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Hopf Bifurcation for a Maturity Structured Population Dynamic Model

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Abstract

This article is devoted to investigate some dynamical properties of a structured population dynamic model with random walk on (0,+∞). This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and use integrated semigroup theory to study such a partial differential equation. Moreover, a Hopf bifurcation theorem is given for this model.

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References

  • Ackleh, A.S., Deng, K.: A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle. SIAM J. Appl. Math. 69, 1644–1661 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Amann, H.: Hopf bifurcation in quasilinear reaction-diffusion systems. In: Busenberg, S.N., Martelli, M. (eds.) Delay Differential Equations and Dynamical Systems. Lect. Notes Math., vol. 1475, pp. 53–63. Springer, Berlin (1991)

    Chapter  Google Scholar 

  • Arendt, W.: Vector valued Laplace transforms and Cauchy problems. Isr. J. Math. 59, 327–352 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  • Arino, O.: A survey of structured cell population dynamics. Acta Biotheor. 43, 3–25 (1995)

    Article  Google Scholar 

  • Arino, O., Sanchez, E.: A survey of cell population dynamics. J. Theor. Med. 1, 35–51 (1997)

    Article  MATH  Google Scholar 

  • Bellomo, N., Bellouquid, A., Nieto, J., Soler, J.: Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems. Math. Models Methods Appl. Sci. 20, 1179–1207 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Bertoni, S.: Periodic solutions for non-linear equations of structure populations. J. Math. Anal. Appl. 220, 250–267 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Calsina, A., Ripoll, J.: Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers. J. Math. Biol. 45, 22–33 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Calsina, A., Saldana, J.: Global dynamics and optimal life history of a structured population model. SIAM J. Appl. Math. 59, 1667–1685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Calsina, A., Sanchón, M.: Stability and instability of equilibria of an equation of size structured population dynamics. J. Math. Anal. Appl. 286, 435–452 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu, W.M.: Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27, 233–258 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Chalub, F.A., Dolak-Struss, Y., Markowich, P., Oeltz, D., Schmeiser, C., Soref, A.: Model hierarchies for cell aggregation by chemotaxis. Math. Models Methods Appl. Sci. 16, 1173–1198 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Chu, J., Ducrot, A., Magal, P., Ruan, S.: Hopf bifurcation in a size structured population dynamic model with random growth. J. Differ. Equ. 247, 956–1000 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Crandall, M.G., Rabinowitz, P.H.: The Hopf bifurcation theorem in infinite dimensions. Arch. Ration. Mech. Anal. 67, 53–72 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Cushing, J.M.: Model stability and instability in age structured populations. J. Theor. Biol. 86, 709–730 (1980)

    Article  MathSciNet  Google Scholar 

  • Cushing, J.M.: Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics. Comput. Math. Appl. 9, 459–478 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Da Prato, G., Lunardi, A.: Hopf bifurcation for fully nonlinear equations in Banach space. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 315–329 (1986)

    MATH  Google Scholar 

  • Ducrot, A., Liu, Z., Magal, P.: Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems. J. Math. Anal. Appl. 341, 501–518 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Dyson, J., Villella-Bressan, R., Webb, G.F.: The steady state of a maturity structured tumor cord cell population. Discrete Contin. Dyn. Syst., Ser. B 4, 115–134 (2004)

    MathSciNet  MATH  Google Scholar 

  • Dyson, J., Villella-Bressan, R., Webb, G.F.: A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes. Math. Model. Nat. Phenom. 2, 69–100 (2007)

    Article  MathSciNet  Google Scholar 

  • Eftimie, R., de Vries, G., Lewis, M.A.: Modeling group formation and activity patterns in self-organizing collectives of individuals. Bull. Math. Biol. 69, 1537–1566 (2007a)

    Article  MathSciNet  MATH  Google Scholar 

  • Eftimie, R., de Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. USA 104, 6974–6979 (2007b)

    Article  MathSciNet  MATH  Google Scholar 

  • Eftimie, R., de Vries, G., Lewis, M.A.: Weakly nonlinear analysis of a hyperbolic model for animal group formation. J. Math. Biol. 59, 37–74 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Engel, K.-J., Nagel, R.: One Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    MATH  Google Scholar 

  • Guidotti, P., Merino, S.: Hopf bifurcation in a scalar reaction diffusion equation. J. Differ. Equ. 140, 209–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Gyllenberg, M., Webb, G.F.: A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol. 28, 671–694 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Hadeler, K.P.: Reaction transport systems in biological modelling. In: Diekmann, O., et al. (eds.) Mathematics Inspired by Biology. Lect. Notes Math., vol. 1714, pp. 95–150. Springer, Berlin (1999)

    Chapter  Google Scholar 

  • Hillen, T.: Hyperbolic models for chemosensitive movement. Math. Models Methods Appl. Sci. 12, 1007–1034 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Hillen, T., Stevens, A.: Hyperbolic models for chemotaxis in 1-d. Nonlinear Anal., Real World Appl. 1, 409–433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Hillen, T., Rohde, C., Lutscher, F.: Existence of weak solutions for a hyperbolic model for chemosensitive movement. J. Math. Anal. Appl. 260, 173–199 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Inaba, H.: Mathematical analysis for an evolutionary epidemic model. In: Horn, M.A., Simonett, G., Webb, G.F. (eds.) Mathematical Models in Medical and Health Sciences, pp. 213–236. Vanderbilt Univ. Press, Nashville (1998)

    Google Scholar 

  • Inaba, H.: Endemic threshold and stability in an evolutionary epidemic model. In: Castillo-Chavez, C., et al. (eds.) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, pp. 337–359. Springer, New York (2002)

    Google Scholar 

  • Kac, M.: A stochastic model related to the telegrapher’s equation. Rocky Mt. J. Math. 4, 497–509 (1956) (Reprint 1974)

    Article  Google Scholar 

  • Kellermann, H., Hieber, M.: Integrated semigroups. J. Funct. Anal. 84, 160–180 (1989)

    Article  MathSciNet  Google Scholar 

  • Koch, H., Antman, S.S.: Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations. SIAM J. Math. Anal. 32, 360–384 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Kostova, T., Li, J.: Oscillations and stability due to juvenile competitive effects on adult fertility. Comput. Math. Appl. 32, 57–70 (1996)

    Article  MATH  Google Scholar 

  • Liu, Z., Magal, P., Ruan, S.: Hopf bifurcation for non-densely defined Cauchy problems. Z. Angew. Math. Phys. (2010). doi:10.1007/s00033-010-0088-x

    Google Scholar 

  • Magal, P.: Compact attractors for time-periodic age structured population models. Electron. J. Differ. Equ. 2001, 1–35 (2001)

    MathSciNet  Google Scholar 

  • Magal, P., Ruan, S.: On integrated semigroups and age structured models in L p spaces. Differ. Integral Equ. 20, 197–239 (2007)

    MathSciNet  MATH  Google Scholar 

  • Magal, P., Ruan, S.: On semilinear Cauchy problems with non-dense domain. Adv. Differ. Equ. 14, 1041–1084 (2009a)

    MathSciNet  MATH  Google Scholar 

  • Magal, P., Ruan, S.: Center manifold theorem for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models. Mem. Am. Math. Soc. 202(951) (2009b)

  • Magal, P., Ruan, S.: Sustained oscillations in an evolutionary epidemiological model of influenza a drift. Proc. R. Soc. A 466, 965–992 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Prüss, J.: On the qualitative behavior of populations with age-specific interactions. Comput. Math. Appl. 9, 327–339 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Ricker, W.E.: Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623 (1954)

    Article  Google Scholar 

  • Ricker, W.E.: Computation and interpretation of biological studies of fish populations. Bull. Fish. Res. Board Can. 191 (1975)

  • Sandstede, B., Scheel, A.: Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. 39, 2033–2052 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Simonett, G.: Hopf bifurcation and stability for a quasilinear reaction-diffusion system. In: Ferreyra, G., Goldstein, G., Neubrander, F. (eds.) Evolution Equations. Lect. Notes Pure and Appl. Math., vol. 168, pp. 407–418. Dekker, New York (1995)

    Google Scholar 

  • Swart, J.H.: Hopf bifurcation and the stability of non-linear age-dependent population models. Comput. Math. Appl. 15, 555–564 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme, H.R.: Semiflows generated by Lipschitz perturbations of non-densely defined operators. Differ. Integral Equ. 3, 1035–1066 (1990a)

    MathSciNet  MATH  Google Scholar 

  • Thieme, H.R.: “Integrated semigroups” and integrated solutions to abstract Cauchy problems. J. Math. Anal. Appl. 152, 416–447 (1990b)

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme, H.R.: Quasi-compact semigroups via bounded perturbation. In: Arino, O., Axelrod, D., Kimmel, M. (eds.) Advances in Mathematical Population Dynamics: Molecules, Cells and Man, pp. 691–713. World Sci. Publ., River Edge (1997)

    Google Scholar 

  • Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Dekker, New York (1985)

    MATH  Google Scholar 

  • Webb, G.F.: An operator-theoretic formulation of asynchronous exponential growth. Trans. Am. Math. Soc. 303, 155–164 (1987)

    Article  Google Scholar 

  • Webb, G.F.: Population models structured by age, size, and spatial position. In: Magal, P., Ruan, S. (eds.) Structured Population Models in Biology and Epidemiology. Lecture Notes in Math., vol. 1936, pp. 1–49. Springer, Berlin (2008)

    Chapter  Google Scholar 

  • Zhang, P., Feng, Z., Milner, F.: A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies. Math. Biosci. 205, 83–107 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Jixun Chu & Rong Yuan

  2. UMR CNRS 5251 IBM & INRIA sud-ouest Anubis, Université Victor Segalen Bordeaux 2, 3ter place de la victoire, Bat E 2 eme étage, 33000, Bordeaux, France

    Pierre Magal

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  1. Jixun Chu
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  2. Pierre Magal
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  3. Rong Yuan
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Correspondence to Pierre Magal.

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Communicated by P.K. Maini.

Research was partially supported by the French Ministry of Foreign and European Affairs program France–China PFCC EGIDE (20932UL).

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Chu, J., Magal, P. & Yuan, R. Hopf Bifurcation for a Maturity Structured Population Dynamic Model. J Nonlinear Sci 21, 521–562 (2011). https://doi.org/10.1007/s00332-010-9091-9

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  • DOI: https://doi.org/10.1007/s00332-010-9091-9

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