phytoplancton
Fr´ ed´ eric Barraquand, Coralie Picoche
Institut de Math´ ematiques de Bordeaux
8 Octobre 2019
frederic.barraquand@u-bordeaux.fr
1 - Qu’est-ce qui g´ en` ere les efflorescences ou “blooms”?
Croissance exponentielle de la biomasse sur des surfaces importantes (ha ` a km 2 et plus) Des raisons toujours d´ ebattues Importance pour l’´ ecologie : pompe ` a carbone, eutrophisation, toxines et intoxications
(alimentaires) induites...
Phaeocystis globosa blooming in Boulogne (IFREMER)
Chlorophyll a in the North Atlantic
(May 21 st , 2016, PREVIMER)
F. Barraquand, C. Picoche Le phytoplancton et ses maths
2 - Qu’est-ce qui maintient la diversit´ e du plancton?
The paradox of the plankton
[...] how it is possible for a number of species to coexist in a relatively isotropic or unstructured environment all competing for the same
sorts of materials. — Hutchinson, 1961
Blooms : les “mar´ ees rouges” – red tides
Bloom de Karenia Brevis, Floride. Credit: NOAA.
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Deux protagonistes et leur histoire
Lawrence Basil Slobodkin
Tout juste sorti de th` ese, ´ ecologue int´ eress´ e par la mod´ elisation du vivant, autour de 1950.
Karenia Brevis
Entre 20 et 40 µm, 2 flagelles.
Credit: FWC Fish and Wildlife
Research Institute.
Pr´ eambule : la diffusion comme mod` ele de mouvement
a) flux laminaire; b) flux turbulent
Approximation diffusive – mouve- ment al´ eatoire des particules
∂p
∂t = D ∂ 2 p
∂x 2 (1)
pour une densit´ e d’organismes p(x, t) dans l’eau. Approxime :
La turbulence du fluide Le comportement actif des organismes
R´ eorientation apr` es chocs (mouvement Brownien original)
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Observations et conjectures
Tir´ e de Okubo and Levin – Diffusion and Ecological Problems: Modern Perspectives (2001)
∃ Taille minimale des “red tides”
Probablement associ´ ees aux masses d’eau de faible salinit´ e
(NB masses d’eau = f(temp´ erature, salinit´ e) → densit´ e )
The size of water masses containing plankton blooms
Kierstad and Slobodkin, Journal of Marine Research, 1953. Adapt´ e de Okubo & Levin 2001.
∂S
∂t = D ∂ 2 S
∂x 2 + rS (2)
avec
S (x, t) = 0 pour x < 0 et x > L S (x, 0) = f (x) > 0 sauf aux bornes qui a pour solution
S (x, t) = A n
∞
X
n=1
sin ( nπx
L ) exp (r − Dn 2 π 2 /L 2 )t (3) avec A n = 2 L R 0 L f (x) sin ( nπx L )dx .
F. Barraquand, C. Picoche Le phytoplancton et ses maths
The size of water masses containing plankton blooms
De ce mod` ele on d´ eduit
L c = π s
D
r (4)
et en 2 dimensions
R c = 2.4048 s
D
r (5)
L c Longeur et R c rayon critique D diffusivit´ e horizontale
r taux de croissance per capita
En r´ ealit´ e, c’est plus compliqu´ e...
Distribution verticale
D´ emographie complexe des esp` eces toxiques Pr´ edateurs, parasites, etc.
Fischer et al., 2014. Sixty years of Sverdrup: A retrospective of progress in the study of phytoplankton blooms.
Oceanography.
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Mod` ele de distribution verticale (s = profondeur)
∂ω
∂t (s , t) = (p(I(s , t)) − l d )ω(s , t) + ν ∂ω
∂s (s, t) + D ∂ 2 ω
∂ 2 s (s , t) I(s , t) = I 0 exp
− Z z max
0
k ω(σ, t)d σ − K bg z max
νω(s , t) + D ∂ω
∂s (s , t) = 0 ∀t, s ∈ {0, z max }
(6)
ω(s, t) plankton abundance
p(I(s, t)) production related to light intensity l d mortality and sinking loss
ν laminar flow (advection)
D diffusion (turbulence)
k light interception
K bg background turbidity
Huisman & Weissing : comp´ etition pour la lumi` ere
∂ω i
∂t (s , t) = (p i (I(s , t))−l i,d )ω i (s, t)+ dD i ds (s) ∂ω i
∂s (s , t)+D i (s ) ∂ 2 ω i
∂ 2 s (s, t) R´ eponses ` a la lumi` ere esp` ece-sp´ ecifique
Huisman & Weissing, 1999. Species and dynamics in Phytoplank- ton Blooms: Incomplete Mixing and Competition for Light. The American Naturalist.
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Huisman & Weissing : comp´ etition pour la lumi` ere
∂ω i
∂t (s , t) = (p i (I(s , t))−l i,d )ω i (s, t)+ dD i ds (s) ∂ω i
∂s (s , t)+D i (s ) ∂ 2 ω i
∂ 2 s (s, t) Effet de l’environnement
Huisman & Weissing, 1999. Species and dynamics in Phytoplankton Blooms: Incomplete Mixing and Competition
for Light. The American Naturalist.
2 - Qu’est-ce qui maintient la diversit´ e du plancton?
Coexistence ` a petite ´ echelle spatiale, e.g. dans les mˆ emes 10 mL.
F. Barraquand, C. Picoche Le phytoplancton et ses maths
La coexistence dans des mod` eles de Lotka-Volterra (1926)
dN i
dt = r i N i −
n
X
j =1
α ij N j N i (7)
N i > 0, N j > 0 si α ii > α ij (et vice versa) pour n = 2
α ii >> α ij pour n grand (Barabas et al. American Naturalist 2016)
α ii >> α ij traduit le fait que les esp` eces ne rentrent pas beaucoup en comp´ etition parce qu’elles ont des
Ressources (nutriments, lumi` eres) diff´ erentes Pr´ edateurs diff´ erents
Habitats diff´ erents
Th´ eorie neutre de la biodiversit´ e (2000) - esp` eces indistinguables (` a un mˆ eme niveau trophique)
Si les esp` eces sont indistinguables, α ii = α ij par d´ efinition.
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Test des m´ ecanismes de coexistence sur donn´ ees r´ eelles
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2 3 4 5 6 7
Chaetoceros spp.
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Teychan Buoy 7
2000 2005 2010 2015
Log ab undance
1987 ` a maintenant Deux sites
tous les 15 jours
>300 taxa (≈ esp` eces) Variables
“environnementales”
0102030405060
Nitrogen/Silicon (µM/l)
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● Teychan Buoy 7 Nitrogen Silicon
Simplification du syst` eme phytoplancton
>300 taxa → 12 groupes au niveau genre
G´ en´ eralement la mˆ eme taille Comp´ etiteurs possible
Log10 Bio v olume ( µ m 3 ) AST NIT PSE SKE CHA GUI LEP RHI GYM PRP CR Y EUG
−2 0 1 2 3 4 5 6 7
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Hypoth` eses des mod` eles Lotka-Volterra
dN i
dt = r i N i −
n
X
j =1
α ij N j N i (8)
Effet lin´ eaire des densit´ es sur les taux de croissance par individu 1
N i dN i
dt = r i −
n
X
j =1
α ij N j (9)
Mod` eles alternatifs
Effet nonlin´ eaire g (N) 1 N i
dN i
dt = r i −
n
X
j=1
α ij g (N j ) (10)
Log
1 N i
dN i
dt = r i −
n
X
j =1
α ij ln(N j ) (11)
Astuce
d ln(N i )
dt = r i −
n
X
j=1
α ij ln(N j ) (12)
et finalement avec x = ln(N) dx i
dt = r i −
n
X
j =1
α ij x j (13)
F. Barraquand, C. Picoche Le phytoplancton et ses maths
Mod` eles autor´ egressifs multivari´ es (statistiques)
x t = (x i,t ) i=1,...,12 , x t+1 = x t + Ax t + Cu t + e t , e t ∼ N 12 (0, Σ) (14)
EUGCRYPRPGYMRHILEPGUICHASKEPSENITAST
AST NIT PSESKECHAGUI LEP RHIGYMPRPCRYEUG Sal. Irr. WindSeas.
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A ` a gauche, C ` a droite; u t environnement (lumi` ere, vent, ...).
Mod` eles autor´ egressifs multivari´ es (statistiques)
x t = (x i,t ) i=1,...,12 , x t+1 = x t + Ax t + Cu t + e t , e t ∼ N 12 (0, Σ) (14)
EUGCRYPRPGYMRHILEPGUICHASKEPSENITAST
AST NIT PSESKECHA GUI LEP RHIGYMPRPCRYEUG Sal. Irr. WindSeas.
