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Original Articles

A didactical note on the advantage of using two parameters in Hopf bifurcation studies

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Pages 21-30 | Received 16 May 2012, Accepted 14 Dec 2012, Published online: 17 Jan 2013

References

  • M. Adimy, F. Crauste, and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math. 65(4) (2005), pp. 1328–1352. doi: 10.1137/040604698
  • D. Breda, S. Maset, and R. Vermiglio, TRACE-DDE: A Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, Lecture Notes in Control and Information Sciences Vol. 338, Springer-Verlag, Berlin, 2009.
  • D. Breda, O. Diekmann, W.F. de Graaf, A. Pugliese, and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn. 6 (2012), pp. 103–117. doi: 10.1080/17513758.2012.716454
  • H.W. Broer, V. Naudot, R. Roussarie, K. Saleh, and F.O.O. Wagener, Organising centres in the semi-global analysis of dynamical systems, Int. J. Appl. Math. Stat. 12 (2007), pp. 7–36.
  • J.M. Cushing, Bifurcation of periodic solutions of integro-differential equations with applications to time delay models in population dynamics, SIAM J. Appl. Math. 33 (1977), pp. 640–654. doi: 10.1137/0133045
  • J.M. Cushing, Nontrivial periodic solutions of integro-differential equations, J. Integr. Equ. 1 (1979), pp. 165–181.
  • J.M. Cushing, Nontrivial Periodic Solutions of Some Volterra Integral Equations, Lecture Notes in Mathematics Vol. 737, Springer-Verlag, Berlin, 1979.
  • J.M. Cushing, Bifurcation of periodic solutions of nonlinear equations in age-structured population dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982.
  • J.M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl. 9 (1983), pp. 459–478. doi: 10.1016/0898-1221(83)90060-3
  • A.M. de Roos, O. Diekmann, P. Getto, and M.A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, Bull. Math. Biol. 72 (2010), pp. 259–297. doi: 10.1007/s11538-009-9445-3
  • O. Diekmann and M. Gyllenberg, Equations with infinite delay: Blending the abstract and the concrete, J. Differ. Equ. 252 (2012), pp. 819–851. doi: 10.1016/j.jde.2011.09.038
  • O. Diekmann, R.M. Nisbet, W.S.C. Gurney, and F. van den Bosch, Simple mathematical models for cannibalism: A critique and a new approach, Math. Biosci. 78 (1986), pp. 21–46. doi: 10.1016/0025-5564(86)90029-5
  • O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
  • O. Diekmann, Ph. Getto, and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal. 39 (2007), pp. 1023–1069. doi: 10.1137/060659211
  • O. Diekmann, M. Gyllenberg, J.A.J. Metz, S. Nakaoka, and A.M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol. 61 (2010), pp. 277–318. doi: 10.1007/s00285-009-0299-y
  • G. Enciso and E.D. Sontag, On the stability of a model of testosterone dynamics, J. Math. Biol. 49 (2004), pp. 627–634. doi: 10.1007/s00285-004-0291-5
  • T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences Vol. 3, Springer, Berlin, 2009.
  • W.S.C. Gurney, S.P. Blythe, and R.M. Nisbet, Nicholson's blowflies revisited, Nature 287 (1980), pp. 17–21. doi: 10.1038/287017a0
  • J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences Vol. 99, Springer-Verlag, Berlin, 1993.
  • M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Applied Mathematics Monographs Vol. 7, Giardini editori e stampatori, Pisa, Italy, 1995.
  • Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, Berlin, 2004.
  • M.C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood 51 (1978), pp. 941–956.
  • A.J. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool. 2 (1954), pp. 9–65. doi: 10.1071/ZO9540009
  • A.J. Nicholson, The self-adjustment of populations to change, Cold Spring Harb. Symp. Quant. Biol. 22 (1957), pp. 153–173. doi: 10.1101/SQB.1957.022.01.017
  • L. Pujo-Menjouet and M.C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biol. 3 (2004), pp. 235–244. doi: 10.1016/j.crvi.2003.05.004
  • L. Pujo-Menjouet, S. Bernard, and M.C. Mackey, Long period oscillations in a g0 model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst. 4 (2005), pp. 312–332. doi: 10.1137/030600473
  • P.L. Simon, H. Farkas, and M. Wittmann, Constructing global bifurcation diagrams by the parametric representation method, J. Comput. Appl. Math. 108 (1999), pp. 157–176. doi: 10.1016/S0377-0427(99)00108-9
  • H.L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics Vol. 57, Springer, Berlin, 2011.
  • G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Equations, Longman Scientific and Technical, Essex, UK, 1989.