Abstract
Survival analyses, investigations of extinction and persistence, are executed for populations represented by a nonautonomous differential equation model. The population is assumed governed by density dependent and time varying density independent demographic parameters. While traditional approaches to extinction postulate extinction on an infinite time horizon and at zero abundance level, survival analysis is developed not only for this traditional setting but also on a finite time horizon and at a nonzero threshold level. A main conclusion is that extinction of a temporally stressed population is determined by a totality of density independent and density dependent factors.
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References
Clark, C. W.: Mathematical bioeconomics. New York: Wiley 1976
Freedman, H. I., Waltman, P.: Mathematical analysis of some three-species food-chain models. Math. Biosci. 33, 257–276 (1977)
Freedman, H. I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984)
Freedman, H. I., Waltman, P.: Persistence in a model of three competitive populations. Math. Biosci. 73, 89–101 (1985)
Gard, T. C.: Persistence in food webs. In: Levin, S. A., Hallam, T. G. (eds.) Mathematical ecology (Proceedings, Trieste 1982, pp. 208–219) Lecture Notes in Biomathematics, Vol. 54. Berlin Heidelberg New York Tokyo: Springer 1984
Gard, T. C., Hallam, T. G.: Persistence in food webs: I. Lotka-Volterra food chains. Bull. Math. Biol. 41, 877–891 (1979)
Gause, G. F.: The struggle for existence. New York: Hafner Press 1934 (Reprinted in 1964)
Goh, B. S.: Nonvulnerability of ecosystems in unpredictable environments. Theor. Popul. Biol. 10, 83–95 (1976).
Hallam, T. G.: Population dynamics in a homogeneous environment. In: Hallam, T. G., Levin, S. A. (eds.) Mathematical ecology: An introduction. Berlin Heidelberg New York Tokyo: Springer 1986
Hallam, T. G., Clark, C. E., Jordan, G. S.: Effects of toxicants on populations: A qualitative approach II. First order kinetics. J. Math. Biol. 18, 25–37 (1983)
Hallam, T. G., deLuna, J. T.: Effects of toxicants on populations: A qualitative approach III. Environmental and food chain pathways. J. Theor. Biol. 109, 411–429 (1984)
Hallam, T. G., Ma Zhien: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327–339 (1986)
Malthus, T. R.: An essay on the principle of population as it affects the future improvement of society, with remarks on the speculations of Mr. Godwin, M. Condorcet, and other writers. London: J. Johnson 1798
Lakshmikanthan, V., Leela, S.: Differential and integral inequalities. New York: Academic Press 1969
Levin, S. A.: Ecosystem analysis and prediction. Philadelphia: SIAM, 1975
May, R. M.: Theoretical ecology: Principles and applications. In: May, R. M. (ed.) Philadelphia: Saunders 1976
Roughgarden, J.: Theory of population genetics and evolutionary ecology: An introduction. New York: Macmillan 1977
Smith, F. E.: Population dynamics in Daphnia magna and a new model for population growth. Ecology 44, 651–663 (1963)
Steele, H. J.: The structure of marine ecosystems. Cambridge: Harvard 1974
Verhulst, P. F.: Notice sur la loi que la population suit dans son accroissement. Correspondences Mathematiques et Physiques 10, 113–121 (1838)
Verhulst, P. F.: Deuxieme mémoire sur la loi d'accroissement de la population. Mem. Acad. Roy. Belg. 20, (no. 3), 1–32 (1847)
Weiss, L., Infante, E. F.: On the stability of systems defined over a finite time interval. Proc. Nat. Acad. Sci. USA 54, 44–48 (1965)
Wiegert, R. C.: Mathematical representation of ecological interactions. In: Levin, S. A. (ed.) Ecosystem analysis and prediction, pp. 43–54. Philadelphia: SIAM-SIMS 1975
Woodwell, G. M.: The threshold problem in ecosystems. In: Levin, S. A. (ed.) Ecosystem analysis and prediction, pp. 9–21. Philadelphia: SIAM-SIMS 1975
Wu, L. S. Y.: On the stability of ecosystems. In: Levin, S. A. (ed.) Ecosystem analysis and prediction, pp. 155–165. Philadelphia: SIAM 1975
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Hallam, T.G., Zhien, M. On density and extinction in continuous population models. J. Math. Biology 25, 191–201 (1987). https://doi.org/10.1007/BF00276389
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DOI: https://doi.org/10.1007/BF00276389