American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent
  • DCDS-B Home
  • This Issue
  • Next Article
    Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system
August  2016, 21(6): 1975-1998. doi: 10.3934/dcdsb.2016032

Asymptotic analysis of a size-structured cannibalism population model with delayed birth process

Dongxue Yan 1, , Yu Cao 1, and  Xianlong Fu 2,

1. 

Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
2. 

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

Received  November 2014 Revised  February 2016 Published  June 2016

In this paper, we study a size-structured cannibalism model with environment feedback and delayed birth process. Our focus is on the asymptotic behavior of the system, particularly on the effect of cannibalism and time lag on the long-term dynamics. To this end, we formally linearize the system around a steady state and study the linearized system by $C_0$-semigroup framework and spectral analysis methods. These analytical results allow us to achieve linearized stability, instability and asynchronous exponential growth results under some conditions. Finally, some examples are presented and simulated to illustrate the obtained stability conclusions.
Keywords: delayed boundary condition, $C_0$-semigroup, linearized stability, asynchronous exponential growth., cannibalism, Size-structured population model.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: 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
References:
[1]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population,, J. Diff. Eq., 217 (2005), 431.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[2]

M. Boulanouar, The asymptotic behavior of a structured cell population,, J. Evol. Eq., 11 (2011), 531.  doi: 10.1007/s00028-011-0100-8.  Google Scholar

[3]

Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups,, North-Holland, (1987).   Google Scholar

[4]

J. M. Cushing, A size-structured model for cannibalism,, Theoret. Population Biol., 42 (1992), 347.  doi: 10.1016/0040-5809(92)90020-T.  Google Scholar

[5]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279.  doi: 10.1016/0025-5564(79)90073-7.  Google Scholar

[6]

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.  doi: 10.1137/060659211.  Google Scholar

[7]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., (2007), 187.  doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[8]

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

[9]

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

[10]

L. R. Fox, Cannibalism in natural populations,, Annu. Rev. Ecol. Syst., 6 (1975), 87.  doi: 10.1146/annurev.es.06.110175.000511.  Google Scholar

[11]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, J. Math. Anal. Appl., 328 (2007), 119.  doi: 10.1016/j.jmaa.2006.05.032.  Google Scholar

[12]

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.  doi: 10.1080/00036810701545634.  Google Scholar

[13]

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

[14]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback,, Commun. Pure Appl. Anal., 8 (2009), 1825.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[15]

G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 7 (2007), 735.  doi: 10.3934/dcdsb.2007.7.735.  Google Scholar

[16]

X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process,, Discr. Cont. Dyn. Syst. B, 18 (2013), 109.  doi: 10.3934/dcdsb.2013.18.109.  Google Scholar

[17]

X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model,, Discr. Cont. Dyn. Syst. B, 19 (2014), 391.  doi: 10.3934/dcdsb.2014.19.391.  Google Scholar

[18]

M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, J. Math. Anal. Appl., 167 (1992), 443.  doi: 10.1016/0022-247X(92)90218-3.  Google Scholar

[19]

Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism,, J. Math. Biol., 51 (2005), 695.  doi: 10.1007/s00285-005-0342-6.  Google Scholar

[20]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, Lect. Notes in Math., 1076 (1984), 86.  doi: 10.1007/BFb0072769.  Google Scholar

[21]

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

[22]

B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809.  doi: 10.1080/03605308908820630.  Google Scholar

[23]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393.  doi: 10.1006/jmaa.1999.6708.  Google Scholar

[24]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431.  doi: 10.1006/jmaa.2000.7089.  Google Scholar

[25]

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

[26]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291.  doi: 10.1016/0022-1236(90)90096-4.  Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[28]

S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427.  doi: 10.1002/mma.462.  Google Scholar

[29]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, J. Evol. Equ., 5 (2005), 61.  doi: 10.1007/s00028-004-0159-6.  Google Scholar

[30]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484.  doi: 10.1137/0132040.  Google Scholar

[31]

K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, SIAM J. Math. Anal., 11 (1980), 901.  doi: 10.1137/0511080.  Google Scholar

[32]

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

show all references

References:
[1]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population,, J. Diff. Eq., 217 (2005), 431.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[2]

M. Boulanouar, The asymptotic behavior of a structured cell population,, J. Evol. Eq., 11 (2011), 531.  doi: 10.1007/s00028-011-0100-8.  Google Scholar

[3]

Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups,, North-Holland, (1987).   Google Scholar

[4]

J. M. Cushing, A size-structured model for cannibalism,, Theoret. Population Biol., 42 (1992), 347.  doi: 10.1016/0040-5809(92)90020-T.  Google Scholar

[5]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279.  doi: 10.1016/0025-5564(79)90073-7.  Google Scholar

[6]

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.  doi: 10.1137/060659211.  Google Scholar

[7]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., (2007), 187.  doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[8]

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

[9]

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

[10]

L. R. Fox, Cannibalism in natural populations,, Annu. Rev. Ecol. Syst., 6 (1975), 87.  doi: 10.1146/annurev.es.06.110175.000511.  Google Scholar

[11]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, J. Math. Anal. Appl., 328 (2007), 119.  doi: 10.1016/j.jmaa.2006.05.032.  Google Scholar

[12]

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.  doi: 10.1080/00036810701545634.  Google Scholar

[13]

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

[14]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback,, Commun. Pure Appl. Anal., 8 (2009), 1825.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[15]

G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 7 (2007), 735.  doi: 10.3934/dcdsb.2007.7.735.  Google Scholar

[16]

X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process,, Discr. Cont. Dyn. Syst. B, 18 (2013), 109.  doi: 10.3934/dcdsb.2013.18.109.  Google Scholar

[17]

X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model,, Discr. Cont. Dyn. Syst. B, 19 (2014), 391.  doi: 10.3934/dcdsb.2014.19.391.  Google Scholar

[18]

M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, J. Math. Anal. Appl., 167 (1992), 443.  doi: 10.1016/0022-247X(92)90218-3.  Google Scholar

[19]

Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism,, J. Math. Biol., 51 (2005), 695.  doi: 10.1007/s00285-005-0342-6.  Google Scholar

[20]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, Lect. Notes in Math., 1076 (1984), 86.  doi: 10.1007/BFb0072769.  Google Scholar

[21]

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

[22]

B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809.  doi: 10.1080/03605308908820630.  Google Scholar

[23]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393.  doi: 10.1006/jmaa.1999.6708.  Google Scholar

[24]

T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431.  doi: 10.1006/jmaa.2000.7089.  Google Scholar

[25]

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

[26]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291.  doi: 10.1016/0022-1236(90)90096-4.  Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[28]

S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427.  doi: 10.1002/mma.462.  Google Scholar

[29]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, J. Evol. Equ., 5 (2005), 61.  doi: 10.1007/s00028-004-0159-6.  Google Scholar

[30]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484.  doi: 10.1137/0132040.  Google Scholar

[31]

K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, SIAM J. Math. Anal., 11 (1980), 901.  doi: 10.1137/0511080.  Google Scholar

[32]

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

[1]

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
[2]

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
[3]

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
[4]

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
[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 - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[9]

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
[10]

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
[11]

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
[12]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563
[13]

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
[14]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577
[15]

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
[16]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[17]

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
[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]

Fadia Bekkal-Brikci, Khalid Boushaba, Ovide Arino. Nonlinear age structured model with cannibalism. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 201-218. doi: 10.3934/dcdsb.2007.7.201
[20]

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

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences