Bulletin of Mathematical Biology

, Volume 49, Issue 6, pp 671–696 | Cite as

Equilibrium and extinction in stochastic population dynamics

  • H. Roozen


Stochastic models of interacting biological populations, with birth and death rates depending on the population size are studied in the quasi-stationary state. Confidence regions in the state space are constructed by a new method for the numerical, solution of the ray equations. The concept of extinction time, which is closely related to the concept of stability for stochastic systems, is discussed. Results of numerical calculations for two-dimensional stochastic population models are presented.


Equilibrium Point Stochastic System Confidence Region Competition Model Exit Time 
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  1. Abramowitz, M. and I. A. Stegun. 1972.Handbook of Mathematical Functions, New York: Dover Publications.Google Scholar
  2. Batschelet, E. 1981.Circular Statistics in Biology, Mathematics in Biology. New York: Academic Press.MATHGoogle Scholar
  3. Feller, W. 1952. “The Parabolic Differential Equations and the Associated Semi-groups of Transformations.”Ann. Math. 55; 468–519.MATHMathSciNetCrossRefGoogle Scholar
  4. Gardiner, C. W., 1983.Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer series in synergetics, Vol. 13. Springer.Google Scholar
  5. Hogg, R. V. and A. T. Craig. 1970.Introduction to Mathematical Statistics 3rd edn. London: Macmillan.MATHGoogle Scholar
  6. Karlin, S. and H. M. Taylor. 1975/1981.A First/Second Course in Stochastic Processes, New York: Academic Press.Google Scholar
  7. Knessl, C., M. Mangel, B. J. Matkowsky, Z. Schuss and C. Tier. 1984. “Solution of Kramers-Moyal Equations for Problems in Chemical Physics.”J. Chem. Phys. 81 (3).Google Scholar
  8. Knessl, C., B. J. Matkowsky, Z. Schuss and C. Tier. 1985. “An Asymptotic Theory of Large Deviations for Markov Jump Processes.”SIAM J. appl. Math. 46 (6).Google Scholar
  9. Ludwig, D. 1975. “Persistence of Dynamical Systems under Random Perturbations”SIAM Rev. 17 (4).Google Scholar
  10. Matkowsky, B. J. and Z. Schuss. 1977. “The Exit Problem for Randomly Perturbed Dynamical Systems”.SIAM J. appl. Math. 33 (2).Google Scholar
  11. May, R. M. 1974.Stability and Complexity in Model Ecosystems, 2nd edn. Princeton University Press.Google Scholar
  12. Nisbet, R. M. and W. S. C. Gurney. 1982.Modelling Fluctuating Populations. Wiley-Interscience.Google Scholar
  13. Roozen, H. 1986. “Numerical, Construction of Rays and Confidence Contours in Stochastic Population Dynamics”. Note AM-N8602, Centre for Mathematics and Computer Science, Amsterdam.Google Scholar
  14. Roughgarden, J. 1979.Theory of Population Genetics and Evolutionary Ecology: an Introduction. Macmillan.Google Scholar
  15. Schuss, Z. 1980.Theory and Applications of Stochastic Differential Equations. Wiley Series in Probability and Mathematical Statistics. Wiley.Google Scholar
  16. Van Kampen, N. G. 1981.Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland.MATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 1987

Authors and Affiliations

  • H. Roozen
    • 1
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

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