Abstract
The negative binomial distribution often describes the distribution of arthropods within agroecosystems. When mobile arthropods are disrupted, such as may occur during chemical applications or changes in climatic conditions, they tend to move until they are again in a negative binomial distribution. While arthropod movement continues, the distribution remains negative binomial. This paper gives a mathematical model for this behavior. The movement of each arthropod is modeled using a Markov process based on a common transition function which is doubly stochastic and irreducible. The arthropods move independently except that the time until an arthropod moves is exponentially distributed with a mean proportional to the number of arthropods on the plant. Using this type of interaction, the Bose-Einstein distribution (which assigns equal probability to distinguishable spatial patterns) is the equilibrium distribution. If the number of arthropods and plants are each as large as 100, this distribution is closely approximated by the geometric distribution which is the limiting distribution. The geometric is a special case of the negative binomial distribution (k = 1) and is often observed when the sampling unit is the minor habitat of the arthropod. Parallels are drawn between the study of particles in a phase space and the distribution of arthropods in agroecosystems.
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References
Anscombe, F. J. 1949. The statistical analysis of insect counts based on the negative binomial distribution. Biometrics5: 165 – 173.
Bliss, C. I. 1971. The aggregation of species within spatial units, pp. 311–336. InG. P. Patii, E. C. Pielou, W. E. Waters [eds.], Statistical Ecology, Vol. 1. The Pennsylvania State University Press, University Park, PA.
Boswell, M. T. & G. P. Patii. 1970. Chance mechanisms generating the negative binomial distribution, pp. 3–22. InG. P. Patii, [ed.], Random Counts in Scientific Work, Vol. 1. The Pennsylvania State University Press, University Park, PA.
Dingle, H. 1974. The experimental analysis of migration and life history strategies in insects, pp. 329–342. InL. Barton—Browne [ed.], Experimental Analysis of Insect Behaviour. Springer—Verlag, Berlin.
Hamilton, W. D. & R. M. May. 1977. Dispersal in stable habitats. Nature269: 578 – 581.
Hasseil, M. P. 1978. The dynamics of arthropod predator—prey systems. Princeton Univ. Press, Princeton, N.J.
Hassell, M. P. & R. M. May. 1986. Generalist and specialist natural enemies in insect predator-prey interactions. J. Anim. Ecol55: 923 – 940.
Leslie, P. H. and J. C. Gower. 1960. The properties of a stochastic model for the predator—prey type of interaction between two species. Biometrika47: 219 – 234.
Motro, U. 1982a. Optimal rates of dispersal I. Haploid populations. Theor. Pop. Biol23: 394 – 411
Motro, U. 1982b. Optimal rates of dispersal II. Diploid populations. Theor. Pop. Biol.21: 412 – 429.
Motro, U. 1983. Optimal rates of dispersal III. Parent—Offspring conflict. Theor. Pop. Biol.23: 159 – 168.
Murdoch, W. W. & A. Oaten. 1975. Prédation and population stability, pp. 2–131. InA. MacFadyen [ed.]. Adv. Ecol Res. Vol. 9. Academic Press, London.
Pathria, R. K. 1972. Statistical Mechanics. Pergamon Press, Braunschweig.
Pianka, E. R. 1981. Competition and niche theory, pp. 167–196. InR. M. May [ed.], Theoretical Ecology: Principles and Applications, 2nd Ed. Sinauer Associated, Inc., Sunderland, MA.
Rankin, M. A. & M. C. Singer. 1984. Insect movement: Mechanisms and effects, pp. 185–216. InC. B. Huffaker and R. L. Rabb [eds.], Ecological Entomology. John Wiley and Sons, New York.
Skellam, J. G. 1973. The formulation and interpretation of mathematical models of diffusionary processes in population biology, pp. 63–85. InM. S. Bartlett and R. W. Hiorns [eds.], The Mathematical Theory of the Dynamics of Biological Populations. Academic Press, New York.
Spitzer, F. 1970. Interaction of Markov processes. Advances in Math. 5: 246 – 290.
Taylor, L. R. 1984. Assessing and interpreting the spatial distribution of insect populations. Ann. Rev. Entomol.29: 321 – 357.
Waymire, E. 1980. Zero—range interaction at Bose—Einstein speeds under a positive recurrent single particle law. Ann. Prob.8: 441 – 450.
Willson, Linda J., J. L. Folk & J. H. Young. 1984. Multistage estimation compared with fixed—sample—size estimation of the negative binomial parameter k. Biometrics40: 109 – 117.
Willson, Linda J. & J. H. Young. 1983. Sequential estimation of insect population densitites with a fixed coefficient of variation. Environ. Entomol.12: 669 – 672.
Willson, L. J., J. H. Young & J. L. Folks. 1987. A biological application of Bose—Einstein statistics. Commun. Statist. Theor. Meth.16: 445 – 459.
Young, J. H. & L. J. Willson. 1987. The use of Bose—Einstein statistics in population dynamics models of arthropods. Ecol. Model.36: 89 – 99.
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© 1989 Springer-Verlag Berlin Heidelberg
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Young, L.J.W., Young, J.H. (1989). A Model of Arthropod Movement Within Agroecosystems. 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_27
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DOI: https://doi.org/10.1007/978-1-4612-3664-1_27
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