Delays in physiological systems

SummaryIn comparison to most physical or chemical systems, biological systems are of extreme complexity. In addition the time needed for transport or processing of chemical components or signals may be of considerable length. Thus temporal delays have to be incorporated into models leading to differential-difference and functional differential equations rather than ordinary differential equations. A number of examples, on different levels of biological organization, demonstrate that delays can have an influence on the qualitative behavior of biological systems: The existence or non-existence of instabilities and periodic or even chaotic oscillations can entirely depend on the presence or absence of delays with appropriate duration.

[1]  U. Heiden.  Periodic solutions of a nonlinear second order differential equation with delay , 1979 .

[2]  Fritz Schürer.  Zur Theorie des Balancierens , 1948 .

[3]  K. P. Hadeler,et al.  Delay equations in biology , 1979 .

[4]  Shui-Nee Chow,et al.  Existence of periodic solutions of autonomous functional differential equations , 1974 .

[5]  R M May,et al.  A note on difference-delay equations. , 1976, Theoretical population biology.

[6]  A. Hatley.  Mathematics in Science and Engineering , Volume 6: Differential- Difference Equations. Richard Bellman and Kenneth L. Cooke. Academic Press, New York and London. 462 pp. 114s. 6d. , 1963, The Journal of the Royal Aeronautical Society.

[7]  Bruce W. Knight,et al.  A Quantitative Description of the Dynamics of Excitation and Inhibition in the Eye of Limulus , 1970, The Journal of general physiology.

[8]  W Reichardt,et al.  Visual control of orientation behaviour in the fly: Part I. A quantitative analysis , 1976, Quarterly Reviews of Biophysics.

[9]  B. D. Coleman,et al.  Theory of the response of the limulus retina to periodic excitation , 1976, Journal of Mathematical Biology.

[10]  John J. Tyson,et al.  Existence of periodic solutions for negative feedback cellular control systems , 1977 .

[11]  B W Knight,et al.  On tuning and amplification by lateral inhibition. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[12]  N. Macdonald,et al.  Time lag in a model of a biochemical reaction sequence with end product inhibition. , 1977, Journal of theoretical biology.

[13]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[14]  W. Alt.  Some periodicity criteria for functional differential equations , 1978 .

[15]  K. Laidler.  The chemical kinetics of enzyme action , 1973 .

[16]  Harvey Thomas Banks,et al.  DELAY SYSTEMS IN BIOLOGICAL MODELS: APPROXIMATION TECHNIQUES , 1977 .

[17]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[18]  K. Hadeler.  Periodic solutions of ẋ(t) = –f(x(t), x(t – 1)) , 1979 .

[19]  Robert M. May,et al.  Time delays, density-dependence and single-species oscillations , 1974 .

[20]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[21]  K. P. Hadeler,et al.  Periodic solutions of difference-differential equations , 1977 .

[22]  James A. Yorke,et al.  On the nonlinear differential delay equation x′(t) = −f(x(t), x(t − 1)) , 1977 .

[23]  B. D. Coleman,et al.  Periodic Solutions of Certain Nonlinear Integral Equations with a Time Lag , 1976 .

[24]  N. Macdonald.  Time lags in biological models , 1978 .

[25]  John J. Tyson,et al.  The Dynamics of Feedback Control Circuits in Biochemical Pathways , 1978 .

[26]  C. WEHRHAHN,et al.  Real-time delayed tracking in flies , 1976, Nature.

[27]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[28]  K. Hadeler.  Some Aspects of the Mathematics of Limulus , 1977 .

[29]  R. D. Driver.  Introduction to the Theory and Application of Differential Equations with Deviating Arguments (L. E. El’sgol’ts and S. B. Norkin) , 1975 .

[30]  H D Landahl,et al.  Some conditions for sustained oscillations in biochemical chains with feedback inhibition. , 1969, The Bulletin of mathematical biophysics.

[31]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[32]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics , 1977 .