Skip to main content

Delay equations in biology

Part of the Lecture Notes in Mathematics book series (LNM,volume 730)

Keywords

  • Periodic Solution
  • Lateral Inhibition
  • Delay Equation
  • Population Growth Model
  • Retarded Argument

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alt, W., Some periodicity criteria for functional differential equations, Manuscripta mathematica 23, 295–318 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Andersen, H., and Johnsson, A., Entrainment of geotropic oscillations in hypocotyls of Helianthus annuus — an experimental and theoretical investigation I, II. Phys. Plant. 26, 44–51, 52–61 (1972).

    CrossRef  Google Scholar 

  • Andersen, H., A mathematical model for circum-nutations, Thesis, Report 2/1976, Department of Electrical Measurements, Lund Institute of Technology.

    Google Scholar 

  • Banks, H.T., Delay systems in biological models: approximation techniques. In: V. Lakshmikantham (ed.), Nonlinear systems and applications. Forc. Conf. Arlington 1976, Academic Press, N.Y. (1977).

    Google Scholar 

  • Bellman, R., and Cooke, K.L., Differential-difference equations, Acad. Press, N.Y. (1963).

    MATH  Google Scholar 

  • Browder, F.E., A further generalization of the Schauder fixed-point theorem. Duke Math. J. 32, 575–578 (1965).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Chow, S.-N., Existence of periodic solutions of autonomous functional differential equations, J. Diff. Equ. 15, 350–378 (1974).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Chow, S.-N., and Hale, J., Periodic solutions of autonomous equations. J. Math. Anal. Appl. 66, 495–506 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Theory of delayed lateral inhibition in the compound eye of Limulus, Proceedings Nat. Acad.Sc. 71, 2887–2891 (1974).

    CrossRef  MATH  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Consequences of delayed lateral inhibition in the retina of Limulus I. Elementary theory of spatially uniform fields, J. Theor. Biol. 51, 243–265 (1975).

    CrossRef  MathSciNet  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Consequences of delayed lateral inhibition in the retina of Limulus II. Theory of spatially uniform fields, assuming the 4-point property, J. Theor. Biol. 51, 267–291 (1975).

    CrossRef  MathSciNet  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Periodic solutions of a nonlinear functional equation describing neural action, Istituto Lombardo Acad. Sci. Lett. Rend. A, 109, 91–111 (1975).

    MathSciNet  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Periodic solutions of certain nonlinear integral equations with a time-lag, SIAM J. Appl. Math. 31, 111–120 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Coleman, B.D., and Renninger, G.H., Theory of response of the LIMULUS retina to periodic excitation. J. Math. Biol. 3, 103–120 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cunningham, W.J., A nonlinear differential difference equation of growth, Proc. Nat. Acad. Sci. U.S.A. 40, 709–713 (1954).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cunningham, W.J., and Wangersky, P.I., On time lags in equations of growth. Proc. Acad. Sci. U.S.A. 42, 699–702 (1956).

    CrossRef  MATH  Google Scholar 

  • Cushing, J.M., Bifurcation of periodic oscillations due to delays in single species growth models, J. Math. Biol. 6, 145–161 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cushing, J.M., Periodic solutions of two species interaction models with lags, Math. Biosc. 31, 143–156 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cushing, J.M., Integrodifferential equations and delay models in population dynamics, Lecture Notes in Biomath. 20, Springer-Verlag (1978).

    Google Scholar 

  • Dunkel, G.M., Some mathematical models for population growth with lags. Univ. of Maryland, Technical Note BN-548 (1968).

    Google Scholar 

  • Dunkel, G.M., Single-species model for population growth depending on past history. In: Seminar on Differential Equations and Dynamical Systems, 92–99, Lecture Notes in Mathematics 60, Springer (1968).

    Google Scholar 

  • Grafton, R.B., A periodicity theorem for autonomous functional differential equations, J. Diff. Equ. 87–109 (1969).

    Google Scholar 

  • Grafton, R.B., Periodic solutions of certain Liénard equations with delay, J. Diff. Equ. 11, 519–527 (1972).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hadeler, K.P., On the stability of the stationary state of a population growth model with time-lag, J. Math. Biol. 3, 197–201 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hadeler, K.P., Some aspects of the mathematics of LIMULUS. In:Optimal estimation in approximation theory, edited by Ch.A. Micchelli and Th.J. Rivlin, Plenum Press 1977, p. 241–257.

    Google Scholar 

  • Hadeler, K.P., and Tomiuk, J., Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal. 65, 87–95 (1977).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hadeler, K.P., Periodic solutions of \(\dot x(t) = - f(x(t),{\mathbf{ }}x(t - 1))\) Math. Meth. Appl. Sciences 1, 62–69 (1979).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hassard, B., and Wan, Y.H., Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63, 297–312 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • an der Heiden, U., Stability properties of neural and cellular control systems, Math. Biosciences 31, 275–283 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • an der Heiden, U., Delay-induced biochemical oscillations. In: Bioph. and Bioch. Information Transfer and Recogn. Varna Conf.Proc., 533–540, Plenum Press NY 1978.

    Google Scholar 

  • an der Heiden, U., Chaotisches Verhalten in zellulären Kontrollprozessen. In: Zelluläre Kommunikations-und Kontrollmechanismen (ed. L.Rensing, G.Roth) Universitätsverlag Bremen 1978.

    Google Scholar 

  • an der Heiden, U., Periodic solutions of a non-linear second order differential equation with delay, J.Math.Anal. Appl. to appear

    Google Scholar 

  • an der Heiden, U., Delays in physiological systems. In: Lecture Notes in Biomathematics (ed.R.Berger) Springer Verlag 1979.

    Google Scholar 

  • Israelsson, D., and Johnsson, A., A theory for circumnutations in Helianthus annuus, Physiol. Plant. 20, 957–976 (1967).

    CrossRef  Google Scholar 

  • Johnsson, A., and Israelsson, D., Application ofa theory for circumnutations to geotropic movements, Physiol. Plant. 21, 282–291 (1968).

    CrossRef  Google Scholar 

  • Johnsson, A., and Israelsson, D., Phase-shift in geotropical oscillations—a theoretical and experimental study, Physiol. Plant. 22, 1226–1237 (1969).

    CrossRef  Google Scholar 

  • Johnsson, A., Geotropic responses in Helianthus and the dependence on the auxin ratio.— With a refined mathematical description of the course of geotropic movements. Physiol. Plant. 24, 419–425 (1971).

    CrossRef  Google Scholar 

  • Johnsson, A., Gravitational stimulations inhibit oscillatory growth movements of plants, Z. Naturforsch. 29c, 717–724 (1974).

    Google Scholar 

  • Kaplan, J.L. and Yorke, J.A., Existence and stability of periodic solutions of x′(t)=−f(x(t),x(t−1)), in: Cesari Lamberto (ed.) Dynamical Systems I,II Providence (1974).

    Google Scholar 

  • Kaplan, J.L., Sorg,M., and Yorke, J.A., Solutions of x′(t)=f(x(t),x(t−L)) have limits when f is an order relation., to appear.

    Google Scholar 

  • Kazarinoff, N.D., van den Driessche, P., and Wan, Y.-H., Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, J.Inst.Math.Appl. 21,461–477 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Levin, S.A., and May, R.M., A note on difference-delay equations, Theor. Pop. Biol. 9, 178–187 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Li, T.Y., and Yorke, J.A., Period three implies chaos, Am. Math. Monthly 82, 985–992 (1975).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Mackey, M.C., A unified hypothesis for the origin of aplastic anemia and periodic haematopoiesis, Blood, to appear.

    Google Scholar 

  • Mackey, M.C., and Glass, L., Oscillation and chaos in physiological control systems, Science 197, 287–289 (1977).

    CrossRef  Google Scholar 

  • May, R.M., Conway, G.R., Hassell, M.P., and Southwood, T.R.E., Time delays, density-dependence and single-species oscillations, J.Anim.Ecol. 43, 747–770 (1974).

    CrossRef  Google Scholar 

  • MacDonald, N., Time delay in prey-predator models, Math.Biosc. 28, 321–330 (1976), II. Bifurcation theory Math. Biosc. 33, 227–234 (1977).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • MacDonald, N., Time lag in a model of a biochemical reaction sequence with end product inhibition, J. theor.Biol. 67, 549–556 (1977).

    CrossRef  MathSciNet  Google Scholar 

  • May, R.M., Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).

    CrossRef  Google Scholar 

  • Maynard Smith, J., Mathematical Ideas in Biology, Cambridge University Press, London 1968.

    CrossRef  Google Scholar 

  • Maynard Smith, J., Models in Ecology, Cambridge University Press, London 1974.

    Google Scholar 

  • Nussbaum, R.D., Periodic solutions of some nonlinear autonomous functional differential equations II, J. Diff. Equ. 14, 360–394 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Nussbaum, R.D., Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. 101, 263–306 (1974).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Nussbaum, R.D., A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal. 19, 319–338 (1975).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Pesin, J.B., On the behavior of a strongly nonlinear differential equation with retarded argument. Differentsial'nye uravnenija 10, 1025–1036 (1974).

    MathSciNet  MATH  Google Scholar 

  • Perez, J.F., Malta, C.P., and Coutinho, F.A.B., Qualitative analysis of oscillations in isolated populations of flies, J. Theor. Biol. 71, 505–514 (1978).

    CrossRef  MathSciNet  Google Scholar 

  • Schürer, F., Zur Theorie des Balancierens, Math. Nachr. 1, 295–331 (1948).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Somolinos, A.S., Periodic solutions of the sunflower equation: \(\ddot x + (a/r){\mathbf{ }}\dot x{\mathbf{ }} + {\mathbf{ }}(b/r)\) sin x (t−r)=0, Quart. of Appl.Math. 35, 465–477 (1978).

    MathSciNet  MATH  Google Scholar 

  • Stech, H.W., The effect of time lags on the stability of the equilibrium state of a population growth equation, J. Math. Biol. 5, 115–120 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Walther, H.-O., On a transcendental equation in the stability analysis of a population growth model, J. Math. Biol. 3, 187–195 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Walther, H.-O., Über Ejektivität und periodische Lösungen bei autonomen Funktionaldifferentialgleichungen mit verteilter Verzögerung, Habilitationsschrift München 1977.

    Google Scholar 

  • Walther, H.-O., On instability, ω-limit sets and periodic solutions of nonlinear autonomous delay equations, these proceedings

    Google Scholar 

  • Wehrhahn, C., and Poggio, T., Real-time delayed tracking in flies. Nature 261, 43–44 (1976).

    CrossRef  Google Scholar 

  • Wörz-Busekros, A., Global stability in ecological systems with continuous time delays, SIAM J. Appl. Math. 35, 123–134 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Walther, H.-O., A theorem on the amplitudes of periodic solutions of differential delay equations with application to bifurcation. J.Diff. Equ. 29, 396–404 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1979 Springer-Verlag

About this paper

Cite this paper

Hadeler, K.P. (1979). Delay equations in biology. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064316

Download citation

  • DOI: https://doi.org/10.1007/BFb0064316

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09518-7

  • Online ISBN: 978-3-540-35129-0

  • eBook Packages: Springer Book Archive