Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T03:31:27.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating the Stability Region for a Differential Equationwith a Distributed Delay

Published online by Cambridge University Press:  26 March 2009

S. A. Campbell  and
R. Jessop
S. A. Campbell*
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada
R. Jessop
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada
*
sacampbell@uwaterloo.ca
Get access

Abstract

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

Keywords

delay differential equationsdistributed delaylinear stabilitydelay independent stability
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 1 - 27
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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