American Institute of Mathematical Sciences

Advanced
July  1999, 5(3): 663-683. doi: 10.3934/dcds.1999.5.663

Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$

Abdelaziz Rhandi 1, and  Roland Schnaubelt 2,

1. 

Départment de Mathématiques, Faculté des Sciences Semlalia, B.P.: S.15 40000 Marrakech, Maroc, Morocco
2. 

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received  January 1998 Revised  May 1999 Published  May 1999

We prove existence and uniquences of positive solutions of an age-structured population equation of McKendrick type with spatial diffusion in $L^1$. The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [19], [24], [30], and [31] to the non-autonomous situation. We use the theory of evolution semigroups and extrapolation spaces.
Keywords: resolvent positive operator, Hillc-Yosida operator, Age-structured population equation, irreducibility, quasi-compact semigroup, essential spectral radius, balanced exponential growth, Miyadera perturbaion, extrapolated semigroup, positive evolution family, evolution semigroup..
Mathematics Subject Classification: Primary: 47D06, 92D25; Secondary: 34G10, 47A1.
Citation: Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
[1]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
[2]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[3]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[4]

Viorel Nitica, Andrei Török. On a semigroup problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2365-2377. doi: 10.3934/dcdss.2019148
[5]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[6]

Luisa Arlotti, Bertrand Lods. Transport semigroup associated to positive boundary conditions of unit norm: A Dyson-Phillips approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2739-2766. doi: 10.3934/dcdsb.2014.19.2739
[7]

J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
[8]

Joseph Auslander, Xiongping Dai. Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4647-4711. doi: 10.3934/dcds.2019190
[9]

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
[10]

Andrzej Biś. Entropies of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 639-648. doi: 10.3934/dcds.2004.11.639
[11]

Michael Blank. Recurrence for measurable semigroup actions. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335
[12]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[13]

Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences & Engineering, 2013, 10 (1) : 1-17. doi: 10.3934/mbe.2013.10.1
[14]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580
[15]

Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs. Networks & Heterogeneous Media, 2017, 12 (4) : 643-662. doi: 10.3934/nhm.2017026
[16]

Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
[17]

Z.-R. He, M.-S. Wang, Z.-E. Ma. Optimal birth control problems for nonlinear age-structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 589-594. doi: 10.3934/dcdsb.2004.4.589
[18]

Diène Ngom, A. Iggidir, Aboudramane Guiro, Abderrahim Ouahbi. An observer for a nonlinear age-structured model of a harvested fish population. Mathematical Biosciences & Engineering, 2008, 5 (2) : 337-354. doi: 10.3934/mbe.2008.5.337
[19]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[20]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences