Quiescent phases with distributed exit times

Karl-Peter Hadeler; Frithjof Lutscher

doi:10.3934/dcdsb.2012.17.849

Article Contents

2012, Volume 17, Issue 3: 849-869. Doi: 10.3934/dcdsb.2012.17.849

This issue Previous Article Gravitational and electromagnetic properties of almost standing fields Next Article Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential

Quiescent phases with distributed exit times

Karl-Peter Hadeler^1, and
Frithjof Lutscher^2,

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287
2.
Department of Mathematics, University of Ottawa, Ottawa, ON K1N6N5

Received: September 30, 2010

Revised: April 30, 2011

Published: April 2012

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Abstract

Diffusive coupling of a dynamical system to a quiescent (zero) phase, with the same rates for all variables, stabilizes against oscillations. When the coupling rates are increased then, at a stationary point, the eigenvalues of the Jacobian matrix with positive real parts and large imaginary parts may move towards the imaginary axis of the complex plane and eventually enter the left half-plane. Diffusive coupling means that holding times in the active and in the quiescent phase are exponentially distributed. Here, we ask whether similar phenomena occur if the exponential distributions are replaced by other distributions. A general stability result can be shown for arbitrary distributions, and several more specific results for Gamma distributions and delta peaks (leading to delay equations). Some of the results apply to traveling fronts in reaction diffusion equations with quiescent phase.

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Mathematics Subject Classification: Primary: 34K08, 34K20, 45D05; Secondary: 92B05.

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