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Reduction of Discrete Dynamical Systems with Applications toDynamics Population Models

Published online by Cambridge University Press:  28 November 2013

R. Bravo de la Parra ,
M. Marvá ,
E. Sánchez  and
L. Sanz
R. Bravo de la Parra*
Affiliation:
Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
M. Marvá
Affiliation:
Departamento de Física y Matemáticas, Universidad de Alcalá 28871 Alcalá de Henares (Madrid), Spain
E. Sánchez
Affiliation:
Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain
L. Sanz
Affiliation:
Departamento de Matemática Aplicada, ETSI Industriales Universidad Politécnica de Madrid José Gutiérrez Abascal 2, 28006 Madrid, Spain
*
⋆Corresponding author. E-mail: rafael.bravo@uah.es
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Abstract

In this work we review the aggregation of variables method for discrete dynamicalsystems. These methods consist of describing the asymptotic behaviour of a complex systeminvolving many coupled variables through the asymptotic behaviour of a reduced systemformulated in terms of a few global variables. We consider population dynamics modelsincluding two processes acting at different time scales. Each process has associated a mapdescribing its effect along its specific time unit. The discrete system encompassing bothprocesses is expressed in the slow time scale composing the map associated to the slow oneand the k-th iterate of the map associated to the fast one. In the linear case a result isstated showing the relationship between the corresponding asymptotic elements of bothsystems, initial and reduced. In the nonlinear case, the reduction result establishes theexistence, stability and basins of attraction of steady states and periodic solutions ofthe original system with the help of the same elements of the corresponding reducedsystem. Several models looking over the main applications of the method to populationsdynamics are collected to illustrate the general results.

Keywords

discrete modelsaggregation of variablestime scalespopulation dynamic.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 107 - 129
Copyright
© EDP Sciences, 2013

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References

Ahmed, E., Hegazi, S.A.. On some variants of dynamical systems. Chaos Soliton. Fract., 12 (2001), No. 11, 2003–2008. CrossRefGoogle Scholar
P. Auger. Dynamics and thermodynamics in hierarchically organized systems. Pergamon Press, Oxford, 1989.
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.
Auger, P., Bravo de la Parra, R., Poggiale, J.-C., Sánchez, E., Sanz, L.. Aggregation methods in dynamical systems and applications in population and community dynamics. Phys. Life. Rev., 5 (2008), No. 2, 79–105. CrossRefGoogle Scholar
Auger, P., Poggiale, J.-C.. Aggregation and emergence in systems of ordinary differential equations. Math. Comput. Model., 27 (1998), No. 4, 1–21. CrossRefGoogle Scholar
Auger, P., Poggiale, J.-C., Sánchez, E.. A review on spatial aggregation methods involving several time scales. Ecol. Complex., 10 (2012), No. 1, 12–25. CrossRefGoogle Scholar
Auger, P., Roussarie, R.. Complex ecological models with simple dynamics: From individuals to populations. Acta Biotheor., 42 (1994), No. 2-3, 111–136. CrossRefGoogle Scholar
Bravo de la Parra, R., Auger, P., Sánchez, E.. Aggregation methods in discrete models. J. Biol. Syst., 3 (1995), No. 2, 603–612. CrossRefGoogle Scholar
Bravo de la Parra, R., Sánchez, E., Arino, O., Auger, P.. A Discrete Model with Density Dependent Fast Migration. Math. Biosci., 157 (1999), No. 1, 91–110. CrossRefGoogle Scholar
Bravo de la Parra, R., Sánchez, E., Auger, P.. Time scales in density dependent discrete models. J. Biol. Syst., 5 (1997), No. 1, 111–129. CrossRefGoogle Scholar
H. Caswell. Matrix Population Models: Construction, Analysis and Interpretation, second ed. Sinauer Associates Inc., Sunderland, 2001.
J.M. Cushing. An Introduction to Structured Population Dynamics. SIAM, Philadelphia, 1998.
Dubreuil, E., Auger, P., Gaillard, J.M., Khaladi, M.. Effect of aggressive behavior on age-structured population dynamics. Ecol. Model., 193 (2006), No. 3-4, 777–786. CrossRefGoogle Scholar
Fenichel, N.. Persistence and Smoothness of Invariant Manifolds for Flows. Indiana U. Math. J., 21 (1972), No. 3, 193–226. CrossRefGoogle Scholar
J. Hofbauer, K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, 1998.
Iwasa, Y., Andreasen, V., Levin, S.. Aggregation in model ecosystems I: Perfect Aggregation. Ecol. Model., 37 (1987), No. 3-4, 287–302. CrossRefGoogle Scholar
Iwasa, Y., Levin, S., Andreasen, V.. Aggregation in model ecosystems. II. Approximate Aggregation. J. Math. Appl. Med. Biol., 6 (1989), No. 1, 1–23. CrossRefGoogle Scholar
Lett, P., Auger, P., Bravo de la Parra, R.. Migration Frequency and the Persistence of Host-Parasitoid Interactions. J. Theor. Bio., 221 (2003), No. 4, 639–654. CrossRefGoogle Scholar
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.
Luckyanov, N.K., Svirezhev, Yu.M., Voronkova, O.V.. Aggregation of variables in simulation models of water ecosystems. Ecol. Model., 18 (1983), No. 3-4, 235–240. CrossRefGoogle Scholar
Marvá, M., Moussaoui, A., Bravo de la Parra, R., Auger, P.. A density dependent model describing age structured population dynamics using hawk-dove tactics. J. Differ. Equ. Appl., 19 (2013), No. 6, 1022–1034. CrossRefGoogle Scholar
Marvá, M., Sánchez, E., Bravo de la Parra, R., Sanz, L.. Reduction of slow–fast discrete models coupling migration and demography. J. Theor. Biol., 258 (2009), No. 3, 371–379. CrossRefGoogle Scholar
M. Marvá. Approximate aggregation on nonlinear dynamical systems. Ph.D. Thesis, Universidad de Alcalá, Spain, 2011.
Neubert, M.G., Caswell, H.. Density-dependent vital rates and their population dynamic consequences. J. Math. Biol., 41 (2000), No. 2, 103–121. CrossRefGoogle ScholarPubMed
Nguyen Huu, T., Auger, P., Lett, C., Marvá, M.. Emergence of global behaviour in a host-parasitoid model with density-dependent dispersal in a chain of patches. Ecol. Complex., 5 (2008), No. 1, 9–21. CrossRefGoogle Scholar
Nguyen Huu, T., Bravo de la Parra, R., Auger, P.. Approximate aggregation of linear discrete models with two time-scales: re-scaling slow processes to the fast scale. J. Differ. Equ. Appl., 17 (2011), No. 4, 621–635. Google Scholar
Sánchez, E., Bravo de la Parra, R., Auger, P.. Discrete Models with Different Time-Scales. Acta Biotheor., 43 (1995), No. 4, 465–479. CrossRefGoogle Scholar
Sanz, L., Alonso, J.A.. Approximate Aggregation Methods in Discrete Time Stochastic Population Models. Math. Model. Nat. Phenom., 5 (2010), No. 6, 38–69. CrossRefGoogle Scholar
Sanz, L., Blasco, A., Bravo de la Parra, R.. Approximate reduction of multi-type Galton-Watson processes with two time scales. Math. Mod. Meth. Appl. S., 13 (2003), No. 4, 491–525. CrossRefGoogle Scholar
Sanz, L., Bravo de la Parra, R.. Variables aggregation in a time discrete linear model. Math. Biosci., 157 (1999), No. 1, 111–146. CrossRefGoogle Scholar
Sanz, L., Bravo de la Parra, R.. Time scales in stochastic multiregional models. Nonlinear Anal-Real., 1 (2000), No. 1, 89–122. CrossRefGoogle Scholar
Sanz, L., Bravo de la Parra, R.. Time scales in a non autonomous linear discrete model. Math. Mod. Meth. Appl. S., 11 (2001), No. 7, 1203–1235. CrossRefGoogle Scholar
Sanz, L., Bravo de la Parra, R., Sánchez, E.. Two time scales non-linear discrete models approximate reduction. J. Differ. Equ. Appl., 14 (2008), No. 6, 607–627. CrossRefGoogle Scholar
E. Seneta. Non-Negative Matrices and Markov Chains. Springer Verlag, New York, 1981.
G.W. Stewart, J.I. Guang Sun. Matrix Perturbation Theory. Boston Academic Press, Boston, 1990.
Taylor, A.D.. Heterogeneity in hostparasitoid interactions: aggregation of risk and the CV2 > 1 rule. Trends Ecol. Evol., 8 (1993), No. 11, 400–405. CrossRefGoogle Scholar