Chaos and population disappearances in simple ecological models

Journal of Mathematical Biology

Volumn 42, Issue 3, 2001, Pages 239-260

  Schreiber, Sebastian J ^a  

^a WESTERN WASHINGTON UNIVERSITY   (United States)

Author keywords

Bifurcations; Chaos; Extinction; Population disappearance; Population dynamics; Unimodal maps

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; ECOLOGY; NONLINEAR SYSTEM; POPULATION DYNAMICS;

ANIMALS; ECOLOGY; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS; POPULATION DYNAMICS;

EID: 0035294379     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002850000070     Document Type: Article

References (28)

1. Abraham, R.H., Stewart, H.B.: A chaotic blue sky catastrophee in forced relaxation oscillators, Physica D, 21, 394-400 (1986)
2. Beeton, A.H.: Changes in the environment and biota of the great lakes, Eutrophication: Causes, Conseqeunces, Correctives (Washington, D. C.), National Academy of Sciences, pp. 150-187 (1969)
3. Bellows, T.S.: The descriptive properties of some models for density dependence, J. Anim. Ecol., 26, 129-156 (1981)
4. Beverton, R.J.H., Holt, S.J.: On the dynamics of exploited fish populations, Fisheries Investigation Series, vol. 2, Ministry of Agriculture, Fisheries, and Food, London (1957)
5. Brauer, F., Sanchez, D.A.: Constant rate population harvesting: equilibrium and stability, Theor. Popul. Biol., 8, 12-30 (1975)
6. Clark, C.: Mathematical bioeconomics, John Wiley (1976)
7. de Melo, W., van Strien, S.: One dimensional dynamics, Springer-Verlag, Berlin (1993)
8. Getz, W.M.: A hypothesis regarding the abruptness of density dependence and the growth rate of populations, Ecology, 77, 2014-2026 (1996)
9. Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos, Phys. D, 7, 181-200 (1983)
10. Gyllenberg, M., Osipov, A.V., Söderbacka, G.: Bifurcation analysis of a metapopulation model with sources and sinks, J. Nonlinear Sci., 6, 329-366 (1996)
11. Hastings, A.: Complex interactions between dispersal and dynamics: Lessons from coupled Logistic equations, Ecology, 74, 1362-1372 (1993)
12. Hastings, A., Higgins, K.: Persistence of transients in spatially structured ecological models, Science, 263, 1133-1136 (1994)
13. Hille, E.: Lectures on ordinary differential equations, Addison-Wesley, Reading, MA (1968)
14. Holling, C.S.: Resilience and stability of ecological systems, Ann. Rev. Ecol. Sys., 4, 1-23 (1973)
15. Karr, J.R.: Population variability and extinction in the avifauna of a tropical land bridge island, Ecology, 63, 1975-1978 (1982)
16. Katok, A., Hasselblatt, B.: Modern theory of dynamical systems, Cambridge University Press (1995)
17. Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100, 495-524 (1987)
18. May, R.M.: Thresholds and breakpoints in ecosystems with a multiplicity of stable steady states, Nature, 269, 471-477 (1977)
19. McCann, K., Yodzis, P.: Nonlinear dynamics and population dissappearances, Amer. Nat., 144, 873-879 (1994)
20. Melbourne, I.: An example of a non-asymptotically stable attractor, Nonlinearity, 4, 835-844 (1991)
21. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53, 17-51 (1981)
22. Potts, G.R., Aebischer, N.J.: Population dynamics of the grey partridge perdix perdix 1793-1993: Monitoring, modelling and management, Ibis, 137, S29-S37 (1995)
23. Ricker, W.E.: Stock and recruitment, J. Fish. Res. Board. Can., 11, 559-623 (1954)
24. Rosenzweig, M.L.: Paradox of enrichment: Destabization of exploitation ecosystems in ecological time, Science, 171, 385-387 (1971)
25. Singer, D.: Stable orbits and bifurcation of maps of the interval, SIAM, J. Appl. Math., 35, 260-267 (1978)
26. Sinha, S., Parthasarathy, S.: Unusual dynamics of extinction in a simple ecological model, Proc. Nat. Acad. Sci., 93, 1504-1508 (1996)
27. Stone, L.: Period-doubling reversals and chaos in simple ecological models, Nature, 617-620 (1993)
28. Vandermeer, J., Yodzis, P.: Basin boundary collision as a model of discontinuous change in ecosystems, Ecology, 80, 1817-1827 (1999)