Bulletin of Mathematical Biology

, Volume 54, Issue 5, pp 827–837 | Cite as

Stochastic models for toxicant-stressed populations

  • Thomas C. Gard


We obtain conditions for the existence of an invariant distribution on (0, ∞) for stochastic growth models of Ito type. We interpret the results in the case where the intrinsic growth rate is adjusted to account for the impact of a toxicant on the population. Comparisons with related results for ODE models by Hallamet al. are given, and consequences of taking the Stratonovich interpretation for the stochastic models are mentioned.


Stochastic Differential Equation Positive Equilibrium Intrinsic Growth Rate Invariant Distribution Differential Equation Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Butler, G., H. I. Freedman and P. Waltman. 1986. Uniformly persistent systems.Proc. Am. math. Soc. 96, 425–430.MATHMathSciNetCrossRefGoogle Scholar
  2. DeAngelis, D. L., R. A. Goldstein and R. V. O'Neill. 1975. A model for trophic interaction.Ecology 56, 881–982.CrossRefGoogle Scholar
  3. Gallopin, G. C. 1971. A generalized model of a resource-population system: I. General properties. II. Stability analysis.Oecologia 7, 382–413;7, 414–432.CrossRefGoogle Scholar
  4. Gard, T. C. 1988.Introductions to Stochastic Differential Equations. New York: Marcel Dekker.Google Scholar
  5. Gard, T. C. 1990. A stochastic model for the effects of toxicants on population.Ecol. Modelling 51, 273–280.CrossRefGoogle Scholar
  6. Goel, N. S. S. C. Maitra and E. W. Montroll. 1971 On the Volterra and other nonlinear models of interacting populations.Rev. Mod. Phys. 43, 231–276.MathSciNetCrossRefGoogle Scholar
  7. Gompertz, B. 1925. On the nature of the function expressive of the law of human mortability.Phil. Trans. 115, 513–585.Google Scholar
  8. Hallam, T. G. 1986. Population dynamics in a homogeneous environment. InMathematical Ecology, T. G. Hallam and S. A. Levin (Eds). Berlin: Springer-Verlag.Google Scholar
  9. Hallam, T. G. and Ma Zhien. 1986. Persistence in population models with demographic fluctuations.J. math. Biol. 24, 327–339.MATHMathSciNetGoogle Scholar
  10. Ma Zhien Song Baojun and T. G. Hallam. 1989. The threshold of survival for systems in a fluctuating environment.Bull. math. Biol. 51 311–323.CrossRefGoogle Scholar
  11. Rosenzweig, M. 1971. The paradox of enrichment: destabilization of exploitation ecosystems in ecological time.Science 171, 385–387.Google Scholar
  12. Smith, F. E. 1963. Population dynamics in Daphnia magna and a new model for population growth.Ecology 44, 651–663.CrossRefGoogle Scholar
  13. Vance, R. R. 1990. Population growth in a time-varying environment.Theor. Pop. Biol. 37, 438–454.MATHMathSciNetCrossRefGoogle Scholar
  14. Vance R. R. and E. A. Coddington. 1989. A nonautonomous model of population growth.J. math. Biol. 27, 491–506.MATHMathSciNetCrossRefGoogle Scholar
  15. Wong, E. and M. Zakai. 1965. On the convergence of ordinary integrals to stochastic integrals.Ann. math. Stat. 36, 1560–1564.MATHMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 1992

Authors and Affiliations

  • Thomas C. Gard
    • 1
  1. 1.Department of MathematicsThe University of GeorgiaAthensUSA

Personalised recommendations