Multiple attractors via CUSP bifurcation in periodically varying environments

Journal of Difference Equations and Applications

Volumn 11, Issue 4-5, 2005, Pages 365-377

  Franke, John E ^a   Yakubu, Abdul Aziz ^b  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b HOWARD UNIVERSITY   (United States)

Author keywords

Multiple attractors; Periodic dynamical systems; Semi conjugacy; Tangent bifurcation

Indexed keywords

EID: 22544487784     PISSN: 10236198     EISSN: None     Source Type: Journal    
DOI: 10.1080/10236190412331335436     Document Type: Article

References (39)

1. Alexander, J.C., Yorke, J.A., You, Z. and Kan, I., 1992, Riddled Basins, International Journal of Bifurcation and Chaos, 2(4), 795-813.
2. Begon, M., Harper, J.L. and Townsend, C.R., 1996, Ecology: Individuals, Populations and Communities (Blackwell Science Ltd).
3. Brauer, F. and Castillo-Chavez, C., 2001, Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, Vol. 40 (New York: Springer Verlag).
4. Costantino, R.F., Cushing, J.M., Dennis, B. and Desharnais, R.A., 1998, Resonant popopulation cycles in temporarily fluctuating habitats, Bulletin of Mathematical Biology, 60, 247-273.
5. Cushing, J.M., 1977, Periodic time-dependent predator-prey systems, SIAM Journal of Applied Mathematics, 32, 82-95.
6. Elaydi, S.N., 2000, Discrete Chaos (Boca Raton, FL: Chapman & Hall/CRC).
7. Elaydi, S.N. and Sacker, R.J., Global stability of periodic orbits of nonautonomous difference equations and population biology, Journal of Differential Equations (in press).
8. Elaydi, S.N. and Sacker, R.J., Global stability of periodic orbits of nonautonomous difference equations in population biology and Cushing-Henson conjectures, Journal of Differential Equations (in press).
9. Elaydi, S.N. and Yakubu, A.-A., 2002, Global stability of cycles: Lotka-Volterra competition model with stocking, Journal of Difference Equations and Applications, 8(6), 537-549.
10. Franke, J.E. and Selgrade, J.F., 2003, Attractor for Periodic Dynamical Systems, Journal of Mathematical Analysis and Applications, 286, 64-79.
11. Franke, J.E. and Yakubu, A.-A., 1996, Extinction and persistence of species in discrete competitive systems with a safe refuge, Journal of Mathematical Analysis and Applications, 23, 746-761.
12. Franke, J.E. and Yakubu, A.-A., 1992, Geometry of exclusion principles in discrete systems, Journal of Mathematical Analysis and Applications, 168, 385-400.
13. Franke, J.E. and Yakubu, A.-A., 1991, Mutual exclusion versus coexistence for discrete competitive systems, Journal of Mathematical Biology, 30, 161-168.
14. Guémez, J. and Matías, M.A., 1993, Control of chaos in unidimensional maps, Physics Letters A, 181, 29-32.
15. Hanski, I., 1991, Single-species metapopulation dynamics-concepts, models and observations, Biological Journal of Linnean Society, 42, 17-38.
16. Hanski, I.A. and Gilpin, M.E., 1997, Metapopulation Biology: Ecology, Genetics, and Evolution (San Diego, California: Academic Press Ltd).
17. Hassell, M.P., 1976, The dynamics of competition and predation. Studies in Biol, Vol. 72 (The Camelot Press Ltd).
18. Hassell, M.P., Lawton, J.H. and May, R.M., 1976, Patterns of dynamical behavior in single species populations, Journal of Animal Ecology, 45, 471-486.
19. Henson, S.M., 2000, Multiple attractors and resonance in periodically forced population models, Physica D, 140, 33-49.
20. Henson, S.M., Costantino, R.F., Cushing, J.M., Dennis, B. and Desharnais, R.A., 1999, Multiple attractors, saddles, and population dynamics in periodic habitats, Bulletin of Mathematical Biology, 61, 1121-1149.
21. S.M. Henson, R.F. Costantino, R.A. Desharnais, J.M. Cushing, B. Dennis, Basins of attraction: population dynamics with two stable 4-cycles (Preprint)
22. Henson, S.M. and Gushing, J.M., 1996, Hierarchical models of intraspecific competition: scramble versus contest, Journal of Mathematical Biology, 34, 755-772.
23. Henson, S.M. and Cushing, J.M., 1997, The effect of periodic habitat fluctuations on a nonlinear insect population model, Journal of Mathematical Biology, 36, 201-226.
24. Jillson, D., 1980, Insect populations respond to fluctuating environments, Nature, 288, 699-700.
25. Kocic, V.L. and Ladas, G., 1993, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (Boston: Kluwer Academic).
26. Kuznetsov, Y.A., 1995, Elements of Applied Bifurcation Theory (Springer-Verlag).
27. Li, J., 1992, Periodic solutions of population models in a periodically fluctuating environment, Mathematical Biosciences, 110, 17-25.
28. May, R.M. and Oster, G.F., 1976, Bifurcations and dynamic complexity in simple ecological models, American Naturalist, 110, 573-579.
29. May, R.M., 1977, Simple mathematical models with very complicated dynamics, Nature, 261, 459-469.
30. May, R.M., 1974, Stability and Complexity in Model Ecosystems (Princeton University Press).
31. May, R.M., Hassell, M.P., Anderson, R.M. and Tonkyn, D.W., 1981, Density dependence in host-parasitoid models, Journal of Animal Ecology, 50, 855-865.
32. Maynard Smith, J. and Slatkin, M., 1973, The stability of predator-prey systems, Ecology, 54, 384-391.
33. Milton, J.G. and Belair, J., 1990, Chaos, noise, and extinction in models of population growth, Theoretical Population Biology, 37, 273-290.
34. Nicholson, A.J., 1954, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian Journal of Zoology, 2, 1-65.
35. Nusse, H.E. and Yorke, J.A., 1997, Dynamics: Numerical Explorations (New York: Springer-Verlag).
36. Ricker, W.E., 1954, Stock and recruitment, Journal of Fisheries Research Board of Canada, 11(5), 559-623.
37. Selgrade, J.F. and Roberds, H.D., 2001, On the structure of attractors for discrete, periodically forced systems with applications to population models, Physica D, 158, 69-82.
38. Yakubu, A.-A., 2003, Multiple attractors in juvenile-adult single species models, Journal of Difference Equations and Applications, 9(12), 1083-1093.
39. Yodzis, P., 1989, Introduction to Theoretical Ecology (New York: Harper & Row, Publishers).