# American Institute of Mathematical Sciences

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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