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

Bissell, J.J.; Straughan, S.

Abstract

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

Bissell, J., & Straughan, S. (2014). Discontinuity waves as tipping points: Applications to biological & sociological systems. Discrete and Continuous Dynamical Systems - Series B, 19(7), 1911-1934. https://doi.org/10.3934/dcdsb.2014.19.1911

• Journal Article Type: Article
• Publication Date: Sep 1, 2014
• Deposit Date: Aug 19, 2014
• Journal: Discrete and Continuous Dynamical Systems - Series B
• Print ISSN: 1531-3492
• Electronic ISSN: 1553-524X
• Publisher: American Institute of Mathematical Sciences (AIMS)
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 7
• Pages: 1911-1934
• DOI: https://doi.org/10.3934/dcdsb.2014.19.1911
• Keywords: Discontinuity waves, Shocks, Traffic modelling, Crowd dynamics, Hantavirus, SIS epidemic model, Hyperbolic reaction-diffusion equations, `Tipping point'.
• Public URL: https://durham-repository.worktribe.com/output/1455389