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
References:
[1]

O. Arino, E. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence,, J. Math. Anal. Appl., 215 (1997), 499. doi: 10.1006/jmaa.1997.5654. Google Scholar

[2]

L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations,, J. Biol. Dynamics, 3 (2009), 196. Google Scholar

[3]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis, and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000). Google Scholar

[4]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28 (1990), 671. doi: 10.1007/BF00160231. Google Scholar

[5]

K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent phases,, in, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar

[6]

K. P. Hadeler, Quiescent phases and stability,, Lin. Alg. Appl., 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar

[7]

K. P. Hadeler, Homogeneous systems with a quiescent phase,, Math. Models Natur. Phenom., 3 (2008), 115. doi: 10.1051/mmnp:2008044. Google Scholar

[8]

K. P. Hadeler, Neutral delay equations from and for population dynamics,, in, 8 (2008). Google Scholar

[9]

K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases,, in, (2010). Google Scholar

[10]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Q., 10 (2002), 473. Google Scholar

[11]

T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar

[12]

W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Math. Biosci., 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar

[13]

M. Kot, "Elements of Mathematical Ecology,'', Cambridge University Press, (2001). Google Scholar

[14]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis,, Forma, 11 (1996), 1. Google Scholar

[15]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. Google Scholar

[16]

A. Maler and F. Lutscher, Cell cycle times and the tumour control probability,, Mathematics in Medicine and Biology, 27 (2010), 313. doi: 10.1093/imammb/dqp024. Google Scholar

[17]

T. Malik and H. Smith, A resource-based model of microbial quiescence,, J. Math. Biol., 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar

[18]

M. G. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theor. Popul. Biol., 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar

[19]

J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441. doi: 10.1006/jdeq.1996.0037. Google Scholar

[20]

E. Pachepsky, F. Lutscher, R. Nisbet and M. A. Lewis, Persistence, spread and the drift paradox,, Theoretical Population Biology, 67 (2005), 61. doi: 10.1016/j.tpb.2004.09.001. Google Scholar

[21]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

[22]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1029. doi: 10.1098/rspa.2006.1806. Google Scholar

show all references

References:
[1]

O. Arino, E. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence,, J. Math. Anal. Appl., 215 (1997), 499. doi: 10.1006/jmaa.1997.5654. Google Scholar

[2]

L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations,, J. Biol. Dynamics, 3 (2009), 196. Google Scholar

[3]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis, and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000). Google Scholar

[4]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28 (1990), 671. doi: 10.1007/BF00160231. Google Scholar

[5]

K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent phases,, in, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar

[6]

K. P. Hadeler, Quiescent phases and stability,, Lin. Alg. Appl., 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar

[7]

K. P. Hadeler, Homogeneous systems with a quiescent phase,, Math. Models Natur. Phenom., 3 (2008), 115. doi: 10.1051/mmnp:2008044. Google Scholar

[8]

K. P. Hadeler, Neutral delay equations from and for population dynamics,, in, 8 (2008). Google Scholar

[9]

K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases,, in, (2010). Google Scholar

[10]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Q., 10 (2002), 473. Google Scholar

[11]

T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar

[12]

W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Math. Biosci., 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar

[13]

M. Kot, "Elements of Mathematical Ecology,'', Cambridge University Press, (2001). Google Scholar

[14]

M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis,, Forma, 11 (1996), 1. Google Scholar

[15]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. Google Scholar

[16]

A. Maler and F. Lutscher, Cell cycle times and the tumour control probability,, Mathematics in Medicine and Biology, 27 (2010), 313. doi: 10.1093/imammb/dqp024. Google Scholar

[17]

T. Malik and H. Smith, A resource-based model of microbial quiescence,, J. Math. Biol., 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar

[18]

M. G. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theor. Popul. Biol., 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar

[19]

J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441. doi: 10.1006/jdeq.1996.0037. Google Scholar

[20]

E. Pachepsky, F. Lutscher, R. Nisbet and M. A. Lewis, Persistence, spread and the drift paradox,, Theoretical Population Biology, 67 (2005), 61. doi: 10.1016/j.tpb.2004.09.001. Google Scholar

[21]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

[22]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1029. doi: 10.1098/rspa.2006.1806. Google Scholar

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