Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Diagonal dominance and harmless off-diagonal delays

Authors: Josef Hofbauer and Joseph W.-H. So
Journal: Proc. Amer. Math. Soc. 128 (2000), 2675-2682
MSC (1991): Primary 34K20
Published electronically: February 28, 2000
MathSciNet review: 1707519
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Systems of linear differential equations with constant coefficients, as well as Lotka-Volterra equations, with delays in the off-diagonal terms are considered. Such systems are shown to be asymptotically stable for any choice of delays if and only if the matrix has a negative weakly dominant diagonal.

References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] Abraham Berman and Robert J. Plemmons, Nonnegative matrices in the mathematical sciences, Classics in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original. MR 1298430
  • [3] L. E. Èl′sgol′ts and S. B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Translated from the Russian by John L. Casti; Mathematics in Science and Engineering, Vol. 105. MR 0352647
  • [4] Miroslav Fiedler, Special matrices and their applications in numerical mathematics, Martinus Nijhoff Publishers, Dordrecht, 1986. Translated from the Czech by Petr Přikryl and Karel Segeth. MR 1105955
  • [5] K. Gopalsamy, Harmless delays in a periodic ecosystem, J. Austral. Math. Soc. Ser. B 25 (1984), no. 3, 349–365. MR 729591, 10.1017/S0334270000004112
  • [6] R. P. Agarwal (ed.), Recent trends in differential equations, World Scientific Series in Applicable Analysis, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1180096
  • [7] Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878
  • [8] Josef Hofbauer and Karl Sigmund, The theory of evolution and dynamical systems, London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge, 1988. Mathematical aspects of selection; Translated from the German. MR 1071180
  • [9] Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
  • [10] Peter Lancaster and Miron Tismenetsky, The theory of matrices, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR 792300
  • [11] Zheng Yi Lu and Yasuhiro Takeuchi, Global dynamic behavior for Lotka-Volterra systems with a reducible interaction matrix, J. Math. Anal. Appl. 193 (1995), no. 2, 559–572. MR 1338722, 10.1006/jmaa.1995.1253
  • [12] Zhengyi Lu and Wendi Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208 (1997), no. 1, 277–280. MR 1440357, 10.1006/jmaa.1997.5301
  • [13] Ray Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Differential Equations 82 (1989), no. 2, 251–268. MR 1027969, 10.1016/0022-0396(89)90133-2

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34K20

Retrieve articles in all journals with MSC (1991): 34K20

Additional Information

Josef Hofbauer
Affiliation: Institut für Mathematik, Universität Wien, A-1090 Wien, Austria

Joseph W.-H. So
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Received by editor(s): October 23, 1998
Published electronically: February 28, 2000
Additional Notes: This research was partially supported by NSERC of Canada, grant number OGP36475.
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society