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Hunting French Ducks in Population Dynamics

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Applied Non-Linear Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 93))

Abstract

Equations with periodic coefficients for singularly perturbed growth can be analysed by using fast and slow timescales in the framework of Fenichel geometric singular perturbation theory and its extensions. The analysis is restricted to one-dimensional time-periodic ordinary differential equations and shows the presence of slow manifolds, canards and the dynamical exchanges between several slow manifolds. There exist permanent (or periodic) canards and periodic solutions containing canards.

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References

  1. Bernadete, D.M., Noonburg, V.W., Pollina, B.: Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation. MAA Monthly 115, 202–219 (2008)

    Google Scholar 

  2. Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. Asymptotic Analysis II. Lect. Notes Math.985, 449–494 (1983)

    Article  MathSciNet  Google Scholar 

  3. Jones, C.K.R.T.: Geometric singular perturbation theory. Dynamical Systems, Lect. Notes Math. Montecatini Terme 1609, 44–118 (1994)

    Google Scholar 

  4. Kaper, T.J.: An introduction to geometric methods and dynamical systems theory for singular perturbation problems. Proc. Symp. Appl. Math. 56 (1999)

    Google Scholar 

  5. Kaper, T.J., Jones, C.K.R.T.: A primer on the exchange lemma for fast-slow systems. IMA Vols. Math. Appl. 122 65–88 (2000)

    Article  MathSciNet  Google Scholar 

  6. Krupa, M., Szmolyan, P.: Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14, 1473–1491 (2001a)

    Article  MathSciNet  MATH  Google Scholar 

  7. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Diff. Equat.174, 312–368 (2001b)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kuznetsov, Y.A., Levitin, V.V.: A multiplatform environment for analysing dynamical systems. Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, Amsterdam, (1997) (http://www.math.uu.nl/people/kuznet/)

  9. Kuznetsov, Y.A., Muratori, S., Rinaldi, S.: Homoclinic bifurcations in slow-fast second order systems. Nonlinear Anal. Theory, Methods Appl. 25, 747–762 (1995)

    Google Scholar 

  10. Neihstadt, A.I.: Asymptotic investigation of the loss of stability by an equilibrium as a pair of imaginary eigenvalues slowly cross the imaginary axis. Usp. Mat. Nauk 40, 190–191 (1985)

    Google Scholar 

  11. Pliss, V.A. Nonlocal problems of the theory of oscillations. Academic Press, NewYork (1966)

    MATH  Google Scholar 

  12. Rinaldi, S., Muratori, S., Kuznetsov, Y.A.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull. Math. Biol. 55, 15–35 (1993)

    Article  MATH  Google Scholar 

  13. Rogovchenko, S.P., Rogovchenko, Y.V.: Effect of periodic environmental fluctuations on the Pearl-Verhulst model. Chaos, Sol. Fract. 39, 1169–1181 (2009)

    Google Scholar 

  14. Sonneveld, P., van Kan, J.: On a conjecture about the periodic solution of the logistic equation. J. Math. Biol. 8, 285–289 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tikhonov, A.N.: Systems of differential equations containing a small parameter multiplying the derivative. Mat. Sb. 31(73), 575–586 (1952)

    Google Scholar 

  16. Verhulst, F.: Methods and Applications of Singular Perturbations. Springer, NewYork (2005)

    Book  MATH  Google Scholar 

  17. Verhulst, F.: Periodic solutions and slow manifolds. Int. J. Bifur. Chaos. 17(8) 2533–2540 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Verhulst, F., Bakri, T.: The dynamics of slow manifolds. J. Indones. Math. Soc. (MIHMI) 13, 73–90 (2007)

    Google Scholar 

  19. Verhulst, P.F.: Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique 10, 113–121 (1838)

    Google Scholar 

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Acknowledgements

Taoufik Bakri kindly introduced me to the use of Content. Comments by Odo Diekmann on the interpretation of the time-varying P.F. Verhulst model are gratefully acknowledged.

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Authors and Affiliations

  1. Mathematisch Instituut, University of Utrecht, 80.010, 3508 TA, Utrecht, The Netherlands

    Ferdinand Verhulst

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  1. Ferdinand Verhulst
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Correspondence to Ferdinand Verhulst .

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Editors and Affiliations

  1. Department of Automation, Biomechanics and Mechatronics, Łódź University of Technology, Łódź, Poland

    Jan Awrejcewicz

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Verhulst, F. (2014). Hunting French Ducks in Population Dynamics. In: Awrejcewicz, J. (eds) Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-319-08266-0_23

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