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
DOI: https://doi.org/10.1090/S0002-9939-00-05564-7
Published electronically: February 28, 2000
MathSciNet review: 1707519
Full-text PDF

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] L. V. Ahlfors, Complex analysis: an introduction to the theory of analytic functions of one complex variable (3rd ed.), McGraw-Hill, New York, 1979. MR 80c:30001
  • [2] A. Berman and R. Plemmons, Nonnegative matrices in the mathematical sciences, SIAM, Philadelphia, 1994. MR 95e:15013
  • [3] L. E. El'sgol'ts and S.B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press, New York, 1973. MR 50:5134
  • [4] M. Fiedler, Special matrices and their applications in numerical mathematics, Martinus Nijhoff Publ. (Kluwer), Dordrecht, 1986. MR 92b:15003
  • [5] K. Gopalsamy, Harmless delays in a periodic ecosystem, J. Austral. Math. Soc. B 25 (1984), 349-365. MR 85c:92031
  • [6] I. Gyori, Stability in a class of integrodifferential systems, Agarwal, R. P. (ed.), Recent trends in differential equations., World Sci. Ser. Appl. Anal. 1, Singapore: World Scientific Publishing., 1992, pp. 269-284. MR 93e:34005
  • [7] J. K. Hale and S.M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993. MR 94m:34169
  • [8] J. Hofbauer, K. Sigmund, The theory of evolution and dynamical systems, Cambridge University Press, Cambridge, 1988. MR 91h:92019
  • [9] Y. Kuang, Delay Differential equations with applications in population dynamics, Academic Press, New York, 1993. MR 94f:34001
  • [10] P. Lancaster and M. Tismenetsky, The theory of matrices (2nd ed.), Academic Press, New York, 1985. MR 87a:15001
  • [11] Z. Lu and Y. Takeuchi, Global dynamic behavior for Lotka-Volterra systems with a reducible interaction matrix, J. Math. Anal. Appl. 193 (1995), 559-572. MR 96f:34071
  • [12] Z. Lu and W. Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208 (1997), 277-280. MR 97m:34152
  • [13] R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Diff. Equations 82 (1989), 251-268. MR 91f:34058

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
Email: Josef.Hofbauer@univie.ac.at

Joseph W.-H. So
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: joseph.so@ualberta.ca

DOI: https://doi.org/10.1090/S0002-9939-00-05564-7
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

American Mathematical Society