doi: 10.3934/eect.2021030
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Long-time behavior of a size-structured population model with diffusion and delayed birth process

1. 

School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

2. 

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

* Corresponding author: Xianlong Fu

Received  November 2020 Revised  March 2021 Early access July 2021

Fund Project: This work is supported by NNSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000), Natural Science Foundation of Jiangsu Province of China (grant No. BK20200749), Nanjing University of Posts and Telecommunications Science Foundation (rant No. NY220093)

This work focuses on the long time behavior for a size-dependent population system with diffusion and Riker type birth function. Some dynamical properties of the considered system is investigated by using $ C_0 $-semigroup theory and spectral analysis arguments. Some sufficient conditions are obtained respectively for asymptotical stability, asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system. In the end several examples and their simulations are also provided to illustrate the achieved results.

Citation: Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations & Control Theory, doi: 10.3934/eect.2021030
References:
[1]

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

[2]

P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250146, 16 pp. doi: 10.1142/S0218127412501465.  Google Scholar

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J. ChuA. DucrotP. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003.  Google Scholar

[5]

K. CookeP. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.  doi: 10.1007/s002850050194.  Google Scholar

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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.  Google Scholar

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O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations, Birkhäuser, Basel, 2008,187–200. doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[8]

A. DucrotP. Magal and and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, J. Appl. Anal. Comput., 1 (2011), 373-395.  doi: 10.11948/2011026.  Google Scholar

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

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

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X. FuZ. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657.  Google Scholar

[12]

X. Fu and Q. Wu, Asymptotic behaviors of a size-structured population model, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 1025-1042.  doi: 10.1007/s10255-017-0717-7.  Google Scholar

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G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, Lect. Notes in Math., 1076 (1984), 86-100.  doi: 10.1007/BFb0072769.  Google Scholar

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

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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

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P. GettoM. GyllenbergY. Nakata and F. Scarabel, Stability analysis of a state dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0.  Google Scholar

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Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[20]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

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P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 965-992.  doi: 10.1098/rspa.2009.0435.  Google Scholar

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R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Vol. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in $L^1$, Discrete Contin. Dynam. Systems, 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.  Google Scholar

[28]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). Google Scholar

[29]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.  Google Scholar

[30]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.  Google Scholar

[31]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.  Google Scholar

[32]

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

[33]

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

[34]

X. Wang, H. Wang and M. Y. Li, $R_0$ and sensitivity analysis of a predator-prey model with seasonality and maturation delay, Math. Biosci., 315 (2019), 108225, 11 pp. doi: 10.1016/j.mbs.2019.108225.  Google Scholar

[35]

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

[36]

D. Yan and X. Fu, Long-time behavior of spatially and size-structured population dynamics with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750032, 23 pp. doi: 10.1142/S0218127417500328.  Google Scholar

[37]

C. Zheng, F. Zhang and J. Li, Stability analysis of a population model with maturation delay and Ricker birth function, Abstr. Appl. Anal., (2014), Art. ID 136707, 1–8. doi: 10.1155/2014/136707.  Google Scholar

show all references

References:
[1]

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

[2]

P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250146, 16 pp. doi: 10.1142/S0218127412501465.  Google Scholar

[3]

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

[4]

J. ChuA. DucrotP. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003.  Google Scholar

[5]

K. CookeP. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.  doi: 10.1007/s002850050194.  Google Scholar

[6]

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.  Google Scholar

[7]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations, Birkhäuser, Basel, 2008,187–200. doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[8]

A. DucrotP. Magal and and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, J. Appl. Anal. Comput., 1 (2011), 373-395.  doi: 10.11948/2011026.  Google Scholar

[9]

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

[10]

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

[11]

X. FuZ. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657.  Google Scholar

[12]

X. Fu and Q. Wu, Asymptotic behaviors of a size-structured population model, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 1025-1042.  doi: 10.1007/s10255-017-0717-7.  Google Scholar

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, Lect. Notes in Math., 1076 (1984), 86-100.  doi: 10.1007/BFb0072769.  Google Scholar

[14]

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

[15]

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

[16]

P. GettoM. GyllenbergY. Nakata and F. Scarabel, Stability analysis of a state dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0.  Google Scholar

[17]

Z. HeD. Ni and S. Wang, Existence and stability of steady states for hierarchical age-structured population models, Electron. J. Differ. Equ., 124 (2019), 1-14.   Google Scholar

[18] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Math. Soc. Lect. Note Ser., Vol. 41, Cambridge Univ. Press, Cambridge-New York, 1981.   Google Scholar
[19]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[20]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[21]

P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 965-992.  doi: 10.1098/rspa.2009.0435.  Google Scholar

[22]

R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Vol. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in $L^1$, Discrete Contin. Dynam. Systems, 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.  Google Scholar

[28]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). Google Scholar

[29]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.  Google Scholar

[30]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.  Google Scholar

[31]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.  Google Scholar

[32]

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

[33]

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

[34]

X. Wang, H. Wang and M. Y. Li, $R_0$ and sensitivity analysis of a predator-prey model with seasonality and maturation delay, Math. Biosci., 315 (2019), 108225, 11 pp. doi: 10.1016/j.mbs.2019.108225.  Google Scholar

[35]

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

[36]

D. Yan and X. Fu, Long-time behavior of spatially and size-structured population dynamics with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750032, 23 pp. doi: 10.1142/S0218127417500328.  Google Scholar

[37]

C. Zheng, F. Zhang and J. Li, Stability analysis of a population model with maturation delay and Ricker birth function, Abstr. Appl. Anal., (2014), Art. ID 136707, 1–8. doi: 10.1155/2014/136707.  Google Scholar

$ b $ and $ c $ are: (a) $ u_0(x) = \frac{1}{1+x^3}-0.01 $; (b) $ u_0(x) = \frac{1}{1+x^3}+0.05 $;">Figure 1.  "a" represents the stationary solution, the initial conditions corresponding to curves $ b $ and $ c $ are: (a) $ u_0(x) = \frac{1}{1+x^3}-0.01 $; (b) $ u_0(x) = \frac{1}{1+x^3}+0.05 $;
Figure 2.  The case of asynchronous exponential growth
Figure 3.  The case of Hopf bifurcation
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