Stochastic models and statistical inference for plant pollen dispersal

Laredo, Catherine; Grimaud, Agnès

Journal de la Société française de statistique & Revue de statistique appliquée, Tome 148 (2007) no. 1, pp. 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.

Keywords: pollen dispersal, field experiment, cross-pollination rates, maize, airborne pollen, meteorological parameters, deconvolution, parametric inference, quasilikelihood, hypothesis testing, , individual pollen dispersal function, mechanistic models, lagrangian stochastic models, Langevin equations, hitting times
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

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