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

S. A. Campbell¹ ; R. Jessop¹

¹ Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

Mathematical modelling of natural phenomena, Tome 4 (2009) no. 2, pp. 1-27.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20094201

Bibliographie

[1] M. Adimy, F. Crauste, S. Ruan SIAM J. Appl. Math. 2005 1328 1352

[2] M. Adimy, F. Crauste, S. Ruan Nonl. Anal.: Real World Appl. 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 Phys. Rev. Lett. 2003 094101

[5] F.M. Atay J. Diff. Eqs. 2006 190 209

[6] F.M. Atay DCDS 2008 197 205

[7] S. Bernard, J. Bélair, M.C. Mackey DCDS 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 Neur. Net. 2002 867 871

[11] Y. Chen IEEE Trans. Circuits Syst.-I 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 J. Math. Anal. Appl. 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 J. Diff. Eqs. 2008 1049 1079

[16] K. Gopalsamy. Stability and oscillations in delay differential equations of population dynamics. Kluwer, Dordrecht, 1992.

[17] K. Gopalsamy, X.-Z. He Physica D 1994 344 358

[18] R.V. Hogg and A.T. Craig. Introduction to mathematical statistics. Prentice Hall, United States, 1995.

[19] G.E. Hutchinson Ann. N.Y. Acad. Sci. 1948 221 246

[20] V.K. Jirsa, M. Ding Phys. Rev. Lett. 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 Physica D 2001 123 141

[24]

[25] N. MacDonald. Biological delay systems: linear stability theory. Cambridge University Press, Cambridge, 1989.

[26] M.C. Mackey J. Math. Biol. 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 Physica D 2004 323 342

[30] A. Thiel, H. Schwegler, C.W. Eurich Complexity 2003 102 108

[31] G.S.K. Wolkowicz, H. Xia, S. Ruan SIAM J. Appl. Math. 1997 1281 1310

[32] G.S.K. Wolkowicz, H. Xia, J. Wu J. Math. Biol. 1999 285 316

[33] P. Yan J. Theoret. Biol. 2008 238 252