985), 379384. doi: 10.1214/aop/1176992997. 
[31] 
D. Sherrington and S. Kirkpatrick, Solvable model of a spinglass, Spin Glass Theory and Beyond, 35 (1986), 104108. doi: 10.1142/9789812799371_0010. 
[32] 
H. Sompolinsky, A. Crisanti and H.J. Sommers, Chaos in random neural networks, Physical Review Letters, 61 (1988), 259262. doi: 10.1103/PhysRevLett.61.259. 
[33]  S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. 
[34] 
A.S. Sznitman, Topics in propagation of chaos, In Ecole d'Eté de Probabilités de SaintFlour XIX, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169. 
[35] 
J. Touboul, Limits and dynamics of stochastic neuronal networks with random heterogeneous delays, Journal of Statistical Physics, 149 (2012), 569597. doi: 10.1007/s1095501206076. 
[36] 
J. Touboul, The hipster effect: When anticonformists all look the same, arXiv: 1410.8001, 2014. 
[37] 
J. Touboul, G. Hermann and O. Faugeras, Noiseinduced behaviors in neural mean field dynamics, SIAM Journal on Applied Dynamical Systems, 11 (2012), 4981. doi: 10.1137/110832392. 
show all references
References:
[1] 
L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush University of California Press, BerkeleyLos Angeles, Calif. 1964. 
[2] 
N. Brunel and V. Hakim, Fast global oscillations in networks of integrateandfire neurons with low firing rates, Neural Computation, 11 (1999), 16211671. doi: 10.1162/089976699300016179. 
[3] 
J. Carr, Applications of Centre Manifold Theory, SpringerVerlag, 1981. 
[4] 
D. Challet, M. Marsili and Y.Ch. Zhang, Modeling market mechanism with minority game, Physica A: Statistical Mechanics and its Applications, 276 (2000), 284315. doi: 10.1016/S03784371(99)00446X. 
[5] 
D. Challet, M. Marsili and Y.Ch. Zhang, Minority Games: Interacting Agents in Financial Markets, OUP Catalogue, 2013. 
[6] 
A. Clauset, C. R. Shalizi and M. E. J. Newman, Powerlaw distributions in empirical data, SIAM review, 51 (2009), 661703. doi: 10.1137/070710111. 
[7] 
F. Collet, M. Formentin and D. Tovazzi, Rhythmic behavior in a twopopulation meanfield ising model, Physical Review E, 94 (2016), 042139, 7pp. doi: 10.1103/PhysRevE.94.042139. 
[8] 
A. Crisanti and H. Sompolinsky, Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model, Physical Review A, 36 (1987), 49224939. doi: 10.1103/PhysRevA.36.4922. 
[9] 
A. Crisanti and H. Sompolinsky, Dynamics of spin systems with randomly asymmetric bounds: Ising spins and glauber dynamics, Phys. Review A, 37 (1987), 48654874. doi: 10.1103/PhysRevA.37.4865. 
[10] 
P. Dai Pra, M. Fischer and D. Regoli, A curieweiss model with dissipation, Journal of Statistical Physics, 152 (2013), 3753. doi: 10.1007/s1095501307562. 
[11] 
P. Dai Pra, E. Sartori and M. Tolotti, Rhytmic behavior in large scale systems: a model related to meanfield games, preprint, http://www.math.unipd.it/ daipra/draftGamePaolo.pdf, 2017. 
[12] 
P. D. Pra, E. Sartori and M. Tolotti, Climb on the bandwagon: Consensus and periodicity in a lifetime utility model with strategic interactions, arXiv: 1804.07469, 2018. 
[13] 
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom, A. M. Riet and B. Sautois, Matcont and cl Matcont: Continuation Toolboxes in Matlab, Universiteit Gent, Belgium and Utrecht University, The Netherlands, 2006. 
[14] 
A. Dhooge, W. Govaerts and Y. A. Kuznetsov, Matcont: A matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 141164. doi: 10.1145/779359.779362. 
[15] 
B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14., Siam, 2002. doi: 10.1137/1.9780898718195. 
[16] 
T. Faria, On a planar system modelling a neuro network with memory, Journal of Differential Equations, 168 (2000), 129149. doi: 10.1006/jdeq.2000.3881. 
[17] 
S. Galam, Sociophysics: A Physicist's Modeling of PsychoPolitical Phenomena, Springer, 2012. doi: 10.1007/9781461420323. 
[18] 
F. Giannakopoulos and A. Zapp, Local and global hopf bifurcation in a scalar delay differential equation, Journal of Mathematical Analysis and Applications, 237 (1999), 425450. doi: 10.1006/jmaa.1999.6431. 
[19] 
J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, volume 42 of Applied mathematical sciences, Springer, 1983. doi: 10.1007/9781461211402. 
[20] 
J. Haidt, Disagreeing virtuously, Disagreeing Virtuously, page 138, 2017. 
[21] 
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Verlag, 1993. doi: 10.1007/9781461243427. 
[22] 
G. Hermann and J. Touboul, Heterogeneous connections induce oscillations in largescale networks, Physical Review Letters, 109 (2012), 018702. doi: 10.1103/PhysRevLett.109.018702. 
[23] 
A. Jackson and D. Ladley, Market ecologies: The effect of information on the interaction and profitability of technical trading strategies, International Review of Financial Analysis, 47 (2016), 270280. doi: 10.1016/j.irfa.2016.02.007. 
[24] 
J. S. Juul and M. A. Porter, Hipsters on networks: How a small group of individuals can lead to an antiestablishment majority, arXiv: 1707.07187, 2017. 
[25] 
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, volume 112., SpringerVerlag, New York, 1995. doi: 10.1007/9781475724219. 
[26]  
[27] 
C. Quiñinao and J. Touboul, Limits and dynamics of randomly connected neuronal networks, Acta Applicandae Mathematicae, 136 (2015), 167192. doi: 10.1007/s1044001499455. 
[28] 
A. Roxin, N. Brunel and D. Hansel, Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks, Physical Review Letters, 94 (2005), 238103. doi: 10.1103/PhysRevLett.94.238103. 
[29] 
M. Scheutzow, Noise can create periodic behavior and stabilize nonlinear diffusions, Stochastic Processes and Their Applications, 20 (1985), 323331. doi: 10.1016/03044149(85)902194. 
[30] 
M. Scheutzow, Some examples of nonlinear diffusion processes having a timeperiodic law, The Annals of Probability, 13 (1985), 379384. doi: 10.1214/aop/1176992997. 
[31] 
D. Sherrington and S. Kirkpatrick, Solvable model of a spinglass, Spin Glass Theory and Beyond, 35 (1986), 104108. doi: 10.1142/9789812799371_0010. 
[32] 
H. Sompolinsky, A. Crisanti and H.J. Sommers, Chaos in random neural networks, Physical Review Letters, 61 (1988), 259262. doi: 10.1103/PhysRevLett.61.259. 
[33]  S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. 
[34] 
A.S. Sznitman, Topics in propagation of chaos, In Ecole d'Eté de Probabilités de SaintFlour XIX, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169. 
[35] 
J. Touboul, Limits and dynamics of stochastic neuronal networks with random heterogeneous delays, Journal of Statistical Physics, 149 (2012), 569597. doi: 10.1007/s1095501206076. 
[36] 
J. Touboul, The hipster effect: When anticonformists all look the same, arXiv: 1410.8001, 2014. 
[37] 
J. Touboul, G. Hermann and O. Faugeras, Noiseinduced behaviors in neural mean field dynamics, SIAM Journal on Applied Dynamical Systems, 11 (2012), 4981. doi: 10.1137/110832392. 
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