Approximating the Stability Region for a Differential Equation with a Distributed Delay

S. A. Campbell; R. Jessop

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 2, page 1-27
  • ISSN: 0973-5348

Abstract

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We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.

How to cite

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Campbell, S. A., and Jessop, R.. "Approximating the Stability Region for a Differential Equation with a Distributed Delay." Mathematical Modelling of Natural Phenomena 4.2 (2009): 1-27. <http://eudml.org/doc/222208>.

@article{Campbell2009,
abstract = { We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used. },
author = {Campbell, S. A., Jessop, R.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {delay differential equations; distributed delay; linear stability; delay independent stability; delay independent stability; approximation of stability region},
language = {eng},
month = {3},
number = {2},
pages = {1-27},
publisher = {EDP Sciences},
title = {Approximating the Stability Region for a Differential Equation with a Distributed Delay},
url = {http://eudml.org/doc/222208},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Campbell, S. A.
AU - Jessop, R.
TI - Approximating the Stability Region for a Differential Equation with a Distributed Delay
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/3//
PB - EDP Sciences
VL - 4
IS - 2
SP - 1
EP - 27
AB - We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.
LA - eng
KW - delay differential equations; distributed delay; linear stability; delay independent stability; delay independent stability; approximation of stability region
UR - http://eudml.org/doc/222208
ER -

References

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  1. M. Adimy, F. Crauste, S. Ruan. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM J. Appl. Math., 65 (2005), 1328–1352.  
  2. M. Adimy, F. Crauste, S. Ruan. Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics. Nonl. Anal.: Real World Appl., 6 (2005), 651–670.  
  3. J. Arino, P. van den Driessche. Time delays in epidemic models: modeling and numerical considerations, in Delay differential equations and applications, chapter 13, 539–558. Springer, Dordrecht, 2006.  
  4. F.M. Atay. Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett., 91 (2003), 094101.  
  5. F.M. Atay. Oscillator death in coupled functional differential equations near Hopf bifurcation. J. Diff. Eqs., 221 (2006), 190–209.  
  6. F.M. Atay. Delayed feedback control near Hopf bifurcation. DCDS, 1 (2008), 197–205.  
  7. S. Bernard, J. Bélair, M.C. Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. DCDS, 1B (2001), 233–256.  
  8. F. Brauer, C. Castillo-Chávez. Mathematical models in population biology and epidemiology. Springer, New York, 2001.  
  9. S.A. Campbell, I. Ncube. Some effects of gamma distribution on the dynamics of a scalar delay differential equation. Preprint, (2009).  
  10. Y. Chen. Global stability of neural networks with distributed delays. Neur. Net., 15 (2002), 867–871.  
  11. Y. Chen. Global stability of delayed Cohen-Grossberg neural networks. IEEE Trans. Circuits Syst.-I, 53 (2006), 351–357.  
  12. R.V. Churchill, J.W. Brown. Complex variables and applications. McGraw-Hill, New York, 1984.  
  13. K.L. Cooke, Z. Grossman. Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl., 86 (1982), 592–627.  
  14. J.M. Cushing. Integrodifferential equations and delay models in population dynamics, Vol. 20 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin, New York, 1977.  
  15. T. Faria, J.J. Oliveira. Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. J. Diff. Eqs., 244 (2008), 1049–1079.  
  16. K. Gopalsamy. Stability and oscillations in delay differential equations of population dynamics. Kluwer, Dordrecht, 1992.  
  17. K. Gopalsamy and X.-Z. He. Stability in asymmetric Hopfield nets with transmission delays. Physica D, 76 (1994), 344–358.  
  18. R.V. Hogg and A.T. Craig. Introduction to mathematical statistics. Prentice Hall, United States, 1995.  
  19. G.E. Hutchinson. Circular cause systems in ecology. Ann. N.Y. Acad. Sci., 50 (1948), 221–246.  
  20. V.K. Jirsa, M. Ding. Will a large complex system with delays be stable?. Phys. Rev. Lett., 93 (2004), 070602.  
  21. K. Koch. Biophysics of computation: information processing in single neurons. Oxford University Press, New York, 1999.  
  22. Y. Kuang. Delay differential equations: with applications in population dynamics, Vol. 191 of Mathematics in Science and Engineering. Academic Press, New York, 1993.  
  23. X. Liao, K.-W. Wong, Z. Wu. Bifurcation analysis on a two-neuron system with distributed delays. Physica D, 149 (2001), 123–141.  
  24. N. MacDonald. Time lags in biological models, Vol. 27 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin; New York, 1978.  
  25. N. MacDonald. Biological delay systems: linear stability theory. Cambridge University Press, Cambridge, 1989.  
  26. M.C. Mackey, U. an der Heiden. The dynamics of recurrent inhibition. J. Math. Biol., 19 (1984), 211–225.  
  27. J.M. Milton. Dynamics of small neural populations, Vol. 7 of CRM monograph series. American Mathematical Society, Providence, 1996.  
  28. S. Ruan. Delay differential equations for single species dynamics, in Delay differential equations and applications, chapter 11, 477–515. Springer, Dordrecht, 2006.  
  29. S. Ruan, R.S. Filfil. Dynamics of a two-neuron system with discrete and distributed delays. Physica D, 191 (2004), 323–342.  
  30. A. Thiel, H. Schwegler, C.W. Eurich. Complex dynamics is abolished in delayed recurrent systems with distributed feedback times. Complexity, 8 (2003), 102–108.  
  31. G.S.K. Wolkowicz, H. Xia, S. Ruan. Competition in the chemostat: A distributed delay model and its global asymptotic behaviour. SIAM J. Appl. Math., 57 (1997), 1281–1310.  
  32. G.S.K. Wolkowicz, H. Xia, J. Wu. Global dynamics of a chemostat competition model with distributed delay. J. Math. Biol., 38 (1999), 285–316.  
  33. P. Yan. Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks. J. Theoret. Biol., 251 (2008), 238–252.  

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