January  2013, 18(1): 109-131. doi: 10.3934/dcdsb.2013.18.109

Stability results for a size-structured population model with delayed birth process

1. 

Department of Mathematics, East China Normal University, Shanghai, 200241, China, China

Received  October 2011 Revised  July 2012 Published  September 2012

In this paper, we discuss the qualitative behavior of an age/size-structured population equation with delay in the birth process. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results allow us to derive linearized stability and instability results under some conditions. The principal stability criterions are given in terms of a modified net reproduction rate. Finally, two examples are presented and simulated to illustrated the obtained conclusions.
Citation: Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
References:
[1]

D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations,, J. Franklin Inst., 297 (1974), 345.   Google Scholar

[2]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279.   Google Scholar

[3]

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), 1023.   Google Scholar

[4]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., 47 (2008), 187.   Google Scholar

[5]

K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61.   Google Scholar

[6]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer, (2000).   Google Scholar

[7]

M. Farkas, On the stability of stationary age distributions,, Appl. Math. Comp., 131 (2002), 107.   Google Scholar

[8]

J. Z. Farkas, Stability conditions for a nonlinear size structured model,, Nonl. Anal. (RWA), 6 (2005), 962.   Google Scholar

[9]

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

[10]

J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow,, Appl. Anal., 86 (2007), 1087.   Google Scholar

[11]

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

[12]

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

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation,, Lect. Notes in Math., 1076 (1984), 86.   Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[15]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809.   Google Scholar

[16]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393.   Google Scholar

[17]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431.   Google Scholar

[18]

T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, Diff. Int. Equ., 14 (2001), 19.   Google Scholar

[19]

M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics,", Giardini Editori, (1994).   Google Scholar

[20]

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

[21]

A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations,", Springer, (1986).   Google Scholar

[22]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291.   Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983).   Google Scholar

[24]

S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427.   Google Scholar

[25]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, J. Evol. Equ., 5 (2005), 61.   Google Scholar

[26]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910.   Google Scholar

[27]

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

[28]

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

[29]

G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics,", Marcell Dekker, (1985).   Google Scholar

show all references

References:
[1]

D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations,, J. Franklin Inst., 297 (1974), 345.   Google Scholar

[2]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279.   Google Scholar

[3]

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), 1023.   Google Scholar

[4]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., 47 (2008), 187.   Google Scholar

[5]

K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61.   Google Scholar

[6]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer, (2000).   Google Scholar

[7]

M. Farkas, On the stability of stationary age distributions,, Appl. Math. Comp., 131 (2002), 107.   Google Scholar

[8]

J. Z. Farkas, Stability conditions for a nonlinear size structured model,, Nonl. Anal. (RWA), 6 (2005), 962.   Google Scholar

[9]

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

[10]

J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow,, Appl. Anal., 86 (2007), 1087.   Google Scholar

[11]

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

[12]

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

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation,, Lect. Notes in Math., 1076 (1984), 86.   Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[15]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809.   Google Scholar

[16]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393.   Google Scholar

[17]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431.   Google Scholar

[18]

T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, Diff. Int. Equ., 14 (2001), 19.   Google Scholar

[19]

M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics,", Giardini Editori, (1994).   Google Scholar

[20]

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

[21]

A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations,", Springer, (1986).   Google Scholar

[22]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291.   Google Scholar

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983).   Google Scholar

[24]

S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427.   Google Scholar

[25]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, J. Evol. Equ., 5 (2005), 61.   Google Scholar

[26]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910.   Google Scholar

[27]

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

[28]

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

[29]

G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics,", Marcell Dekker, (1985).   Google Scholar

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