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

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.
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 and Continuous Dynamical Systems, 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), 408-436.

[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), 137-150. doi: 10.1016/S0895-7177(98)00013-2.

[3]

N. T. J. Bailey, The Elements of Stochastic Processes, Wiley, New York, 1964.

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer, London, 2006.

[5]

J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems, in Multiscale Problems in Biomathematics, Physics and Mechanics: Modelling, Analysis and Numerics (eds A. Abdulle, J. Banasiak, A. Damlamian and M. Sango), GAKUTO Internat. Ser. Math. Sci. Appl. 31, Gakkotosho, Tokyo, 2009, 221-255.

[6]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7.

[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), 121-154. doi: 10.1007/s00028-010-0086-7.

[8]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.

[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), 533-559. doi: 10.1007/s00009-013-0319-4.

[10]

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

[11]

A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127.

[12]

A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327. doi: 10.1007/BF02573493.

[13]

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

[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), 17-26. doi: 10.1016/S0895-7177(00)00017-0.

[15]

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation, 2nd edition, Sinauer Associates, Inc., Sunderland, 2001.

[16]

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

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[18]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, New York, 1959.

[19]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications, 31, AMS, Providence, 1957.

[20]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process, Mathematical Population Studies, 1 (1988), 49-77. doi: 10.1080/08898488809525260.

[21]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X.

[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), 153-186. doi: 10.1016/j.mbs.2005.02.006.

[23]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[24]

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

[25]

E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer Series in Statistics, Springer-Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[26]

M. Sova, Convergence d'opérations lineaires non bornées, Rev. Roum. Math. Pures et App., 12 (1967), 373-389.

[27]

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

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), 408-436.

[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), 137-150. doi: 10.1016/S0895-7177(98)00013-2.

[3]

N. T. J. Bailey, The Elements of Stochastic Processes, Wiley, New York, 1964.

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer, London, 2006.

[5]

J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems, in Multiscale Problems in Biomathematics, Physics and Mechanics: Modelling, Analysis and Numerics (eds A. Abdulle, J. Banasiak, A. Damlamian and M. Sango), GAKUTO Internat. Ser. Math. Sci. Appl. 31, Gakkotosho, Tokyo, 2009, 221-255.

[6]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7.

[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), 121-154. doi: 10.1007/s00028-010-0086-7.

[8]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.

[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), 533-559. doi: 10.1007/s00009-013-0319-4.

[10]

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

[11]

A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127.

[12]

A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327. doi: 10.1007/BF02573493.

[13]

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

[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), 17-26. doi: 10.1016/S0895-7177(00)00017-0.

[15]

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation, 2nd edition, Sinauer Associates, Inc., Sunderland, 2001.

[16]

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

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[18]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, New York, 1959.

[19]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications, 31, AMS, Providence, 1957.

[20]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multi state stable population process, Mathematical Population Studies, 1 (1988), 49-77. doi: 10.1080/08898488809525260.

[21]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X.

[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), 153-186. doi: 10.1016/j.mbs.2005.02.006.

[23]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[24]

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

[25]

E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer Series in Statistics, Springer-Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[26]

M. Sova, Convergence d'opérations lineaires non bornées, Rev. Roum. Math. Pures et App., 12 (1967), 373-389.

[27]

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

[1]

Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032

[9]

Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055

[10]

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

[11]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]