ModlisationModlisationModlisationModlisation mathmatiquemathmatiquemathmatiquemathmatique en en en en cologiecologiecologiecologie ::::

de de de de lindividulindividulindividulindividu la population et la la population et la la population et la la population et la

communautcommunautcommunautcommunaut

Pierre Auger

IRD, UR Geodes et MOMISCO

Centre de recherche IRD de lIle de France, Bondy

E-mail: pierre.auger@bondy.ird.fr

Universit de Rennes 1, 26 mars 2008

Outline of the presentation

Predator-prey model in a patchy environment

Effects of density dependent migrations

Aggregation methods in time discrete models

Aggregation methods for ODEs : emergence

Ecological levels of organization

Individual level

Population level

Community level

Time scales and levels of organization

Individual level : Day

Population level : Year

Community level : Evolutionary time scales

Take advantage of time scales to aggregate :

Build an aggregated (reduced) model governinga few global variables at a slow time scale.

Aggregation of variables

More realistic models, i.e. structured population models : behaviour, age, species

Predator-prey model in a two patch

environment with fast migration

A patchy environment

Two patches

R. R. MchichMchich, P. Auger and N. , P. Auger and N. RaRassissi, 2000. , 2000. The dynamics of a fish stock exploited on two fishing zonesThe dynamics of a fish stock exploited on two fishing zones . . ActaActa BiotheoreticaBiotheoretica. Vol. 48, N. . Vol. 48, N. , pp. 207, pp. 207--218. 218.

Fast dispersal

( )

( )

1 12 1 1 1 1 1 1

1

2 21 2 2 2 2 2 2

2

12 1 1 1 1 1 1

21 2 2 2 2 2 2

' 1

' 1

'

'

dx xkx k x r x a x p

d K

dx xk x kx r x a x p

d K

dpmp m p b x p d p

ddp

m p mp b x p d pd

= +

= +

= +

= +

The fast model

1 2

1 2

x x x

p p p

= += +Total prey and predator densities are constant at fast time scale:

12 1

21 2

12 1

21 2

'

'

'

'

dxkx k x

ddx

k x kxddp

mp m pddp

m p mpd

=

=

=

=

Fast and stable equilibrium

1 1

1 1

( ) ' 0

( ) ' 0

k x x k x

m p p m p

= =

*1 1

*2 2

''

'

kxx x

k kk x

x xk k

= =+

= =+

*1 1

*2 2

''

'

mpp p

m mm p

p pm m

= =+

= =+

The aggregated model

1dx x

rx axpdt K

dpbxp dp

dt

=

= ( ) ( )

1 1 2 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 2 2

2 2

1 1 2 2

1 2

r r r

a a a

b b b

d d d

rK

r r

K K

= += += += +

=

+

with

Condition for predator persistence

bK d> bK d

AggregatedAggregated model model similarsimilar to to completecomplete modelmodel

Complete model Aggregated model

( )

( )

1 12 1 1 1 1 1 1

1

2 21 2 2 2 2 2 2

2

12 1 1 1 1 1 1

21 2 2 2 2 2 2

' 1

' 1

'

'

dx xkx k x r x a x p

d K

dx xk x kx r x a x p

d K

dpmp m p b x p d p

ddp

m p mp b x p d pd

= +

= +

= +

= +

1dx x

rx axpdt K

dpbxp dp

dt

=

=

When the fast derivation

method fails

1 2

( )

( )

=

+=

+=

11

222121212

1111212121

NebPd

dP

NrNmNmd

dN

ParNNmNmd

dN

=

=

PeaNPdt

dP

aNPrNdt

dN

N

Nr

N

Nrr

*2

2

*1

1 += NN

aa*

11=

2112

21*2

2112

12*1 mm

NmN

mm

NmN

+=

+=

Reduction of the model

( )

( )

=

=

=

0

;;

;;

d

d

yxgd

dy

yxfd

dx

( );yhx =

( )( )( ) ( ) ( )

+=

=

OyGyG

yyhgdt

dy

0;;

;;;

Thorme de Fnichel (1971)

( ) ( )( )21122

121*

2*

11 ;

mmN

ParrNNPN

+=

=

=

PeaNPdt

dP

aNPrNdt

dN

( )( )

( )

+=

+=

PNNPeaPeaNPdt

dP

ParrPNNaNPrNdt

dN

;

;

11

1211

Fenichel center manifold theorem

The fast derivation method is valid :

If the aggregated model is structurally stable

If

References on aggregation methods

PDEs et DDEs :

ODEs, JC Poggiale, PhD thesis (1994) :

Time discrete systems : L. Sanz, PhD thesis (1998) and A. Blasco, PhD thesis (2000) :

Perfect and approximate aggregation :

Y. Iwasa, V. Andreasen, S.A. Levin. Aggregation in model ecosystems I. Perfect aggregation. Ecol Model. 37 (1987) 287.

Y. Iwasa Y., S.A. Levin, V. Andreasen. Aggregation in model ecosystems II. Approximate aggregation. IMA. J. Math. Appl. Med. Bio. 6 (1989) 1.

P. Auger et R. Roussarie, Complex ecological models with simple dynamics : From individuals to populations, Acta Biotheoretica, 42 (1994) 111.P. Auger & J.C. Poggiale. Aggregation and Emergence in Systems of Ordinary Differential Equations. Math. Computer Modelling. Special issue "Aggregation and Emergence in Population Dynamics". P. Antonelli & P. Auger guest-Editors. 27 (1998) 1.P. Auger and R. Bravo de la Parra. Methods of aggregation of variables in population dynamics. Comptes-Rendus de lAcadmie des Sciences, Sciences de la Vie, 323 (2000) 665.P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sanchez, L. Sanz, T. Nguyen Huu, Chapter in a book, Le Havre Conference, Lecture notes in Mathematics, Mathematical Biosciences sub-series, Springer, to appear.

O. Arino, E. Sanchez, R. Bravo de la Parra and P. Auger. A model of an age-structured population with two time scales. SIAM J Appl. Math., 60, 408-436, 1999.

Sanchez E., Bravo de la Parra R., Auger P. and Gomez-Mourelo P. Time scales in linear delayed differential equations. Journal of Mathematical Analysis and Applications, 323, pp. 680-699, 2006.

R. Bravo de la Parra, P. Auger & E. Sanchez, Aggregation methods in discrete models, Journal of Biological Systems, (1995), 3(2), 603-612.E. Sanchez, R. Bravo de la Parra & P. Auger. Linear Discrete Models with Different Time Scales. Acta Biotheoretica, 43, pp. 465-479, 1995.R. Bravo de la Parra, E. Sanchez, O. Arino and P. Auger. A discrete model with density dependent fast migration.Mathematical Biosciences, 157, 91-109, 1999.

D-D dispersal in a patchy environment

P. Amarasekare. Interactions between Local Dynamics and Dispersal: Insights from Single Species Models. Theoretical Population Biology, 53 (1998) 44.

P. Amarasekare. Spatial variation and density-dependent dispersal in competitive coexistence. Proc. R. Soc. Lond. B 271 (2004) 1497.

P. Amarasekare. The role of density-dependent dispersal in sourcesink dynamics. Journal of Theoretical Biology, 226 (2004) 159.

DensityDensity dependentdependent migration of migration of predatorspredators

R. Mchich, P. Auger, R. Bravo de la Parra and N. Rassi, 2002. Ecological Modelling.

11 0

1'( )m x

x =

+

22 0

1( )m x

x =

+

( )

( )

1 12 1 1 1 1 1 1

1

2 21 2 2 2 2 2 2

2

12 2 1 1 1 1 1 1 1

21 1 2 2 2 2 2 2 2

' 1

' 1

( ) '( )

'( ) ( )

dx xkx k x r x a x p

d K

dx xk x kx r x a x p

d K

dpm x p m x p b x p d p

ddp

m x p m x p b x p d pd

= +

= +

= +

= +

Fast and stable equilibrium

*1 1

*2 2

''

'

kxx x

k kk x

x xk k

= =+

= =+

( )( )

( )( )

1 0*1 1

1 2 0

2 0*2 2

1 2 0

( )2

( )2

x pp x p

x x

x pp x p

x x

+= =

+ +

+= =

+ +

1 ( )

( ) ( )

dx xrx a x xp

dt K

dpb x xp d x p

dt

=

= ( ) ( )

1 1 2 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 2 2

2 2

1 1 2 2

1 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

r r r

a x a x a x

b x b x b x

d x d x d x

rK

r r

K K

= +

= +

= +

= +

=

+

Aggregated model

EmergenceEmergence

The complete model The aggregated model

1 ( )

( ) ( )

dx xrx a x xp

dt K

dpb x xp d x p

dt

=

= ( )

( )

1 12 1 1 1 1 1 1

1

2 21 2 2 2 2 2 2

2

12 2 1 1 1 1 1 1 1

21 1 2 2 2 2 2 2 2

' 1

' 1

( ) '( )

'( ) ( )

dx xkx k x r x a x p

d K

dx xk x kx r x a x p

d K

dpm x p m x p b x p d p

ddp

m x p m x p b x p d pd

= +

= +

= +

= +

EmergenceEmergence

Functional emergence : The mathematical functions of the aggregated and local models are different

Dynamical emergence : The dynamics of the aggregated model isqualitatively different from the dynamics of the local model

The predator-prey model with prey D-D predator dispersal and predator D-D prey dispersal

1 12 2 0 2 1 1 0 1 1 1 1 1 1

1

2 21 1 0 1 2 2 0 2 2 2 2 2 2

2

1 2 11 1 1 1 1

2 2 0 1 1 0

2 1 2

1 1 0 2 2 0

( ) ( ) 1

( ) ( ) 1

[ ]

dn np n p n rn a n p

d K

dn np n p n r n a n p

d K

dp p pp b n p

d n n

dp p p

d n n

= + + +

= + + +

= + + + +

= + + + 2 2 2 2 2[ ]p b n p

+

ELABDELLAOUI A., AUGER P., BRAVO DE LA PARRA R., KOOI B. & MCHICH R. Mathematical Biosciences, (2007).

Fast equilibrium

2 2 0 2 1 1 0 1

2 1

2 2 0 1 1 0

( ) ( )p n p n

p p

n n

+ = + = + +

2 1 0 1 1 1 0 1

1 1 1 0 1 2 1 0

( ( ) )( ) ( )

( )( ) ( ( ) )

p p n n p n

p p n p n n

+ = + + = +

Fast equilibrium

1 1 2 1 2 1

AD B CD En p n n n p p p

+ += , = , = , = ,

2 2 1 0 2 2 1

0 2 0 2 0 2

0 2 1 2 1 2 1

0 0 2 1 0 2

( ) (2 )( )

(2 )( )

(2 )( ) ( )

(2 ) ( )

A n n

B np n p n

C p p

E p p np p

= +

= + +

= +

= + +

with

0 2 0 2 2 1(2 )(2 )p n np = + +

The aggregated model

1 11 1 2 1 1 1 1 2 1 1

1 2

1 1 2 1 1 1 1 2 1 1

1 ( ) 1 ( ( )( ))

( ( )) ( )( )

n n ndnrn r n n a n p a n n p p

dt K K

dpp p p bn p b n n p p

dt

= + + = + + +

Bifurcation analysis (aggregated)

Bifurcation analysis (complete model)

= 0.01

Bifurcation analysis (aggregated)

Systme hte-parasitode

Edelstein-Keshet, 1989

Population dhtes linstant t : NtPopulation de parasitodes linstant t : Pt

Modle de Nicholson

Bailey classique

Modle( )exp ta P 1t tN N+ =

Un point dquilibre positif

( )( )

( )

ln*

1

ln*

Na c

Pa

=

=1> instable !

( )( )1 1 expt t tP c N a P+ =

Time Series

0 5 10 15 20 250

160

320

480

640

800

a = 0.15, c = 1, = 2

Nt Pt

Simulations numriques

Modle spatialis

Grille de dimension A x A:

Dynamique locale

Phase de dplacement:

k vnements de dispersion

k=1

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

k=3

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

k=7

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

k=15

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Effet du paramtre k

= 0.5

Dynamique locale de Nicholson-Bailey

( )( )( )

1 exp

1 exp

t t t

t t t

N N aP

P cN aP

+ =

=

H

PMobilit des htes

Mobilit des parasites

( )

( )

1voisins

1voisins

18

18

Ht H t t

Pt P t t

N N N

P P P

+

+

= +

= +

Migration

Dynamique spatiale

Spirales

H=1, P=0.89, a=1, =2, c=1

H=0.2, P=0.89, a=1, =2, c=1

Chaos spatial Structures cristallines

H=0.05, P=1, a=1, =2, c=1

k >> 1: agrgation spatiale

2

2

1

1 (1 )

t

t

Pa

At t

Pa

At t

N N e

P cN e

+

+

=

=

* * 1

i i A = =

Modle agrg: rpartition spatiale uniforme chaque gnration:

Modle obtenu: mme forme, paramtres diffrents

Simulations numriques

0

50000

100000

150000

200000

250000

300000

350000

400000

0 200 400 600 800 1000

t

den

sity

NP

k=1, A=50

k faible : persistence

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 20 40 60 80

td

ensi

ty

NP

k=4, A=50

k fort : instabilit

Le modle agrg est valide pour de faibles valeurs de k

Modle de Nicholson Bailey

avec quation logistique

( )( )1 exp 1 /t tN r N K+ = ( )exp taP( )( )1 1 expt t tP cN aP+ =

Population dhtes linstant t : NtPopulation de parasitodes linstant t : Pt

Iteration Map

5 7 9 11 13 152

3,6

5,2

6,8

8,4

10

y

Pt

Nt

Point dquilibre stable

a=0.15, r=2, K=20, c=1

Types de dynamique

Iteration Map

0 12 24 36 48 60

x

0

3

6

9

12

15

Courbe invariante

a=0.2, r=2.6, K=50, c=0.4

Nt

PtIteration Map

0 20 40 60 80 1000

10

20

30

40

50

Nt

Pt

a=0.15, r=2.5, K=50, c=1

Attracteur chaotique

Dynamique locale: modle de Nicholson-Bailey avec croissance logistique des htes

( )

( )

1voisins

1voisins

18

18

Ht H t t

Pt P t t

N N N

P P P

+

+

= +

= +

PH Mobilit des htes

Mobilit des parasitodes

Migration (rpte k fois)

( )

( )( )1

1

exp 1 exp

1 exp

tt t t

t t t

NN N r aP

K

P cN aP

+

+

=

=

k=1 k=2

k=3 k=4

Modle complet. La densit des htes estreprsente en bleu.

a = 0.2, r = 2.6, K = 50,

c = 0.4,

n= p= 1 et A = 200.

k >> 1: agrgation spatiale

1 2 2

2

exp 1 exp

1 exp

t tt t

tt t

N aPN N r

KA A

aPP cN

A

+ =

=

* * 1

i i A = =

Modle agrg: rpartition spatiale uniforme chaque gnration:

Modle obtenu: mme forme, paramtres diffrents

a=0.2, r=2.6, K=50, c=0.4, n=p=1 et A=50. (b)-(c)-(d) Modle agreg (a) en rouge, et le modle

complet pour k=1 en bleu pour 0

Modle agrg (rouge) avec k=1, A=100 (a) pour 1000

Sensibilit aux conditions

initiales

(a) (b) (c)

++----CI(c)

+++++-CI(b)

+++++-CI(a)

987421k

a=0.2, r=2.6, K=50, c=0.4, A=100 and n=p=1, +(resp. -) : la distribution des hteset des parasitodes tend (resp. ne tend pas) vers luniformit spatiale

Effet du paramtre k : A=100

Valeur seuil de k

k 10832

2001005030A

Rsultats

Le modle agrg est utilisable pour de faibles valeurs de k mthodes dagrgation des variables utilisables sur des modles appliqus

Pas dmergence

Le synchronisme spatial peut tre d la dispersion des individus (alternative effet Moran)

Un modle de dynamique dune

population de carabe forestier

en milieu fragment

Pierre Auger*, Franoise Burel**,

Jean-Baptiste Pichancourt**

*IRD UR Geodes, Centre dle de France

**UMR CNRS Ecobio, Universit de Rennes

B Bois, Fort, bosquets

M Matrice agricole : Mas

H Haies de bords de champs1 . . ttN L M N+ =

A.ModleA.ModleA.ModleA.Modle de Leslie spatialisde Leslie spatialisde Leslie spatialisde Leslie spatialis

Matrice de diffusion (stepping stone)

Fuit M B, CC ou H

Quitte peu B

Ne distingue pas B et CC

Fcondit (f) : f(B) et f(CC)

Survie (s): s(B)=s(CC) > s(H) > s(M)

Matrice dmographique de Leslie

L A1 A2

S21 S32S33

F13

CC chemin creux bord de haies

paysage en grille

B. Modle de paysage & modle de mouvement

0.2

1

0.5

0.5

H

0.20.50.5H

111M

0.110.5CC

0.050.51B

MCCBElments

transitions entre lments (qij)

mij = pj . qij

Martin (2001); Pichancourt et al. Ecological Modelling (2006)

Paysage en diagramme

%%B

M

H CC

Simplification du modle de mouvement

1k

t tN LM N+ =

k augmente

Fragmentation = diminution de la quantit dhabitat favorable (exemple : bois) + volution des grandes taches vers des taches plus petites et loignes

Agrgation de variables Agrgation de variables Agrgation de variables Agrgation de variables

(10% CC)(10% CC)(10% CC)(10% CC)

95% 100%agrgcomplet

< qui lui est diffomorphe et qui est localement invariante sous laction

du flot dfini par le systme diffrentiel.

.est alors ,est Si kk CMCf

Bifurcation diagram

Cost window for stability

Stability for models I and II

Domain with two limit cycles

Prey-predator model with predator density

dependent migration of prey

MCHICH R., AUGER P. and POGGIALE J-C. Effect of predator density dependent dispersal of prey on stability of a predator-prey system. Mathematical Biosciences. 206(2), pp. 343-356, 2007.

( ) ( ) ( )

( ) ( ) ( )

( )

( )222222121111112

1

222222021012

111111012021

'

'

pnbnmppmd

dp

pnbnpmmpd

dp

pnanrnpnpd

dn

pnanrnpnpd

dn

++=

++=

+++=

+++=

The aggregated model :

( )

( )

2

0

2

0

1

2

1

2

dnrn anp bnp

dt p

dpMp bnp cnp

dt p

= + +

= + ++

1 2 = +

Prey-predator model with predator density

dependent migration of prey

02 c b

Similar Patches : Coexistence = 1 2

Prey-predator model with predator density

dependent migration of prey

Fast equilibrium : adding S, D and F

( )

( )

=

=

AuuAucpd

dp

AuuAucpddp

T

DDDD

T

HDHH

)(

)(

Replicator equations

Game dynamics and time scales

Game dynamics : evolutionary time scales

Behavioural plasticity : fast time scale

Domestic cats

P. AUGER & D. PONTIER. Mathematical Biosciences, 1998.

D. PONTIER, P. AUGER, R. BRAVO DE LA PARRA, E. SANCHEZ. Ecological Modelling, 2000.

Hierarchical structure

Fast intra-group dynamics

Slow inter-group dynamics

Complete model

A groups and N sub-groups

Fast part

( )

Nii xxxf

d

dx,...,,

21=

1

Which global variables ?A groups and N sub-groups

Conservative fast dynamics

Aggregated variables

( )

Nii xxxf

d

dx,...,, 21=

t=

Y

Fast time

First integral

=i

ixY

0=

Building an aggregated model

Fast stable equilibrium

( ) 0,...,, 21 ==

Nii xxxf

d

dx

( ) ( ) ( ) ( )( )

=

i NNiYxYxYxYxf

dt

dY **1

**

1,...,,,...,

( ) Yxi *

Substitution of the fast equilibrium

The dynamics of the aggregated model is an approximation of the dynamics of the complete model

Two models

( ) ( ) ( ) ( )( )

=

iNNi

YxYxYxYxfdt

dY **1

**

1,...,,...,,..., )(O+

Aggregated model

( )

Nii xxxf

d

dx,...,,

21= ( )

+

NNi

xxxxxxf ,...,,,,...,,2121

Complete model

: paramtre positif dcrivant la rapidit de lajustement du prix sur le march

A : capacit limite du march >0

Equilibre rapide:

R. Mchich, P. Auger, N. Rassi & B. Koii, 2008. Stability of a fishery bio-economic model with price equation. Theoretical Population Biology. Submitted.

Effet du prix variable : contexte pcherieEffet du prix variable : contexte pcherie

Points dquilibre:

(0,0): E0 point selle

(K,0): EK stable si K(3A/K)

Modle agrgModle agrg

Changement de variablesChangement de variablesChangement de variablesChangement de variablesChangement de variablesChangement de variablesChangement de variablesChangement de variables

LOCBIF & DsTool

Surexploitation: (n1,E1)

Pcherie durable: (n3,E3)

Comparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibresComparaison des prix des deux quilibres

Two fisheries

An over exploited fishery : The fishery allows a large fishing effort at a satisfyingmarket price but the fish density is maintained at a low level with extinction risk.

A traditional fishery : The fishery maintains the fish stock at a desirable levelfar from extinction but with small fishing effort and market price.

Zone

2 (C

)

Zone

1 (C

entr

ale)

Cap Boujdor

Cap Juby

Cap Ghir

Cap Cantin

Cap Blanc

Zones des pcheries de la sardine en atlantique marocain

Deux flottes de pche

Classe 2 (e)Classe 1 (E)

396284Puissance Moyenne

(CV)

[70-120][25-70]T.J.B

( )

( )

( ) ( )

( ) ( )

1 11 1 1 1 1 1 1 1

1

2 22 2 2 2 2 2 2 2

2

11 1 1 1 1

22 2 2 2 2

12 1 1 1 1 1 1

21 2 2 2 2 2 2

1

1

dn nr n a n e a n E

d K

dn nr n a n e a n E

d K

dEc E pa n E

ddE

c E pa n Edde

me me c e pa n edde

me me c e pa n ed

=

=

= +

= +

= + +

= + +

Un modle avec deux flottes sur deux zones

E1, E2 : efforts de pche des petits vaisseaux

n1, n2 : densit de poissons sur les deux zones

e1, e2 : efforts de pche des petits vaisseaux

Paramtres estimer :

Paramtres relatifs la dynamique et lexploitation du stock :

0.00350.025Coefficient de capturabilitq

57231703Capacit de charge K

1.141.53Taux de croissance r

Zone CZone

(Centrale)

Informations utilises pour lestimation :

Sries de biomasses de la sardines de 1995 2005(Rapports de campagnes acoustiques Dr Fridtjof Nansen)

Sries de captures de la sardine dans les ports de 1995 2005(Statistiques de lOffice National de Pche)

Sries des efforts de pche de 1995 2005(Donnes de la FAO)

Paramtres relatifs lexploitation et la commercialisation de la sardine:

Classe 2Classe 1

1.08Prix moyen (Dh/Kg)

110147237Cot par unit deffort

29%Conserve+Conglation

2%Appts

22%Consommation locale

47%Sous produits

Proportion annuelleDestination

Destinations de la sardine pche:

Source: Daprs A. Kamili (2006); Mmoire de master intilul: BIO-ECONOMIE ET GESTION DE LA PECHERIE DES PETITS PELAGIQUES: Cas de lAtlantique Centre Marocain

Prix par unit de poisson (Kg) et cot par unit deffort (Jours de pche) :