American Institute of Mathematical Sciences

Advanced
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

Xianlong Fu 1, and  Dongmei Zhu 1,

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.
Keywords: Size-structured population model, $C_0$-semigroup, linearized stability., delayed boundary condition, spectral analysis.
Mathematics Subject Classification: Primary: 34G20, 92D25; Secondary: 34D20, 35F3.
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), 345-376. Google Scholar

[2]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.  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), 1023-1096.  Google Scholar

[4]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200.  Google Scholar

[5]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar

[6]

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

[7]

M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.  Google Scholar

[8]

J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962-969. 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), 119-136.  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), 1087-1103.  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), 249-266.  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), 735-754.  Google Scholar

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Lect. Notes in Math., 1076 (1984), 86-100.  Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  Google Scholar

[15]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832.  Google Scholar

[16]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.  Google Scholar

[17]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. Google Scholar

[18]

T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 2001, 19-36.  Google Scholar

[19]

M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics," Giardini Editori, Pisa, 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), 665-675.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. 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), 61-77.  Google Scholar

[26]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. Google Scholar

[27]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.  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), 901-910.  Google Scholar

[29]

G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Marcell Dekker, New York, 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), 345-376. Google Scholar

[2]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.  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), 1023-1096.  Google Scholar

[4]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200.  Google Scholar

[5]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar

[6]

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

[7]

M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.  Google Scholar

[8]

J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962-969. 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), 119-136.  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), 1087-1103.  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), 249-266.  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), 735-754.  Google Scholar

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Lect. Notes in Math., 1076 (1984), 86-100.  Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  Google Scholar

[15]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832.  Google Scholar

[16]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.  Google Scholar

[17]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. Google Scholar

[18]

T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 2001, 19-36.  Google Scholar

[19]

M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics," Giardini Editori, Pisa, 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), 665-675.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. 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), 61-77.  Google Scholar

[26]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. Google Scholar

[27]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.  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), 901-910.  Google Scholar

[29]

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

[1]

Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391
[2]

Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637
[3]

Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032
[4]

Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021030
[5]

Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015
[6]

Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3111-3133. doi: 10.3934/dcdsb.2020053
[7]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[8]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[9]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048
[10]

Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015
[11]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233
[12]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[13]

József Z. Farkas, Thomas Hagen. Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1825-1839. doi: 10.3934/cpaa.2009.8.1825
[14]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758
[15]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577
[16]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[17]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[18]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719
[19]

José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195
[20]

Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy