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Asymptotic methods in mathematical biology

Part 1 Survey Paper

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Part of the Lecture Notes in Mathematics book series (LNM,volume 985)

Abstract

In this contribution we study asymptotic methods for differential equation models of physiological and ecological phenomena. In a survey of the literature special attention is given to the Hopf bifurcation, almost linear oscillations, relaxation oscillations, nonlinear reaction-diffusion and to the change in stability of an ecological system due to periodic forcing.

Keywords

  • Hopf Bifurcation
  • Singular Perturbation
  • Asymptotic Method
  • Relaxation Oscillator
  • Linear Oscillation

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.

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References

  1. R. ARIS., The mathematical theory of diffusion and reaction in permeable catalysts, vol. I & II, Clavendon Press, Oxford, 1975.

    MATH  Google Scholar 

  2. D.G. ARONSON & H.F. WEINBERGER, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), 33–76.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M. ASHKENAZI & H.G. OTHMER, Spatial patherns in coupled biochemical oscillators, J. Math. Biology 5 (1977), 1–31.

    CrossRef  MathSciNet  Google Scholar 

  4. E.J. BISSET, Interaction of steady and periodic bifurcating modes with imperfection effects in reaction diffusion systems, SIAM J. Appl. Math. 40 (1981), 224–241.

    CrossRef  MathSciNet  Google Scholar 

  5. J.G. BLOM, R. DE BRUIN, J. GRASMAN & J.G. VERWER, Forced prey-predictor oscillations, J. Math. Biology 12 (1981), 141–152.

    CrossRef  MATH  Google Scholar 

  6. N.N. BOGOLUIBOV & Y.A. MITROPOLSKY, Asymptotic Methods in the Theory of Non-linear Oscillations, Hindustan Publ. Corps, New Delhi, 1961.

    Google Scholar 

  7. G. BOJADZIEV, The KBM method applied to models of population dynamics, Bull. Math. Biology 40 (1979), 335–345.

    MathSciNet  Google Scholar 

  8. S.N. BUSENBERG & K.L. COOKE, Differential Equations and Applications in Ecology, Epidemics, and Population Problems, Acad. Press. New York, 1981.

    MATH  Google Scholar 

  9. J. COSTE, J. PEYRAUD & P. COULLET, Asymptotic behavior in the dynamics of competing species, SIAM J. Appl. Math. 36 (1979), 516–543.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. B. CHANCE, E.K. PYE, A.K. GOSH & B. HESS (eds.), Biological and biochemical oscillators, Acad. Press, New York, 1973.

    Google Scholar 

  11. O. COPE, Stability of limit cycle solutions of reaction-diffusion equations, SIAM J. Appl. Math. 38 (1980), 457–479.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. J.M. CUSHING., Periodic time dependent predator-prey systems, SIAM J. Appl. Math. 32 (1977), 82–95.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. J.M. CUSHING., Two species competition in a periodic environment, J. Math. Biology 10 (1980), 385–400.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J.A. DESIMONE & J.A. PENNLINE, A new asymptotic analysis of the n-th order reaction-diffusion problem: analytical and numerical studies, Math. Biosci 40 (1978), 303–318.

    CrossRef  MATH  Google Scholar 

  15. M.R. DUFFY, N.F. BRITTON & J.D. MURRAY, Special wave solutions of practical reaction diffusion systems, 39 (1980), 3–13.

    MathSciNet  Google Scholar 

  16. J.W. ECKHAUS., Asymptotic Analysis of Singular Perturbations, North-Holland Publ. Amsterdam, 1979.

    MATH  Google Scholar 

  17. P.C. FIFE., Singular perturbation and wave front techniques in reaction-diffusion problems, in Proc. of the AMS-SIAM Symposium on Asymptotic Methods, R.E. O’MALLEY (ed.) (1976), 23–50.

    Google Scholar 

  18. P.C. FIFE., Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomath. 28, Springer Verlag Berlin, 1977.

    MATH  Google Scholar 

  19. H.I. FREEDMAN., Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in pure and appl. math., vol. 57, M. Dekker Inc. New York, 1980.

    Google Scholar 

  20. H.I. FREEDMAN & P. WALTMAN, Perturbation of two-dimensional predator-prey equations with an unperturbed critical point, SIAM J. Appl. Math. 29 (1975), 719–733.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. J. GRASMAN & M.J.W. JANSEN, Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology, J. Math. Biology 7 (1979), 171–197.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. J. GRASMAN & E.J.M. VELING, Asymptotic methods for the Volterra-Lotka equations, in Asymptotic Analysis from Theory to Application, F. Verhulst (ed.), Lecture Notes in Math. 711, Springer Verlag Berlin, 1979, 146–157.

    Google Scholar 

  23. J.K. HALE., Remarks on bifurcation theory in differential equations, in New Approaches to Nonlinear Problems in Dynamics, P.J. Holmes (ed.), SIAM Philadelphia, 1980, 379–391.

    Google Scholar 

  24. A.L. HODGKIN & A.F. HUXLEY, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol, 117 (1952), 500–544.

    CrossRef  Google Scholar 

  25. P.J. HOLMES & J.E. MARSDEN, Dynamical systems and invariant manifolds, in New Approaches to Nonlinear Problems in Dynamics, SIAM Philadelphia, 1980, 3–25.

    Google Scholar 

  26. J. HOPFBAUER., On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal. T, M&A 5 (1981) 1003–1007.

    CrossRef  MathSciNet  Google Scholar 

  27. F.C. HOPPENSTEADT (ed), Mathematical aspects of physiology, Lectures in Appl. Math. vol. 19, Am. Math. Soc., Providence, 1981.

    MATH  Google Scholar 

  28. F.C. HOPPENSTEADT (ed)., Nonlinear oscillations in biology, Seminar on Appl. Math. by Am. Math. Soc. and SIAM, AMS Lectures in Appl. Math 17, 1979.

    Google Scholar 

  29. G. IOOSS & D.D. JOSEPH, Elementary stability and bifurcation theory, Springer Verlag, Berlin, 1980.

    CrossRef  MATH  Google Scholar 

  30. J.P. KEENER., Activators and inhibitors in patterns formation, Studies in Appl. Math. 59 (1978) 1,23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. D.A. LINKENS., Analytical solutions of large numbers of mutually coupled nearly sinusoidal oscillators, IEEE Trans. on Circuits and Systems vol. CAS 21 (1974), 294–300.

    CrossRef  Google Scholar 

  32. D.A. LINKENS., Modelling of gastro-intestinal electrical rhythms, in Biological systems, modelling and control, D.A. Linkens (ed.). The Inst. of Electr. Engin., London (1979), 202–241.

    Google Scholar 

  33. M. MIMURA & Y. NISHUIRA, Spatial patterns for an interaction-diffusion equation in morphogenesis, J. Math. Biology 7 (1979), 243–263.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. N. MINORSKY., Nonlinear oscillations, Nostrand, Princeton, 1962.

    MATH  Google Scholar 

  35. E.F. MISHCHENKO., Asymptotic calculation of periodic solutions of systems of differential equations containing small parameters in the derivations, Am. Math. Soc. Transl., series 2, 18 (1961), 199–230.

    CrossRef  MATH  Google Scholar 

  36. P. DE MOTTONI, A SCHIAFFINO, Competition systems with periodic coefficients: a geometric approach, J. Math. Biology 11 (1981), 319–335.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. J.D. MURRAY., Lectures on nonlinear-differential-equation models in biology, Clarendon Press, Oxford, 1977.

    MATH  Google Scholar 

  38. J.C. NEU., Large populations of coupled chemical oscillators, SIAM J. Appl. Math. 38 (1980), 305–316.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. G. NICOLIS & I. PRIGOGINE, Self-organisation in Nonequilibrium Systems, Wiley-Interscience Publ., New York, 1977.

    MATH  Google Scholar 

  40. R.E. O’MALLEY., Introduction to singular perturbation theory, Acad. Press, New York., 1974.

    MATH  Google Scholar 

  41. E. OTT., Strange attractors and chaotic motions of dynamical systems, Rev. Mod. Phys. 53 (1981), 655–671.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. A.S. PERELSON & L.A. SEGEL, A singular perturbation approach to diffusion reaction equations containing a point source with application to the hemolytic plaque assay, J. Math. Biology 6 (1978), 75–85.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. L.S. PONTRYAGIN., Asymptotic behaviour of solutions of systems of differential equations with small parameter in the higher derivatives, Am. Math. Soc. Transl., series 2, 18 (1961), 295–320.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. S. ROSENBLAT., Population models in a periodically fluctuating environment, J. Math. Biology 9 (1980), 23–36.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. K.E. SWINK., A nonlinear model for human population dynamics, SIAM. J. Appl. Math. 40 (1981), 266–278.

    CrossRef  MathSciNet  Google Scholar 

  46. V. TORRE., Synchronisation of non-linear biochemical oscillators coupled by diffusion, Biol. Cybern. 17 (1975), 137–144.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. J.J. TYSON., The Belousov-Zhabotinskii Reaction, Lecture Notes in Biomath. 10, 1976.

    Google Scholar 

  48. B. VAN DER POL & J. VAN DER MARK, The heartbeat considered as a relaxation oscillation and an electrical model of the heart, Phil. Mag. 4, (1928), 763–773.

    Google Scholar 

  49. A.T. WINFREE., The geometry of biological time, Biomathematics vol. 8, Springer Verlag, Berlin, 1980.

    CrossRef  MATH  Google Scholar 

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© 1983 Springer-Verlag

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Grasman, J. (1983). Asymptotic methods in mathematical biology. In: Verhulst, F. (eds) Asymptotic Analysis II —. Lecture Notes in Mathematics, vol 985. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062363

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  • DOI: https://doi.org/10.1007/BFb0062363

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