Reduction of discrete dynamical systems with applications to dynamics population models

Mathematical Modelling of Natural Phenomena

Volumn 8, Issue 6, 2013, Pages 107-129

  Morozov, A ^c   Tyutyunov, Yu V ^c   Bravo De La Parra, R ^a   Marva M ^a   Sanchez E ^b   Sanz, L ^b  

^a UNIVERSITY OF ALCALÁ   (Spain)

^b UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

^c NONE

Author keywords

Aggregation of variables; Discrete models; Population dynamic.; Time scales

Indexed keywords

ASYMPTOTIC BEHAVIOUR; BASINS OF ATTRACTION; DIFFERENT TIME SCALE; DISCRETE DYNAMICAL SYSTEMS; DISCRETE MODELS; PERIODIC SOLUTION; POPULATION DYNAMICS MODELS; TIME-SCALES;

ASYMPTOTIC ANALYSIS; POPULATION DYNAMICS; TIME MEASUREMENT;

DYNAMICAL SYSTEMS;

EID: 84889867704     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20138608     Document Type: Article

References (36)

1. E. Ahmed, S.A. Hegazi. On some variants of dynamical systems. Chaos Soliton. Fract., 12 (2001), No. 11, 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. Aggregation methods in dynamical systems and applications in population and community dynamics. Phys. Life. Rev., 5 (2008), No. 2, 79-105.
5. P. Auger, J.-C. Poggiale. Aggregation and emergence in systems of ordinary differential equations. Math. Comput. Model., 27 (1998), No. 4, 1-21. (Pubitemid 28118278)
6. P. Auger, J.-C. Poggiale, E. Sánchez. A review on spatial aggregation methods involving several time scales. Ecol. Complex., 10 (2012), No. 1, 12-25.
7. P. Auger, R. Roussarie. Complex ecological models with simple dynamics: From individuals to populations. Acta Biotheor., 42 (1994), No. 2-3, 111-136.
8. R. Bravo de la Parra, P. Auger, E. Sánchez. Aggregation methods in discrete models. J. Biol. Syst., 3 (1995), No. 2, 603-612.
9. R. Bravo de la Parra, E. Sánchez, O. Arino, P. Auger. A Discrete Model with Density Dependent Fast Migration. Math. Biosci., 157 (1999), No. 1, 91-110.
10. R. Bravo de la Parra, E. Sánchez, P. Auger. Time scales in density dependent discrete models. J. Biol. Syst., 5 (1997), No. 1, 111-129. (Pubitemid 127532220)
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. Effect of aggressive behavior on age-structured population dynamics. Ecol. Model., 193 (2006), No. 3-4, 777-786. (Pubitemid 43247832)
14. N. Fenichel. Persistence and Smoothness of Invariant Manifolds for Flows. Indiana U. Math. J., 21 (1972), No. 3, 193-226.
15. J. Hofbauer, K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, 1998.
16. Y. Iwasa, V. Andreasen, S. Levin. Aggregation in model ecosystems I: Perfect Aggregation. Ecol. Model., 37 (1987), No. 3-4, 287-302.
17. Y. Iwasa, S. Levin, V. Andreasen. Aggregation in model ecosystems. II. Approximate Aggregation. J. Math. Appl. Med. Biol., 6 (1989), No. 1, 1-23.
18. P. Lett, P. Auger, R. Bravo de la Parra. Migration Frequency and the Persistence of Host-Parasitoid Interactions. J. Theor. Bio., 221 (2003), No. 4, 639-654. (Pubitemid 36444531)
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. Aggregation of variables in simulation models of water ecosystems. Ecol. Model., 18 (1983), No. 3-4, 235-240. (Pubitemid 14274028)
21. M. Marvá, A. Moussaoui, R. Bravo de la Parra, P. Auger. A density dependent model describing age structured population dynamics using hawk-dove tactics. J. Differ. Equ. Appl., 19 (2013), No. 6, 1022-1034.
22. M. Marvá, E. Sánchez, R. Bravo de la Parra, L. Sanz. Reduction of slow-fast discrete models coupling migration and demography. J. Theor. Biol., 258 (2009), No. 3, 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. Density-dependent vital rates and their population dynamic consequences. J. Math. Biol., 41 (2000), No. 2, 103-121.
25. T. Nguyen Huu, P. Auger, C. Lett, M. Marvá. 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. (Pubitemid 350309722)
26. T. Nguyen Huu, R. Bravo de la Parra, P. Auger. 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.
27. E. Sánchez, R. Bravo de la Parra, P. Auger. Discrete Models with Different Time-Scales. Acta Biotheor., 43 (1995), No. 4, 465-479.
28. L. Sanz, J.A. Alonso. Approximate Aggregation Methods in Discrete Time Stochastic Population Models. Math. Model. Nat. Phenom., 5 (2010), No. 6, 38-69.
29. L. Sanz, A. Blasco, R. Bravo de la Parra. Approximate reduction of multi-type Galton-Watson processes with two time scales. Math. Mod. Meth. Appl. S., 13 (2003), No. 4, 491-525. (Pubitemid 36725602)
30. L. Sanz, R. Bravo de la Parra. Variables aggregation in a time discrete linear model. Math. Biosci., 157 (1999), No. 1, 111-146. (Pubitemid 29165064)
31. L. Sanz, R. Bravo de la Parra. Time scales in stochastic multiregional models. Nonlinear Anal-Real., 1 (2000), No. 1, 89-122.
32. L. Sanz, R. Bravo de la Parra. Time scales in a non autonomous linear discrete model. Math. Mod. Meth. Appl. S., 11 (2001), No. 7, 1203-1235. (Pubitemid 33686544)
33. L. Sanz, R. Bravo de la Parra, E. Sánchez. Two time scales non-linear discrete models approximate reduction. J. Differ. Equ. Appl., 14 (2008), No. 6, 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. 2 > 1 rule. Trends Ecol. Evol., 8 (1993), No. 11, 400-405. (Pubitemid 24410280)