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
References:
[1]

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

[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. doi: 10.1007/s11538-005-9034-4. Google Scholar

[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. doi: 10.1016/j.jtbi.2009.07.009. Google Scholar

[4]

A. Aw and M. Rascale, Resurrection of "Second Order'' models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099. Google Scholar

[5]

E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction diffusion model for the hantavirus infection,, Mathematical Methods in the Applied Sciences, 31 (2008), 481. doi: 10.1002/mma.929. Google Scholar

[6]

A. D. Barnosky, E. A. Hadly, J. Bascompte, E. L. Berlow, J. H. Brown, M. Fortelius, W. M. Getz, J. Harte, A. Hastings, P. A. Marquet, N. D. Martinez, A. Mooers, P. Roopnarine, G. Vermeij, J. W. Williams, R. Gillespie, J. Kitzes, C. Marshall, N. Matzke, D. P. Mindell, E. Revilla and A. B. Smith, Approaching a state shift in Earth's biosphere,, Nature, 486 (2012), 52. doi: 10.1038/nature11018. Google Scholar

[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. doi: 10.1142/S0218202508003054. Google Scholar

[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. doi: 10.1137/090746677. Google Scholar

[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. doi: 10.3934/nhm.2011.6.383. Google Scholar

[10]

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

[11]

R. A. Bentley and M. J. O'Brien, Tipping points, animal culture, and social learning,, Current Zoology, 58 (2012), 298. Google Scholar

[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). doi: 10.3389/fpsyg.2012.00569. Google Scholar

[13]

P. J. Chen, Growth and decay of waves in solids,, in Handbuch der Physik, VIa/3 (1973), 303. Google Scholar

[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]

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. doi: 10.1080/01495739.2010.517674. Google Scholar

[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. doi: 10.1016/j.physleta.2005.12.101. Google Scholar

[17]

I. Christov, P. M. Jordan and C. I. Christov, Modelling weakly nonlinear acoustic wave propagation,, Quart. Jl. Mech. Appl. Math., 60 (2007), 473. 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. 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. 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. 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, (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. 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. doi: 10.1016/j.ijnonlinmec.2011.11.007. Google Scholar

[24]

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

[25]

J. A. Foley, Tipping Points in the Tundra,, Science, 310 (2005), 627. 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. 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. 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, (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, (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. 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. 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. doi: 10.1098/rspa.2007.1895. Google Scholar

[34]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Math. Computers Simulation, 80 (2009), 202. 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. 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. 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., 34 (2012), 56. doi: 10.1016/j.euromechflu.2012.01.016. Google Scholar

[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. doi: 10.1098/rspa.2012.0321. Google Scholar

[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. doi: 10.1038/nature06111. Google Scholar

[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. doi: 10.1007/BF00276179. Google Scholar

[41]

R. S. Mani, V. Ravi, A. Desai and S. N. Madhusudana, Emerging Viral Infections in India,, Proceedings of the National Academy of Sciences, 82 (2012), 5. doi: 10.1007/s40011-011-0001-1. Google Scholar

[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. doi: 10.1016/j.mcm.2008.07.037. Google Scholar

[43]

A. Marasco, Second - order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media,, Int. J. Engng. Sci., 47 (2009), 499. doi: 10.1016/j.ijengsci.2008.08.009. Google Scholar

[44]

A. Marasco and A. Romano, On the acceleration waves in second - order elastic, isotropic, compressible, and homogeneous materials,, Mathematical and Computer Modelling, 49 (2009), 1504. doi: 10.1016/j.mcm.2008.06.005. Google Scholar

[45]

V. Méndez and J. Camacho, Dynamics and thermodynamics of delayed population growth,, Physical Review E, 55 (1997), 6476. Google Scholar

[46]

V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations,, Physical Review E, 60 (1999), 5231. doi: 10.1103/PhysRevE.60.5231. Google Scholar

[47]

J. N. Mills, T. L. Yates, T. G. Ksiazek, C. J. Peters and J. E.. Childs, Long-Term studies of hantavirus reservoir populations in the southwestern united states: Rationale, potential, and methods,, Emerging Infectious Diseases, 5 (1999), 95. doi: 10.3201/eid0501.990111. Google Scholar

[48]

A. Miranville, A phase-field model Based on a three-phase-lag heat conduction,, Applied Mathematics and Optimization, 63 (2011), 133. doi: 10.1007/s00245-010-9114-9. Google Scholar

[49]

A. Miranville, On a phase-field model with a logarithmic nonlinearity,, Applications of Mathematics, 57 (2012), 215. doi: 10.1007/s10492-012-0014-y. Google Scholar

[50]

M. Ostoja-Starzewski and J. Trebicki, Stochastic dynamics of acceleration waves in random media,, Mechanics of Materials, 38 (2006), 840. doi: 10.1016/j.mechmat.2005.06.022. Google Scholar

[51]

P. Paoletti, Acceleration waves in complex materials,, Discrete Continuous Dyn. Systems B, 17 (2012), 637. doi: 10.3934/dcdsb.2012.17.637. Google Scholar

[52]

M. Rietkerk, S. C. Dekker, P. C. de Ruiter and J. van de Koppel, Self-Organized patchiness and catastrophic shifts in ecosystems,, Science, 305 (2004), 1926. doi: 10.1126/science.1101867. Google Scholar

[53]

F. Sauvage, M. Langlais, N. G. Yoccoz and D. Pontier, Modelling hantavirus in fluctuating populations of bank voles: The role of indirect transmission on virus persistence,, Journal of Animal Ecology, 72 (2003), 1. doi: 10.1046/j.1365-2656.2003.00675.x. Google Scholar

[54]

M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions,, Nature, 461 (2009), 53. doi: 10.1038/nature08227. Google Scholar

[55]

M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bascompte, W. Brock, V. Dakos, J. van de Koppel, I. A. van de Leemput, S. A. Levin, E. H. van Nes, M. Pascual and J. Vandermeer, Anticipating critical transitions,, Science, 338 (2012), 344. doi: 10.1126/science.1225244. Google Scholar

[56]

C. Schmaljohn and B. Hjelle, Hantaviruses: A global disease problem,, Emerging Infectious Diseases, 3 (1997), 95. doi: 10.3201/eid0302.970202. Google Scholar

[57]

V. D. Sharma and R. Venkatraman, Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows,, Int. J. Non-linear Mech., 47 (2012), 918. doi: 10.1016/j.ijnonlinmec.2012.06.001. Google Scholar

[58]

F. M. F. Simoes, J. A. C. Martins and B. Loret, Instabilities in elastic-plastic fluid-saturated porous media: Harmonic wave versus acceleration wave analyses,, Int. J. Solids Structures, 36 (1999), 1277. doi: 10.1016/S0020-7683(98)00002-X. Google Scholar

[59]

B. Straughan, Stability, and Wave Motion in Porous Media,, volume 165 of Appl. Math. Sci., (2008). Google Scholar

[60]

B. Straughan, Heat Waves,, volume 177 of Appl. Math. Sci., (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar

[61]

B. Straughan, Tipping points in Cattaneo-Christov thermohaline convection,, Proc. Roy. Soc. London A, 467 (2011), 7. doi: 10.1098/rspa.2010.0104. Google Scholar

[62]

B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids,, in Encyclopedia of thermal stresses, (2013). Google Scholar

[63]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/3/033001. Google Scholar

[64]

C. Truesdell and R. Toupin, The Classical Field Theories,, article in Handbuch der Physik, (1960). Google Scholar

[65]

C. A. Truesdel and W. Noll, The Non-Linear Field Theories of Mechanics,, Springer, (1992). doi: 10.1115/1.3625229. Google Scholar

[66]

G. Walker, The tipping point of the iceberg,, Nature, 441 (2006), 802. Google Scholar

[67]

D. H. Wall, Global change tipping points: above- and below-ground biotic interactions in a low diversity ecosystem,, Philosophical Transactions of the Royal Society B, 362 (2007), 2291. doi: 10.1098/rstb.2006.1950. Google Scholar

[68]

G. B. Whitham, Linear and Non-Linear Waves,, Wiley, (1974). Google Scholar

[69]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behaviour,, Transportation Research Pat B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

show all references

References:
[1]

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

[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. doi: 10.1007/s11538-005-9034-4. Google Scholar

[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. doi: 10.1016/j.jtbi.2009.07.009. Google Scholar

[4]

A. Aw and M. Rascale, Resurrection of "Second Order'' models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099. Google Scholar

[5]

E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction diffusion model for the hantavirus infection,, Mathematical Methods in the Applied Sciences, 31 (2008), 481. doi: 10.1002/mma.929. Google Scholar

[6]

A. D. Barnosky, E. A. Hadly, J. Bascompte, E. L. Berlow, J. H. Brown, M. Fortelius, W. M. Getz, J. Harte, A. Hastings, P. A. Marquet, N. D. Martinez, A. Mooers, P. Roopnarine, G. Vermeij, J. W. Williams, R. Gillespie, J. Kitzes, C. Marshall, N. Matzke, D. P. Mindell, E. Revilla and A. B. Smith, Approaching a state shift in Earth's biosphere,, Nature, 486 (2012), 52. doi: 10.1038/nature11018. Google Scholar

[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. doi: 10.1142/S0218202508003054. Google Scholar

[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. doi: 10.1137/090746677. Google Scholar

[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. doi: 10.3934/nhm.2011.6.383. Google Scholar

[10]

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

[11]

R. A. Bentley and M. J. O'Brien, Tipping points, animal culture, and social learning,, Current Zoology, 58 (2012), 298. Google Scholar

[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). doi: 10.3389/fpsyg.2012.00569. Google Scholar

[13]

P. J. Chen, Growth and decay of waves in solids,, in Handbuch der Physik, VIa/3 (1973), 303. Google Scholar

[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]

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. doi: 10.1080/01495739.2010.517674. Google Scholar

[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. doi: 10.1016/j.physleta.2005.12.101. Google Scholar

[17]

I. Christov, P. M. Jordan and C. I. Christov, Modelling weakly nonlinear acoustic wave propagation,, Quart. Jl. Mech. Appl. Math., 60 (2007), 473. 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. 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. 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. 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, (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. 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. doi: 10.1016/j.ijnonlinmec.2011.11.007. Google Scholar

[24]

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

[25]

J. A. Foley, Tipping Points in the Tundra,, Science, 310 (2005), 627. 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. 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. 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, (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, (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. 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. 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. doi: 10.1098/rspa.2007.1895. Google Scholar

[34]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Math. Computers Simulation, 80 (2009), 202. 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. 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. 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., 34 (2012), 56. doi: 10.1016/j.euromechflu.2012.01.016. Google Scholar

[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. doi: 10.1098/rspa.2012.0321. Google Scholar

[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. doi: 10.1038/nature06111. Google Scholar

[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. doi: 10.1007/BF00276179. Google Scholar

[41]

R. S. Mani, V. Ravi, A. Desai and S. N. Madhusudana, Emerging Viral Infections in India,, Proceedings of the National Academy of Sciences, 82 (2012), 5. doi: 10.1007/s40011-011-0001-1. Google Scholar

[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. doi: 10.1016/j.mcm.2008.07.037. Google Scholar

[43]

A. Marasco, Second - order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media,, Int. J. Engng. Sci., 47 (2009), 499. doi: 10.1016/j.ijengsci.2008.08.009. Google Scholar

[44]

A. Marasco and A. Romano, On the acceleration waves in second - order elastic, isotropic, compressible, and homogeneous materials,, Mathematical and Computer Modelling, 49 (2009), 1504. doi: 10.1016/j.mcm.2008.06.005. Google Scholar

[45]

V. Méndez and J. Camacho, Dynamics and thermodynamics of delayed population growth,, Physical Review E, 55 (1997), 6476. Google Scholar

[46]

V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations,, Physical Review E, 60 (1999), 5231. doi: 10.1103/PhysRevE.60.5231. Google Scholar

[47]

J. N. Mills, T. L. Yates, T. G. Ksiazek, C. J. Peters and J. E.. Childs, Long-Term studies of hantavirus reservoir populations in the southwestern united states: Rationale, potential, and methods,, Emerging Infectious Diseases, 5 (1999), 95. doi: 10.3201/eid0501.990111. Google Scholar

[48]

A. Miranville, A phase-field model Based on a three-phase-lag heat conduction,, Applied Mathematics and Optimization, 63 (2011), 133. doi: 10.1007/s00245-010-9114-9. Google Scholar

[49]

A. Miranville, On a phase-field model with a logarithmic nonlinearity,, Applications of Mathematics, 57 (2012), 215. doi: 10.1007/s10492-012-0014-y. Google Scholar

[50]

M. Ostoja-Starzewski and J. Trebicki, Stochastic dynamics of acceleration waves in random media,, Mechanics of Materials, 38 (2006), 840. doi: 10.1016/j.mechmat.2005.06.022. Google Scholar

[51]

P. Paoletti, Acceleration waves in complex materials,, Discrete Continuous Dyn. Systems B, 17 (2012), 637. doi: 10.3934/dcdsb.2012.17.637. Google Scholar

[52]

M. Rietkerk, S. C. Dekker, P. C. de Ruiter and J. van de Koppel, Self-Organized patchiness and catastrophic shifts in ecosystems,, Science, 305 (2004), 1926. doi: 10.1126/science.1101867. Google Scholar

[53]

F. Sauvage, M. Langlais, N. G. Yoccoz and D. Pontier, Modelling hantavirus in fluctuating populations of bank voles: The role of indirect transmission on virus persistence,, Journal of Animal Ecology, 72 (2003), 1. doi: 10.1046/j.1365-2656.2003.00675.x. Google Scholar

[54]

M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions,, Nature, 461 (2009), 53. doi: 10.1038/nature08227. Google Scholar

[55]

M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bascompte, W. Brock, V. Dakos, J. van de Koppel, I. A. van de Leemput, S. A. Levin, E. H. van Nes, M. Pascual and J. Vandermeer, Anticipating critical transitions,, Science, 338 (2012), 344. doi: 10.1126/science.1225244. Google Scholar

[56]

C. Schmaljohn and B. Hjelle, Hantaviruses: A global disease problem,, Emerging Infectious Diseases, 3 (1997), 95. doi: 10.3201/eid0302.970202. Google Scholar

[57]

V. D. Sharma and R. Venkatraman, Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows,, Int. J. Non-linear Mech., 47 (2012), 918. doi: 10.1016/j.ijnonlinmec.2012.06.001. Google Scholar

[58]

F. M. F. Simoes, J. A. C. Martins and B. Loret, Instabilities in elastic-plastic fluid-saturated porous media: Harmonic wave versus acceleration wave analyses,, Int. J. Solids Structures, 36 (1999), 1277. doi: 10.1016/S0020-7683(98)00002-X. Google Scholar

[59]

B. Straughan, Stability, and Wave Motion in Porous Media,, volume 165 of Appl. Math. Sci., (2008). Google Scholar

[60]

B. Straughan, Heat Waves,, volume 177 of Appl. Math. Sci., (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar

[61]

B. Straughan, Tipping points in Cattaneo-Christov thermohaline convection,, Proc. Roy. Soc. London A, 467 (2011), 7. doi: 10.1098/rspa.2010.0104. Google Scholar

[62]

B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids,, in Encyclopedia of thermal stresses, (2013). Google Scholar

[63]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/3/033001. Google Scholar

[64]

C. Truesdell and R. Toupin, The Classical Field Theories,, article in Handbuch der Physik, (1960). Google Scholar

[65]

C. A. Truesdel and W. Noll, The Non-Linear Field Theories of Mechanics,, Springer, (1992). doi: 10.1115/1.3625229. Google Scholar

[66]

G. Walker, The tipping point of the iceberg,, Nature, 441 (2006), 802. Google Scholar

[67]

D. H. Wall, Global change tipping points: above- and below-ground biotic interactions in a low diversity ecosystem,, Philosophical Transactions of the Royal Society B, 362 (2007), 2291. doi: 10.1098/rstb.2006.1950. Google Scholar

[68]

G. B. Whitham, Linear and Non-Linear Waves,, Wiley, (1974). Google Scholar

[69]

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behaviour,, Transportation Research Pat B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

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