Global stability of a biological model with time delay
Authors:
Suzanne M. Lenhart and Curtis C. Travis
Journal:
Proc. Amer. Math. Soc. 96 (1986), 7578
MSC:
Primary 34K20; Secondary 92A15
MathSciNet review:
813814
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay.
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 [1]
 H. R. Bailey and M. Z. Williams, Some results on the differential difference equation , J. Math. Anal. Appl. 15 (1966), 569587. MR 0197886 (33:6046)
 [2]
 J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci. 27 (1975), 109117.
 [3]
 S. N. Busenberg and C. C. Travis, On the use of reduciblefunctional differential equations in biological models, J. Math. Anal. Appl. 89 (1982), 4666. MR 672188 (84f:34102)
 [4]
 D. S. Cohen, E. Coutsias, and J. Neu, Stable oscillations in single species growth models with hereditary effects, Math. Biosci. 44 (1979), 255267. MR 532312 (80c:92015)
 [5]
 K. L. Cooke and J. M. Ferreira, Stability conditions for linear retarded functional differential equations, preprint. MR 719331 (84k:34079)
 [6]
 J. M. Cushing, Integrodifferential equations and delay models in population dynamics, Lecture Notes in Biomathematics, no. 20, SpringerVerlag, Berlin, 1977. MR 0496838 (58:15300)
 [7]
 R. Datko, A procedure for determination of the exponential stability of certain differentialdifference equations, Quart. Appl. Math. 36 (1978), 279292. MR 508772 (80c:34080)
 [8]
 D. M. Fargue, Reducibilité des sytèmes hereditaires a des systèmes dynamiques, C. R. Acad. Sci. Paris, Sér. B. 277 (1973), 471473.
 [9]
 J. K. Hale, Sufficient conditions for stability and instability of autonomous functionaldifferential equations, J. Differential Equations 1 (1965), 452482. MR 0183938 (32:1414)
 [10]
 , Theory of functional differential equations, SpringerVerlag, New York, 1977.
 [11]
 J. K. Hale, E. F. Infante, and F. P. Tsen, Stability in linear delay equations, Lefschetz Center for Dynamical Systems, Brown University Report #8223.
 [12]
 N. D. Hayes, Roots of the transcendental equation associated with a certain differentialdifference equation, J. London Math. Soc. (2) 25 (1950), 226232. MR 0036426 (12:106d)
 [13]
 U. S. Koslesov, Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with the age structure taken into account, Math. USSRSb. 45 (1983), 91100.
 [14]
 S. M. Lenhart and C. C. Travis, Stability of functional partial differential equations, J. Differential Equations (to appear). MR 794769 (87g:45011)
 [15]
 A. Mazarov, On the differentialdifference growth equation, Search 4 (1973), 199201.
 [16]
 N. McDonald, Time lags in biological models, Lecture Notes in Biomathematics, no. 27, SpringerVerlag, Berlin, 1978. MR 521439 (80j:92010)
 [17]
 R. V. Plemmons, matrix characterizations. I: Nonsingular matrices, Linear Algebra Appl. 18 (1977), 175188. MR 0444681 (56:3031)
 [18]
 W. M. Post and C. C. Travis, Global stability in ecological models with continuous time delays, Integral and Functional Differential Equations (T. Herdman, H. Stech, and S. Rankin, Eds.), Dekker, New York, 1981, pp. 241249. MR 617054 (82j:92056)
 [19]
 A. WorzBusekros, Global stability in ecological systems with continuous time delays, SIAM J. Appl. Math. 35 (1978), 123134. MR 0490069 (58:9429)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608138143
PII:
S 00029939(1986)08138143
Keywords:
LotkaVolterra delay differential equation,
global stability,
Liapunov function
Article copyright:
© Copyright 1986
American Mathematical Society
