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A Stochastic partial differential equation forphytoplankton aggregationN. El SaadiGEODES-IRD 32, avenue Henri VaragnatF-93143 Bondy cedex, France.ABSTRACT. The aim of this work is to provide a stochastic mathematical model of aggregation inphytoplankton, from the point of view of modelling a system of a large but finite number of phytoplanktonparticles that are subject to random dispersal modeled as Brownian motions, mutual interactionsallowing the particles motions some dependance and branching (creation or annihilation). We presentthe passage from the ”microscopic” description to the ”macroscopic” one, when the initial total numberof particles tends to infinity (large phytoplankton populations). The limit of the system is then anextension of the Dawson-Watanabe superprocess: it is a superprocess with spatial interactions whichcan be described by a stochastic partial differential equation.RÉSUMÉ.KEYWORDS : Phytoplankton aggregation, Interacting branching diffusion process, Martingale problem,Weak convergence, Dawson-Watanabe superprocess, Stochastic partial differential equation.MOTS-CLÉS :TAMTAM –Tunis– 2005 526
SPDE for phytoplankton aggregatios 5271. IntroductionThe role of aggregates in marine food webs and vertical transport processes is nowwell recognized ([1], [9]). Recent attention has been devoted to modeling studies of themechanisms by which aggregates form and the dynamics governing their formation.Coagulation theory has more recently been applied to describe aggregation of marineparticles and specifically phytoplankton aggregation. Many laboratory experiments,mesocosm experiments and field observations have all demonstrated that coagulation theoryat times provides an accurate description of phytoplankton aggregate formation (see[12]). Aggregation by physical coagulation requires that primary particles collide bysome physical process and stick together upon collision. However, studies of marineaggregates at small-scales have emphasized biological mechanisms for their formation.Indeed, some planktonic species (algae, bacteria, dinoflagellates) have chemosensoryabilities ([8],[15]), that is, the capability of sensing the chemical field generated by thepresence of others particles. Dinoflagellates and more generally algae are known to leakorganic matter into solution [13] and this leakage creates a zone around individual cellscalled the ” phycosphere”, where extracellular products exist in enhanced concentrationsover background [2]. The released products as amino-acids and sugar attract algae orbacteria that are present in a suitable neighborhood.The goal of this work is to develop a mathematical rigorous model of aggregationin phytoplankton that takes into account the small-scales biological interactions abovementioned.For this purpose, we start with a microscopic description of a finite systemof phytoplankton cells and pass after rescaling to a continuous limit (a macroscopic description).The dynamics of the system at small scale are: a random dispersal due to theenvironment turbulence, attraction mechanisms between cells due to the chemosensorybehavior and a random branching modeling cell division and natural death. Our modelcan be considered as an extension of the class of models reviewed in [14].In [14], a background for the modelling of interacting systems is given. In this paper,we add branching mechanisms for our interacting particle system, that is the particles canreproduce and die. We do not consider other processes for the planktonic particles suchus growth, gravitational sinking or grazing by higher animals. The limit obtained in ourcase is a measure-valued process. It is a ” superprocess with spatial interactions”.We stress on the fact that, owing to the branching, the approach in [14] for the derivationof the continuum model fails in our situation. Instead, we use a martingale approach.The paper is structured as follows: in section 2, we describe the dynamics of thesystem of phytoplankton particles (spatial motion, spatial interactions and branching).In section 3, we define the interacting branching-diffusion process to describe thespatial and temporal distribution of the system. We set the martingale problem associatedto this process and characterize the latter as the unique solution to this martingale problem.Section 4 is devoted to the limit of the system when the initial number of particlestends to infinity. We prove the weak convergence of the rescaled interacting branching-TAMTAM –Tunis– 2005
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PréfaceOrganiser la seconde éditi
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Comité d’organisationMohamed Abd
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SOMMAIRE / CONTENTSI Conférences I
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Equilibre dans un système dynamiqu
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Arlequin method: Practical impacts
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Application de la méthode de Gauss
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Conférences InvitéesInvited Prese
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4 Amara et al.1. IntroductionWe are
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6 Amara et al.2.2. The nonlinear Na
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8 Amara et al.4. Numerical resultsW
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10 Auger et al.1. Introduction :Dan
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12 Auger et al.La condition de stab
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14 Auger et al.3.1. Taux de migrati
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16 Auger et al.L’équilibre rapid
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18 Auger et al.[7] BRAVO DE LA PARR
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Quelques aspects mathématiques etn
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22 El Alaoui Talibi M.- Modèle d
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24 El Alaoui Talibi M.[3] M. Chamba
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26 Habbal1. IntroductionAngiogenesi
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28 HabbalThe pressure drop denoted
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30 Habbaltheorem 1 There exists a N
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32 HabbalFigure 2. Second Nash loop
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34 Hauray et al.1. IntroductionWe a
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36 Hauray et al.minimal time interv
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38 Hauray et al.From this last theo
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Approximation de l’opérateur d
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42 LemrabetLorsque δ → 0, l’é
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44 Lemrabetavec[∆ ∗ (δ)c = [I
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46 Lemrabet6. Bibliographie[1] N. B
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48 Jaafar-Belaid et al.1. Introduct
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50 Jaafar-Belaid et al.where f(ρ)
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52 Jaafar-Belaid et al.202040406060
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IIContrôleControl55
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58 Arfaoui1. IntroductionCe travail
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60 ArfaouiLe système adjoint assoc
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62 Arfaoui[3] H. ARFAOUI , « Comma
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64 Benzekri et al.1. Global control
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66 Benzekri et al.ẏ = ∂H∂x =
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0 1 2 3 4 5 668 Benzekri et al.32.5
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70 Akian et al.1. IntroductionWe co
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72 Akian et al.We prove that v t+δ
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.74 Akian et al.0.0−1.0Exact solu
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76 Bokanowski et al.1. General fram
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78 Bokanowski et al.One interesting
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80 Bokanowski et al.4. Numerical ex
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Sur les problèmes de contrôle opt
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84 Metouileur lorsque u est dans L
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86 MetouiThéorème 2 Soit ū ε le
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88 Metoui[5] L.S. HOU, S.S. RAVINDR
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Système d’information intégré
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Gestion et modélisation des ressou
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Random perturbations of reduced gra
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Global Optimization of Water Resour
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Global Optimization of Water Resour
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IVEquations DifférentiellesDiffere
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114 B. Abdelkader1. IntroductionTel
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116 B. AbdelkaderNotons que les con
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118 B. AbdelkaderAlors pour tout X
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120 Bouzahir1. IntroductionWe consi
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122 Bouzahirwhere the linear operat
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Goursat boundary value problem forh
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126 A. Guezane-Lakoud et al.3. Func
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128 A. Guezane-Lakoud et al.5. Exis
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ACP neuronale : application à l’
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132 Chitroubl’approximation de la
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134 Chitroubcompression égal à 3.
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136 Chitroub[4] J. P. Nougier, Mét
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138 Ferchichi et al.1. Introduction
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140 Ferchichi et al.soit T −1 = (
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142 Ferchichi et al.[4] G.I. BISCHI
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144 Hafidi1. IntroductionOn étudie
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146 Hafidia ) ∫ Ωpdx → +∞ qu
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148 HafidiLEMME 3.2. On a lim R(θ
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150 Lefebvre1. IntroductionSoit (X
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152 LefebvreOn peut aussi écrire q
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154 Lefebvre5. BibliographieD.R. Co
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156 Meskine1. IntroductionDans le p
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158 Meskinefaibles σ( ∏ L M ,
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160 Meskine[12] E. YA. KHRUSLOV, L.
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162 K. Saoudi1. IntroductionL’ét
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164 K. Saoudice qui montre que la s
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166 K. Saoudieton aboutit au résul
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168 K. Saoudietc 1 + c 2 = 1 ∫(u
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170 K. SaoudiRemarque 3.1. Du lemme
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172 K. Saoudiet∫v(t, x) dxΩest c
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VEstimation d’ErreursError Estima
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178 Achchab et al.1. IntroductionDu
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180 Achchab et al.such that}Xh 2 ={
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182 Achchab et al.[2] B.ACHCHAB, M.
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184 Alla et al.1. IntroductionLes e
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186 Alla et al.3. Position du probl
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IsoValue-0.00070533-0.000604452-0.0
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190 Achchab et al.1. IntroductionLe
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192 Achchab et al.où T hi/2 est le
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IsoValue0.000933570.002800710.00840
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196 El Dabaghi et al.1. Motivation
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198 El Dabaghi et al.0 ≤ λ i ≤
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200 El Dabaghi et al.Figure 4. Mail
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Residual error estimators for the t
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204 Kharrat et al.The step (3) need
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206 Kharrat et al.following estimat
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VIMécanique des FluidesFluid Mecha
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212 Abdelwahed et al.1. Introductio
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214 Abdelwahed et al.Ainsi, suivre
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216 Abdelwahed et al.Figure 3. Isov
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Procédure spectrale avec diagonali
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220 El Guarmah et al.où Ra et P r
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222 El Guarmah et al.M 10 20 25 35
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Simulation de l’onde de crue via
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226 El Dabaghi et al.de R 2 , de fr
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228 El Dabaghi et al.INRIA-MODULEFm
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INRIA-MODULEF230 El Dabaghi et al.I
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232 Amoura et al.1. IntroductionFor
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234 Amoura et al.where u = rot r ψ
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236 Amoura et al.CURRENT FUNCTION5
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238 Frih et al.1. IntroductionLa mo
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240 Frih et al.et dans la fracture,
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242 Frih et al.6. Bibliographie[1]
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244 Hasnaoui et al.1. IntroductionL
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246 Hasnaoui et al.On obtient alors
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248 Hasnaoui et al.5. Conclusion et
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250 Hernane-Boukari1. IntroductionW
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252 Hernane-Boukarianda 3 =−11 +
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254 Kloucha et al.1. Motivation et
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256 Kloucha et al.Si le domaine Λ
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INRIA-MODULEF0INRIA-MODULEF.250INRI
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Dimensions fractales : attracteurs
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262 Lamrini Uahabi et al.4. Attract
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264 Lamrini Uahabi et al.l 0 = λ
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266 Lamrini Uahabi et al.En conséq
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Numerical analysis for the problem
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270 El-Kyal et al.d(u) = 1 2(∇u +
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272 El-Kyal et al.Theorem 3.1 There
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274 Marchand et al.1. IntroductionD
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276 Marchand et al.On remarque en p
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278 Marchand et al.dans la formulat
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280 Mezali et al.1. Motivation et m
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282 Mezali et al.s’annulant sur c
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284 Mezali et al.Temps CPUTemps Ela
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R. Touihri & S. Ghnimi *Méthode de
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288 Touihri et al.Les équations go
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290 Touihri et al.Tol N It (Newt) t
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292 Touihri et al.5. Bibliographie[
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Décomposition de domaine pour un m
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DDM pour les fractures 297et−→
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DDM pour les fractures 299dont l’
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Arlequin method: Practical impacts
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Arlequin method 303of the Arlequin
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Arlequin method 3052.2. Penalized A
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Arlequin method 307refinements of t
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Reduction methods and uncertaintypr
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Reduction methods and uncertainty p
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Reduction methods and uncertainty p
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0.00e+00 1.00e−11 2.00e−11 3.00
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Homogénéisation d’un problème
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Problème de conduction-rayonnement
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Problème de conduction-rayonnement
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Equation d’Hamilton-Jacobi 3231.
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Equation d’Hamilton-Jacobi 325qui
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0Equation d’Hamilton-Jacobi 3272
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Détermination du nombre de région
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Simulation des courants de Foucault
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Une méthode d’accélération de
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Schéma SRNHS 3491. IntroductionUn
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Schéma SRNHS 3513. Application du
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Schéma SRNHS 3535. ConclusionLes r
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Smoothing and compressing surfaces
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Problème de convection en milieu p
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
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Condensation de masse pour la méth
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Condensation de masse 367montre que
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Condensation de masse 369Le terme h
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Condensation de masse 371Aet Qile f
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VIIIModèles CinétiquesKinetic Mod
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376 Ben Abdallah et al.1. Introduct
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378 Ben Abdallah et al.where B ∓
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380 Ben Abdallah et al.ture that th
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382 Degond et al.1. IntroductionOn
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384 Degond et al.q φ s (x, t)/(|q|
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386 Degond et al.Enfin l’opérate
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Les métaheuristiques : application
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Les métaheuristiques 391simulé fu
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Les métaheuristiques 393Fig. 1 Dia
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Les métaheuristiques 395Fig.4 : R
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Prices after capacity addition in m
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Multi-user elastic demand communica
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Shape optimization 4051. Introducti
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Shape optimization 4072.1. Variatio
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Shape optimization 409(a)(b)(c)(d)F
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Problèmes d’optimisation en éco
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Assimilation de données pour l’e
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Écart à la Réciprocité et ident
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Identification de fissures planes 4
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Algorithme de Kohn et Vogelius 4331
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Algorithme de Kohn et Vogelius 435l
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Algorithme de Kohn et Vogelius 4375
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Résolution d’un problème de Cau
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Problème inverse géométrique 445
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Identification de fissures 4511. In
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−0.5 −0.4 −0.3 −0.2 −0.1
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Application de l’algorithme deNeu
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Complétion de données en élastic
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Complétion de données en élastic
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Analytic extensions on an annulus:a
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Analytic extensions on an annulus 4
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Analytic extensions on an annulus 4
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−1 −0.8 −0.6 −0.4 −0.2 0
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Cauchy problem for Laplace’s equa
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Page 495 and 496: Modélisation asymptotique des onde
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Page 503 and 504: Régularisation pour l’aéroacous
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Page 527 and 528: ¨§¨ §IRAM method 509To overcome
Page 529 and 530: IRAM method 511The application of t
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Page 534 and 535: 516 Achchab1. IntroductionDans la m
Page 536 and 537: 518 AchchabNous montrons dans le pa
Page 538 and 539: 520 Achchab5. Bibliographie[1 ] AIE
Page 540 and 541: 522 Derbel et al.1. IntroductionLa
Page 542 and 543: 524 Derbel et al.On se propose de d
Page 546 and 547: 528 El Saadidiffusion process to a
Page 548 and 549: 530 El Saadiwhere δ Xi(t) is the D
Page 550 and 551: 532 El Saadi5. References[1] A. L.
Page 552 and 553: 534 Ben Miled et al.1. Introduction
Page 554 and 555: 536 Ben Miled et al.nature de l’e
Page 556 and 557: 538 Ben Miled et al.Ensuite on étu
Page 558 and 559: Optimal spatial distribution of a b
Page 560 and 561: 542 Mchich et al.at time t. The com
Page 562 and 563: 544 Mchich et al.3∑where r = r i
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Page 568 and 569: 550 Ayadi1. IntroductionConsider a
Page 570 and 571: 552 AyadiFor all 0 ≤ p ≤ i, P p
Page 572 and 573: 554 AyadiFigure 3. The curves above
Page 574 and 575: Analyse mathématique et numérique
Page 576 and 577: 558 Chamekh et al.2. Modèle de tig
Page 578 and 579: 560 Chamekh et al.Γ sr(s)r(σ)Figu
Page 580 and 581: Modélisation de l’effet dynamiqu
Page 582 and 583: 564 RahmaniI 2 étant l’identité
Page 584 and 585: 566 Rahmani• ( u 1δ1∫+ ,0u 1δ
Page 586 and 587: Résolution d’un systéme d’ela
Page 588 and 589: 570 Saidoù u vérifie :n∑n∑j=1
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576 Achchab et al.On définit : F +
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578 Achchab et al.3. Estimation a p
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Modélisation d’un pieuR. Aboulai
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582 Aboulaich et al.Figure 1. Struc
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584 Aboulaich et al.5. Résultats n
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XIVThéorie des GraphesGraph Theory
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590 Hlaoui1. IntroductionMore and m
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592 Hlaoui3. The New Graph Represen
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594 Hlaoui1 1 2 3 4 1 1 3 411 421 4
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596 M. Nekri et al.1. IntroductionL
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598 M. Nekri et al.proposés. Ces m
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600 M. Nekri et al.cycle de longueu
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