June 2018, 7(2): 293-316. doi: 10.3934/eect.2018015

Asymptotic behavior of a hierarchical size-structured population model

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

* Corresponding author: Xianlong Fu

Received  October 2017 Revised  December 2017 Published  May 2018

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11771075), STCSM (No. 18dz2271000) and Shanghai Leading Academic Discipline Project (No. B407)

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by $C_0-{\rm{semigroup}}$ framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

Citation: 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
References:
[1]

A. S. AcklehK. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59. doi: 10.1007/s00245-004-0806-2.

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Equ., 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013.

[3]

A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20. doi: 10.1006/jmaa.2001.6705.

[4]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7.

[5]

M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552. doi: 10.1007/s00028-011-0100-8.

[6]

À. Calsina and J. Saldaña, Asymptotic behavior of a model of hierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987. doi: 10.1007/s002850050085.

[7]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023.

[8]

J. M. Cushing and S. R.-J. Jang, Dynamics of hierarchical models in discrete time, J. Diff. Equ. Appl., 11 (2005), 95-115. doi: 10.1080/10236190512331328343.

[9]

O. DiekmannPh. 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), 1023-1069. doi: 10.1137/060659211.

[10]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations (Eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, 2008, 187-200. doi: 10.1007/978-3-7643-7794-6_12.

[11]

O. DiekmannM. Gyllenberg and J. A. J. Metz, Steady state analysis of structured population models, Theoret. Population Biol., 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8.

[12]

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

[13]

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

[14]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discr. Impuls. Syst. Ser. A, 17 (2010), 639-657.

[15]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032.

[16]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Comm. Pure Appl. Anal., 8 (2009), 1825-1839. doi: 10.3934/cpaa.2009.8.1825.

[17]

G. FragnelliA. 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. doi: 10.3934/dcdsb.2007.7.735.

[18]

X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131. doi: 10.3934/dcdsb.2013.18.109.

[19]

X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model, Discr. Cont. Dyn. Syst. B, 19 (2014), 391-417. doi: 10.3934/dcdsb.2014.19.391.

[20]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100. doi: 10.1007/BFb0072769.

[21]

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

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Mathematics Applied to Science(New Orleans, La., 1986), Academic Press, 1988, 79-105.

[23]

M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467. doi: 10.1016/0022-247X(92)90218-3.

[24]

E. A. Kraev, Existence and uniqueness results for height structured hierarchical population models, Natur. Resource Modeling, 14 (2001), 45-70. doi: 10.1111/j.1939-7445.2001.tb00050.x.

[25]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[26]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. doi: 10.1016/0022-1236(90)90096-4.

[27]

R. NagelG. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100. doi: 10.1080/16073606.1996.9631827.

[28]

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

[29]

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

[30]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.

[31]

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. doi: 10.1007/s00028-004-0159-6.

[32]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683. doi: 10.3934/dcds.1999.5.663.

[33]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. doi: 10.1137/0132040.

[34]

K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. doi: 10.1137/0511080.

[35]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161. doi: 10.1007/BF01303264.

[36]

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

[37]

D. Yan and X. Fu, The asymptotic behavior of an age-cycle structured cell model with delay, J. Dyn. Cont. Syst., 22 (2016), 441-458. doi: 10.1007/s10883-015-9285-4.

[38]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delyaed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655. doi: 10.3934/cpaa.2016.15.637.

show all references

References:
[1]

A. S. AcklehK. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59. doi: 10.1007/s00245-004-0806-2.

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Equ., 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013.

[3]

A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20. doi: 10.1006/jmaa.2001.6705.

[4]

G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7.

[5]

M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552. doi: 10.1007/s00028-011-0100-8.

[6]

À. Calsina and J. Saldaña, Asymptotic behavior of a model of hierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987. doi: 10.1007/s002850050085.

[7]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023.

[8]

J. M. Cushing and S. R.-J. Jang, Dynamics of hierarchical models in discrete time, J. Diff. Equ. Appl., 11 (2005), 95-115. doi: 10.1080/10236190512331328343.

[9]

O. DiekmannPh. 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), 1023-1069. doi: 10.1137/060659211.

[10]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations (Eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, 2008, 187-200. doi: 10.1007/978-3-7643-7794-6_12.

[11]

O. DiekmannM. Gyllenberg and J. A. J. Metz, Steady state analysis of structured population models, Theoret. Population Biol., 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8.

[12]

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

[13]

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

[14]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discr. Impuls. Syst. Ser. A, 17 (2010), 639-657.

[15]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032.

[16]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Comm. Pure Appl. Anal., 8 (2009), 1825-1839. doi: 10.3934/cpaa.2009.8.1825.

[17]

G. FragnelliA. 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. doi: 10.3934/dcdsb.2007.7.735.

[18]

X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131. doi: 10.3934/dcdsb.2013.18.109.

[19]

X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model, Discr. Cont. Dyn. Syst. B, 19 (2014), 391-417. doi: 10.3934/dcdsb.2014.19.391.

[20]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100. doi: 10.1007/BFb0072769.

[21]

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

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Mathematics Applied to Science(New Orleans, La., 1986), Academic Press, 1988, 79-105.

[23]

M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467. doi: 10.1016/0022-247X(92)90218-3.

[24]

E. A. Kraev, Existence and uniqueness results for height structured hierarchical population models, Natur. Resource Modeling, 14 (2001), 45-70. doi: 10.1111/j.1939-7445.2001.tb00050.x.

[25]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[26]

R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. doi: 10.1016/0022-1236(90)90096-4.

[27]

R. NagelG. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100. doi: 10.1080/16073606.1996.9631827.

[28]

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

[29]

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

[30]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.

[31]

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. doi: 10.1007/s00028-004-0159-6.

[32]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683. doi: 10.3934/dcds.1999.5.663.

[33]

K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. doi: 10.1137/0132040.

[34]

K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. doi: 10.1137/0511080.

[35]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161. doi: 10.1007/BF01303264.

[36]

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

[37]

D. Yan and X. Fu, The asymptotic behavior of an age-cycle structured cell model with delay, J. Dyn. Cont. Syst., 22 (2016), 441-458. doi: 10.1007/s10883-015-9285-4.

[38]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delyaed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655. doi: 10.3934/cpaa.2016.15.637.

Figure 1.  Choosing the parameters with $m = 10, ~~T = 120$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Figure 2.  Choosing the parameters with $m = 10, ~~T = 160$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Figure 3.  Choosing the parameter $m = 100, ~~T = 1050$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Figure 4.  Choosing the parameter $m = 80, ~~T = 1200$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Figure 5.  Choosing the parameters with $m = 40$, $T = 100$. The initial condition corresponding to the curve is $u_0(s) = s(16.5-s), ~0\leq s \leq 16.5$, $-1\leq t \leq100$; $u_0(s) = (s-16.5)(39-s), ~16.5\leq s \leq 39$, $-1\leq t \leq100$
[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

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