Aggregation in age and space structured population models: an asymptotic analysis approach

Abstract

In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).

References

  1. 1.

    Arendt W., Batty Ch.J.K., Hieber M., Neubrander F. (2001) Vector-valued Laplace transforms and Cauchy problems. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  2. 2.

    Arino O., Sánchez E., Bravodela Parra R., Auger P. (1999) A singular perturbation in an age-structured population model. SIAM Journal on Applied Mathematics, 60(2): 408–436

    Google Scholar 

  3. 3.

    Auger, P., Bravo de la Parra, R., Poggiale, J.-C., Sánchez, E. and Nguyen-Huu, T., Aggregation of Variables and Application to Population Dynamics, in: Magal, P. and Ruan, S. (eds), Structured Population Models in Biology and Epidemiology, LMN 1936, Springer Verlag, Berlin, 2008, 209–263.

  4. 4.

    Banasiak J., Arlotti L. (2006) Perturbations of positive semigroups with applications. Springer, London

    MATH  Google Scholar 

  5. 5.

    Banasiak, J., Asymptotic analysis of singularly perturbed dynamical systems, in: A. Abdulle, J. Banasiak, A. Damlamian and M. Sango (eds), Multiscale Problems in Biomathematics, Physics and Mechanics: Modelling, Analysis and Numerics, GAKUTO Internat. Ser. Math. Sci. Appl. 31, Gakkotosho, Tokyo, 2009, 221–255.

  6. 6.

    Banasiak J., Bobrowski A. (2009) Interplay between degenerate convergence of semigroups and asymptotic analysis. J. Evol. Equ., 9(2): 293–314

    MathSciNet  Article  Google Scholar 

  7. 7.

    Banasiak J., Shindin S. (2010) Chapman-Enskog asymptotic procedure in structured population dynamics. Il Nuovo Cimento C, 33(1): 31–38

    Google Scholar 

  8. 8.

    Bobrowski A. (2005) Functional analysis for probability and stochastic processes. Cambridge University Press, Cambridge

    MATH  Book  Google Scholar 

  9. 9.

    Bravodela Parra R., Arino O., Sánchez E., Auger P. (2000) A model for an age-structured population with two time scales. Math. Comput. Modelling 31(4–5): 17–26

    Google Scholar 

  10. 10.

    Dautrey, R. and Lions J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6, Springer Verlag, Berlin, 2000.

  11. 11.

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

    MATH  Google Scholar 

  12. 12.

    Iannelli, M., Mathematical theory of age-structured population dynamics, Applied Mathematics Monographs 7, Consiglio Nazionale delle Ricerche (C.N.R.), Giardini, Pisa, 1995.

  13. 13.

    Kato T. (1984) Perturbation Theory for Linear Operators, 2nd ed. Springer Verlag, Berlin

    MATH  Google Scholar 

  14. 14.

    Lions P.L., Toscani G. (1997) Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoamericana 13(3): 473–513

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Lisi M., Totaro S. (2005) The Chapman-Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections. Math. Biosci., 196(2): 153–186

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Martin R.H. (1976) Nonlinear Operators & Differential Equations in Banach Spaces. Wiley, New York

    MATH  Google Scholar 

  17. 17.

    MikaJ.R. Banasiak J. (1995) Singularly Perturbed Evolution Equations with Applications in KineticTheory. World Sci., Singapore

    Google Scholar 

  18. 18.

    Seneta, E., Nonnegative matrices and Markov chains 2nd ed., Springer Series in Statistics. Springer-Verlag, New York, 1981.

  19. 19.

    Webb G.F. (1985) Theory of Nonlinear Age-dependent Population Dynamics. Marcel Dekker, New York

    MATH  Google Scholar 

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Correspondence to J. Banasiak.

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The work of all authors was supported by the National Research Foundation of South Africa under grant FA2007030300001 and the University of KwaZulu-Natal Research Fund.

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Banasiak, J., Goswami, A. & Shindin, S. Aggregation in age and space structured population models: an asymptotic analysis approach. J. Evol. Equ. 11, 121–154 (2011). https://doi.org/10.1007/s00028-010-0086-7

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Mathematics Subject Classification (2000)

  • 92D25
  • 35B25
  • 35Q80
  • 47D06
  • 47N20

Keywords

  • Structured population models
  • Aggregation
  • Singular perturbation
  • Asymptotic analysis
  • Semigroups