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Asymptotic behaviors of a size-structured population model

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Abstract

In this paper we devote ourselves to the study of the asymptotic behavior of a size-structured population dynamics with random diffusion and delayed birth process. Within a semigroup framework, we discuss the local stability and asynchrony respectively for the considered population system under some conditions. We use for our discussion the techniques of operator matrices, Hille-Yosida operators, positivity, spectral analysis as well as Perron-Frobenius theory.

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References

  1. Bátkai A., Piazzera, S. Semigroups and linear partial differential equations with delay. J. Math. Anal. Appl., 264: 1–20 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boulanouar, M. The asymptotic behavior of a structured cell population. J. Evol. Equ., 11: 531–552 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslander, D.M., Oster, G.F., Huffaker, C.B. Dynamics of interacting populations. J. Franklin Inst., 297: 345–376 (1974)

    Article  Google Scholar 

  4. Di Blasio, G. Nonlinear age-dependent population growth with history-dependent birth rate. Math. Biosci., 46: 279–291 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Diekmann, O., Getto, Ph., Gyllenberg, M. Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal., 39: 1023–1096 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diekmann, O., Gyllenberg, M. Abstract delay equations inspired by population dynamics. Fun. Anal. Evol. Eq., 47: 187–200 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Engel, K.J. Operator matrices and systems of evolution equations. RIMS Kokyuroku, 966: 61–80 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Engel, K.J., Nagel, R. A Short Course on Operator Semigroups. Springer, New York, 2006

    MATH  Google Scholar 

  11. Farkas, M. On the stability of stationary age distributions. Appl. Math. Comp., 131: 107–123 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Farkas, J.Z. Stability conditions for a nonlinear size-structured model. Nonl. Anal. (RWA), 6: 962–969 (2005)

    Article  MATH  Google Scholar 

  13. Farkas, J.Z., Hagen, T. Stability and regularity results for a size-structured population model. J. Math. Anal. Appl., 328: 119–136 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Farkas, J.Z., Hagen, T. Asymptotic behavior of size-structured populations via juvenile-adult interaction. Discr. Cont. Dyn. Syst. B, 9: 249–266 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Fragnelli, G., Idrissi, A., Maniar, L. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discr. Cont. Dyn. Syst. B, 7: 735–754 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fu, X., Zhu, D. Stability results for a size-structured population model with delayed birth process. Discr. Cont. Dyn. Syst. B, 18: 109–131 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Greiner, G. A typical Perron-Frobenius theorem with applications to an age-dependent populationequation. Lect. Notes in Math., 1076: 86–100 (1984)

    Article  Google Scholar 

  18. Greiner, G. Perturbing the boundary conditions of a generator. Houston J. Math., 13: 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Guo, B., Chan, W. A semigroup approach to age dependent population dynamics with time delay. Comm. PDEs, 14: 809–832 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gyllenberg, M., Webb, G.F. Asynchronous exponential growth of semigroups of nonlinear operators. J. Math. Anal. Appl., 167: 443–467 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hagen, T. Eigenvalue asymptotics in isothermal forced elongation. J. Math. Anal. Appl., 224: 393–407 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hagen, T., Renardy, M. Eigenvalue asymptotics in nonisothermal elongational flow. J. Math. Anal. Appl., 252: 431–443 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hagen, T., Renardy, M. Studies on the linear equations of melt-spinning of viscous fluids. Diff. Int. Eq., 14: 19–36 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Iannelli, M. Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori, Pisa, 1994

    Google Scholar 

  25. Inaba, H. Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model. J. Math. Biol., 54: 101–146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu Y., He, Z. Stability results for a size-structured population model with resources-dependence and inflow. J. Math. Anal. Appl., 360: 665–675 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  28. Metz, A.J., Diekmann, O. The Dynamics of Psyiologically Structured Populations. Springer, Berlin, 1986

    Book  MATH  Google Scholar 

  29. Nagel, R., (ed.) One-Parameter Semigroups of Positive Operators. Lect. Notes in Math. vol. 1184, Springer-Verlag, Berlin, 1986

  30. Nagel, R. The spectrum of unbounded operator matrices with non-diagonal domain. J. Funct. Anal., 89: 291–302 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nagel, R., Nickel, G., Romanelli, S. Identification of extrapolation spaces for unbounded operators. Quaestiones Math., 19: 83–100 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nickel, G., Rhandi, A. On the essential spectral radius of semigroups generated by perturbations of Hille-Yosida operators. Tuübinger Berichte zur Funktionalanalysis, 4: 207–220 (1994/95)

    Google Scholar 

  33. Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983

    Book  MATH  Google Scholar 

  34. Pizzera, S. An age dependent population equation with delayed birth process. Math. Meth. Appl. Sci., 27: 427–439 (2004)

    Article  MathSciNet  Google Scholar 

  35. Pizzera, S., Tonetto, L. Asynchronous exponential growth for an age dependent population equation with delayed birth process. J. Evol. Equ., 5: 61–77 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rhandi, A., Schnaubelt, R. Asymptotic behaviour of a non-autonomous population equation with diffusion in L1. Discr. Cont. Dyn. Syst., 5: 663–683 (1999)

    Article  MATH  Google Scholar 

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

  38. Swick, K.E. A nonlinear age-dependent model of single species population dynamics. SIAM J. Appl. Math., 32: 484–498 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  39. Swick, K.E. Periodic solutions of a nonlinear age-dependent model of single species population dynamics. SIAM J. Math. Anal., 11: 901–910 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  40. Webb, G.F. Theory of Nonlinear Age-Dependent Population Dynamics. Marcell Dekker, New York, 1985

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

    Xian-long Fu & Qiong Wu

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  1. Xian-long Fu
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  2. Qiong Wu
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Correspondence to Xian-long Fu.

Additional information

Supported by the National Natural Science Foundation of China (No. 11671142 and 11371087), STCSM (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407).

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Fu, Xl., Wu, Q. Asymptotic behaviors of a size-structured population model. Acta Math. Appl. Sin. Engl. Ser. 33, 1025–1042 (2017). https://doi.org/10.1007/s10255-017-0717-7

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  • DOI: https://doi.org/10.1007/s10255-017-0717-7

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