# zbMATH — the first resource for mathematics

Stochastic dispersal processes in plant populations. (English) Zbl 0889.92026
Summary: A dispersal model for airborne pollen based on assumptions about wind directionality, gravity, and a wind threshold at which pollen is taken by the wind is developed, using a three dimensional diffusion approximation. The bivariate probability distribution of pollen receipt by flowers at the same height as the pollen source is derived. Gravity, vertical random movements, and vegetation density turn out to have similar effects on this distribution. Maximum likelihood methods for estimating the combined parameters from data with multiple point or continuous pollen sources, and one or more plant varieties, are developed. Using an example data set from the literature, it is shown that our model gives a better fit than more traditional descriptive dispersal models of the form $$e^{-a{r^b}}$$. We also show that estimates of important properties of the dispersal distribution, such as the variances, become considerably smaller using our model than for the more traditional models. Finally, we discuss potential extensions and evolutionary implications of these types of models.

##### MSC:
 92D40 Ecology 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 62P10 Applications of statistics to biology and medical sciences; meta analysis
bootstrap; Maple
Full Text:
##### References:
 [1] Bateman, A. J., Contamination in seed crops. III. relation with isolation distance, Heredity, 1, 303-336, (1947) [2] Bateman, A. J., Contamination of seed crops. II. wind pollination, Heredity, 1, 235-246, (1947) [3] Batschelet, E., Circular Statistics in Biology, (1981), Academic Press London · Zbl 0524.62104 [4] Char, B. W., Maple Reference Manual, (1988) [5] Efron, B.; Tibshirani, R. J., An Introduction to the Bootstrap, (1993), Chapman & Hall London · Zbl 0835.62038 [6] Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugen. London, 7, 355-369, (1937) · JFM 63.1111.04 [7] Gliddon, C., The impact of hybrids between genetically modified crop plants and their related species: biological models and theoretical perspectives, Molecular Ecol., 3, 41-44, (1994) [8] Greene, D. F.; Johnson, E. A., A model of wind dispersal of winged or plumed seeds, Ecology, 70, 339-347, (1989) [9] Gregory, P. H., The Microbiology of the Atmosphere, (1973), Wiley New York [10] Haldane, J. B.S., The theory of a cline, J. Genetics, 48, 277-284, (1948) · Zbl 0032.03601 [11] Kareiva, P.; Morris, W.; Jacobi, C. M., Studying and managing the risk of cross-fertilization between transgenic crops and wild relatives, Molecular Ecol., 3, 15-21, (1994) [12] Karlin, S.; Taylor, H. M., A Second Course in Stochastic Processes, (1981), Academic Press New York [13] Kendall, M.; Stuart, A.; Ord, J. K., Kendall’s Advanced Theory of Statistics, (1983), Griffin London [14] Kolmogorov, A.; Petrovskii, I.; Piskunov, N., Applicable Mathematics of Non-physical Phenomena, (1937), Wiley New York [15] Lenski, R. E., Quantifying fitness and gene stability in microorganisms, (Ginzburg, L. R., Assessing Ecological Risks of Biotechnology, (1991), Butterworth-Heinemann Boston), 173-192 [16] Levin, D. A.; Kerster, H. W., Gene flow in seed plants, Evol. Biol., 7, 139-220, (1974) [17] Luna, R. E.; Church, H. W., Estimation of longterm concentrations using a “universal” wind speed distribution, J. Appl. Meteorology, 13, 910-916, (1974) [18] Malécot, G., The Mathematics of Heredity, (1969), Freeman San Fransisco [19] Mollison, D., Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B, 39, 283-326, (1977) · Zbl 0374.60110 [20] Morris, W. F., Predicting the consequences of plant spacing and biased movement for pollen dispersal by honey bees, Ecology, 74, 493-500, (1993) [21] Morris, W. F.; Kareiva, P. M.; Raymer, P. L., Do barren zones and pollen traps reduce gene escape from transgenic crops?, Ecol. Appl., 4, 157-165, (1994) [22] Nagylaki, T., Clines with asymmetric migration, Genetics, 88, 813-827, (1978) [23] Nagylaki, T.; Lucier, B., Numerical analysis of random drift in a cline, Genetics, 94, 497-517, (1979) [24] Okubo, A.; Levin, S. A., A theoretical framework for data analysis of wind dispersal of seeds and pollen, Ecology, 70, 329-338, (1989) [25] Persson, B.; Ståhl, E., Survival and yield ofpinus sylvestris, Scand. J. For. Res., 5, 381-395, (1990) [26] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes in Pascal, (1986), Cambridge Univ. Press Cambridge · Zbl 0587.65005 [27] Regal, P. J., The adaptive potential of genetically engineered organisms in nature, Trends Ecol. and Evol., 3, 36-38, (1988) [28] Slatkin, M., Gene flow and selection in a cline, Genetics, 75, 733-756, (1973) [29] Weinberger, H. F., Long-time behaviour of a class of biological models, SIAM J. Math. Anal., 13, 353-396, (1982) · Zbl 0529.92010 [30] Woodell, S. R.J., Directionality in bumblebees in relation to environmental factors, (Richards, A. J., The Pollination of Flowers by Insects, (1978), Academic Press New York) [31] Wright, S., Isolation by distance, Genetics, 28, 114-138, (1943) [32] Wright, S., Dispersion in drosophila pseudoobscura, Amer. Naturalist, 102, 923, (1968) [33] Wright, S., Evolution and the Genetics of Populations, (1969), Univ. of Chicago Press London [34] Zannetti, P., Air Pollution Modelling, (1990), Van Nostrand Reinhold New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.