e c o l o g i c a l m o d e l l i n g 197 (2006) 274–280

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

Optimal spatial distribution of the fishing effort in a multifishing zone model

Rachid Mchicha,∗, Najib Charoukib, Pierre Auger c, Nadia Raıssid, Omar Ettahiri e

a Ecole Nationale de Commerce et de Gestion, B.P. 1255, 90000 Tanger, Moroccob Laboratoire des Approches et Methodologies, INRH, 2 rue de Tiznit, Casablanca, Moroccoc UR GEODES, IRD, Centre de Recherche d’Ile de France, 32 av. Henri Varagnat, 93143 Bondy cedex, Franced Laboratoire SIANO, Faculte des Sciences, Universite Ibn Tofaıl, B.P. 133, Kenitra, Moroccoe Laboratoire d’Oceanographie Biologique, INRH, 2 rue de Tiznit, Casablanca, Morocco

a r t i c l e i n f o

Article history:

Published on line 5 June 2006

Keywords:

Fishery model

Aggregation of variables

Optimal spatial distribution of

a b s t r a c t

This work presents a stock-effort dynamical model of a fishery subdivided on several fishing

zones. The stock corresponds to a fish population moving between different zones, on which

they are harvested by fishing fleets. The bio-economical model is a set of ODE’s governing

the fishing efforts and the stocks on the different fishing zones. We take profit from the

existence of two time scales (a fast one for fish migration and fleets movements, and a slow

one for fish growth and mortality and fleets revenue) to construct a reduced (aggregated)

model. The aggregated model describes the global evolution of the harvested stock as well

fishing effort

Management measures

as the total fishing effort. The mathematical analysis of the model allows the optimization

of the spatial distribution of the fishing effort and the identification of an efficient set of

management measures, which corresponds in one hand to set an appropriate system of tax

and/or subsidies, and on the other hand to control the displacement of the fleets between

orde

scribe the dynamics of fisheries (one can refer for example to

the fishing zones, in

1. Introduction

The overexploitation of marine resources and the continu-ous high pressure on fish stocks leads inexorably to their ex-tinction and to a global and worrying decrease in the levelof catch. For example, in Millischer et al. (1999), authors studythe evolution of Brittany’s industrial fleets and show an impor-tant decrease of their overall fishing power in the eighties forsaithe, cod, haddock and whiting in the West of Scotland area.

An other important case is also that of the pelagic re-sources, which represent 75% of the Atlantic fish stock inNorth-West Africa (see (Ressources Halieutiques Marocaines,

2002)). In Morocco, the exploitation of this stock and partic-ulary the sardine represents an important social economicalactivity. Its distribution extends from Cap Blanc (21N) to Cap

∗ Corresponding author. Tel.: +212 63 03 62 38; fax: +212 39 31 34 93.E-mail address: racmchich@yahoo.com (R. Mchich).

0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2006.03.026

r to increase the total activity.

© 2006 Elsevier B.V. All rights reserved.

Spartel (35.45N) where three stocks are recognized (Belveze,1984). The northern stock is situated between Casablanca andCap Spartel (3330′N–36N), the central from Laayoune to CapCantin-Safi (27N–32.30N) and the southern between Cap Blancand Cap Boujdor (21N–26N). Drastic fluctuations of abundancehave been observed for sardines and other pelagic during thelast 50 years. This situation interpellates the responsible offishery management, for making appropriate decisions, ascontrolling the repartition of the fleets on different fishingzones, in order to avoid a potential slump in the sector.

Many mathematical models have been developed to de-

Charles et al., 2000; Haddon, 2001; Quinn and Deriso, 1999), andmany other works included economic factors (see the booksby Clark (1990) and Cohen (1987) and the works of Clarke and

g 1

N (N > 3) fishing zones. Section 5 is devoted to a discussion ofthe obtained results and to some perspectives. We also try toadapt the interpretation of our results to the case of Moroc-

)

97 (2006) 274–280 275

can sardine’s fisheries. We will see that assumptions of ourmodel correspond to the case of the central stock, which isdistributed in three closed areas through which the fish car-ries out seasonal migrations (see the report of the F.A.O. (1985)).This stock is exploited by a fleet based in the principal portsof the region.

2. The three fishing zones model withfishing activity independent from fishing zones

2.1. Presentation of the complete model

We consider a fish population harvested by a fishing fleet. Fish-ing boats are allowed to exploit the resource on three fishingzones. Let N(t) be the fish population density at time t

N(t) = (n1(t), n2(t), n3(t))T (2.1)

The upper subscript T denotes the transpose vector.n1(t) (respectively, n2(t) and n3(t)) is the fish population den-

sity on fishing area 1, (respectively, 2 and 3), at time t. Similarly,the fishing effort is subdivided into three components on eachfishing zone

E(t) = (E1(t), E2(t), E3(t))T (2.2)

E1(t) (respectively, E2(t) and E3(t)) is the fishing effort on fish-ing area 1, (respectively, 2 and 3), at time t. The complete modelis a system of six ODEs governing the previous variables. Threeequations describe the evolution of the fish densities on thethree zones⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

εdn1

dt= (kn2 − kn1) + ε

[r1n1

(1 − n1

K1

)− an1E1

]ε

dn2

dt= (kn1 + kn3 − (k + k)n2) + ε

[r2n2

(1 − n2

K2

)− an2E2

]ε

dn3

dt= (kn2 − kn3) + ε

[r3n3

(1 − n3

K3

)− an3E3

](2.3

where k, k, k and k are fish migration rates between the differ-ent fishing zones, see Fig. 1. On each zone i, i ∈ [1, 3], the fishpopulation grows logistically with growth rate ri and carryingcapacity Ki. We assume the mass action law for fish capture oneach zone. a is the catchability coefficient assumed indepen-dent on zone i. ε is a small dimensionless parameter. There-fore, two time scales are involved in model (2.3). The fishingzones are assumed to be relatively close of each other and fishmigration between the fishing areas is thus fast in compari-son with fish growth and mortality due to capture by fishingon each zone.

Three equations describe the evolution of the fishing ef-forts on the three zones⎧⎪⎪⎪⎨⎪⎪⎪⎩

εdE1

dt= (mE2 − mE1) + ε (bn1 − c)E1

εdE2

dt= (mE1 + mE3 − (m + m)E2) + ε (bn2 − c)E2

dE3

(2.4)

εdt

= (mE2 − mE3) + ε (bn3 − c)E3

e c o l o g i c a l m o d e l l i n

Munro (1991) and Raıssi (2001)). This paper is situated in thisgeneral context and illustrates a fishery management modelon several fishing zones. It generalizes previous works Mchichet al. (2002, 2000, 2005), in which authors built and studied dif-ferent models about the fishing activity on two different spa-tial zones. The aim of this work is to study the optimal spatialdistribution of the fishing effort in a multi fishing zone, andto give some efficient management measures, by setting anappropriate system of tax and/or subsidies, and on the otherhand to control the displacement of the fleets between thefishing zones, in order to increase the total activity.

In Section 2 we present the complete fishing activity modelon three zones. It consists in a system of six ordinary differ-ential equations governing the three local fishing stocks andthe three fishing efforts on three adjacent fishing zones. Weassume two time scales: a fast one associated to the quickmovements between different zones, and a slow one whichcorresponds to the growth of fish populations and the vari-ation of the fleets revenue. The growth of the fish is relatedto the natural demography rate less the mortality rate dueto the fishery, whereas the fishery fleet’s growth is functionof the profit generated by the activity, which could be in-terpreted as an investment (one can refer to Clark (1990),Clark et al. (1979) and Mchich et al. (2002, 2005)). We re-call that we take advantage of the two time scales to re-duce the complete models by use of aggregation methods(one can see the review article on aggregation techniques byAuger and Bravo de la Parra (2000)).

The sufficient conditions for a system to be perfectly aswell as approximately aggregated have been investigated inthe frame of general population models by Iwasa et al. (1987,1989) and Levin and Pacala (1997). Some aggregation methodspermit to reduce a system with a large number of variables in-volving different time scales into an aggregated system witha few global variables. The method is based on perturbationtechnics and on the application of an adequate version (onecan see Auger and Poggiale, 1996; Auger and Roussarie, 1994;Michalski et al., 1997; Poggiale, 1994) of the Center ManifoldTheorem (see Fenichel, 1971). The aggregation of the completemodel consists in supposing that the fast dynamics has at-tained a stable equilibrium and in substituting this fast equi-librium into the equations of the complete model. Therefore,we obtain a reduced model, which describes the dynamics ofthe global population of preys and predators, and has the ad-vantage that it is possible to perform its complete qualitativeanalysis.

The analysis of the corresponding aggregated model leadsto a level of total sustainable fishing activity function of thespatial distribution of the fishing effort. That is the spatialdistribution could be chosen in order to maximize this totalequilibrium level.

In Section 3, we generalized this study to the case wherethe fishing efficiency is zone dependent. In this case, the re-sulted total maximum sustainable fishing effort depends onspatial distribution of the fishing effort, as well as the fishingcost of each fishery and its catchability coefficient. In the Sec-tion 4, we generalize the model of Section 2 to a model with

m, m, m and m are boat migration rates between the differentfishing zones, see Fig. 2. On each zone i, boats capture fish. Thecapture rate in Eq. (2.4) is equal to b = pa where p is the price

276 e c o l o g i c a l m o d e l l i n g 197 (2006) 274–280

betw

Fig. 1 – Fish migration flows

per unit of fish density assumed constant. Fishing boats pay acost per unit of fishing effort noted c on each zone. We assumethat cost as well as capture rate are zone independent. Twotime scales are also involved in model (2.4). Boats movementbetween the fishing areas is fast in comparison with fishingactivity revenue (bni − c)Ei, on each zone i. The model assumesthat in the long term, if the benefit is larger than the cost,new boats are entering in fishing activity. Otherwise, boatsare stopping fishing activity or move to another fishery. Boatsare assumed to move fast from zone to zone (for exampleeach day or week) but capture a relatively small proportion ofthe total resource on each zone (each day or week).

2.2. Fast equilibrium

The first step consists in setting ε = 0 into Eq. (2.3). This modelis the fast model for fish⎧⎪⎪⎪⎨⎪⎪⎪⎩

dn1

d�= (kn2 − kn1)

dn2

d�= (kn1 + kn3 − (k + k)n2)

dn3

d�= (kn2 − kn3)

(2.5)

where � denotes the fast time, t the slow time with t = ε�. Thefast model (2.5) is conservative. At the fast time scale, the totalfish density n(t) = n1(t) + n2(t) + n3(t) remains constant.

Similarly, setting ε = 0 into Eq. (2.4) defines the fast modelfor fishing efforts⎧⎪⎪⎪⎨⎪⎪⎪⎩

dE1

d�= (mE2 − mE1)

dE2

d�= (mE1 + mE3 − (m + m)E2)

dE3

d�= (mE2 − mE3)

(2.6)

The fast model (2.6) is also conservative. At the fast time

scale, the total fishing effort E(t) = E1(t) + E2(t) + E3(t) remainsconstant.

A simple calculation shows that there exists a single pos-itive and stable equilibrium for any positive initial condition.

Fig. 2 – Boats migration flows betw

een the three fishing zones.

This fast equilibrium for fish is given by the following expres-sions:⎧⎪⎨⎪⎩

n∗1 = �∗

1n

n∗2 = �∗

2n

n∗3 = �∗

3n

(2.7)

where the fish proportions �∗i

in each zone i are constant andgiven by the next expressions⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

�∗1 = 1

1 + (k/k) + (kk/kk)

�∗2 = k/k

1 + (k/k) + (kk/kk)

�∗3 = (kk/kk)

1 + (k/k) + (kk/kk)

(2.8)

A similar positive and stable fast equilibrium holds for fish-ing efforts⎧⎪⎨⎪⎩

E∗1 = �∗

1E

E∗2 = �∗

2E

E∗3 = �∗

3E

(2.9)

where the fish proportions �∗i

in each zone i are constant andgiven by the next expressions⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�∗1 = 1

1 + (m/m) + (mm/mm)

�∗2 = m/m

1 + (m/m) + (mm/mm)

�∗3 = mm/mm

1 + (m/m) + (mm/mm)

(2.10)

It can easily be shown that the fast equilibrium for fish andboat is globally asymptotically stable in the positive quadrantfor any positive initial condition.

een the three fishing zones.

2.3. The aggregated model

The next step consists in the substitution of the fast equilibria(2.7) and (2.9) into the equations of the complete model given

n g 1

bir⎧⎨⎩ts⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

b

Psgtgdi

2

Ae

••

•

maf

••

•

e c o l o g i c a l m o d e l l i

y (2.3) and (2.4), and addition of fish and boat equations lead-ng to a reduced model, called the “aggregated model”, whicheads as follows:

dn

dt= rn(1 − n

K) − anE

dE

dt= (bn − c)E

(2.11)

Global fish growth rate r and carrying capacity K, are linkedo parameters of the complete model through the next expres-ions

r =3∑

i=1

ri�∗i

rK =

3∑i=1

ri(�∗i)2

Ki

(2.12)

For the global fishing rates, we have

a = a

3∑i=1

�∗i �∗

i

b = b

3∑i=1

�∗i �∗

i

(2.13)

Using the previous expression b = pa, we have

˜ = pa (2.14)

For aggregation methods, we refer to Auger and Bravo de laarra (2000), Auger and Poggiale (1996, 1998), Auger and Rous-arie (1994), Michalski et al. (1997), Poggiale (1994). The aggre-ated model is obtained by an approximation. It is valid whenhe small parameter is small enough (ε � 1) and when the ag-regated model is structurally stable. This aggregated modelescribes the time evolution of the total fish density and fish-

ng efforts at the slow time scale.

.4. Study of the aggregated model

simple study shows that the aggregated model has threequilibria:

The origin (0, 0).(K, 0) which is an equilibrium without sustainable fishingactivity.(n∗, p∗) which is an equilibrium with sustainable fishing ac-tivity.

Local stability analysis is not shown because the aggregatedodel is a classical model, see for example the analysis inpredator–prey context (Bazykin, 1998). Main results are the

ollowing ones:

The origin is a saddle.When (n∗, p∗) does not belong to the positive quadrant, (K, 0)is stable for any initial condition in the positive quadrant.When (n∗, p∗) is positive, it is stable for any initial condi-

tion in the positive quadrant and the following condition isverified:

n∗ = c

pa< K (2.15)

97 (2006) 274–280 277

In that case, the sustainable total fishing activity is givenby the next relation

E∗ = r

a(1 − c

pKa) (2.16)

In the expression (2.16), the only parameter dependingon the spatial equilibrium distribution �∗

iof the fishing ef-

forts is the parameter a. Other parameters depend on fishspatial distribution but not on fishing effort spatial distri-bution.

2.5. Optimal spatial distribution of the fishing effort

The total sustainable fishing activity is given by the next rela-tion

E∗(a) = r

a(1 − c

pKa) (2.17)

Parameter a incorporates the spatial distribution of thefishing effort.

It is not possible to change the spatial distribution of thefishes but the fishermen can decide to subdivide the total fish-ing effort between the three fishing zones in a desired way. Asimple calculation shows that function E∗(a) has a maximumfor a spatial distribution of the fishing efforts chosen as fol-lows:

aopt = 2c

pK(2.18)

It must be noted that this relation is in agreement withEq. (2.15) which is needed to ensure a sustainable fishingactivity. Eq. (2.18) shows that if the spatial distribution of thefish is known as well as the fishing cost, the price and thetotal fish carrying capacity, then the spatial distribution ofthe fishing efforts must be chosen in order to verify the nextrelation

3∑i=1

�∗i �∗

i = 2c

pKa(2.19)

Therefore, the choice of a particular spatial distribution ofthe fishing effort allows to maximize the total fishing activity.In that case, the optimal total sustainable fishing activity isgiven by the next relation

E∗(aopt) = r

aopt(1 − c

pKaopt) = rpK

4c(2.20)

In the next section, we generalize the model studied in Sec-tion 2, to a model where the cost and the catchability coeffi-cient are depending on fishing zone.

3. The model with zone dependent fishingefficiency

In the previous section, capture rates as well as costs arezone independent. A more general case corresponds to catch-ability coefficients and costs depending on the fishing areas.

l i n g 197 (2006) 274–280

Fig. 3 – Shape of the function E∗(a, c) = ra

(1 − c

pKa

)in 3D,

with parameters as follows: r = 0.5, p = 0.25 and K = 1. c is

4. The model with N fishing zones

The previous model of Section 2 for three zones can easilybe extended to N fishing zones, with N > 3. In that case, the

278 e c o l o g i c a l m o d e l

In this case, the complete model reads as follows:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

εdn1

dt= (kn2 − kn1) + ε [r1n1(1 − n1

K1) − a1n1E1]

εdn2

dt= (kn1 + kn3 − (k + k)n2) + ε [r2n2(1 − n2

K2) − a2n2E2]

εdn3

dt= (kn2 − kn3) + ε [r3n3(1 − n3

K3) − a3n3E3]

(3.1)

where ai is the catchability coefficient now depending on zonei. Other parameters are the same as in Section 2. For fishingefforts, the model reads⎧⎪⎪⎪⎨⎪⎪⎪⎩

εdE1

dt= (mE2 − mE1) + ε (b1n1 − c1)E1

εdE2

dt= (mE1 + mE3 − (m + m)E2) + ε (b2n2 − c2)E2

εdE3

dt= (mE2 − mE3) + ε (b3n3 − c3)E3

(3.2)

On each zone i (i ∈ [1, 3]), the terms bi = pai are now de-pending on the fishing zone, where p is the price per unit offish density still assumed constant. Fishing boats pay a costper unit of fishing effort, noted ci and also depending on zonei. This term ci can be rewritten as follows: ci = c + mi, where c isthe fishing cost and which will be considered the same for allthe fleets, while mi is a tax (when mi > 0) or a subsidy (whenmi < 0) for the participation to the fishing activity.

The aggregated model now reads as follows:⎧⎨⎩

dn

dt= rn(1 − n

K) − anE

dE

dt= (bn − c)E

(3.3)

where global parameters do not change with respect to Section2 except

c =3∑

i=1

ci�∗i and a =

3∑i=1

ai�∗i �∗

i (3.4)

Contrary to Section 2, the global cost now depends on thespatial distribution of the fishing effort on the three zones. In-deed, in the previous case, the cost was zone independent andtherefore the global cost was equal to the local one because ofEq. (3.5)

If ci = c, ∀i, then

3∑i=1

ci�∗i = c

3∑i=1

�∗i = c (3.5)

This is no more the case and this simplification cannot bemade in the present section. Therefore, the total sustainablefishing activity now depends on the fishing effort distributionvia two global parameters a and c. Thus, it can be consideredas a two variables function

E∗(a, c) = r

a(1 − c

pKa) (3.6)

We are going to make a further assumption. The cost oneach zone has a minimal value

∀i, 0 ≤ cmin ≤ ci (3.7)

Assumption (3.7) is realistic because fishing activity can-not be realized without a minimum cost per unit of time. For

varying between 0.2 and 0.25 while a is varying between0.7 and 30. We see clearly that there is a maximum forE∗(a, c) for c minimal.

example, boat needs energy, maintenance of the material. . .,etc., which has a cost.

Looking at Eq. (3.6), it is obvious that E∗(a, c) is a decreasingfunction with respect to the cost and that for a fixed costvalue, the function has a maximum, see Figs. 3 and 4. Thus, inthe bounded domain of the cost, there is a global maximumgiven by

aopt = 2cmin

pK(3.8)

Then, the spatial distribution of the fishing effort must bechosen in order to verify Eq. (3.8).

Fig. 4 – Shape of the function E∗(a) = ra

(1 − cmin

pKa

)with

parameters as follows: r = 0.5, cmin = 0.2, p = 0.25 andK = 1. We see clearly that there is a maximum for E∗(a) foraopt = 2cmin

pK = 1.6 minimal.

e c o l o g i c a l m o d e l l i n g 197 (2006) 274–280 279

pilch

a⎧⎨⎩w⎧⎪⎨⎪⎩a{

fip

∑

5

icbLmtt(b

Fig. 5 – Seasonal distribution of Sardina

ggregated model reads

dn

dt= rn(1 − n

K) − anE

dE

dt= (bn − c)E

(4.1)

here we have the next generalized expressions

r =∑N

i=1 ri�∗i

r

K=

N∑i=1

ri(�∗i)2

Ki

(4.2)

nd

a = a∑N

i=1 �∗i�∗

i

b = b∑N

i=1 �∗i�∗

i

(4.3)

There is also an optimal spatial distribution of the totalshing effort on the fishing zones which verify the next ex-ression

N

i=1

�∗i �∗

i = 2c

pKa(4.4)

. Discussion and conclusion

Let us interpret the result obtained in the analysis of thentroduced example of the spatial distribution of Moroccanentral sardine fisheries (see Fig. 5). This stock was studiedy Furnestin and Furnestin (1970), Belveze (1972, 1984) andamboeuf (1977), and was the object of hypothesis of seasonaligrations. These works concluded that the sardine popula-

ion is situated into three closed zones in central stock. In win-er, the sardine population migrates to the south of Sidi Ifnizone 1) for reproduction, while in the spring, it invades theay of Agadir (zone 2). In summer, the sardine stocks move to

ardus along Moroccan Atlantic coasts.

the waters of Essaouira and Safi (zone 3), following the devel-opment of the summery upwelling. In autumn, adults movesouthwards to Agadir. Belveze and Erzini (1983) concluded thatthese migrations correspond to seasonal phenomena of exten-sion and concentration of the population, in relation with thedevelopment of the upwelling between Cap Cantin and CapGhir and not a migration of the totality of the sardine popu-lation. Values of the stock’s spatial distributions �i (i = 1, 2, 3)can be computed by means of direct evaluation using acoustictechniques (one can see the Cruise report, 2003). So, the as-sumption of Section 3 made on the catchability coefficient ai

and the unit fishing effort cost zones-dependent ci (i = 1, 2, 3),seems well adapted to this context. Indeed, the fishing activityis primarily estival in zone 3, while the capture are almost nullin winters. In zone 1, captures are more important in winterperiods, due to the regrouping of adults for the reproduction.

In the light of the present analysis, the authority responsi-ble of fishery management, that is the Moroccan government,will act on two parameters: the fishery’s spatial distribution�i (i = 1, 2, 3) and the value of ci (which is equal to c + mi) inorder to steer the total fishing effort E to the level of what welabelled the maximum sustainable fishing activity. In fact, themanager must find the values of �i(i = 1, 2, 3) and mi solutionof the following optimization problem:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

max E∗(�i, mi) = r(pK∑3

i=1 ai�i�i −∑3

i=1(c + mi)�i)

pK(∑3

i=1 ai�i�i)2

0 ≤ cmin ≤ c + mi ≤ cmax, i = 1, 2, 33∑

i=1

�i = 1

(5.1)

In practice the choice of suitable optimal values of � is in

i

the action’s field of the government, recall in the Eq. (2.9), pa-rameters �i appear as the proportion of the fishery in the zone iwith respect to the global fishing effort. Now let us turn to thepossibility of choice of the value of cmin , when the value of

l i n g

Natural Resour. Model. 14 (2), 1–24.

280 e c o l o g i c a l m o d e l

each of ci is giving. This choice is possible as cmin will be theminimum of the sum of a given fixed cost c and the parametermi interpreted as a tax if it’s positive or subsidy if not, free tochoice that is cmin = min (c + m). This management measureare more and more used, it was in the mean of the discus-sion in Raıssi (2001). In conclusion the dynamical model in theheart of this work, will be adapted and proposed as platformto the decision’s maker in the fisheries management. It couldbe improved in order to be applied to more general settingsas for instance if we assume that the rates of fleet’s migrationare stock dependent as it was the case in Mchich et al. (2002).One can also include in the stock’s growth a structured formwith respect to the fish size and or age.

Another interesting perspective of this work is to take intoaccount in the model an “Allee” effect for fish growth and thusthe possibility of stock extinction. We refer to Bazykin (1998)book for prey–predator models and Allee effect.

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Auger, P.M., Roussarie, R., 1994. Complex ecological models withsimple dynamics: from individuals to population. ActaBiotheor. 42, 111–136.

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