Geodesic
Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models
R. Bravo de la Parra 1   ; M. Marvá 1   ; E. Sánchez 2   ; L. Sanz 2
1 Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
2 Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 107-129 Cet article a éte moissonné depuis la source EDP Sciences

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In this work we review the aggregation of variables method for discrete dynamical systems. These methods consist of describing the asymptotic behaviour of a complex system involving many coupled variables through the asymptotic behaviour of a reduced system formulated in terms of a few global variables. We consider population dynamics models including two processes acting at different time scales. Each process has associated a map describing its effect along its specific time unit. The discrete system encompassing both processes is expressed in the slow time scale composing the map associated to the slow one and the k-th iterate of the map associated to the fast one. In the linear case a result is stated showing the relationship between the corresponding asymptotic elements of both systems, initial and reduced. In the nonlinear case, the reduction result establishes the existence, stability and basins of attraction of steady states and periodic solutions of the original system with the help of the same elements of the corresponding reduced system. Several models looking over the main applications of the method to populations dynamics are collected to illustrate the general results.
DOI : 10.1051/mmnp/20138608

R. Bravo de la Parra  1   ; M. Marvá  1   ; E. Sánchez  2   ; L. Sanz  2

1 Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
2 Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain 
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R. Bravo de la Parra; M. Marvá; E. Sánchez; L. Sanz. Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 107-129. doi: 10.1051/mmnp/20138608

[1] E. Ahmed, S.A. Hegazi Chaos Soliton. Fract. 2001 2003 2008

[2] P. Auger. Dynamics and thermodynamics in hierarchically organized systems. Pergamon Press, Oxford, 1989.

[3] P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez, T. Nguyen-Huu. Aggregation of variables and applications to population dynamics. In: P. Magal, S. Ruan (Eds.). Structured Population Models in Biology and Epidemiology. Lecture Notes in Mathematics 1936, Mathematical Biosciences Subseries, Springer Verlag, Berlin, 2008, 209–263.

[4] P. Auger, R. Bravo De La Parra, J.-C. Poggiale, E. Sánchez, L. Sanz Phys. Life. Rev. 2008 79 105

[5] P. Auger, J.-C. Poggiale Math. Comput. Model. 1998 1 21

[6] P. Auger, J.-C. Poggiale, E. Sánchez Ecol. Complex. 2012 12 25

[7] P. Auger, R. Roussarie Acta Biotheor. 1994 111 136

[8] R. Bravo De La Parra, P. Auger, E. Sánchez J. Biol. Syst. 1995 603 612

[9] R. Bravo De La Parra, E. Sánchez, O. Arino, P. Auger Math. Biosci. 1999 91 110

[10] R. Bravo De La Parra, E. Sánchez, P. Auger J. Biol. Syst. 1997 111 129

[11] H. Caswell. Matrix Population Models: Construction, Analysis and Interpretation, second ed. Sinauer Associates Inc., Sunderland, 2001.

[12] J.M. Cushing. An Introduction to Structured Population Dynamics. SIAM, Philadelphia, 1998.

[13] E. Dubreuil, P. Auger, J.M. Gaillard, M. Khaladi Ecol. Model. 2006 777 786

[14] N. Fenichel Indiana U. Math. J. 1972 193 226

[15] J. Hofbauer, K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, 1998.

[16] Y. Iwasa, V. Andreasen, S. Levin Ecol. Model. 1987 287 302

[17] Y. Iwasa, S. Levin, V. Andreasen J. Math. Appl. Med. Biol. 1989 1 23

[18] P. Lett, P. Auger, R. Bravo De La Parra J. Theor. Bio. 2003 639 654

[19] H. Lischke, T.J. Löffler, P.E. Thornton, N.E. Zimmermann. Up-scaling of biological properties and models to the landscape level. In: F. Kienast, S. Ghosh, O. Wildi (Eds.). A Changing World: Challenges for Landscape Research. Landscape Series 8, Springer Verlag, Berlin, 2007, 273–296.

[20] N.K. Luckyanov, Yu.M. Svirezhev, O.V. Voronkova Ecol. Model. 1983 235 240

[21] M. Marvá, A. Moussaoui, R. Bravo De La Parra, P. Auger J. Differ. Equ. Appl. 2013 1022 1034

[22] M. Marvá, E. Sánchez, R. Bravo De La Parra, L. Sanz J. Theor. Biol. 2009 371 379

[23] M. Marvá. Approximate aggregation on nonlinear dynamical systems. Ph.D. Thesis, Universidad de Alcalá, Spain, 2011.

[24] M.G. Neubert, H. Caswell J. Math. Biol. 2000 103 121

[25] T. Nguyen Huu, P. Auger, C. Lett, M. Marvá Ecol. Complex. 2008 9 21

[26] T. Nguyen Huu, R. Bravo De La Parra, P. Auger J. Differ. Equ. Appl. 2011 621 635

[27] E. Sánchez, R. Bravo De La Parra, P. Auger Acta Biotheor. 1995 465 479

[28] L. Sanz, J.A. Alonso Math. Model. Nat. Phenom. 2010 38 69

[29] L. Sanz, A. Blasco, R. Bravo De La Parra Math. Mod. Meth. Appl. S. 2003 491 525

[30] L. Sanz, R. Bravo De La Parra Math. Biosci. 1999 111 146

[31] L. Sanz, R. Bravo De La Parra Nonlinear Anal-Real. 2000 89 122

[32] L. Sanz, R. Bravo De La Parra Math. Mod. Meth. Appl. S. 2001 1203 1235

[33] L. Sanz, R. Bravo De La Parra, E. Sánchez J. Differ. Equ. Appl. 2008 607 627

[34] E. Seneta. Non-Negative Matrices and Markov Chains. Springer Verlag, New York, 1981.

[35] G.W. Stewart, J.I. Guang Sun. Matrix Perturbation Theory. Boston Academic Press, Boston, 1990.

[36] A.D. Taylor Trends Ecol. Evol. 1993 400 405

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