Advertisement

Stochastic Differential Equations As Insect Population Models

  • Brian Dennis
Part of the Lecture Notes in Statistics book series (LNS, volume 55)

Abstract

Stochastic differential equations are a potentially important class of models for describing insect population dynamics. Their advantages include ease of use, relative tractability, ease of understanding, and the potential for approximating many types of stochastic variation affecting insect populations. This paper is an exposition for quantitative ecologists on parameter estimation for one-dimensional stochastic differential equations. Stochastic versions of the exponential growth model and the logistic model are developed in detail as examples. Topics discussed include transition distributions and moments, stationary distributions, maximum likelihood estimates, conditional least squares estimates, maximum quasi-likelihood estimates, jackknifing, multiple stable and unstable equilbria, and deterministic chaos.

Keywords

Stationary Distribution Stochastic Differential Equation Population Abundance Insect Population Deterministic Chaos 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allee, W. C., A. E. Emerson, O. Park, T. Park & K. P. Schmidt. 1949. Principles of Animal Ecology. W. B. Saunders, Philadelphia.Google Scholar
  2. Andrewartha, H. G. & L. C. Birch. 1954. The Distribution and Abundance of Animals. University of Chicago Press, Chicago.Google Scholar
  3. Basawa, I. V. & B. L. S. Prakasa Rao. 1980. Statistical Inference for Stochastic Processes. Academic Press, New York.MATHGoogle Scholar
  4. Berryman, A. A. 1978. Towards a Theory of Insect Epidemiology. Res. Pop. Ecol.19: 181 – 196.CrossRefGoogle Scholar
  5. Braumann, C. A. 1983a. Population Growth in Random Environments. Bull. Math. Biol.45: 635 – 641.MATHGoogle Scholar
  6. Braumann, C. A. 1983b. Population Extinction Probabilities and Methods of Estimation for Population Stochastic Differential Equation Models. Pp. 553–559. InR.S. Buey and J.M.F. Moura [eds.], Nonlinear Stochastic Problems. D. Reidel, Dordrecht, Holland.Google Scholar
  7. Campbell, R. W. & R. J. Sloan. 1978. Numerical Bimodality among North American Gypsy Moth Populations. Environ. Entomol7: 641 – 646.Google Scholar
  8. Campbell, R. W. & R. J. Sloan. 1978. Numerical Bimodality among North American Gypsy Moth Populations. Environ. Entomol7: 641 – 646.Google Scholar
  9. Dennis, B. & G. P. Patii. 1984. The Gamma Distribution and Weighted Multimodal Gamma Distributions as Models of Population Abundance. Math. Biosci.68: 187 – 212.MathSciNetMATHCrossRefGoogle Scholar
  10. Dennis, B. & G. P. Patii. 1988. Applications in Ecology. Chapter 12 pp. 303–330. In E.L. Crow and K. Shimizu [eds.], Lognormal Distributions: Theory and Applications. Marcel Dekker, New York.Google Scholar
  11. Feldman, M. W. & J. Roughgarden. 1975. A Population’s Stationary Distribution and Chance of Extinction in a Stochastic Environment with Remarks on the Theory of Species Packing. Theor. Popul. Biol.7: 197 – 207.MathSciNetCrossRefGoogle Scholar
  12. Garcia, O. 1983. A Stochastic Differential Equation Model for the Height Growth of Forest Stands. Biometrics39: 1059 – 1072.CrossRefGoogle Scholar
  13. Gardiner, C. W. 1985. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Second edition. Springer—Verlag, Berlin.Google Scholar
  14. Gleser, L. J. & D. S. Moore. 1985. The Effect of Positive Dependence on Chi—squared Tests for Categorical Data. J. R. Stat. Soc.B47: 459 – 465.MathSciNetGoogle Scholar
  15. Godambe, V. P. 1985. The Foundations of Finite Sample Estimation in Stochastic Processes. Biometrika72: 419 – 428.MathSciNetMATHCrossRefGoogle Scholar
  16. Goel, N. S. & N. Richter—Dyn. 1974. Stochastic Models in Biology. Academic Press, New York.Google Scholar
  17. Graybill, F. A. 1976. Theory and Application of the Linear Model. Wads worth, Belmont, California.Google Scholar
  18. Grebogi, C., E. Ott & J. A. Yorke. 1987. Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics. Science238: 632 – 638.MathSciNetCrossRefGoogle Scholar
  19. Hamada, Y. 1981. Dynamics of the Noise—induced Phase Transition of the Verhuist Model. Progr. Theor. Phys.65: 850 – 860.MathSciNetMATHCrossRefGoogle Scholar
  20. Horsthemke, W. & R. Lefever. 1984. Noise—induced Transitions. Springer—Verlag, Berlin.MATHGoogle Scholar
  21. Jennrich, R. I. & R. H. Moore. 1975. Maximum Likelihood Estimation by Means of Nonlinear Least Squares. Proc. Stat. Comp. Am. Stat. Assoc.52 – 65.Google Scholar
  22. Karlin, S. & H. M. Taylor. 1981. A Second Course in Stochastic Processes. Academic Press, New York.MATHGoogle Scholar
  23. Klimko, L. A. & P. I. Nelson. 1978. On Conditional Least Squares Estimation for Stochastic Processes. Ann. Stat.6: 629 – 642.MathSciNetMATHCrossRefGoogle Scholar
  24. Lasota, A. & M. C. Mackey. 1985. Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge.MATHGoogle Scholar
  25. Lasota, A. & M. C. Mackey. 1985. Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge.MATHGoogle Scholar
  26. Ludwig, D., D. D. Jones & C. S. Holling. 1978. Qualitative Analysis of Insect Outbreak Systems: the Spruce Budworm and Forest. J. Anim. Ecol.47: 315 – 332.CrossRefGoogle Scholar
  27. May, R. M. 1974a. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, New Jersey.Google Scholar
  28. McCiillagh, P. & J. A. Neider. 1983. Generalized Linear Models. Chapman and Hall, London.Google Scholar
  29. Nisbet, R. M. & W. S. C. Gurney. 1982. Modelling Fluctuating Populations. John Wiley & Sons, New York.Google Scholar
  30. Pielou, E. C. 1977. Mathematical Ecology. John Wiley & Sons, New York.Google Scholar
  31. Prajneshu, Time-dependent Solution of the Logistic Model for Population Growth in Random Environment. J. Appi. Prob.17: 1083 – 1086.MathSciNetMATHCrossRefGoogle Scholar
  32. Press, W. H., B. P. Flannery, S. A. Teukolsky & W. T. Vetterling. 1986. Numerical Recipes. Cambridge University Press, Cambridge.Google Scholar
  33. Ricciardi, L. M. 1977. Diffusion Processes and Related Topics in Biology. Springer—Verlag, Berlin.MATHGoogle Scholar
  34. Risken, H. 1984. The Fokker-Planck Equation. Springer—Verlag, Berlin.MATHGoogle Scholar
  35. Schaffer, W. M. & M. Kot. 1986. Differential Systems in Ecology and Epidemiology. Chapter 8 pp. 158–178. InA. V. Holden [ed.], Chaos. Princeton University Press, Princeton, New Jersey.Google Scholar
  36. Soong, T. T. 1973. Random Differential Equations in Science and Engineering. Academic Press, New York.MATHGoogle Scholar
  37. Takahashi, F. 1964. Reproduction Curve with Two Equilibrium Points: a Consideration of the Fluctuation of Insect Population. Res. Pop. Ecol.6: 28 – 36.CrossRefGoogle Scholar
  38. Turelli, M. 1977. Random Environments and Stochastic Calculus. Theor. Pop. Biol.12: 140 – 178.MathSciNetMATHCrossRefGoogle Scholar
  39. Wiesak, K. 1988. Asymptotic Solution of a Stochastic Logistic Equation with a Small Diffusion Coefficient. Ph.D. Thesis, University of Idaho, Moscow, Idaho.Google Scholar
  40. Wong, E. 1964. The Construction of a Class of Stationary Markoff Processes. Pp. 264–276. InR. Bellman [ed.], Stochastic processes in Mathematical Physics and Engineering. American Mathematical Society, Providence, Rhode Island.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Brian Dennis
    • 1
  1. 1.College of Forestry, Wildlife and Range SciencesUniversity of IdahoMoscowUSA

Personalised recommendations