American Institute of Mathematical Sciences

Advanced
February  2015, 35(2): 617-635. doi: 10.3934/dcds.2015.35.617

Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model

Jacek Banasiak 1, and  Amartya Goswami 2,

1. 

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban
2. 

Department of Mathematical Sciences, University of Zululand, South Africa

Received  January 2013 Revised  January 2014 Published  September 2014

Multiple time scales are common in population models with age and space structure, where they are a reflection of often different rates of demographic and migratory processes. This makes the models singularly perturbed and allows for their aggregation which, while significantly reducing their complexity, does not alter their essential dynamic properties. There are several methods of aggregation of such models. In this paper we shall show how the Trotter-Kato-Sova-Kurtz theory developed to analyze convergence of $C_0$-semigroups can be used in this field. The paper also extends some of the previous results by considering reducible migration matrices which are important in modelling populations living in geographically patched areas with restricted communication between the patches.
Keywords: reducible matrices., Trotter-Kato theorem, semigroups, convergence of semigroups, singular perturbation, asymptotic analysis, Structured population models, multiple time scales, aggregation.
Mathematics Subject Classification: Primary: 92D25, 35B25, 47D06; Secondary: 35Q92, 47N2.
Citation: Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617
References:
[1]

O. Arino, E. Sánchez, R. Bravo de la Parra and P. Auger, A singular perturbation in an age-structured population model,, SIAM Journal on Applied Mathematics, 60 (1999), 408.   Google Scholar

[2]

O. Arino, E. Sánchez and R. Bravo De La Parra, A model of an age-structured population in a multipatch environment,, Math. Compt. Modelling, 27 (1998), 137.  doi: 10.1016/S0895-7177(98)00013-2.  Google Scholar

[3]

N. T. J. Bailey, The Elements of Stochastic Processes,, Wiley, (1964).   Google Scholar

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications,, Springer, (2006).   Google Scholar

[5]

J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems,, in Multiscale Problems in Biomathematics, (2009), 221.   Google Scholar

[6]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis,, J. Evol. Equ., 9 (2009), 293.  doi: 10.1007/s00028-009-0009-7.  Google Scholar

[7]

J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: an asymptotic analysis approach,, J. Evol. Equ., 11 (2011), 121.  doi: 10.1007/s00028-010-0086-7.  Google Scholar

[8]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices,, Appl. Math. Lett., 25 (2012), 2193.  doi: 10.1016/j.aml.2012.06.001.  Google Scholar

[9]

J. Banasiak, A. Goswami and S. Shindin, Singularly perturbed population models with reducible migration matrix. 2. Asymptotic analysis and numerical simulations,, Mediterr. J. Math., 11 (2014), 533.  doi: 10.1007/s00009-013-0319-4.  Google Scholar

[10]

A. Bobrowski, Convergence of One-Parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere,, Cambridge University Press, ().   Google Scholar

[11]

A. Bobrowski, A note on convergence of semigroups,, Ann. Polon. Math., 69 (1998), 107.   Google Scholar

[12]

A. Bobrowski, Degenerate convergence of semigroups,, Semigroup Forum, 49 (1994), 303.  doi: 10.1007/BF02573493.  Google Scholar

[13]

A. Bobrowski, Functional Analysis for Probability and Stochastic Processes,, Cambridge University Press, (2005).  doi: 10.1017/CBO9780511614583.  Google Scholar

[14]

R. Bravo de la Parra, O. Arino, E. Sánchez and P. Auger, A model for an age-structured population with two time scales,, Math. Comput. Modelling, 31 (2000), 17.  doi: 10.1016/S0895-7177(00)00017-0.  Google Scholar

[15]

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation,, 2nd edition, (2001).   Google Scholar

[16]

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

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence,, Wiley, (1986).  doi: 10.1002/9780470316658.  Google Scholar

[18]

F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).   Google Scholar

[19]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (1957).   Google Scholar

[20]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process,, Mathematical Population Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar

[21]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions,, J. Funct. Anal., 12 (1973), 55.  doi: 10.1016/0022-1236(73)90089-X.  Google Scholar

[22]

M. Lisi and S. Totaro, The Chapman-Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections,, Math. Biosci., 196 (2005), 153.  doi: 10.1016/j.mbs.2005.02.006.  Google Scholar

[23]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[24]

J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications in Kinetic Theory,, World Sci., (1995).  doi: 10.1142/9789812831248.  Google Scholar

[25]

E. Seneta, Nonnegative Matrices and Markov Chains,, 2nd edition, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[26]

M. Sova, Convergence d'opérations lineaires non bornées,, Rev. Roum. Math. Pures et App., 12 (1967), 373.   Google Scholar

[27]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985).   Google Scholar

show all references

References:
[1]

O. Arino, E. Sánchez, R. Bravo de la Parra and P. Auger, A singular perturbation in an age-structured population model,, SIAM Journal on Applied Mathematics, 60 (1999), 408.   Google Scholar

[2]

O. Arino, E. Sánchez and R. Bravo De La Parra, A model of an age-structured population in a multipatch environment,, Math. Compt. Modelling, 27 (1998), 137.  doi: 10.1016/S0895-7177(98)00013-2.  Google Scholar

[3]

N. T. J. Bailey, The Elements of Stochastic Processes,, Wiley, (1964).   Google Scholar

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications,, Springer, (2006).   Google Scholar

[5]

J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems,, in Multiscale Problems in Biomathematics, (2009), 221.   Google Scholar

[6]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis,, J. Evol. Equ., 9 (2009), 293.  doi: 10.1007/s00028-009-0009-7.  Google Scholar

[7]

J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: an asymptotic analysis approach,, J. Evol. Equ., 11 (2011), 121.  doi: 10.1007/s00028-010-0086-7.  Google Scholar

[8]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices,, Appl. Math. Lett., 25 (2012), 2193.  doi: 10.1016/j.aml.2012.06.001.  Google Scholar

[9]

J. Banasiak, A. Goswami and S. Shindin, Singularly perturbed population models with reducible migration matrix. 2. Asymptotic analysis and numerical simulations,, Mediterr. J. Math., 11 (2014), 533.  doi: 10.1007/s00009-013-0319-4.  Google Scholar

[10]

A. Bobrowski, Convergence of One-Parameter Operator Semigroups. In Models of Mathematical Biology and Elsewhere,, Cambridge University Press, ().   Google Scholar

[11]

A. Bobrowski, A note on convergence of semigroups,, Ann. Polon. Math., 69 (1998), 107.   Google Scholar

[12]

A. Bobrowski, Degenerate convergence of semigroups,, Semigroup Forum, 49 (1994), 303.  doi: 10.1007/BF02573493.  Google Scholar

[13]

A. Bobrowski, Functional Analysis for Probability and Stochastic Processes,, Cambridge University Press, (2005).  doi: 10.1017/CBO9780511614583.  Google Scholar

[14]

R. Bravo de la Parra, O. Arino, E. Sánchez and P. Auger, A model for an age-structured population with two time scales,, Math. Comput. Modelling, 31 (2000), 17.  doi: 10.1016/S0895-7177(00)00017-0.  Google Scholar

[15]

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation,, 2nd edition, (2001).   Google Scholar

[16]

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

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence,, Wiley, (1986).  doi: 10.1002/9780470316658.  Google Scholar

[18]

F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).   Google Scholar

[19]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (1957).   Google Scholar

[20]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process,, Mathematical Population Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar

[21]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions,, J. Funct. Anal., 12 (1973), 55.  doi: 10.1016/0022-1236(73)90089-X.  Google Scholar

[22]

M. Lisi and S. Totaro, The Chapman-Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections,, Math. Biosci., 196 (2005), 153.  doi: 10.1016/j.mbs.2005.02.006.  Google Scholar

[23]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[24]

J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications in Kinetic Theory,, World Sci., (1995).  doi: 10.1142/9789812831248.  Google Scholar

[25]

E. Seneta, Nonnegative Matrices and Markov Chains,, 2nd edition, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[26]

M. Sova, Convergence d'opérations lineaires non bornées,, Rev. Roum. Math. Pures et App., 12 (1967), 373.   Google Scholar

[27]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985).   Google Scholar

[1]

Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267
[2]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034
[3]

Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099
[4]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[5]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[6]

Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735
[7]

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
[8]

Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032
[9]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457
[10]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[11]

B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558
[12]

Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311
[13]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[14]

José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[15]

Min He. On continuity in parameters of integrated semigroups. Conference Publications, 2003, 2003 (Special) : 403-412. doi: 10.3934/proc.2003.2003.403
[16]

Alastair Fletcher. Quasiregular semigroups with examples. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2157-2172. doi: 10.3934/dcds.2019090
[17]

Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020238
[18]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[19]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
[20]

John R. Tucker. Attractors and kernels: Linking nonlinear PDE semigroups to harmonic analysis state-space decomposition. Conference Publications, 2001, 2001 (Special) : 366-370. doi: 10.3934/proc.2001.2001.366

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences