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Analytic semigroups and some degenerate evolution equations defined on domains with corners
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 |
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. |
[3] |
N. T. J. Bailey, The Elements of Stochastic Processes,, Wiley, (1964).
|
[4] |
J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications,, Springer, (2006).
|
[5] |
J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems,, in Multiscale Problems in Biomathematics, (2009), 221.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[12] |
A. Bobrowski, Degenerate convergence of semigroups,, Semigroup Forum, 49 (1994), 303.
doi: 10.1007/BF02573493. |
[13] |
A. Bobrowski, Functional Analysis for Probability and Stochastic Processes,, Cambridge University Press, (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.
doi: 10.1016/S0895-7177(00)00017-0. |
[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).
|
[17] |
S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence,, Wiley, (1986).
doi: 10.1002/9780470316658. |
[18] |
F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).
|
[19] |
E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (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.
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.
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.
doi: 10.1016/j.mbs.2005.02.006. |
[23] |
C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).
doi: 10.1137/1.9780898719512. |
[24] |
J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications in Kinetic Theory,, World Sci., (1995).
doi: 10.1142/9789812831248. |
[25] |
E. Seneta, Nonnegative Matrices and Markov Chains,, 2nd edition, (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.
|
[27] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (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. 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. |
[3] |
N. T. J. Bailey, The Elements of Stochastic Processes,, Wiley, (1964).
|
[4] |
J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications,, Springer, (2006).
|
[5] |
J. Banasiak, Asymptotic analysis of singularly perturbed dynamical systems,, in Multiscale Problems in Biomathematics, (2009), 221.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[12] |
A. Bobrowski, Degenerate convergence of semigroups,, Semigroup Forum, 49 (1994), 303.
doi: 10.1007/BF02573493. |
[13] |
A. Bobrowski, Functional Analysis for Probability and Stochastic Processes,, Cambridge University Press, (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.
doi: 10.1016/S0895-7177(00)00017-0. |
[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).
|
[17] |
S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence,, Wiley, (1986).
doi: 10.1002/9780470316658. |
[18] |
F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).
|
[19] |
E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Colloquium Publications, (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.
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.
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.
doi: 10.1016/j.mbs.2005.02.006. |
[23] |
C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).
doi: 10.1137/1.9780898719512. |
[24] |
J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications in Kinetic Theory,, World Sci., (1995).
doi: 10.1142/9789812831248. |
[25] |
E. Seneta, Nonnegative Matrices and Markov Chains,, 2nd edition, (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.
|
[27] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985).
|
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