American Institute of Mathematical Sciences

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

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

John Bissell 1, and  Brian Straughan 1,

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.
Keywords: hyperbolic reaction-diffusion equations, `tipping point'., traffic modelling, Discontinuity waves, SIS epidemic model, crowd dynamics, Hantavirus, shocks.
Mathematics Subject Classification: Primary: 35L60, 35K57, 35L67; Secondary: 92B9.
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:
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[9]

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[12]

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[13]

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[15]

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[16]

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[17]

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[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]

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[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

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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

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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

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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

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