On the problem of linearization

for state-dependent delay differential equations

Authors:
Kenneth L. Cooke and Wenzhang Huang

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1417-1426

MSC (1991):
Primary 34K20

DOI:
https://doi.org/10.1090/S0002-9939-96-03437-5

MathSciNet review:
1340381

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Abstract | References | Similar Articles | Additional Information

Abstract: The local stability of the equilibrium for a general class of state-dependent delay equations of the form

has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay (except the value of near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay as a constant value at the equilibrium.

**[1]**W. G. Aiello, H. I. Freedman, and J. Wu,*Analysis of a model representing stage-structured population growth with state-dependent time delay*, SIAM J. Appl. Math.**52**, 1992, 855--869. MR**93j:92025****[2]**W. Alt,*Periodic solutions of some autonomous differential equations with variable time delay*, Lecture Notes in Mathematics, Vol. 730, Springer-Verlag, 1979. MR**80i:34118****[3]**J. Belair,*Population models with state-dependent delays*, In Mathematical Population Dynamics (O. Arino, D. E. Axelrod, and M. Kimmel, Eds.), Marcel Dekker, Inc., New York-Basel-Hong Kong, 1991, 165--176.MR**94e:92022****[4]**S. P. Blythe, R. M. Nisbet, and W. S. C. Gurney,*The dynamics of population models with distributed maturation periods*, Theor. Pop. Biology**25**, 1984, 289--311.MR**85h:92030****[5]**K. L. Cooke,*Functional differential equations: some models and perturbation problems*, In Differential Equations and Dynamical Systems (J. K. Hale and J. P. LaSalle, Eds.), Academic Press, New York, 1967. MR**36:5461****[6]**K. L. Cooke and W. Huang,*A theorem of George Seifert and an equation with state-dependent delay*, In Delay and Differential Equations (A. M. Fink, R. K. Miller, and W. Kliemann, Eds.), World Scientific, Singapore, 1992, 65--77. MR**93d:34128****[7]**J. A. Gatica and P. Waltman,*Existence and uniqueness of solutions of a functional differential equation modeling thresholds*, Nonlinear Analysis T.M.A.**8**, 1984, 1215--1222. MR**86h:34080****[8]**J. K. Hale,*Theory of Functional Differential Equations*, Springer-Verlag, New York, 1977. MR**58:22904****[9]**F. C. Hoppensteadt and P. Waltman,*A flow mediated control model of respiration*, In Some Mathematical Questions in Biology, Vol. 12, Lectures on Mathematics in the Life Sciences, 1979, 211--218. MR**82k:92026****[10]**Y. Kuang and H. L. Smith,*Slowly oscillating periodic solutions of autonomous state-dependent delay equations*, IMA Preprint Series #754, Minneapolis, 1990.**[11]**J. A. J. Metz and O. Diekmann,*The Dynamics of Physiologically Structured Populations*, Lecture Notes in Biomathematics 68, Springer-Verlag, 1986. MR**88b:92049****[12]**H. L. Smith,*Threshold delay differential equations are equivalent to FDE's*, preprint.**[13]**H. L. Smith,*Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study*, Math. Biosci. 113 (1993), 1--23. MR**93k:20054**

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Additional Information

**Kenneth L. Cooke**

Affiliation:
Department of Mathematics, Pomona College, Claremont, California 91711

Email:
kcooke@pomona.edu

**Wenzhang Huang**

Affiliation:
Department of Mathematics, University of Alabama in Huntsville, Huntsville, Alabama 35899

Email:
huang@math.uah.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03437-5

Received by editor(s):
December 3, 1993

Additional Notes:
The first author’s research was supported in part by NSF grant DMS 9208818

The second author’s research was supported in part by NSF grant DEB 925370 to Carlos Castillo-Chavez and by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1996
American Mathematical Society