Un modèle spatialisé de la population d'anchois et sa dynamique de 1990 à 2001

Sandrine Vaz & Pierre PetitgasIfremer, DRV / RH / Laboratoire d’Ecologie Halieutique,

Centre de Nantes

La population d’anchois dans le Golfe de Gascogne présente une variabilité importante

Thématique

Dynamique de la population nécessite• Cycle de vie• Interaction avec l’environnement

• Distribution spatiale (frayères)

Différents habitats sont occupés par différentes sous population

• Projection de l’abondance

Les modèles démographiques pour populations sous divisées

• Dynamique de la population • Distribution spatiale et préférences d ’habitat

Effet des variations des paramètres démographiques sur la croissance, l ’abondance et la distribution spatiale de la population

Distribution des longueurs, poids et âges moyens de la population d ’anchois

A

E

DC

B

Caractérisation and distribution spatiale des sous-populations ayant différents traits démographiques

Definition de 5 zones

Structure du modèle

1 2 3 4

Classe 1 Classe 2 Classe 3

P1

F3F2

P2

F1

N(t) = (n1, n2, n3)

n1(t+1) = F1n1(t) + F2n2(t) + F3n3(t)

n2(t+1) = P1n1(t) n3(t+1) = P2n2(t)

Modèles matriciels - Principe

fa1 fa2 fa3 m0ba1 m0ba2 m0ba3 m0ca1 m0ca2 m0ca3 m0da1 m0da2 m0da3 m0ea1 m0ea2 m0ea3pa1 0 0 m1ba 0 0 m1ca 0 0 m1da 0 0 m1ea 0 00 pa2 0 0 m2ba 0 0 m2ca 0 0 m2da 0 0 m2ea 0

m0ab1 m0ab2 m0ab3 fb1 fb2 fb3 m0cb1 m0cb2 m0cb3 m0db1 m0db2 m0db3 m0eb1 m0eb2 m0eb3m1ab 0 0 pb1 0 0 m1cb 0 0 m1db 0 0 m1eb 0 00 m2ab 0 0 pb2 0 0 m2ca 0 0 m2db 0 0 m2eb 0

m0ac1 m0ac2 m0ac3 m0bc1 m0bc2 m0bc3 fc1 fc2 fc3 m0dc1 m0dc2 m0dc3 m0ec1 m0ec2 m0ec3m1ac 0 0 m1bc 0 0 pc1 0 0 m1dc 0 0 m1ec 0 00 m2ac 0 0 m2bc 0 0 pc2 0 0 m2dc 0 0 m2ec 0

m0ad1 m0ad2 m0ad3 m0bd1 m0bd2 m0bd3 m0cd1 m0cd2 m0cd3 fd1 fd2 fd3 m0ed1 m0ed2 m0ed3m1ad 0 0 m1bd 0 0 m1cd 0 0 pd1 0 0 m1ed 0 00 m2ad 0 0 m2bd 0 0 m2cd 0 0 pd2 0 0 m2ed 0

m0ae1 m0ae2 m0ae3 m0be1 m0be2 m0be3 m0ce1 m0ce2 m0ce3 m0de1 m0de2 m0de3 fe1 fe2 fe3m1ae 0 0 m1be 0 0 m1ce 0 0 m1de 0 0 pe1 0 00 m2ae 0 0 m2be 0 0 m2ce 0 0 m2de 0 0 pe2 0

A B C D E

A

B

C

D

E

Structure de la matrice

0

1

2

3

4

5

6

7

8

9

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

année

Ab

on

da

nc

e (

mil

lia

rds

d'in

div

idu

s)

A B

C D

E LAG

Données disponibles

Abondance et longueur moyenne par classe d’âge et par zone

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

A B C D E

Zones

Ab

un

da

nc

e (

% o

f to

tal

nu

mb

er)

Age3+

Age2

Age1

• Abondance moyenne de la population sur 11 ans (9 ans d ’observations)

•Population stable (taux de croissance = 1)

• Structure moyenne • Longueur moyenne par classe d ’âge et par zone

Quels paramètres démographiques permettent le renouvellement et la maintenance de la population?

Quelles sont les propriétés et les caractéristiques d ’un tel modèle de population?

Modélisation d ’une situation stable

Fertilité (locale)

Mortalité (globale)

• fertilité net = S0 x Fn• fonction de la longueur et du poids moyen

tntntn NNS ,1,1, • Probabilité de survie

Migration - le schéma de redistribution de la population

Différence dans la distribution spatiale relative d ’une cohorte donnée d ’une année sur l ’autre

Détermination des paramètres

A

B

C

DE

12%

44%

0.5%

8%

Classe Age 1

AB

C

DE

4%

1%

Classe Age 2

10%

7%

AB

C

DE

38%

20%

20%

Classe Age 0

5%

Le schéma de redistribution permettant le maintient de la distribution de la population moyenne

Taux de croissance ~ (ici = 1)

A B C D E1 0.52 0.02 0.08 0.06 0.032 0.08 0.01 0.02 0.003 0.0073 0.004 0.002 0.003 0.001 0.001

Elasticité de aux variations de la fertilité

meta-population0 -9.741 -0.262 -0.03

Variation de la mortalité

A B C D E0 -0.03 -0.02 -0.02 -0.0081 -0.0082 -0.003 <-0.001

Variation de la migration

A B C D EStable structure of newborns 0.1402 0.0991 0.1597 0.5229 0.078Reproductive value of newborns 5.1387 0.3047 0.6724 0.1952 0.5089Mean generation length 1.1578 0.8547 1.1163 0.8967 1.0837

Quelques caractéristiques démographiques

Propriétés de la matrice ULM (Legendre & Clobert, 1995)

La dynamique générale de la population dépend de l ’ordre de succession des dynamiques annuelles

0.52

2.44

0.44

1.101.32

1.09

1.811.811.831.831.83

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

year

Firs

t e

ige

n v

alu

e (

~ g

row

th r

ate

)

observed year

lag year

average growth

stability

1.67

Evolution de dans le temps

L ’étude des propriétés du modèle sur les 11 années de données consécutive a montré que la dynamique de la population est très variable d ’une année sur l ’autre

Conclusion

• La croissance de la population modélisée est déterminée par le recrutement

• Les variations annuelles des paramètres démographiques conditionnent la dynamique générale de la population

Perspectives

• L ’étude de l ’effet de l ’ordre de succession des dynamiques annuelles sur la population à moyen terme

•Relier les variations des taux démographiques et de la distribution spatiale à des variables environnementales

• Simuler des scénarios environnementaux ou d ’exploitation et étudier leurs effets sur la population d ’anchois

The anchovy population in the bay of Biscay displays an important abundance variability

Problematic

0

20000

40000

60000

80000

100000

120000

140000

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Year

To

tal B

iom

ass

(to

ns)

Anchovy total biomasse evaluated from French acoustic surveys over the past 11 years

Population dynamic requires • Life cycle• Interaction with environment

• Spatial distribution (spawning grounds)

Different habitats are occupied bydemographically different sub-population

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Age 2

0.202 (20.2%)

0.760 (76%)

Lg

date

Lat

yeardepthage3

age2

age1

age4

Characterisation of sub-populations

Spatial distribution of sub-populations

Definition of areas with different demographic characteristics

Demographic models for subdivided population

• population dynamics • Spatial distribution and habitat preferences

Effect of varying demographic rates on population growth, abundance and spatial distribution

Characterisation of sub-populations

Spatial distribution of sub-populations

Definition of areas with different demographic characteristics

Definition of zones

1 2 3 4

1 2 3 4

Region A

Region B

P1 (MIGab1) P2 (MIGab2)

P1 (1-MIGab1) P2 (1-MIGab2)

1 2 3 4

1 2 3 4

Region A

Region B

Fn(MIGab0)

Fn(1-MIGab0)

tn

n

n

P

P

FFF

t

n

n

n

3

2

1

2

1

321

3

2

1

00

00)1(

Leslie Matrix

N(t+1) = A N(t)

Multi-site model

B

B

B

A

A

A

n

n

n

n

n

n

regionB

regionAN

3

2

1

3

2

1

B

AB

BA

A

A

M

M

AA

Fertility (local)

Mortality (global)

• Motos (1996) - number of egg per grams

• net fertility (parameter for birth pulse model) = S0 x Fn

class 1 class 2 class 3A 92428 93447 96911B 98810 117802 125701C 98994 117827 128931D 72799 81492 77293E 98035 110548 124188

Total fertility per age classe and per zone• function of the average length/weight

class 0 class 1 class 2Z 11.5 1.9 1.9

Total instantaneous mortality over the 5 zones (Z = -Ln(S))

3

1,,,0

ntntnt NFN

• Number of newborn

tntntn NNS ,1,1, • Survival probability

Migration - the redistribution patterns of the population

Difference in the relative spatial distribution of a given cohort from one year to the next

Determine the amount of individuals lost or gained by each zone

Equivalent to assuming global mortality rates and addressing the observed local differences in term of redistribution

Rates are determined to minimise the distance between zones

Movers from a given zone will move to the geographically closest zone that have gained individuals

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20years

ab

un

da

nc

e (

mill

ion

s o

f in

div

idu

als

)

na

nb

nc

nd

ne

Population projection and matrix properties

1 (~ growth rate) = 1

Population structure isstable

A B C D EStable structure of newborns 0.1402 0.0991 0.1597 0.5229 0.078Reproductive value of newborns 5.1387 0.3047 0.6724 0.1952 0.5089Mean generation length 1.1578 0.8547 1.1163 0.8967 1.0837

A few demographic characteristics

ULM (Legendre & Clobert, 1995)

Matrix elasticity

A B C D E1 0.52 0.02 0.08 0.06 0.032 0.08 0.01 0.02 0.003 0.0073 0.004 0.002 0.003 0.001 0.001

Elasticity of to fertility variation

meta-population0 -9.741 -0.262 -0.03

Elasticity of to mortality variation

A B C D E0 -0.03 -0.02 -0.02 -0.0081 -0.0082 -0.003 <-0.001

Elasticity of to “migration rates” variation

• Model sensitivity to initial conditions

The stabilising role of migration

• Role of newborns redistribution

• Role of adult redistribution

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20

year

ab

un

da

nc

e (

bill

ion

s o

f in

div

idu

als

)

na

nb

nc

nd

ne

t = 0, equal abundance in all zones The population stable structure is restored

• 1 (~ growth rate) = 1.13None

• only zone C is populated { • 1 (~ growth rate) = 1.00

None• only zone A and D are populated {