Stochastic models and statistical inference for plant pollen dispersal
Journal de la société française de statistique, Tome 148 (2007) no. 1, p. 77-105
Il est essentiel de modéliser la dispersion du pollen pour prédire dans différentes conditions environnementales les taux de pollinisations croisées entre plantes d'une même espèce. Pour celà, un outil important est la notion de « fonction individuelle de dispersion de pollen » ou « noyau de dispersion ». Pour les plantes anémophiles, nous proposons ici plusieurs modèles de dispersion de pollen transporté par le vent. Ces modèles reposent sur des hypothèses concernant la direction du vent, la gravité, la « vitesse de décollement » du pollen et peuvent intégrer d'autres paramètres externes ou biologiques. Certains modèles étudiés antérieurement décrivent le transport du pollen par des mouvements browniens avec dérive. Cependant, les modèles de transport de pollen utilisés en aérobiologie utilisent en général l'approche stochastique lagrangienne, dans laquelle les vitesses sont modélisées par des équations différentielles stochastiques ou équations de Langevin. Nous proposons de nouveaux modèles provenant de cette approche. Les trajectoires du pollen sont obtenues en intégrant leurs vitesses. Nous étudions en particulier un modèle où la composante verticale est régie par un processus d'Ornstein-Uhlenbeck intégré. Nous disposons de données de pollinisations croisées issues d'expériences en plein champ de maïs, dans lesquelles la couleur des grains est utilisée comme marqueur phénotypique de la dispersion du pollen. Nous étudions les différentes fonctions de dispersion individuelles associées à ces modèles. Pour estimer et comparer leurs performances sur les données, nous développons un cadre statistique approfondi, qui peut être utilisé pour étudier la plupart des données issues de pollinisations croisées. Ceci nous permet d'analyser successivement anciens et nouveaux modèles. Ce nouveau cadre statistique permet d'améliorer significativement les résultats antérieurs obtenus avec les premiers modèles estimés avec d'autres méthodes. Les performances des modèles stochastiques lagrangiens sont généralement bonnes, mais elles ne sont pas meilleures que celles obtenues avec les premiers modèles analysés dans le cadre statistique introduit ici. Néanmoins, il se peut que ces résultats proviennent de conditions environnementales particulières liées à ces données expérimentales. Les comparaisons des paramètres estimés avec les paramètres physiques ou externes sont très satisfaisantes. L'ensemble de ces résultats montre que les modèles mécanistes sont de bons modèles pour prédire la dispersion du pollen à des distances courtes ou moyennes, ainsi que les taux de pollinisations croisées.
Modelling pollen dispersal is essential to make predictions of cross-pollination rates in various environmental conditions between plants of a cultivated species. An important tool for studying this problem is the “individual pollen dispersal function” or “kernel dispersal”. Various models for airborne pollen dispersal are developed. These models are based on assumptions about wind directionality, gravity, settling velocity and may integrate other biological or external parameters. Some previous approaches have used brownian Motions with drift for modelling pollen trajectories. However, models for pollen transport used in aerobiology are often based on the lagrangian Stochastic approach: velocities of pollen grains satisfy stochastic differential equations or Langevin equations and pollen trajectories are obtained by integrating these velocities. New models based on this approach are introduced. A model where the vertical component is driven by an integrated Ornstein-Uhlenbeck process is studied here. Cross-pollination rates data were obtained from large field experiments of maize using the colour of grains as a phenotypic marker of pollen dispersal. We first studied the various individual dispersal functions associated with these models. Second, a thorough statistical framework was developed in order to estimate and compare their performances on data sets. This framework is quite general and can be used to study many other cross-pollination data. Previous and new models were successively analysed using this framework. This new statistical analysis improved significantly former results which had been obtained on the previous models with other statistical methods. The statistical analyses showed that the performances of Lagrange Stochastic models were good, but not better than the previous mechanistic models analysed using this new statistical framework. These results however might be due to some specific environmental conditions in this experiment. Comparisons with the external parameters were quite good, proving that these models can be used in other environmental conditions. All these results show that mechanistic models are good models for predicting short or medium range pollen dispersal and cross-pollination rates.
Mots clés: dispersion du pollen, expériences en plein champ, taux de pollinisations croisées, maïs ; pollen transporté par le vent, paramètres météorologiques, déconvolution, statistique paramétrique, quasivraisemblance, tests d'hypothèses, fonction de dispersion individuelle, modèles mécanistes, approche stochastique lagrangienne, équations de Langevin, temps d'atteinte
@article{JSFS_2007__148_1_77_0,
     author = {Laredo, Catherine and Grimaud, Agn\`es},
     title = {Stochastic models and statistical inference for plant pollen dispersal},
     journal = {Journal de la soci\'et\'e fran\c caise de statistique},
     publisher = {Soci\'et\'e fran\c caise de statistique},
     volume = {148},
     number = {1},
     year = {2007},
     pages = {77-105},
     language = {en},
     url = {http://http://www.numdam.org/item/JSFS_2007__148_1_77_0}
}
Laredo, Catherine; Grimaud, Agnès. Stochastic models and statistical inference for plant pollen dispersal. Journal de la société française de statistique, Tome 148 (2007) no. 1, pp. 77-105. http://www.numdam.org/item/JSFS_2007__148_1_77_0/

[1] Abramowitz M. and Stegun A.I. (1972) Handbook of Mathematical Functions. Dover Publications, New-York.

[2] Adams W.T., Griffin A.R. and Moran G.F. (1992) Using paternity analysis to measure effective pollen dispersal in plant populations. American Naturalist, 140, 762-780.

[3] Austerlitz F., Dick C.W., Dutech C., Klein E.K., Oddou-Muratorio S., Smouse P.E. and Sork V.L. (2004) Using genetics markers to estimate the pollen dispersal curve. Molecular Ecology, 13, 937-954.

[4] Aylor D.E., Schultes N.P. and Shields E.J. (2003) An aerobiological framework for assessing cross-pollination in maize. Agricultural and Forest Meteorology, 119, 111-129.

[5] Barndorff-Nielsen O.E. (1997) Normal Inverse Gaussian distributions and stochastic volatility modelling. Scand. J. of Statistics, 24, 1-13. | MR 1436619 | Zbl 0934.62109

[6] Bateman A.J. (1947) Contamination of seed crops. II. Wind pollination. Heredity, 1, 235-246.

[7] Bolker P. and Pacala S.W. (1997) Using moment equations to understand stochasticaly driven spatial pattern formation in ecological systems. Theoretical Population Biology, 52, 179-197. | Zbl 0890.92020

[8] Burczyk J. and Koralewski T.E. (2005) Parentage versus two-generation analyses for estimating pollen-mediated gene flow in plant populations. Molecular Ecology, 14, 2525-2537.

[9] Clark J.S., Lewis M. and Horvath L. (2001) Invasion by extremes: population spread with variation in dispersal and reproduction. American Naturalist, 157, 537-554.

[10] Clark J.S., Silman N., Kern R., Macklin E. and Hillerislambers J. (1999) Seed dispersal near and far: patterns across temperate and tropical forests. Ecology, 80, 1475-1494.

[11] Collett D. (1991) Modelling binary data. Chapman and Hall, London. | MR 1999899 | Zbl 1041.62058

[12] Cresswell J.E., Davies T.W., Patrick M.A., Russell F., Pennel C., Vicot M. and Lahoubi M. (2004) Aerodynamics of wind pollination in a zoophilous flower, Brassica Napus. Functional Ecology, 18, 861-866.

[13] Devaux C., Lavigne C., Falentin-Guyomarc'H H., Vautrin S., Lecomte J. and Klein E.K. (2005) High diversity of oilseed rape pollen clouds over an agro-ecosystem indicates long-distance dispersal. Molecular Ecology, 14, 2269-2280.

[14] Devos Y., Reheul D. and De Schrijver A. (2005) The co-existence between transgenic and non-transgenic maize in the European Union: a focus on pollen flow and cross-fertilization. Environ. Biosafety Res., 4, 71-87.

[15] Durbin J. (1992) The first-passage density of the Brownian motion process through a curved boundary. J. Appl. Prob., 29, 291 -304. | MR 1165215 | Zbl 0806.60063

[16] Ellstrand N.C. (1992) Gene flow by pollen: implications for plant conservation genetics. Oikos, 63, 77-86.

[17] Grimaud A. (2005) Modélisation stochastique et estimation de la dispersion du pollen de maïs. Estimation dans des modèles à volatilité stochastique. Ph.D Thesis, Université Paris 7- Denis Diderot.

[18] Grimaud A. and Larédo C. (2006) Parametric models for Corn Pollen dispersal using diffusion processes and statistical estimation. Submitted to publication. 25 p.

[19] Huet S., Bouvier A., Gruet M.A. and Jolivet E. (1996) Statistical tools for nonlinear regression. Spriger-Verlag, New-York. | MR 1416565 | Zbl 0867.62059

[20] Hurvich C.M. and Tsai C.L. (1995) Model selection for extended quasi-likelihood in small samples. Biometrics, 51, 1077-1084. | Zbl 0875.62359

[21] Jorgensen B. (1982) Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics. Springer-Verlag. | MR 648107 | Zbl 0486.62022

[22] Klein E.K., Lavigne C., Foueillassar X., Gouyon P.H. and Larédo C. (2003) Corn Pollen Dispersal: Quasi-mechanistic models and Field Experiments. Ecological Monographs, 73, 131-150.

[23] Klein E.K., Lavigne C., Picault H., Renard M. and Gouyon P.H. (2006) Pollination dispersal of oilseedrape: estimation of the dispersal and effects of field dimension. Journal of Applied Ecology, 43, 141.

[24] Latta R.G., Linhart D., Fleck D. and Elliot M. (1996) Direct and indirect estimates of seeds versus pollen movement within a population of Ponderosa Pine. Evolution, 52, 61-67.

[25] Lavigne C., Klein E.K., Vallee P., Pierre J., Godelle B. and Renard M. (1998) A method to determine the mean dispersal pollen of individuals growing within a large pollen source. Theoretical and Applied Genetics, 96, 886-896.

[26] Lewis M.A. (2000) Spread rate for a nonlinear stochastic invasion J. Math. Biol., 41, 430-454. | MR 1803854 | Zbl 1002.92021

[27] Loos C., Seppelt R., Meier-Bethke S., Schiemann J. and Richter O. (2003) Spatially explicit modelling of transgenic maize pollen dispersal and cross-pollination. Journal of Theoretical Biology, 225, 241-255. | MR 2077391

[28] Mccartney H.A. and Fitt B.D.L. (1985) Construction of dispersal models. Pages 107-143. in C.A. Dilligan editor. Mathematical modeling of crop diseases. Academic Press, London.

[29] Mccartney H. A. and Lacey M.E. (1991) Wind dispersal of pollen from crops of oilseedrape (Brassica Napus L.). Journal of Aerosol Science, 22, 476-477.

[30] Mccullagh H.A. and Nelder J.A. (1989) Generalized Linear Models. 2nd Edition. Chapman and Hall, London. | Zbl 0744.62098

[31] Meagher T.R. and Vassiliadis C. (2003) Spatial geometry determines gene flow in plant populations. Genes in the Environment (eds R. Hails, J. Beringer and H.C.J. Godfray), 76-90. Blackwell Science, Oxford

[32] Milhem H., H. Picault, M. Renard and Huet S. (2006) Effect of a crop discontinuity on the oilseed rape pollen dispersal function. Submitted to publication. 17 p.

[33] Nurminieni M., Tufto J., Nilsson N.O., Rognli O.A. (1998) Spatial models of pollen dispersal in the forage grass meadow fescue. Evolutionary Ecology, 12, 487-502.

[34] Oddou-Muratorio S., Klein E.K. and Austerlitz F. (2005) Pollen flow in the wildservice tree, Sorbus torminalis (L.) Crantz. II. Pollen dispersal and heterogeneity in mating success inferred from parent-offspring analysis. Molecular Ecology, 14, 4441-4452.

[35] Perthame B. and Souganidis P.E. (2004) Front propagation for a jump model arising in spatial ecology. Preprint DMA-UMR8553, Ecole Normale Superieure. Paris. 14 p. | MR 2107777

[36] Raupach (1989) Applying Lagrangian fluid mechanics to infer scalar source distributions from concentration profiles in plant canopies. Agric. Forest Meteorol., 47, 85-108.

[37] Rieger M.A., Lamond M., Preston C., Powles S.B., and Rousch R.T. (2002) Pollen mediated-movement of herbicide resistance between commercial canola fields. Science, 296, 2386-2388.

[38] Rodean H.C. (1996) Stochastic Lagrangian models of turbulent diffusion. American Meteorological Society, 26, No 48.

[39] Seinfeld J.H. and Pandis S.N. (1998) Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. Wiley, New York.

[40] Smouse P.E., Dyer R.J., Westfall R.D. and Sork V.L. (2001) Two-generation analysis of pollen flow across a landscape .I. Male gamete heterogeneity among female. Evolution, 55, 260-271.

[41] Squire G.R., Burn D. and Crawford J.W. (1997) A model for the impact of herbicide tolerance on the performance of oilseedrape as a volunteer weed. Annals of Applied Biology, 131, 315-338.

[42] Treu R. and Emberlin J. (2000) Pollen dispersal in the crop Maize (Zea mays), Oil seed rape (Brassica Napus), Potatoes (Solanum Tuberosum), Sugar beet (Beta vulgaris) and Wheat. http://www.soilassociation.org

[43] Tufto J., Engen S. and Hindar K. (1997) Stochastic Dispersal Processes in Plant Populations. Theoretical Population Biology, 52, 16-26. | Zbl 0889.92026

[44] Weinberger H.F. (1978) Asymptotic behaviour of a model in population genetics. In Chadam J.M.. Lecture Notes in Mathematics: Nonlinear partial differential equations and applications, 648, 47-96. | MR 490066 | Zbl 0383.35034