Asymptotic behavior of a hierarchical size-structured population model

Dongxue Yan; Xianlong Fu

doi:10.3934/eect.2018015

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

Asymptotic behavior of a hierarchical size-structured population model

Dongxue Yan and Xianlong Fu ^,

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.

Keywords: Hierarchical size-structured population, delayed boundary condition, C₀-semigroup, linearized stability, asynchronous exponential growth.

Mathematics Subject Classification: Primary: 35F30, 92D25; Secondary: 34D20.

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. Ackleh, K. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. Google Scholar
[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. Google Scholar
[9]	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), 1023-1069. doi: 10.1137/060659211. Google Scholar
[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. Google Scholar
[11]	O. Diekmann, M. 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. Google Scholar
[12]	K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. Google Scholar
[13]	K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	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. doi: 10.3934/dcdsb.2007.7.735. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. Google Scholar
[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. Google Scholar
[27]	R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100. doi: 10.1080/16073606.1996.9631827. Google Scholar
[28]	R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. 1184, Springer-Verlag, 1986. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[36]	G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	A. S. Ackleh, K. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. Google Scholar
[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. Google Scholar
[9]	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), 1023-1069. doi: 10.1137/060659211. Google Scholar
[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. Google Scholar
[11]	O. Diekmann, M. 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. Google Scholar
[12]	K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. Google Scholar
[13]	K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	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. doi: 10.3934/dcdsb.2007.7.735. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. Google Scholar
[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. Google Scholar
[27]	R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100. doi: 10.1080/16073606.1996.9631827. Google Scholar
[28]	R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. 1184, Springer-Verlag, 1986. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[36]	G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar

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) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$

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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) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$

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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) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$

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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) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$

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Figure 5. Choosing the parameters with $m = 40$, $T = 100$. The initial condition corresponding to the curve is $u_0(s) = s(16.5-s), ~0\leq s \leq 16.5$, $-1\leq t \leq100$; $u_0(s) = (s-16.5)(39-s), ~16.5\leq s \leq 39$, $-1\leq t \leq100$

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