Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-30T18:52:31.236Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distribution of the time to extinction in the stochastic logistic population model

Published online by Cambridge University Press:  01 July 2016

R. H. Norden
R. H. Norden*
Affiliation:
Downside School, Stratton-on-the-Fosse
*
∗Postal address: St. Wulstans, Abbey Road, Chilcompton, Bath BA3 4HY, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to investigate the distribution of the extinction times, T, of the stochastic logistic process from both the numerical and the theoretical standpoint. The problem is approached first by deriving formulae for the moments of T; it is then shown that in most cases T is, very nearly, a gamma variate. Some simulated results are given and these agree well with the theory. Furthermore, a consideration of the process conditioned on non-extinction is shown to be an effective way of obtaining a large t (time) description of the unconditioned process. Finally, a more general form of the model in which the death-rate as well as the birth-rate is ‘density-dependent' is considered, and by comparison with the usual form of the model the effect on T of this additional factor is assessed.

Keywords

DETERMINISTICBIRTHDEATHDENSITY-DEPENDENTCARRYING CAPACITY

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 687 - 708
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Allee, W. C, Emerson, A. E., Park, O., Park, T. and Schmidt, K. P. (1949) Principles of Animal Ecology. Saunders, Philadelphia.Google Scholar
Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
Calhoun, J. B. (1952) The social aspects of population dynamics. J. Mammalogy 24, 139–159.Google Scholar
Christian, J. J. (1971) Population density and reproductive efficiency. Biol. Reprod. 4, 248–294.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Cunningham, W. J. (1954) A non-linear differential-difference equation of growth. Proc. Nat. Acad. Sci. USA 40, 708–713.Google Scholar
Errington, P. L. (1951) Concerning fluctuations in populations of the prolific and widely distributed Muskrat. Amer. Naturalist 85, 273–292.CrossRefGoogle Scholar
Fujita, W. H. and Utida, S. (1953) The effect of population density on the growth of an animal population. Ecology 34, 488–498.Google Scholar
Gause, G. F. (1934) The Struggle for Existence. (Reprinted Dover, New York, 1971.)Google Scholar
Goel, N. S. and Dyn, N. R. (1974) Stochastic Models in Biology. Academic Press, New York.Google Scholar
Hutchinson, G. E. (1954) Theoretical notes on oscillatory populations. J. Wildlife Management 18, 107–109.CrossRefGoogle Scholar
Jensen, A. L. (1975) Comparison of logistic equations for population growth. Biometrics 31, 853–862.CrossRefGoogle ScholarPubMed
Keith, L. B. (1963) Wildlife's Ten-Year Cycle. The University of Wisconsin Press, Madison, WI.Google Scholar
Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
Kendall, D. G. (1949) Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264.Google Scholar
King, J. A. (1955) Social behaviour, social organisation, and population dynamics in a Black-tailed Prairiedog town in the Black Hills of South Dakota. Contrib. Lab. Vert. Biol. Univ. Mich. 67, 1–123.Google Scholar
Ladde, G. S. and Siljak, D. D. (1975) Stability of multispecies communities and randomly varying environments. J. Math. Biol. 2, 165–178.Google Scholar
Levins, R. (1969) The effects of random variations of different types of population growth. Proc. Nat. Acad. Sci. USA 62, 1061–1065.Google Scholar
Lewontin, R. C. and Cohen, D. (1969). On population growth in a randomly varying environment. Proc. Nat. Acad. Sci. USA 62, 1056–1060.Google Scholar
McLaren, I. A., (ed.) (1971) Regulation of Populations. Atherton Press, New York.Google Scholar
Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google Scholar
Pearl, L. and Reed, L. J. (1920) On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Nat. Acad. Sci. USA 6, 275–288.Google Scholar
Picard, P. (1963) Sur les modèles stochastiques logistiques en démographie. Ann. Inst. H. Poincaré B 2, 151–172.Google Scholar
Pielou, E. C. (1969) An Introduction to Mathematical Ecology. Wiley, New York.Google Scholar
Prajneshu, (1976) A stochastic model for two interacting species. Stoch. Proc. Appl. 4, 271–282.Google Scholar
Pratt, D. M. (1943) Analysis of population development in Daphnia at different temperatures. Biol. Bull., Woods Hole 85, 116–140.Google Scholar
Rabinovich, J. E. (1969) The applicability of some population growth models to a single species laboratory population. Ann. Entomol. Soc. Amer. 62, 437–442.Google Scholar
Slobodkin, L. B. (1954) Cycles in animal populations. Amer. Scientist 42, 658–666.Google Scholar
Smith, F. E. (1963) Population dynamics in Daphnia magna and a new model for population growth. Ecology 44, 651–663.CrossRefGoogle Scholar
Turner, M. E., Blumenstein, B. A. and Sebaugh, J. L. (1969) A generalisation of the logistic law of growth. Biometrics 25, 577–580.Google Scholar
Verhulst, P. F. (1838) Notice sur la loi que la population suit dans son accroissement. Corresp. Math. Phys. (Bruxelles) 10, 113–121.Google Scholar
Volterra, V. (1926) Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. R. C. Accad. Lincei (6) 2, 31–113.Google Scholar
Wangersky, P. J. and Cunningham, W. J. (1956) On time lags in equations of growth. Proc. Nat. Acad. Sci. USA 42, 699–702.CrossRefGoogle ScholarPubMed