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

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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

Abstract
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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 and Continuous Dynamical Systems - B, 2014, 19 (7) : 1911-1934. doi: 10.3934/dcdsb.2014.19.1911

References:

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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.
[2]	L. J. S. Allen, R. K. McCormack and C. B. Jonsson, Mathematical models for hantavirus infection in rodents, Bulletin of Mathematical Biology, 68 (2006), 511-524. doi: 10.1007/s11538-005-9034-4.
[3]	L. J. S. Allen, C. L. Wesley, R. D. Owen, D. G. Goodin, D. Koch, C. B. Jonsson, Y. Chu, J. M. S. Hutchinson and R. L. Paige, A habitat-based model for the spread of hantavirus between reservoir and spillover species, Journal of Theoretical Biology, 260 (2009), 510-522. doi: 10.1016/j.jtbi.2009.07.009.
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[7]	N. Bellomo and C. Dogbe, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.
[8]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.
[9]	N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Networks And Heterogeneous Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.
[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.
[11]	R. A. Bentley and M. J. O'Brien, Tipping points, animal culture, and social learning, Current Zoology, 58 (2012), 298-306.
[12]	R. A. Bentley and M. J. O'Brien, Cultural evolutionary tipping points in the storage and transmission of information, Frontiers in Psychology, 3 (2013), 569. doi: 10.3389/fpsyg.2012.00569.
[13]	P. J. Chen, Growth and decay of waves in solids, in Handbuch der Physik, (eds. S. Flügge and C. Truesdell), VIa/3, Springer, Berlin, 1973, 303-402.
[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.
[15]	I. Christov and P. M. Jordan, On the propagation of second sound in nonlinear media: Shock, acceleration and travelling wave results, J. Thermal Stresses, 33 (2010), 1109-1135. doi: 10.1080/01495739.2010.517674.
[16]	I. Christov, P. M. Jordan and C. I. Christov, Nonlinear acoustic propagation in homentropic perfect gases: A numerical study, Physics Letters A, 353 (2006), 273-280. doi: 10.1016/j.physleta.2005.12.101.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
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[25]	J. A. Foley, Tipping Points in the Tundra, Science, 310 (2005), 627-628.
[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.
[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.
[28]	M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
[29]	R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B, 36 (200), 507-535. doi: 10.1016/S0191-2615(01)00015-7.
[30]	D. Iesan and A. Scalia, Thermoelastic Deformations, Kluwer, Dordrecht, 1996. doi: 10.1007/978-94-017-3517-9.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	P. M. Jordan and G. Saccomandi, Compact acoustic travelling waves in a class of fluids with nonlinear material dispersion, Proc. Roy. Soc. London A, 468 (2012), 3441-3457. doi: 10.1098/rspa.2012.0321.
[39]	S. Kefi, M. Rietkerk, C. L. Alados, Y. Pueyo, V. P. Papanastasis, A. ElAich and P. C. de Ruiter, Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems, Nature, 449 (2007), 213-217. doi: 10.1038/nature06111.
[40]	K. A. Lindsay and B. Straughan, Acceleration Waves and Second Sound in a Perfect fluid, Archive for Rational Mechanics and Analysis, 68 (1978), 53-87. doi: 10.1007/BF00276179.
[41]	R. S. Mani, V. Ravi, A. Desai and S. N. Madhusudana, Emerging Viral Infections in India, Proceedings of the National Academy of Sciences, India Section B, 82 (2012), 5-21. doi: 10.1007/s40011-011-0001-1.
[42]	A. Marasco, On the first-order speeds in any direction of acceleration waves in prestressed second - order isotropic, compressible, and homogeneous materials, Mathematical and Computer Modelling, 49 (2009), 1644-1652. doi: 10.1016/j.mcm.2008.07.037.
[43]	A. Marasco, Second - order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media, Int. J. Engng. Sci., 47 (2009), 499-511. doi: 10.1016/j.ijengsci.2008.08.009.
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