Dynamics with riddled basins of attraction in models of interacting populations

Chaos, solitons and fractals

Volumn 12, Issue 2, 2001, Pages 301-311

  Gazelles, Bernard ^a,b  

^a UPMC University Paris 06   (France)

^b UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOS THEORY; MATHEMATICAL MODELS;

CHAOTIC ATTRACTORS;

ECOSYSTEMS;

EID: 0343026675     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(00)00047-3     Document Type: Article

References (40)

1. May RM. Biological populations with nonoverlapping generations: stable points stable, cycles and chaos. Science 1974;186:645-7.
2. May RM. Simple mathematical models with very complicated dynamical behaviour. Nature 1976;269:459-67.
3. Schaffer WN, Kot M. Do strange attractors govern ecological systems?. BioScience 1985;35:342 50.
4. Kot M, Schaffer WN. Chaos in ecological systems: the coals that newcastle forgot. Trends Ecol Evol 1986;1:58 63.
5. Hasting A, Horn CL, Ellner S, Turchin P, Godfray HCJ. Chaos in ecology: is Mother Nature a strange attractor?. Annu Rev Ecol Syst 1993;21:1-33.
6. Ellner S, Turchin P. Chaos in a noisy world: new methods and evidence from time-series analysis. Am Nat 1995;145:343 75.
7. Costantino RF, Desharnais RA, Gushing JM, Dennis B. Chaotic dynamics in an insect population. Science 1997;275:389 91.
8. Ott E. Chaos in dynamical systems. Cambridge: Cambridge University Press; 1993.
9. Buescu J. Exotic attractors. from Liapunov stability to riddled basins. Progress in Mathematics, vol. 153. Basel: Birkhäuser, 1997.
10. Lloyd AL. The coupled logistic map: a simple model for the effects of spatial heterogeneity on populations dynamics. J Theor Biol 1995;173:217 30.
11. Kaitala V, Heino M. Complex non-unique dynamics in simple ecological interactions. Proc R Soc Lond B 1996;263:1011-5.
12. Neubert MG. A simple population model with qualitatively uncertain dynamics. J Theor Biol 1997;189:399-411.
13. Tél T. Transient chaos. In: Bai-Lin Hao, editor. Directions in chaos. Singapore: World Scientific, 1990;3:149-221.
14. Alexander JC, Yorke JA, You Z, Kan I. Riddled basins. Int J Bifurc Chaos 1992;2:795 813.
15. Sommerer JC, Ott E. A physical system with qualitatively uncertain dynamics. Nature 1993;365:138-40.
16. Ott E, Alexander JC, Kan I, Sommerer JC, Yorke JA. The transition to chaotic attractors with riddled basins. Physica D 1994;76:384 410.
17. Lai YC, Grebogi Yorke JA, Venkataramani SC. Riddling bifurcation in chaotic dynamical systems. Phys Rev Lett 1996;77:55 8.
18. Kapitaniak T, Maistrenko YL. Chaos synchronization and riddled basins in two coupled one-dimensional maps. Chaos, Solitons & Fractals 1998;9:271-82.
19. Kapitaniak T, Maistrenko YL, Stefanski A, Brindley J. Bifurcation from locally to globally riddled basins. Phys Rev E 1998;57:R6253 6.
20. Ott E, Sommerer JC. Blowout bifurcations: the occurrence of riddled basins and on-off intermittency. Phys Lett A 1994;188:39 47.
21. Ashwin P, Buescu J, Stewart I. From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 1996;9:706-37.
22. Sommerer JC, Grebogi C. Intermingled basins of attraction: uncomputability in a simple physical system. Phys Lett A 1996;214:243 51.
23. Lai YC, Grebogi C. Intermingled basins and two-state on-off intermittency. Phys Rev E 1995;52:R3313-6.
24. Venkataramani SC, Hunt BR, Ott E. Bubbling transition. Phys Rev E 1996;54:1346-60.
25. Platt N, Spiegel EA, Tresser C. On-off intermittency: a mechanism for bursting. Phys Rev Lett 1993;70:279 82.
26. Heagy JF, Platt N, Hammel SM. Characterization of on-off intermittency. Phys Rev E 1994;49:1140 50.
27. Venkataramani SC, Antonsen TM, Ott E, Sommerer JC. On-off intermittency: power spectrum and fractal properties. Physica D 1996;96:66-99.
28. Warner RR, Chesson PL. Coexistence mediated by recruitment fluctuations: a field guide to the storage effect. Am Nat 1985;25:769 87.
29. Franke JE, Yakubu AA. Mutual exclusion versus coexistence for discrete competitive systems. J Math Biol 1991;30:161-8.
30. Gatto M. The evolutionary optimality of oscillatory and chaotic dynamics in simple population models. Theor Popul Biol 1993;43:310 36.
31. Milnor J. On the concept of attractor. Commun Math Phys 1985;99:177 95.
32. Fernére R, Gatto M. Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theor Popul Biol 1995;48:126-71.
33. Ashwin P, Covas E, Tavakol R. Transverse instability for non-normal parameters. Nonlinearity 1999;12:563 77.
34. Ott E, Sommerer JC, Alexander JC, Kan I, Yorke JA. Scaling behaviour of chaotic systems with riddled basins. Phys Rev Lett 1993;71:4134-7.
35. McDonald SW, Grebogi C, Ott E, Yorke JA. Fractal basin boundaries. Phys Lett A 1985;110:1-9.
36. Grebogi C, McDonald SW, Ott E, Yorke JA. Exterior dimension of fat fractals. Physica D 1985;17:125 53.
37. May RM. Spatial chaos and its role in ecology and evolution. In: Levin SA, editor. Frontiers in mathematical biology. Berlin: Springer, 1994: 326-44.
38. Holt RD, McPeek MA. Chaotic population dynamics favors the evolution of dispersal. Am Nat 1996;148:709-18.
39. Jillson D. Insect populations respond to fluctuating environments. Nature 1980;288:699 700.
40. Begon M, Sait SM, Thompson DJ. Predator-prey cycles with period shifts between two and three-species systems. Nature 1996;381:311-5.