May  2012, 17(3): 849-869. doi: 10.3934/dcdsb.2012.17.849

Quiescent phases with distributed exit times

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States

2. 

Department of Mathematics, University of Ottawa, Ottawa, ON K1N6N5, Canada

Received  October 2010 Revised  May 2011 Published  January 2012

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.
Citation: Karl-Peter Hadeler, Frithjof Lutscher. Quiescent phases with distributed exit times. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 849-869. doi: 10.3934/dcdsb.2012.17.849
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show all references

References:
[1]

J. Math. Anal. Appl., 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654.  Google Scholar

[2]

J. Biol. Dynamics, 3 (2009), 196-208.  Google Scholar

[3]

Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ldt., Chichester, 2000.  Google Scholar

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J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.  Google Scholar

[5]

in "Math Everywhere" (eds. G. Aletti, M. Burger, A. Micheletti and D. Morale), Springer, Berlin, (2007), 7-23. doi: 10.1007/978-3-540-44446-6_2.  Google Scholar

[6]

Lin. Alg. Appl., 428 (2008), 1620-1627. doi: 10.1016/j.laa.2007.10.008.  Google Scholar

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Math. Models Natur. Phenom., 3 (2008), 115-125. doi: 10.1051/mmnp:2008044.  Google Scholar

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in "The 8th Colloquium on the Qualitative Theory of Differential Equations," No. 11, Proc. Colloq. Qual. Theory Differ. Equ., 8, Electronic J. Qualitative Theory of Diff. Equ., Szeged, (2008), 18 pp.  Google Scholar

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Math. Biosci., 119 (1994), 225-239. doi: 10.1016/0025-5564(94)90077-9.  Google Scholar

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Cambridge University Press, Cambridge, 2001.  Google Scholar

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[17]

J. Math. Biol., 53 (2006), 231-252. doi: 10.1007/s00285-006-0003-4.  Google Scholar

[18]

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[21]

J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.  Google Scholar

[22]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.  Google Scholar

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