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August  2020, 25(8): 3111-3133. doi: 10.3934/dcdsb.2020053

On a non-linear size-structured population model

1. 

School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

2. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author: Yunfei Lv

Received  April 2019 Revised  October 2019 Published  February 2020

Fund Project: This research was partially supported by the National Natural Science Foundation of China (Nos. 11871371, 11971023, 11771044)

This paper deals with a size-structured population model consisting of a quasi-linear first-order partial differential equation with nonlinear boundary condition. The existence and uniqueness of solutions are firstly obtained by transforming the system into an equivalent integral equation such that the corresponding integral operator forms a contraction. Furthermore, the existence of global attractor is established by proving the asymptotic smoothness and eventual compactness of the nonlinear semigroup associated with the solutions. Finally, we discuss the uniform persistence and existence of compact attractor contained inside the uniformly persistent set.

Citation: 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
References:
[1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[3]

W. Aiello and H. Freedman, A time-delay model of a single species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.  Google Scholar

[4]

W. AielloH. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.  Google Scholar

[5] H. Andrewartha and L. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954.   Google Scholar
[6]

O. ArinoM. Hbid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: Density-dependence effects, Math. Biosci., 150 (1998), 1-20.  doi: 10.1016/S0025-5564(98)00008-X.  Google Scholar

[7]

M. Banerjee and Y. Takeuchi, Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models, J. Theor. Biol., 412 (2017), 154-171.  doi: 10.1016/j.jtbi.2016.10.016.  Google Scholar

[8]

N. Blakley, Life history signigicance of size-triggered metamorphosis in milkweed bugs (Oncopeltus), Ecology, 62 (1981), 57-64.   Google Scholar

[9]

C. Browne and S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[10]

V. Callier and H. Nijhout, Control of body size by oxygen supply reveals size-dependent and size-independent mechanisms of molting and metamorphosis, Proc. Natl Acad. Sci. USA, 108 (2011), 14664-14669.  doi: 10.1073/pnas.1106556108.  Google Scholar

[11]

J. FangS. Gourley and Y. Lou, Stage-structured models of intra-and inter-specific competition within age classes, J. Differ. Equations, 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048.  Google Scholar

[12]

S. GourleyR. Liu and Y. Lou, Intra-specific competition and insect larval development: A model with time-dependent delay, P. Roy. Soc. Edinb. A., 147A (2017), 353-369.  doi: 10.1017/S0308210516000159.  Google Scholar

[13]

J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr., 25, Am. Math. Soc., Providence, RI, 1988.  Google Scholar

[14]

J. Hale and P. Waltman, Persistence in infinite-dimensional systens, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[15]

M. Hardstone and T. Andreadis, Weak larval competition between the invasive mosquito Aedes japonicus (Diptera: Culicidae) and tree resident containerinhaviting mosquitoes in the lavoratory, J. Med. Entomol, 49 (2012), 277-285.   Google Scholar

[16]

K. LiuY. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Differ. Equations, 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar

[17]

L. LiuJ. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Analysis: RWA, 24 (2015), 18-35.  doi: 10.1016/j.nonrwa.2015.01.001.  Google Scholar

[18]

Y. Lou and X. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[19]

Y. Lv and R. Yuan, Global stability and wavefronts in a cooperation model with state-dependent time delay, J. Math. Anal. Appl., 415 (2014), 543-573.  doi: 10.1016/j.jmaa.2014.01.086.  Google Scholar

[20]

Y. LvR. Yuan and Y. He, Wavefronts of a stage structured model with state-dependent delay, Discret. Contin. Dyn. Syst. Ser. A, 35 (2015), 4931-4954.  doi: 10.3934/dcds.2015.35.4931.  Google Scholar

[21]

Y. LvR. Yuan and Y. Pei, Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equations, 260 (2016), 6201-6231.  doi: 10.1016/j.jde.2015.12.037.  Google Scholar

[22]

Y. LvR. YuanY. Pei and T. Li, Global stability of a competitive model with state-dependent delay, J. Dyn. Diff. Equat., 29 (2017), 501-521.  doi: 10.1007/s10884-015-9475-5.  Google Scholar

[23]

Y. LvY. Pei and R. Yuan, Modeling and analysis of a predator-prey model with state-dependent delay, Int. J. Biomath., 11 (2018), 1-22.  doi: 10.1142/S1793524518500262.  Google Scholar

[24]

Y. LvY. Pei and R. Yuan, Principle of linearized stability and instability for parabolic partial differential equations with state-dependent delay, J. Differ. Equations, 267 (2019), 1671-1704.  doi: 10.1016/j.jde.2019.02.014.  Google Scholar

[25]

P. Magal and X. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[26]

P. MagalC. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[27]

J. MahaffyJ. Bélair and M. Mackey, Hematoppietic model with moving boundary condition and state dependent delay: Applications, J. Theor. Biol., 190 (1998), 135-146.   Google Scholar

[28]

T. Malthus, An Essay on the Principle of Population, Oxford World's Classic reprint, 1798. Google Scholar

[29]

E. McCauleyW. Murdoch and R. Nisbet, Growth, reproduction, and mortality of daphnia pulex leydig: Life at low food, Functional Ecology, 4 (1990), 505-514.  doi: 10.2307/2389318.  Google Scholar

[30]

A. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1926), 98-130.  doi: 10.1017/S0013091500034428.  Google Scholar

[31]

K. Noor-E Jannat and B. Roitverg, Effects of larval density and feeding rates on larval life history traits in Anophelets gambiae s.s (Diptera: Culicidae), J. Vector Ecology, 38 (2013), 120-126.   Google Scholar

[32]

H. Smith, Hopf bifurcation in a system of functional equations modeling the spread of an infectious disease, SIAM J. Appl. Math., 43 (1983), 370-385.  doi: 10.1137/0143025.  Google Scholar

[33]

H. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study, Math. Biosci., 113 (1993), 1-23.  doi: 10.1016/0025-5564(93)90006-V.  Google Scholar

[34]

H. Smith, A structured population model and a related functional differential equation: Global attractors and uniform persistence, J. Dyn. Diff. Equat., 6 (1994), 71-99.  doi: 10.1007/BF02219189.  Google Scholar

[35]

H. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mt. J. Math., 24 (1994), 311-334.  doi: 10.1216/rmjm/1181072468.  Google Scholar

[36]

H. Thieme, Semifows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3 (1990), 1035-1066.   Google Scholar

[37]

P. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121.   Google Scholar

[38]

H. Von Förster, Some remarks on changing populations, In The Kinetics of Cellular Proliferation (ed. F. Stohlman Jr) Grune and Stratton, New York, (1959), 328–407. Google Scholar

[39]

G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.  Google Scholar

show all references

References:
[1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[3]

W. Aiello and H. Freedman, A time-delay model of a single species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.  Google Scholar

[4]

W. AielloH. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.  Google Scholar

[5] H. Andrewartha and L. Birch, The Distribution and Abundance of Animals, University of Chicago Press, Chicago, IL, 1954.   Google Scholar
[6]

O. ArinoM. Hbid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: Density-dependence effects, Math. Biosci., 150 (1998), 1-20.  doi: 10.1016/S0025-5564(98)00008-X.  Google Scholar

[7]

M. Banerjee and Y. Takeuchi, Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models, J. Theor. Biol., 412 (2017), 154-171.  doi: 10.1016/j.jtbi.2016.10.016.  Google Scholar

[8]

N. Blakley, Life history signigicance of size-triggered metamorphosis in milkweed bugs (Oncopeltus), Ecology, 62 (1981), 57-64.   Google Scholar

[9]

C. Browne and S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[10]

V. Callier and H. Nijhout, Control of body size by oxygen supply reveals size-dependent and size-independent mechanisms of molting and metamorphosis, Proc. Natl Acad. Sci. USA, 108 (2011), 14664-14669.  doi: 10.1073/pnas.1106556108.  Google Scholar

[11]

J. FangS. Gourley and Y. Lou, Stage-structured models of intra-and inter-specific competition within age classes, J. Differ. Equations, 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048.  Google Scholar

[12]

S. GourleyR. Liu and Y. Lou, Intra-specific competition and insect larval development: A model with time-dependent delay, P. Roy. Soc. Edinb. A., 147A (2017), 353-369.  doi: 10.1017/S0308210516000159.  Google Scholar

[13]

J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr., 25, Am. Math. Soc., Providence, RI, 1988.  Google Scholar

[14]

J. Hale and P. Waltman, Persistence in infinite-dimensional systens, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[15]

M. Hardstone and T. Andreadis, Weak larval competition between the invasive mosquito Aedes japonicus (Diptera: Culicidae) and tree resident containerinhaviting mosquitoes in the lavoratory, J. Med. Entomol, 49 (2012), 277-285.   Google Scholar

[16]

K. LiuY. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Differ. Equations, 263 (2017), 2078-2112.  doi: 10.1016/j.jde.2017.03.038.  Google Scholar

[17]

L. LiuJ. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Analysis: RWA, 24 (2015), 18-35.  doi: 10.1016/j.nonrwa.2015.01.001.  Google Scholar

[18]

Y. Lou and X. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[19]

Y. Lv and R. Yuan, Global stability and wavefronts in a cooperation model with state-dependent time delay, J. Math. Anal. Appl., 415 (2014), 543-573.  doi: 10.1016/j.jmaa.2014.01.086.  Google Scholar

[20]

Y. LvR. Yuan and Y. He, Wavefronts of a stage structured model with state-dependent delay, Discret. Contin. Dyn. Syst. Ser. A, 35 (2015), 4931-4954.  doi: 10.3934/dcds.2015.35.4931.  Google Scholar

[21]

Y. LvR. Yuan and Y. Pei, Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equations, 260 (2016), 6201-6231.  doi: 10.1016/j.jde.2015.12.037.  Google Scholar

[22]

Y. LvR. YuanY. Pei and T. Li, Global stability of a competitive model with state-dependent delay, J. Dyn. Diff. Equat., 29 (2017), 501-521.  doi: 10.1007/s10884-015-9475-5.  Google Scholar

[23]

Y. LvY. Pei and R. Yuan, Modeling and analysis of a predator-prey model with state-dependent delay, Int. J. Biomath., 11 (2018), 1-22.  doi: 10.1142/S1793524518500262.  Google Scholar

[24]

Y. LvY. Pei and R. Yuan, Principle of linearized stability and instability for parabolic partial differential equations with state-dependent delay, J. Differ. Equations, 267 (2019), 1671-1704.  doi: 10.1016/j.jde.2019.02.014.  Google Scholar

[25]

P. Magal and X. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[26]

P. MagalC. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[27]

J. MahaffyJ. Bélair and M. Mackey, Hematoppietic model with moving boundary condition and state dependent delay: Applications, J. Theor. Biol., 190 (1998), 135-146.   Google Scholar

[28]

T. Malthus, An Essay on the Principle of Population, Oxford World's Classic reprint, 1798. Google Scholar

[29]

E. McCauleyW. Murdoch and R. Nisbet, Growth, reproduction, and mortality of daphnia pulex leydig: Life at low food, Functional Ecology, 4 (1990), 505-514.  doi: 10.2307/2389318.  Google Scholar

[30]

A. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1926), 98-130.  doi: 10.1017/S0013091500034428.  Google Scholar

[31]

K. Noor-E Jannat and B. Roitverg, Effects of larval density and feeding rates on larval life history traits in Anophelets gambiae s.s (Diptera: Culicidae), J. Vector Ecology, 38 (2013), 120-126.   Google Scholar

[32]

H. Smith, Hopf bifurcation in a system of functional equations modeling the spread of an infectious disease, SIAM J. Appl. Math., 43 (1983), 370-385.  doi: 10.1137/0143025.  Google Scholar

[33]

H. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study, Math. Biosci., 113 (1993), 1-23.  doi: 10.1016/0025-5564(93)90006-V.  Google Scholar

[34]

H. Smith, A structured population model and a related functional differential equation: Global attractors and uniform persistence, J. Dyn. Diff. Equat., 6 (1994), 71-99.  doi: 10.1007/BF02219189.  Google Scholar

[35]

H. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mt. J. Math., 24 (1994), 311-334.  doi: 10.1216/rmjm/1181072468.  Google Scholar

[36]

H. Thieme, Semifows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3 (1990), 1035-1066.   Google Scholar

[37]

P. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121.   Google Scholar

[38]

H. Von Förster, Some remarks on changing populations, In The Kinetics of Cellular Proliferation (ed. F. Stohlman Jr) Grune and Stratton, New York, (1959), 328–407. Google Scholar

[39]

G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.  Google Scholar

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