September  2014, 19(7): 1911-1934. doi: 10.3934/dcdsb.2014.19.1911

Discontinuity waves as tipping points: Applications to biological & sociological systems

1. 

Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom, United Kingdom

Received  March 2013 Revised  July 2013 Published  August 2014

The `tipping point' phenomenon is discussed as a mathematical object, and related to the behaviour of non-linear discontinuity waves in the dynamics of topical sociological and biological problems. The theory of such waves is applied to two illustrative systems in particular: a crowd-continuum model of pedestrian (or traffic) flow; and an hyperbolic reaction-diffusion model for the spread of the hantavirus infection (a disease carried by rodents). In the former, we analyse propagating acceleration waves, demonstrating how blow-up of the wave amplitude might indicate formation of a `human-shock', that is, a `tipping point' transition between safe pedestrian flow, and a state of overcrowding. While in the latter, we examine how travelling waves (of both acceleration and shock type) can be used to describe the advance of a hantavirus infection-front. Results from our investigation of crowd models also apply to equivalent descriptions of traffic flow, a context in which acceleration wave blow-up can be interpreted as emergence of the `phantom congestion' phenomenon, and `stop-start' traffic motion obeys a form of wave propagation.
Citation: John Bissell, Brian Straughan. Discontinuity waves as tipping points: Applications to biological & sociological systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1911-1934. doi: 10.3934/dcdsb.2014.19.1911
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show all references

References:
[1]

G. Abramson and V. M. Kenkre, Spatiotemporal patterns in the Hantavirus infection, Physical Review E, 66 (2002), 011912. doi: 10.1103/PhysRevE.66.011912.  Google Scholar

[2]

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[5]

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[6]

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[9]

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[10]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 29 pp. 1230004. doi: 10.1142/S0218202512300049.  Google Scholar

[11]

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[12]

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[13]

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[14]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second sound propagation in moving media, Physical Review Letters, 94 (2005). doi: 10.1103/PhysRevLett.94.154301.  Google Scholar

[15]

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[16]

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[17]

I. Christov, P. M. Jordan and C. I. Christov, Modelling weakly nonlinear acoustic wave propagation, Quart. Jl. Mech. Appl. Math., 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017.  Google Scholar

[18]

M. Ciarletta and B. Straughan, Poroacoustic acceleration waves, Proceedings of the Royal Society A, 462 (2006), 3493-3499. doi: 10.1098/rspa.2006.1730.  Google Scholar

[19]

M. Ciarletta, B. Straughan and V. Zampoli, Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation, Int. J. Engng. Sci., 45 (2007), 736-743. doi: 10.1016/j.ijengsci.2007.05.001.  Google Scholar

[20]

M. Ciarletta, B. Straughan and V. Zampoli, Poroacoustic acceleration waves in a Jordan-Darcy-Cattaneo material, Int. J. Non-linear Mech., 52 (2013), 8-11. doi: 10.1016/j.ijnonlinmec.2013.01.020.  Google Scholar

[21]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundleheren der mathematischen Wissenschaften, Springer, 3rd edition, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[22]

C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math., 44 (1986), 463-474.  Google Scholar

[23]

J. W. Eslick and A. Puri, A dynamical study of the evolution of pressure waves propagating through a semi-infinite region of homogeneous gas combustion subject to a time-harmonic signal at the boundary, Int. J. Non-linear Mech., 47 (2012), 18-28. doi: 10.1016/j.ijnonlinmec.2011.11.007.  Google Scholar

[24]

M. Fabrizio and A. Morro, Electromagnetism of Continuous Media, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198527008.001.0001.  Google Scholar

[25]

J. A. Foley, Tipping Points in the Tundra, Science, 310 (2005), 627-628. Google Scholar

[26]

Y. B. Fu and N. H. Scott, The transistion from acceleration wave to shock wave, Int. J. Engng. Sci., 29 (1991), 617-624. doi: 10.1016/0020-7225(91)90066-C.  Google Scholar

[27]

T. Gultop, B. Alyavuz and M. Kopac, Propagation of acceleration waves in the johnson-segalman fluids, Mech. Res. Communications, 37 (2010), 153-157. doi: 10.1016/j.mechrescom.2009.12.007.  Google Scholar

[28]

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.  Google Scholar

[29]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transportation Research Part B, 36 (): 507.  doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[30]

D. Iesan and A. Scalia, Thermoelastic Deformations, Kluwer, Dordrecht, 1996. doi: 10.1007/978-94-017-3517-9.  Google Scholar

[31]

C. B. Jonsson, L. T. M. Figueiredo and O. Vapalahti, A global perspective on hantavirus ecology, epidemiology, and disease, Clinical Micribiology Reviews, 23 (2010), 412-441. doi: 10.1128/CMR.00062-09.  Google Scholar

[32]

P. M. Jordan, Growth and decay of shock and acceleration waves in a traffic flow model with relaxation, Physica D, 207 (2005), 220-229. doi: 10.1016/j.physd.2005.06.002.  Google Scholar

[33]

P. M. Jordan, Growth, decay and bifurcation of shock amplitudes under the type-II flux law, Proc. Roy. Soc. London A, 463 (2007), 2783-2798. doi: 10.1098/rspa.2007.1895.  Google Scholar

[34]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena, Math. Computers Simulation, 80 (2009), 202-211. doi: 10.1016/j.matcom.2009.06.004.  Google Scholar

[35]

P. M. Jordan, A note on Chrystal's equation, Appl. Math. Computation, 217 (2010), 933-936. doi: 10.1016/j.amc.2010.05.095.  Google Scholar

[36]

P. M. Jordan and J. K. Fulford, A note on poroacoustic travelling waves under Darcy's law: Exact solutions, Applications of Mathematics, 56 (2011), 99-115. doi: 10.1007/s10492-011-0011-6.  Google Scholar

[37]

P. M. Jordan, G. V. Norton, S. A. Chin-Bing and A. Warn-Varnas, On the propagation of nonlinear acoustic waves in viscous and thermoviscous fluids, European J. Mech., B-Fluids, 34 (2012), 56-63. doi: 10.1016/j.euromechflu.2012.01.016.  Google Scholar

[38]

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