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

Previous Article
Non-sharp travelling waves for a dual porous medium equation
CPAA Home
This Issue
Next Article
Some observations on the Green function for the ball in the fractional Laplace framework

March 2016, 15(2): 637-655. doi: 10.3934/cpaa.2016.15.637

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

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 March 2015 Revised November 2015 Published January 2016

Abstract
Related Papers

This paper is devoted to the study of a spatially and size-structured population dynamics model with delayed birth process. Our focus is on the asymptotic behavior of the system, in particular on the effect of the spatial location and the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results respectively for the considered population system under some conditions. For our discussion, we use the approaches concerning operator matrices, Hille-Yosida operators, spectral analysis as well as Perron-Frobenius theory. We also do two numerical simulations to illustrate the obtained stability and asynchrony results.

Keywords: stability, asynchronous exponential growth., delayed boundary condition, $C_0$-semigroup, Population dynamics.

Mathematics Subject Classification: 34G20, 37G15, 35B32, 35K55, 92D2.

Citation: 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

References:

[1]	D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations,, \emph{J. Franklin Inst.}, 297 (1974), 345. Google Scholar
[2]	A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay,, \emph{J. Math. Anal. Appl.}, 264 (2001), 1. doi: 10.1006/jmaa.2001.6705. Google Scholar
[3]	M. Boulanouar, The asymptotic behavior of a structured cell population,, \emph{J. Evol. Equ.}, 11 (2011), 531. doi: 10.1007/s00028-011-0100-8. Google Scholar
[4]	G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, \emph{Math. Biosci.}, 46 (1979), 279. doi: 10.1016/0025-5564(79)90073-7. Google Scholar
[5]	O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars,, \emph{SIAM J. Math. Anal., 39 (2007), 1023. doi: 10.1137/060659211. Google Scholar
[6]	O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, \emph{Fun. Anal. Evol. Eq.}, 47 (2008), 187. doi: 10.1007/978-3-7643-7794-6_12. Google Scholar
[7]	K. J. Engel, Operator matrices and systems of evolution equations,, \emph{RIMS Kokyuroku}, 966 (1996), 61. Google Scholar
[8]	K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). Google Scholar
[9]	M. Farkas, On the stability of stationary age distributions,, \emph{Appl. Math. Comp.}, 131 (2002), 107. doi: 10.1016/S0096-3003(01)00131-X. Google Scholar
[10]	J. Z. Farkas, Stability conditions for a nonlinear size-structured model,, \emph{Nonl. Anal. (RWA)}, 6 (2005), 962. doi: 10.1016/j.nonrwa.2004.06.002. Google Scholar
[11]	J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, \emph{J. Math. Anal. Appl.}, 328 (2007), 119. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar
[12]	X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 1 (2013), 109. doi: 10.3934/dcdsb.2013.18.109. Google Scholar
[13]	G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, \emph{Lect. Notes in Math.}, 1076 (1984), 86. doi: 10.1007/BFb0072769. Google Scholar
[14]	G. Greiner, Perturbing the boundary conditions of a generator,, \emph{Houston J. Math.}, 13 (1987), 213. Google Scholar
[15]	G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators,, Math. Appl. Sci. (New Orleans, (1986), 79. Google Scholar
[16]	B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, \emph{Comm. PDEs}, 14 (1989), 809. doi: 10.1080/03605308908820630. Google Scholar
[17]	M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, \emph{J. Math. Anal. Appl.}, 167 (1992), 443. doi: 10.1016/0022-247X(92)90218-3. Google Scholar
[18]	T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, \emph{J. Math. Anal. Appl.}, 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. Google Scholar
[19]	T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, \emph{J. Math. Anal. Appl.}, 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. Google Scholar
[20]	T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, \emph{Diff. Int. Eq.}, 14 (2001), 19. Google Scholar
[21]	H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model,, \emph{J. Math. Biol.}, 54 (2007), 101. doi: 10.1007/s00285-006-0033-y. Google Scholar
[22]	Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model,, \emph{Discr. Cont. Dyn. Syst. B}, 20 (2015), 1735. doi: 10.3934/dcdsb.2015.20.1735. Google Scholar
[23]	Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow,, \emph{J. Math. Anal. Appl.}, 360 (2009), 665. doi: 10.1016/j.jmaa.2009.07.005. Google Scholar
[24]	R. Nagel ed., One-Parameter Semigroups of Positive Operators,, Lect. Notes in Math. vol. 1184, (1184). Google Scholar
[25]	R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, \emph{J. Funct. Anal.}, 89 (1990), 291. doi: 10.1016/0022-1236(90)90096-4. Google Scholar
[26]	R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators,, \emph{Quaestiones Math.}, 19 (1996), 83. Google Scholar
[27]	J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics,, Jahrbuch Uberblicke Mathematik, (1994), 74. Google Scholar
[28]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[29]	S. Pizzera, An age dependent population equation with delayed birth process,, \emph{Math. Meth. Appl. Sci.}, 27 (2004), 427. doi: 10.1002/mma.462. Google Scholar
[30]	S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, \emph{J. Evol. Equ.}, 5 (2005), 61. doi: 10.1007/s00028-004-0159-6. Google Scholar
[31]	A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$,, \emph{Discr. Cont. Dyn. Syst.}, 5 (1999), 663. doi: 10.3934/dcds.1999.5.663. Google Scholar
[32]	W. E. Ricker, Computation and interpretation of biological studies of fish populations,, \emph{Bull. Fish. Res. Board Can.}, 191 (1975). Google Scholar
[33]	K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, \emph{SIAM J. Appl. Math.}, 32 (1977), 484. Google Scholar
[34]	K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, \emph{SIAM J. Math. Anal.}, 11 (1980), 901. doi: 10.1137/0511080. Google Scholar
[35]	L. Weis, The stability of positive semigroups on $L_p$ spaces,, Proceedings of the American Mathematical Society, 123 (1995), 3089. doi: 10.2307/2160665. Google Scholar
[36]	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,, \emph{J. Franklin Inst.}, 297 (1974), 345. Google Scholar
[2]	A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay,, \emph{J. Math. Anal. Appl.}, 264 (2001), 1. doi: 10.1006/jmaa.2001.6705. Google Scholar
[3]	M. Boulanouar, The asymptotic behavior of a structured cell population,, \emph{J. Evol. Equ.}, 11 (2011), 531. doi: 10.1007/s00028-011-0100-8. Google Scholar
[4]	G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, \emph{Math. Biosci.}, 46 (1979), 279. doi: 10.1016/0025-5564(79)90073-7. Google Scholar
[5]	O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars,, \emph{SIAM J. Math. Anal., 39 (2007), 1023. doi: 10.1137/060659211. Google Scholar
[6]	O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, \emph{Fun. Anal. Evol. Eq.}, 47 (2008), 187. doi: 10.1007/978-3-7643-7794-6_12. Google Scholar
[7]	K. J. Engel, Operator matrices and systems of evolution equations,, \emph{RIMS Kokyuroku}, 966 (1996), 61. Google Scholar
[8]	K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). Google Scholar
[9]	M. Farkas, On the stability of stationary age distributions,, \emph{Appl. Math. Comp.}, 131 (2002), 107. doi: 10.1016/S0096-3003(01)00131-X. Google Scholar
[10]	J. Z. Farkas, Stability conditions for a nonlinear size-structured model,, \emph{Nonl. Anal. (RWA)}, 6 (2005), 962. doi: 10.1016/j.nonrwa.2004.06.002. Google Scholar
[11]	J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, \emph{J. Math. Anal. Appl.}, 328 (2007), 119. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar
[12]	X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 1 (2013), 109. doi: 10.3934/dcdsb.2013.18.109. Google Scholar
[13]	G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, \emph{Lect. Notes in Math.}, 1076 (1984), 86. doi: 10.1007/BFb0072769. Google Scholar
[14]	G. Greiner, Perturbing the boundary conditions of a generator,, \emph{Houston J. Math.}, 13 (1987), 213. Google Scholar
[15]	G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators,, Math. Appl. Sci. (New Orleans, (1986), 79. Google Scholar
[16]	B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, \emph{Comm. PDEs}, 14 (1989), 809. doi: 10.1080/03605308908820630. Google Scholar
[17]	M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, \emph{J. Math. Anal. Appl.}, 167 (1992), 443. doi: 10.1016/0022-247X(92)90218-3. Google Scholar
[18]	T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, \emph{J. Math. Anal. Appl.}, 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. Google Scholar
[19]	T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, \emph{J. Math. Anal. Appl.}, 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. Google Scholar
[20]	T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, \emph{Diff. Int. Eq.}, 14 (2001), 19. Google Scholar
[21]	H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model,, \emph{J. Math. Biol.}, 54 (2007), 101. doi: 10.1007/s00285-006-0033-y. Google Scholar
[22]	Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model,, \emph{Discr. Cont. Dyn. Syst. B}, 20 (2015), 1735. doi: 10.3934/dcdsb.2015.20.1735. Google Scholar
[23]	Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow,, \emph{J. Math. Anal. Appl.}, 360 (2009), 665. doi: 10.1016/j.jmaa.2009.07.005. Google Scholar
[24]	R. Nagel ed., One-Parameter Semigroups of Positive Operators,, Lect. Notes in Math. vol. 1184, (1184). Google Scholar
[25]	R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, \emph{J. Funct. Anal.}, 89 (1990), 291. doi: 10.1016/0022-1236(90)90096-4. Google Scholar
[26]	R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators,, \emph{Quaestiones Math.}, 19 (1996), 83. Google Scholar
[27]	J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics,, Jahrbuch Uberblicke Mathematik, (1994), 74. Google Scholar
[28]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[29]	S. Pizzera, An age dependent population equation with delayed birth process,, \emph{Math. Meth. Appl. Sci.}, 27 (2004), 427. doi: 10.1002/mma.462. Google Scholar
[30]	S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process,, \emph{J. Evol. Equ.}, 5 (2005), 61. doi: 10.1007/s00028-004-0159-6. Google Scholar
[31]	A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$,, \emph{Discr. Cont. Dyn. Syst.}, 5 (1999), 663. doi: 10.3934/dcds.1999.5.663. Google Scholar
[32]	W. E. Ricker, Computation and interpretation of biological studies of fish populations,, \emph{Bull. Fish. Res. Board Can.}, 191 (1975). Google Scholar
[33]	K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, \emph{SIAM J. Appl. Math.}, 32 (1977), 484. Google Scholar
[34]	K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, \emph{SIAM J. Math. Anal.}, 11 (1980), 901. doi: 10.1137/0511080. Google Scholar
[35]	L. Weis, The stability of positive semigroups on $L_p$ spaces,, Proceedings of the American Mathematical Society, 123 (1995), 3089. doi: 10.2307/2160665. Google Scholar
[36]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcell Dekker, (1985). Google Scholar

[1]	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
[2]	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
[3]	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
[4]	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
[5]	Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
[6]	Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223
[7]	Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481
[8]	Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[9]	Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795
[10]	Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505
[11]	Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[12]	Dong Liang, Jianhong Wu, Fan Zhang. Modelling Population Growth with Delayed Nonlocal Reaction in 2-Dimensions. Mathematical Biosciences & Engineering, 2005, 2 (1) : 111-132. doi: 10.3934/mbe.2005.2.111
[13]	Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.
[14]	Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[15]	Peixuan Weng, Xiao-Qiang Zhao. Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 343-366. doi: 10.3934/dcds.2011.29.343
[16]	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
[17]	Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517
[18]	Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[19]	Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
[20]	Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences