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Stochastic Differential Equations As Insect Population Models

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Estimation and Analysis of Insect Populations

Part of the book series: Lecture Notes in Statistics ((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.

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References

  • Allee, W. C., A. E. Emerson, O. Park, T. Park & K. P. Schmidt. 1949. Principles of Animal Ecology. W. B. Saunders, Philadelphia.

    Google Scholar 

  • Andrewartha, H. G. & L. C. Birch. 1954. The Distribution and Abundance of Animals. University of Chicago Press, Chicago.

    Google Scholar 

  • Basawa, I. V. & B. L. S. Prakasa Rao. 1980. Statistical Inference for Stochastic Processes. Academic Press, New York.

    MATH  Google Scholar 

  • Berryman, A. A. 1978. Towards a Theory of Insect Epidemiology. Res. Pop. Ecol.19: 181 – 196.

    Article  Google Scholar 

  • Braumann, C. A. 1983a. Population Growth in Random Environments. Bull. Math. Biol.45: 635 – 641.

    MATH  Google Scholar 

  • 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 

  • Campbell, R. W. & R. J. Sloan. 1978. Numerical Bimodality among North American Gypsy Moth Populations. Environ. Entomol7: 641 – 646.

    Google Scholar 

  • Campbell, R. W. & R. J. Sloan. 1978. Numerical Bimodality among North American Gypsy Moth Populations. Environ. Entomol7: 641 – 646.

    Google Scholar 

  • Dennis, B. & G. P. Patii. 1984. The Gamma Distribution and Weighted Multimodal Gamma Distributions as Models of Population Abundance. Math. Biosci.68: 187 – 212.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Garcia, O. 1983. A Stochastic Differential Equation Model for the Height Growth of Forest Stands. Biometrics39: 1059 – 1072.

    Article  Google Scholar 

  • Gardiner, C. W. 1985. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Second edition. Springer—Verlag, Berlin.

    Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Godambe, V. P. 1985. The Foundations of Finite Sample Estimation in Stochastic Processes. Biometrika72: 419 – 428.

    Article  MathSciNet  MATH  Google Scholar 

  • Goel, N. S. & N. Richter—Dyn. 1974. Stochastic Models in Biology. Academic Press, New York.

    Google Scholar 

  • Graybill, F. A. 1976. Theory and Application of the Linear Model. Wads worth, Belmont, California.

    Google Scholar 

  • Grebogi, C., E. Ott & J. A. Yorke. 1987. Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics. Science238: 632 – 638.

    Article  MathSciNet  Google Scholar 

  • Hamada, Y. 1981. Dynamics of the Noise—induced Phase Transition of the Verhuist Model. Progr. Theor. Phys.65: 850 – 860.

    Article  MathSciNet  MATH  Google Scholar 

  • Horsthemke, W. & R. Lefever. 1984. Noise—induced Transitions. Springer—Verlag, Berlin.

    MATH  Google Scholar 

  • 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 

  • Karlin, S. & H. M. Taylor. 1981. A Second Course in Stochastic Processes. Academic Press, New York.

    MATH  Google Scholar 

  • Klimko, L. A. & P. I. Nelson. 1978. On Conditional Least Squares Estimation for Stochastic Processes. Ann. Stat.6: 629 – 642.

    Article  MathSciNet  MATH  Google Scholar 

  • Lasota, A. & M. C. Mackey. 1985. Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Lasota, A. & M. C. Mackey. 1985. Probabilistic Properties of Deterministic Systems. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • May, R. M. 1974a. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • McCiillagh, P. & J. A. Neider. 1983. Generalized Linear Models. Chapman and Hall, London.

    Google Scholar 

  • Nisbet, R. M. & W. S. C. Gurney. 1982. Modelling Fluctuating Populations. John Wiley & Sons, New York.

    Google Scholar 

  • Pielou, E. C. 1977. Mathematical Ecology. John Wiley & Sons, New York.

    Google Scholar 

  • Prajneshu, Time-dependent Solution of the Logistic Model for Population Growth in Random Environment. J. Appi. Prob.17: 1083 – 1086.

    Article  MathSciNet  MATH  Google Scholar 

  • Press, W. H., B. P. Flannery, S. A. Teukolsky & W. T. Vetterling. 1986. Numerical Recipes. Cambridge University Press, Cambridge.

    Google Scholar 

  • Ricciardi, L. M. 1977. Diffusion Processes and Related Topics in Biology. Springer—Verlag, Berlin.

    MATH  Google Scholar 

  • Risken, H. 1984. The Fokker-Planck Equation. Springer—Verlag, Berlin.

    MATH  Google Scholar 

  • 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 

  • Soong, T. T. 1973. Random Differential Equations in Science and Engineering. Academic Press, New York.

    MATH  Google Scholar 

  • Takahashi, F. 1964. Reproduction Curve with Two Equilibrium Points: a Consideration of the Fluctuation of Insect Population. Res. Pop. Ecol.6: 28 – 36.

    Article  Google Scholar 

  • Turelli, M. 1977. Random Environments and Stochastic Calculus. Theor. Pop. Biol.12: 140 – 178.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

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Author information

Authors and Affiliations

  1. College of Forestry, Wildlife and Range Sciences, University of Idaho, Moscow, Idaho, 83843, USA

    Brian Dennis

Authors
  1. Brian Dennis
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Editor information

Editors and Affiliations

  1. Departments of Statistics and Zoology, University of Wyoming, Laramie, WY, 82071, USA

    Lyman L. McDonald

  2. Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

    Bryan F. J. Manly

  3. Department of Plant, Soil and Insect Sciences, University of Wyoming, Laramie, WY, 82071, USA

    Jeffrey A. Lockwood

  4. Departments of Entomology and Forestry, Virginia Polytechnic Institute, Blacksburg, VA, 24061, USA

    Jesse A. Logan

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© 1989 Springer-Verlag Berlin Heidelberg

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Dennis, B. (1989). Stochastic Differential Equations As Insect Population Models. In: McDonald, L.L., Manly, B.F.J., Lockwood, J.A., Logan, J.A. (eds) Estimation and Analysis of Insect Populations. Lecture Notes in Statistics, vol 55. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3664-1_14

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  • DOI: https://doi.org/10.1007/978-1-4612-3664-1_14

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-96998-5

  • Online ISBN: 978-1-4612-3664-1

  • eBook Packages: Springer Book Archive

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