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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

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