# American Institute of Mathematical Sciences

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.

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##### 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.
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