Competition or cooperation in transboundary fish stocks management: Insight from a dynamical model

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Competition or cooperation in transboundary fish stocks

management: Insight from a dynamical model

Trong Nguyen, Timothée Brochier, Pierre Auger, Viet Duoc Trinh, Patrice

Brehmer

To cite this version:

Trong Nguyen, Timothée Brochier, Pierre Auger, Viet Duoc Trinh, Patrice Brehmer. Competition or

cooperation in transboundary fish stocks management: Insight from a dynamical model. Journal of

Theoretical Biology, Elsevier, 2018, 447, pp.1-11. �10.1016/j.jtbi.2018.03.017�. �hal-01806915�

ContentslistsavailableatScienceDirect

Journal

of

Theoretical

Biology

journalhomepage:www.elsevier.com/locate/jtbi

Competition

or

cooperation

in

transboundary

fish

stocks

management:

Insight

from

a

dynamical

model

Trong

Hieu

Nguyen

a,∗

_,

_Timothée

_Brochier

b,c

_,

_Pierre

_Auger

b,c,d

_,

_Viet

_Duoc

_Trinh

a

_,

Patrice

Brehmer

e,f,g

a Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam b Institut de Recherche pour le Développement (IRD), UMI 209, UMMISCO, IRD France Nord, Bondy, F-93143, France

c Universityé Sorbonne Université, UMI 209, UMMISCO, Paris, F-75005, France d Université Cheikh Anta Diop, UMI 209, UMMISCO, Dakar, Senegal

e Institut de Recherche pour le Développement (IRD), UMR 195 Lemar, Hann, Dakar BP 1386, Senegal

f Institut Sénégalais de Recherches Agricoles (ISRA), Centre de Recherches Océanographiques de Dakar-Thiaroye, PRH, Dakar, BP 2241, Senegal g Institut de

Recherche pour le Développement (IRD), Délégation Régionale France Ouest, UMR Lemar, Campus Ifremer; BP 70, Plouzané 29 280, France

g Institut de Recherche pour le Développement (IRD), Délégation Régionale France Ouest, UMR Lemar, Campus Ifremer; BP 70, Plouzané 29 280, France

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 25 September 2017 Revised 13 January 2018 Accepted 12 March 2018 Available online 13 March 2018

MSC: 34C60 34D20 34D23 34D45 92B05 92D25 Keywords: Shared fish stock Aggregation of variables ODE Pelagic Fisheries management West Africa

a

b

s

t

r

a

c

t

Anidealizedsystemofasharedfishstockassociatedwithdifferentexclusiveeconomiczones (EEZ)is modelled.Parameterswereestimatedforthecaseofthesmallpelagicfisheriessharedbetween South-ernMorocco, MauritaniaandtheSenegambia. Twomodelsoffishingeffortdistributionwereexplored. ThefirstoneconsidersindependentnationalfisheriesineachEEZ,withacostperunitoffishingeffort thatdependsonlocalfisherypolicy.Thesecondoneconsidersthecaseofafullycooperativefishery per-formedbyaninternationalfleetfreelymovingacrosstheborders.Bothmodelsarebasedonasetofsix ordinarydifferentialequationsdescribing thetimeevolutionofthefishbiomassand thefishingeffort. Wetakeadvantageofthetwotimescalestoobtainareducedmodel governingthetotalfishbiomass ofthesystemandfishingeffortsineachzone.Atthefastequilibrium,thefishdistributionfollowsthe idealfreedistributionaccordingtothecarryingcapacityineacharea.Differentequilibriacanbereached accordingtomanagementchoices.Whenfishingfleetsareindependentandnationalfisherypoliciesare notharmonized,inthegeneralcase,competitionleadsafterafewdecadestoascenariowhereonlyone fisheryremainssustainable.Inthecaseofsub-regionalagreementactingontheadjustmentofcostper unitoffishingeffortineachEEZ,wefoundthatalargenumberofequilibriaexists.Inthislastcasethe initialdistributionoffishingeffortstronglyimpacttheoptimalequilibriumthatcanbereached.Lastly, thecountrywiththehighest carryingcapacitydensity may getless landingswhencollaboratingwith othercountries thanifitminimises itsfishingcosts.The secondfullycooperativemodelshowsthat a single internationalfishing fleetmoving freelyinthe fishingareasleads toasustainableequilibrium. Suchfindingsshould fosterregionalfisheriesorganizationstogetpotential newwaysforneighbouring fishstockmanagement.

© 2018TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Exploited fish populations performing transboundary migra-tions is a common situation around the world, as obviously no fish is stopped ator by anycountry border. Documented exam-ples include shared small pelagic fisheries in the California

cur-∗ _{Corresponding author.}

E-mail address: hieunguyentrong@gmail.com (T.H. Nguyen).

rentsystembetweenCanada,USAandMexico(Javoretal., 2011; Loetal., 2011) aswell asin theCanary CurrentSystembetween Morocco, Mauritania, Senegal, Gambia, and Guinea Bissao (Boely et al., 1982; Brochier et al., 2018), which are both East Boarder Upwelling.Reachingfisheryagreementsbetweenthecountries ex-ploitingsamefishstockisaprerequisiteforagood,equitable man-agement(e.g.CampbellandHanich,2015;Pitcheretal., 2002) in waters under national jurisdiction e.g. to avoid fish stock over-exploitation,to understandchangein stockspatialdistribution in

https://doi.org/10.1016/j.jtbi.2018.03.017

thecontextofclimatechangewhichmayquicklydisturbthe exist-ingequilibria(MillerandMunro,2004).

From a theoretical point of view, finding a common supra-national policy allowing a sustainable fishing activity simultane-ously in all the countries exploiting the same fish stock is not a trivial problem and a topic of high political interest. This is-sue is usually studied in the literature using the game theory for the economic analysis of international fisheries agreements (e.g.Aguero andGonzalez,1996;Ishimura etal., 2012;Pintassilgo etal.,2014).Theseapproachesdemonstratedthatthefull coopera-tivemanagementbythedifferentcountriesisnecessarytoachieve sustainablefisheries,andthattheworkstooptimizefisheries man-agementinthecontextofclimatechange(asotherissues)needto be conductedin commonor atleast simultaneously by the con-cernedcountriestobesuccessful(e.g.Ishimuraetal.,2012).Akey messagethat emerges fromthis literature strand is that the self organisationgenerallyleadstoover-exploitationofinternationally sharedfish stocks (Campbell andHanich, 2015;Pintassilgo etal., 2014;Pitcheretal.,2002). Thus,aninternationallegalframework andregulationsmustbedevelopedtohelpthecomplexsystemof shared fisheries to avoid the over-exploitation trap (Feeny et al., 1996) andtoconvergetowardtheoptimumsituationofcollective maximumsustainableyield.

In this work we use a set of ordinary differential equations (ODE)toexplorethesituationsthatcanemergeeitherinthecase ofindependentnationalfishingfleets(eachonefishing onlyinits ownEEZ)inorinthecaseofan uniqueinternationalfishingfleet thatcanfreelymovebetweenEEZs.Inthefirstcase,weconsidera managementactingonthecostperunitoffishingeffort(CUFE) in-steadofcatchquotaintheclassicalapproach.Thebehaviourof de-cisionmakersisnotexplicitlymodelled,butisrepresentedbythe CUFEappliedineachEEZ.Competition occurswheneachcountry tendstominimiseCUFE,butweshowthatfishingagreementscan tendtoharmoniselocalCUFEsinordertolimitthecompetition.In thesecond case, thatwe calledfullycooperative, theCUFEis set constantamongtheEEZ,andthefishingagreementwasassumed tobe anegotiation ontheshare oftheinternationalfishing fleet benefits.

Numerousbio-economicmodelsbasedonODEwereusedto ex-ploretheoptimalharvestingratesinagivenfisheriessystem(e.g.

Crutchfield,1979;Dubeyetal.,2002;Merinoetal.,2007),butless effortwasdevelopedtoapplysuchmodelstothecaseoffisheries managedbydifferentgovernmentsexploitingthesamefish popu-lation.Here weassume anidealsituationconsidering thatallthe fisheries areeffectively regulated (no illegal,unreported, and un-regulated(IUU)Fishing(Agnewetal.,2009)).Weappliedour ap-proachtothecaseoftheNorth–WestAfricafisheriesintheSouth partof the Canary Upwelling System(CUS), using the output of a realistic physical biogeochemical model recently developed for thisregion(Augeretal.,2016)todescribetheenvironmentwhich forcesthefishmigrations.

2. Preliminaries

Weexplorethebehaviourofanidealizedsystemofshared fish-eriesthatsuitsthecurrentknowledgeoftheecosystemintheCUS. Weconsiderthefishstockthatperformmigrationsbetween12.3°N (Senegal) to 26°N (Morocco). We consider 3 political ensembles withinwhichweassume thefisheryregulationcould beuniform, theSouthernMorocco/WesternSahara(20.77–26°N,hereafterzone 1),Mauritania(16.06–20.77°N,hereafterzone2)andthe Senegam-bia(Senegal+theGambia,12.3–16.06°N,hereafterzone3)(Fig.1). The smallpelagicfishcarryingcapacityfromSenegaltoSouth Moroccowasestimated to ∼ 10 million tons, including themain fished species namely Sardina pilchardus, Sardinella aurita and S. maderensis, Trachurus spp., Scomber japonicus, Caranx rhonchus

Fig. 1. Illustration of the shared small pelagic fisheries in the South part of the West African Eastern Boundary Upwelling System. The colourbar indicates the mean chlorophyll ’a’ (1998–2009) which was used as a proxy for the small pelagic fish carrying capacity. The dotted black line shows the limit of the continental shelf which shelter small pelagic fish in this area. The white arrows indicate the seasonal migrations performed by the round sardinella, the main exploited small pelagic fish species in the region ( Boely et al., 1982 ). The dotted white arrow corresponds to the less documented part of the migration (to Guinea Bissau). B i and e i are the biomass

and fishing effort in each zone, respectively, i.e., (1) the Southern Morocco/Western Sahara, (2) Mauritania, (3) Senegal and Gambia. Adapted from Auger et al. (2016) . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(FAO,2012).Thecarryingcapacityisthemaximumbiomassoffish that can feed andgrow in thisregion in average climatic condi-tions (1998–2009),accordingtoavailable food, hereestimatedby phyto-planktonforthe smallpelagic fishconsidered (Aguero and Gonzalez,1996).

Eachof thesespecies hasdistinct environmental preferendum andthus migrationbehaviour(Boelyetal., 1982),butall together these migrations can be represented by as an ideal free distri-bution of the total small pelagic fish biomass according to the environmentcarryingcapacity.Thefishannualgrowth(r) wasset to0.88year−1,asanaverageofsmallpelagicfishgrowthratesin WestAfrica(FAO,2012).

According to observations in the CUS, the small pelagic fish habitat is restricted to the continental shelf (Brehmer, 2004; Brehmer et al., 2006). We computed the surface of the conti-nental shelf, delimited offshore by the 200 m isobath, from the bathymetry used by Auger et al.(2016).We found that the part oftheSouthern Morocco/WesternSaharaconsidered (zone1)has ashelfof41472km2_{,theMauritaniashelf(zone2)is22720km}2

andthe Senegambia shelf (zone3) is 15 040km2_{. The physical}

-biogeochemicaloceansimulationinthisregion(Augeretal.,2016) providesthemeanannualplanktonbiomass:255014tonsinzone 1,105575tonsinzone2and51 632tonsinzone3.Toestimate thecarryingcapacityineachdefinedzone,wedistributedthetotal carryingcapacityestimated(10milliontons;FAO,2012)according totheaverageplanktonbiomassdensity(foodofthesmallpelagic fish) in the small pelagic fish habitat of each zone. We found the average local carrying capacity k1 = 104 tons/km2 in south

Morocco(zone 1), k2 = 45 tons/km2 in Mauritania(zone2) and

k3 =160tons/km2 inSenegambia (zone3).The totalcarrying

zone2(K2=3.3milliontons)andlowestinzone3(K3 =2.4

mil-liontons).

Thedynamicofthesystemismodelledby asystemof6 ordi-nary differentialequations (1). Weassume that fish are perform-ing rapidmigrations acrosszone 1,2and3,suchthat their aver-ageannualbiomass B1(t),B2(t) andB3(t) ineach zone followthe

idealfreedistributionaccordingtothecarryingcapacityK1,K2and

K3.Fishmovementisconsideredasarapidphenomenon(Brehmer

etal.,2000)comparedtothegrowthrateoftheir populationand comparedtothefishingmortalityrate.Asaresult,atthefast equi-librium thereisa fixed proportionofthe totalfish abundancein each zone. Indeed,itisgenerally acceptedthat smallpelagicfish migrationisduetoenvironment,andnotlinkedtofishdensitynor localpolicy infishingstrategy. Therateofmigrationisquantified bytheparameteracorrespondingtothechangeinbiomass den-sityduetofishmovementbetweenthezonesatthefasttimescale ([a]₌milliontons/10days).Twocontrastedcaseswereconsidered forthemovementofthefishingboats.Inonecase,weassumethat the fishing boatscannot freely cross thenational marine borders to lead fishingactivities, andin eachzone, fishing vessels/canoes arecommittedtoaspecificpolicythataffectstheCostperUnitof Fishing Effort (CUFE). Thus, inthiscase we alsoassume that the fishing investments in each country are independent.In the sec-ond case, we assume that fishing boats can freely cross the na-tionalmarinebordersandthattheCUFEpolicyisuniforminthe3 zones.

The fishingefforts ineachcountry aredenotede1(t),e2(t) and

e3(t).The fishing effortisexpressed insurfaceprospected bythe

fishing boats(104 _km2_{). Ifthe surfacefishedexceeds the surface}

ofthearea,itmeansthatthefishingboatsoperatemorethanone time ineach km2_{.Asan indicativerange,inthisidealizedmodel}

we considerthefishingeffortinazonei ofsurfaceSi shouldnot

exceed365 _{× Si},correspondingtoeachkm2_{ofzoneibeingfished}

daily. Let usassume the fishing effortin each zone is controlled through a local policy impacting the cost per unit of effort. Fur-thermore,itisconsideredthateach countryhasa veryhighlevel of governance andmilitary means which ensures theabsence of illegalfishingactivitiesignoringthemaritimeborders.Thereisno precisedataonrealvaluesoffishingeffortsinthethreecountries, indeed even the Copace/FAO workinggroup fromthe united na-tions is still looking for estimations of the fishing efforts andto find an approachfortheir standardization. Thus inthiswork the only alternative approachwasto propose an arbitraryset of val-ues,basedonlocalfisheriesexperiences.

Byanalysingtheaveragefuelconsumption,the“absolute” CUFE can be estimated. Indeed, on the basis of the fuel consumption ofmedium size fishing vessels (∼ 30 m), Sala etal.(2011)found a mean fuel consumption of ∼ 130 l/h during a fishing opera-tion, which occur at _{∼ 4–5} km/h. For this kind of boat, we as-sumethatthemeanradiusoffishdetection(byinstrumentsand/or direct visual) is 50 m around the boat. Based on these assump-tions, we estimate an average fuel consumption of _{∼ 300} l/km2

fished. Considering the fuel price (f) at 1 US$/l, then our esti-mate of the minimum cost per unit of effort for these boats is 300 US$. This is an underestimate of the total cost because we neglect the investment for the boat maintenance and the crew. However we will use this value as a basis in order to estimate whether the governance in each zone must provide subsidiesor tax thelocalfisheries inordertoreachthemaximumsustainable yieldattheregionalscale.Typicalsubsidiesthat impacttheCUFE are reducedpriceoffuelforfishing activity.Inthispaper,we re-flectthe impactoftax orsubsidiesby increasingorreducing the CUFE. The rate ofre-investing benefice into fisheries (

φ

) was ar-bitrary set to 1, which means that all the benefits are invested into fishing effort (in the same zone). Symmetrically, the losses

aredirectlytranslatedintoreducingthefishingeffortinthesame zone.

The fish catchability coefficient (q) is defined here as the fraction of the average fish density that is harvested per each unit effort.Forexample,ifthe fishdensityis1ton/km2_,andthe

catchability 0.5, then the harvest is 0.5 tons. Catchability is the parameterthat isthe harderto estimate fromrealexpert knowl-edgein thecontext ofstrongmodelassumptions on prospection, fishing techniques and fish density distribution (uniform within a zone). Thus, we fix the order of magnitude of this parameter arbitrarilyto0.01such thattheorderofmagnitudeofthefishing effortsremainedinarealisticrange(seeTable1and2).

The fish ex-vessel price is very highly variable on the coast, accordingthe seasonal fluctuations in abundances (Failler, 2014). However, the distant water fishing vessels are more concerned by the global ex-vessel price which is more constant, around 500US$/tons(Swartzetal.,2012).

We shall now present two models corresponding to different strategythatmaybeapplied bydecisionmakers forthe exploita-tionofthesharedfishstock.Thefirstmodelisreferredtoas com-petitive model, witha main assumption that each national fleet onlyoperatesinitsnationalfishingzone.Thesecond modelis re-ferred asthe fullycooperative model, withthe main assumption thatthethreecountriescollaborateandhaveanagreementto con-stituteacommonfleetoperatinginthethreefishingzones.

2.1. Competitivemodel

We now present the equations of the complete competitive model:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

dB1 d

τ

=



_aB 2 K2 − aB1 K1



+

ε



−qB1e1 S1 + rB1



1−B1 K1



dB2 d

τ

=



_aB 1 K1 + aB3 K3 − 2 aB2 K2



+

ε



−qB2e2 S2 + rB2



1−B2 K2



dB3 d

τ

=



_aB2 K2 − aB3 K3



+

ε



−qB3e3 S3 + rB3



1−B3 K3



de1 d

τ

=

ε



φ

c1



_pqB 1e1 S1 − c1 e1



de2 d

τ

=

ε



φ

c2



_pqB 2e2 S2 − c2 e2



de3 d

τ

=

ε



φ

c3



_pqB 3e3 S3 − c3 e3



. (1)

In order to reduce the complete model, we apply aggregation method(Auger etal., 2008). We set

ε

₌0in system(1) andthe simplecalculationleadstothefollowingfastequilibrium

B∗₁=

α

1B, B∗2=

α

2B, B∗3=

α

3B, inwhich

α

1= K1 K,

α

2= K2 K,

α

3= K3 K, K=K1+K2+K3,

α

1,

α

3,

α

3 represent the proportions of fish biomass in zone 1,

2,3atthefastequilibriumrespectively,andB

(

t

)

=B1

(

t

)

+B2

(

t

)

+

B3

(

t

)

isthetotalfishbiomassattheslowtimescalet=

ετ

.

Table 1

List of model parameters identified for small pelagic fish fisheries in North–West Africa ( ∼ 12.3–26 °N). Units were chosen such that most parameters are of the order of one (following the requirement for the aggregation method to be applied), except fish price and catchability coefficient.

Symbol Meaning Unit Value

Time-scale parameters

ε Ratio between fast ( ∼ 10 days) and slow ( ∼ 1 year) processes ( t / τ). Must be ∼ < 0.1 so that the aggregation method can be used ( Auger et al., 2008 )

No dimension 10/365 Environmental and biological parameters for the exploited small pelagic fish

S1 Surface of fish habitat in zone 1: continental shelf from 20.77 °N to 26 °N (South Morocco) 10 4 km 2 4.1472 (a)

S2 Surface of fish habitat in zone 2: continental shelf from 16.06 °N to 20.77 °N (Mauritania) 10 4 km 2 2.2720 (a)

S3 Surface of fish habitat in zone 3: continental shelf from 12.3 °N to 16.06 °N (Senegambia) 10 4 km 2 _{1.5040 (a)}

K1 Total small pelagic fish carrying capacity in zone 1 Megaton (10 6 tons) _{4.3 (b)}

K2 Total small pelagic fish carrying capacity in zone 2 Megaton (10 6 tons) _{3.3 (b)}

K3 Total small pelagic fish carrying capacity in zone 3 Megaton (10 6 tons) _{2.4 (b)}

a Fish migration rate. Define the ”speed” at which the fish tends to the ideal free distribution according to the local carrying capacities

Megatons/10 days 1 (c)

r Fish growth rate (average among small pelagic species in West Africa; FAO (2012) ) year −1 0.88 Economic/efficiency parameters for the small pelagic fish fisheries

f Price of the fuel Millions $/10 0 0 m 3 ( = S/l) 1

q Fish catchability. Represent the fraction of the average fish biomass in given area catch by a fishing vessel trawling on the entire area

No dimension 0.01 (c)

p Fish price. World average ex-vessel price for small pelagic fish Millions $/Megatons 500 (d)

ci Cost for fishing effort in zone i . The value given here correspond to the minimum to ensure boat maintenance,

fuel consumption and crew. National policy may increase or reduce these costs (subsidies or tax)

Millions $/[fishing effort] (or 10 6 $ 10 4 km −2 )

3 × f (e)

φ Rate of re-investing benefits in fisheries 1/year 1 (c)

β 1 st Parameter for the migration of boats in cooperative model: ideal free distribution when B is large _{τ/ [B/S] 1} β0 2 nd Parameter for the migration of boats cooperative in model: uniform distribution when B is small τ/ [S] 1

(a) Surface of the continental shelf, from coast to 200 m isobat (calculated from Etopo2). (b) Total small pelagic Carrying capacity estimated to ∼ 10 million tons ( FAO (2012) , distributed between the zone 1 and 3 according to the mean local plankton density estimated (Eric Machu, IRD, pers. Com). (c) Arbitrary. (d) Swartz et al. (2012) . (e) Estimated from the observations of Sala et al. (2011) .

Table 2

List of states variables used in the complete model.

Symbol Meaning Unit Range

Bi Fish biomass in zone i Megatons 0–K ia

ei Fishing effort in zone i , expressed in unit area fished 10 4 km 2 0–65 × S ib

a K

i = fish carrying capacity in zone i , see Table 1 .

b There is no maximum value for the fishing effort in the model; however we consider that the fishing effort in zone i should be reasonably under 365 × S i , corresponding to each km 2 in

the area being fished daily.

theaggregatedcompetitivemodelreadsasfollows

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

dB dt =



−q

α

1 S1 e1− q

α

2 S2 e2− q

α

3 S3 e3



B+rB



1−B_K



de1 dt =

φ

c1



_pq

_α

1 S1 e1B− c1e1



de2 dt =

φ

c2



_pq

_α

2 S2 e2B− c2e2



de3 dt =

φ

c3



_pq

_α

3 S3 e3B− c3e3



. (2)

To analyse the dynamical behaviour of aggregated competitive model,wewillsurveystabilityofnon-negativeequilibriumpoints

(

B∗,e∗₁,e∗₂,e∗₃

)

of theEq.(2).Weseerightthat two pointsO(0,0, 0,0)andA(K,0,0,0) arenon-negative equilibriumpointsof the equation.BylinearizedprinciplethenthepointOisunstable.The stabilityofpointAisstatedinthefollowingproposition.

Proposition2.1. IfBi=

ciSi

pq

α

i ≥ K

(

⇔

ciSi

pqKi≥ 1

)

foralli=1,3then thepointAisgloballyasymptoticallystableandlim_t→∞e_i

(

t

)

₌0_, i₌

1,3.

Proof.SeeAppendixA. _

When theassumptionofProposition2.1isinfringed,i.e.,there existsBi<K.Bylinearizedprinciple,pointAisunstable.However,

the system (2) has additional non-negative equilibrium points. Withoutlossofgenerality,we assumeBi<K forall i=1,3.With

this assumption we have covered all non-negative equilibrium pointsofsystem(2).Concretely,wehavethefollowingcases

• Case1: IfBi =Bj for i =j then the systemhasto add

equilib-riumpoints P1



B1, rS1

(

K− B1

)

Kq

α

1 , 0,0

,P2



B2,0, rS2

(

K− B2

)

Kq

α

2 , 0

, P3



B3,0,0, rS3

(

K− B3

)

Kq

α

3

.

• Case2:IfonlythereexistsapairBi=Bj =Bkfori,j,kdifferent

thenthesystemhastoaddequilibriumpointP_k(incase1)and a setof equilibriumpoints



i=

{

Mi

}

where B=Bi,ek=0 and

ei=e∗i,ej=e∗jsatisfy qα_i S_i e∗i + qα_j S_j e∗j=r

(

1− B_i K

)

.

• Case3:IfB1=B2=B3=:B∗thenthesystemhastoaddasetof

equilibriumpoints,denotedby



=

{

M

(

B∗,e₁∗,e∗₂,e∗₃

)

}

,inwhich

e∗_1,e∗_2,e∗₃satisfy qα1 S₁ e∗1+ qα2 S₂ e∗2+ qα₃ S₃ e∗3=r

(

1−B ∗ K

)

.

Toinvestigatestabilityofthenon-negativeequilibriumpointsof system(2)ineach theabovecase, wedenoteR4

+=

{

x∈R4:xm≥

0 for m=1,4

}

, R4

++=

{

x∈R4:xm>0 for m=1,4

}

and R4₊m=

{

x_∈R4

+:xm>0

}

with m=1,4. Through analyse stability of the

equilibriumpoint,weobtain asymptoticbehaviourforsolutionof system(2)inthefollowingtheorem.

Theorem2.2. Assume thatB_i= ciSi pq

α

i <

K

(

_⇔ ciSi pqKi<

1

)

forall i₌

a)If Bi =Bj for i =j and Bi is the smallest then point Pi is

globally asymptotically stable, other points are unstable and

limt→∞ek

(

t

)

=0,k =i.

b) IfthereexistsapairBi=Bj =Bkfori,j,kdifferentthenthere

existsatleastonefishingeffortvariabletendtozero.Moreover, if B_k is the smallest thenP_k is globally asymptotically stable andtheequilibriumpointsin



iareunstable. Otherwise,Pkis

unstableand



iis attractionsetforallsolutiontrajectoriesof

thesystem(2).

c) If B1=B2=B3=:B∗ then



is attraction set forall solution

trajectories of the system (2). Moreover, the solutions of the systemwithinitialvaluesin _R4

++willconvergetoequilibrium

pointsin



∩R4 ++.

Proof. SeeAppendixA. 

Ifthereexists isuch thatBi≥ K thenCase3doesn’texist,i.e.,

thesystem(2)hasnotanyequilibriumpointintheset



.Bythe sameargumentofTheorem2.2weobtainthefollowingcorollary.

Corollary 2.3. Assumethatthere existsi, j∈{1,2,3}suchthatK∈

(

B_i,B_j]_, then solution of the system (2) always contains at least a componentfunctionbeingej(t)satisfylimt→∞ej

(

t

)

=0.

Proposition2.1,Theorem2.2,Corollary2.3hasgivenus asymp-toticbehaviourofthesolutionofsystem(2)inallthecases. Sum-mary,wehavethefollowingassertions.

Proposition2.4. PuttingBi= ciSi pq

α

i withi=1,3.Then, a)If Bi≥ K

(

⇔ ciSi pqKi≥ 1

)

with some i∈{1, 2, 3} then

limt→∞ei

(

t

)

=0.

b) If max

{

B1,B2,B3

}

<K and Bi>min

{

B1,B2,B3

}

with some i

thenlim_t→∞ei

(

t

)

=0.

Proposition2.4showstheconditionsfortheextinctionand per-manenceoffishingeffort.Hence,fisherymodelwillbepermanent iftheparametersofmodelhavetosatisfytheconditionsinCase3, i.e.B1=B2=B3=B∗.When thesolutions ofmodeltendto

equi-libria in



, we will findout the conditionsfor the maximum of total catch of the fishery at the equilibrium. The catch per unit oftimeatequilibriumoftheslowaggregatedmodelreadsas fol-lows H:= 3 i=1 qiB ∗ e∗_i =



q

α

1 S1 e∗₁+q

α

2 S2 e∗₂+q

α

3 S3 e∗₃



B∗

where qi= qSα_ii be a local catchability in zone i. Since

(

B∗,e∗₁,e∗₂,e∗₃

)

∈



then H=

(

qα₁ S1 e ∗ 1+ qα₂ S2 e ∗ 2+ qα₃ S3 e ∗ 3

)

B ∗ = r

(

1₋B∗ K

)

B ∗

.ThusHgetsmaximalvalueH∗₌rK 4 when K 2 =B ∗ = c1S1 pq

α

1 = c2S2 pq

α

2 = c3S3 pq

α

3 ⇔c1S1 K1 = c2S2 K2 =c3S3 K3 = pq 2. (3)

The optimal attraction set is



∗=

{

M

(

B∗= K

2,e∗1,e2∗,e∗3

)

}

, where e∗₁,e∗₂,e∗₃ satisfy q

α

1 S1 e∗1+ q

α

2 S2 e∗2+ q

α

3 S3 e∗3= r 2. (4)

Figs. 2 and 3 present numerical simulations of the complete model with two cases, only one fishery is surviving and coexis-tenceofthethreefisheries.OnFig.2,wesuperposethethree pro-jectionsofthetrajectoryofthecompletemodelintheplanes(Bi,

ei)withi=1,3.

2.2.Fullycooperativemodel

Incooperative model,weassume the nationshavea common agreementtoconstituteacommonfishingfleet,eachcountry con-tributingtotheglobalfishingeffort.Theboatsareallowedtomove andtocatch fishinthethreefishingzones. Furthermore,itis as-sumedthatthefishing vesselsmovebetweenthedifferentfishing zones withrespect to local fish biomass. In this way, boats mi-gration ratesaresupposed to befish stockdependent.The larger is the fish biomass in a fishing zone, the less boats leave this areaperunitoftime.Inotherwords,itisassumedthatfleets re-mainlongertimesinzoneswherefishismoreabundant.Wealso suppose that all costs are equal, i.e. c1=c2=c3=c. Then the

equationsofcompletecooperativemodelreadasfollows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

dB1 d

τ

=



_aB 2 K2 − aB1 K1



+

ε



−qB1e1 S1 + rB1



1−B1 K1



dB2 d

τ

=



_aB 1 K1 + aB3 K3 − 2 aB2 K2



+

ε



−qB2e2 S2 + rB2



1−B2 K2



dB3 d

τ

=



_aB 2 K2 − aB3 K3



+

ε



−qB3e3 S3 + rB3



1−B3 K3



de1 d

τ

=



e2

β

B2 S2+

β

0S2 − e1

β

B1 S1+

β

0S1



+

ε



φ

c



_pqB 1e1 S1 − ce1



de2 d

τ

=



e1

β

B1 S1+

β

0S1 + e3

β

B3 S3+

β

0S3 − 2 e2

β

B2 S2 +

β

0S2



+

ε



φ

c



_pqB 2e2 S2 − ce2



de3 d

τ

=



e2

β

B2 S2+

β

0S2 − e3

β

B3 S3+

β

0S3



+

ε



φ

c



_pqB3e3 S3 −ce3



. (5)

Usingthe similar argument,we obtain thesame fastequilibrium

B∗₁₌

α

1B,B∗2=

α

2B,B∗3=

α

3Band e∗_i =

β

αi SiB+

β

0Si

βμ

B+

β

0

ν

e i=1,3, where

μ

₌ 3_i=1αi S_i,

ν

= 3 i=1Siande

(

t

)

=e1

(

t

)

+e2

(

t

)

+e3

(

t

)

.So

wehaveaggregatedcooperativemodel

⎧

⎪

⎪

⎨

⎪

⎪

⎩

dB dt =− q

(

βγ

B+

β

0

δ

)

δ

(

βμ

B+

β

0

ν

)

Be+rB



1−B K



de dt =



c



pq

(

βγ

B+

β

0

δ

)

δ

(

βμ

B+

β

0

ν

)

Be− ce

. (6) inwhich

γ

=

(

α

1S2S3

)

2+

(

α

2S1S3

)

2+

(

α

3S1S2

)

2

δ

,

δ

=S1S2S3.

The dynamical behaviour of (6) is similar to system (9) in

Moussaouietal.(2011).Thesystem(6)hasthreeequilibria(0,0), (K,0)and(B∗,e∗),where B∗=−pq

β

0

δ

+c

δβμ

+



(

pq

β

0

δ

− c

δβμ

)

2+4pq

βγ

c

δβ

0

ν

2pq

βγ

>0, e∗=r



1−B∗ K

δ

(

βμ

B∗+

β

0

ν

)

q

(

βγ

B∗₊

_β

₀

_δ

₎

.

Equilibrium(0,0)isalwaysasaddlenode.Weput

σ

=r

δ

(

βμ

B∗+

β

0

ν

)

2+Kq

ββ

0e∗

(

γ ν

−

δμ

)

.Accordingtovaluesofparameters,we

Fig. 2. Numerical simulations of the complete competitive model, phase portraits with initial values B 1 = 1 . 8 , B 2 = 1 . 2 , B 3 = 0 . 9 , e 1 = 90 , e 2 = 80 , e 3 = 74 and parameters values are in Table 1 . Blue, red and dotted lines correspond with the orbit of zone 1, 2 and 3, respectively. (a): Fishery 3 survives, fisheries 1 and 2 go extinct when the cost per unit of effort among the three zones are uniform, c 1 = c 2 = c 3 = 300 . (b): Coexistence of the 3 Fisheries is stable when total catch gets maximum value, c 1 = 259 . 21 ,

c2 = 363 . 12 , c 3 = 398 . 94 . The biomass are in million tons and the fishing effort in 10 4 km 2 prospected each year; the costs of fishing effort are expressed in US$/km 2 prospected. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Numerical simulations of the complete competitive model, time series. The initial values and parameters values are used in Fig. 2 . Blue, red and dotted lines corre- spond with the trajectory of zone 1, 2 and 3, respectively. (a): Fishery 3 survives, fisheries 1 and 2 go extinct. (b): Coexistence of the 3 Fisheries. The biomass are in million tons and the fishing effort in 10 4 km 2 prospected each year. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

• IfB∗>Kthen(B∗,e∗)<0and(K,0)isastablenode.

• IfB∗<Kand

σ

>0then (K,0)isa saddlenode and(B∗,e∗) is globallyasymptoticallystable.

• IfB∗_<Kand

σ

_<0then(K,0) isasaddlenode,(B∗,e∗) is un-stableandsystem(6)hasalimitcycle.

Thecatchperunitoftimeatequilibriumreads

H∗= q

(

βγ

B∗+

β

0

δ

)

δ

(

βμ

B∗₊

β

₀

ν

₎

B∗e∗=rB∗



1−B∗ K



Thus H∗ has maximum equal to rK₄ when B∗=K

2, i.e. c=c∗=

Kpq

(

K

βγ

+2

β

0

δ

)

2

δ

(

2

β

0

ν

+K

βμ

)

.

3. Discussion

The mathematical model provides some interesting insights about the dynamic of international shared fish stock. First, we interpret the main results of the mathematical model in terms of fisheries management and compare with the results obtained fromotherstudies.Second,wediscusshowthismethodologymay evolvetowardanoperationalmodelthatcanbeusedintheframe ofregionalfishingagreementsbydecisionmakersandmanagers.

Table 3

Cost per Unit of Fishing Effort (CUFE) f or a st andardized industrial fishing boat: model prediction for maximum cost for sustain- able fishery and optimal costs for maximizing total catch according to parameters set provided in Table 1 ; the cost estimated are based on fuel consumption, boat maintenance and crew costs are fixed at 300 US$/km 2 for the three zones, see _{Table 1 .} Unit: cost/km 2 prospected/fished in zone 1, 2, and 3 (must be divided by hundred to convert into millions US$/10 4 km 2 , the unit used in Table 1 ).

Zones Maximum cost for sustainable fisheries c imax ($/km

2 ) _{Optimal cost for maximising total catch c}

iopt ($/km 2 )

1 518.42 259.21

2 726.23 363.12

3 797.87 398.94

3.1. Interpretationofthemainmathematicalresults

3.1.1. Independentnationalfleet:sharedfisheriesasacompetition withonlyonewinnerinthegeneralcase

In each country there is a threshold,maximum value forthe CUFEabovewhichthefisheryisnotviable(Proposition2.4.a).The modelpredictedthatthisvalueisdefinedbytherelationship: cimax=pqki

wherek_iisthelocaldensityofcarryingcapacity: ki=Ki/Si

Thus, thehigherthecarryingcapacitydensityinagivencountry, the highercost per unit ofeffortcan be afforded by the fishery. However, ifthecost exceedsthisthresholdvalue thenthefishery activitygeneratesnotenoughrevenueandtendstodisappear.For theWestAfricacasestudythethresholdcostsaregiveninTable3, accordingto theparameters set inTable 1.Underthe hypothesis ofanabsenceofmanagementpoliciesandorregulationmeasures, then foragiventype offishingvessel theCUFEamongthethree zoneswouldbeuniform,solelyrelyingonfuelconsumption,gears and vessel/canoe maintenance, and crew costs (Table 1). In this case, the ratio ci/ki forzones 1, 2, and 3 is 2.9, 2.1 and 1.9,

re-spectively.Thusinthelongtermtheonlysurvivingfisherywould occur in zone 3, in which there is the highest carrying capacity density.ThislattercaseisillustratedinFigs.2aand3a.However, giventhedifferentnatureofthefisheries inthethreecountriesit is very unlikely to have equals CUFE(see subsection”Toward an operationalmodelofsharedfisheries” ofthesectiondiscussion).

Second,themathematicalanalysisalsoshowedthatingeneral, iftheaveragefishbiomassesaredifferentbetweenthethree coun-tries, only one fishery may actually survive the competition, the onelocatedinthecountrywheretheratiobetweentheCUFEand thecarryingcapacitydensity(c_i/k_i)isthelowest(Theorem2.2.a).

The three countries must find an agreement to ensure sus-tainable fishing in their respective area. There is only one pos-sible agreement that ensures a sustainable fishing activity inthe three countries simultaneously, which is to have equal c/k ratio (Theorem 2.2.c). This case is illustrated in Figs. 2b and 3b. Fur-thermore, inorder to maximize thetotal catch amongthe three countries, the CUFEshouldbe set accordingto thelocalcarrying capacitydensity(Eq.(3)):

ciopt= pqki

2 .

When applyingthe optimalcosts, thetrajectorytends towards one of the stable equilibria according to the initial conditions. Therearenumerousstableequilibriumspossiblewithdifferent dis-tribution of the fishing effort amongcountries (Fig. 4). Although it wasnot demonstrated, the sensitivitytest withrandom distri-bution of the initial fishing effort showedthat the initial fishing effortin a givenarea wassignificantly correlated withthe effort atequilibrium

(

R=0.7,p<0.005

)

.Thissuggeststhatapplyingthe optimalCUFEpolicyatagiventimemaytendtostabilizethe cur-rentdistributionoffishingeffort,whiletheamplitudecanbe mod-ulated,andover-allbenefitsofthefisheriesmightincrease.

Fig. 4. Tri-dimensional view of the positive part of the fishing effort s in the three countries (e1, e2, and e3) in the case of optimal fishing cost. The blue plane corre- sponds to the ensemble of optimal equilibria ( Eq. (4) ). Each coloured level trajec- tory corresponds to a different set of initial conditions, with the circle indicating the equilibrium reached at t = 100 (slow time). Each trajectory converges toward a specific point of the plane, according to the initial conditions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

However, except in case c) of Theorem 2.2, the competitive modelmainlyleadstofleetexclusion.AssoonastheBiarenot

ex-actlyequal,wefallincasea)ofTheorem2.2whereasinglefleet can survivewith extinctionof two other fleetsin thelong term. Indeed,assoon asthe threeBi are not strictly equal,trajectories

are going to move inthe direction ofone of the exclusion equi-librium which is stable, case a). Maintaining the system in case c)ofTheorem2.2wouldassume tochangefrequentlyparameters, whichseemsnotsoeasy inpractice.Thus,itmustbehighlighted that the situation for optimal regional harvesting is a very par-ticular caseof the dynamical system (Theorem 2.2, casec). Any small change in the fish biomass distribution (e.g. from climatic fluctuation)oruncontrolledfishingeffort(e.g.illegalfishing)may causethesystemtoswitchonatrajectoryofthemoregeneralcase whereonefisherytakestheadvantageontheothers(Theorem2.2

casea).Giventheinevitableuncertaintiesonfishstockevaluations andfishing effortcontrol,it isnot realistic forthree countriesto maintainthe exactoptimal cost ascalculated here. However, the optimalcost could be regularly updated accordingto the contin-uous monitoringof the fishing effortandthe fishstockstates in thethree countries(e.g.after one monthorone year). Notethat thesensibilitytestshowedthatitismuchmoreimportanttohave agoodestimationofthefishingeffortinthethreecountriesthan theestimateofthefishbiomass.Theregularupdatingofthelocal optimalfishingcostscouldallowthethreecountriestocollectively adjust the policies (e.g.tax) to get the total catch closest to the maximumsustainableyield.

IfthecaseofinternationalagreementtotunetheCUFEto opti-malvalueineachEEZ,thetotalcatchamongthethreeareas(given bythe harvestfunctionHinSection 2.2) is2.2million tons/year, tobecomparedtotheaveragelandingsofsmallpelagicfishinthis areain2005–2011whichrangesfrom ∼ 1.5to2milliontons/year including several over-exploited species (FAO, 2012). Such over-exploitationratesmaynotbesustainableandincreasethe vulner-abilityoftheecosystemduetotheeffectofclimatechange. How-ever,switchingfromthepresentover-fishingsituationtowardthe maximumcatch byacting onthe costparameters maycausethe fishingeffortandthelandingstolowontheshortterm.

Althoughtheassumptionwemadewereverydifferent,wecan compare our results with game theory models applied to study caseof shared fisheries asfor example the case of Canada, USA and Mexico (Javor et al., 2011; Lo et al., 2011), which are ex-ploiting the same pacific sardine (Sardinops sagax) transbounday stock, with a status ”declining/depleted” in 2011 (Based on US National Marine Fisheries Service). At a first glance, our results are in line with these studies (Ishimura et al., 2012, 2013). As

Ishimuraetal.(2012) wefoundthat thecooperationbetweenthe countriesleadstoahighertotalcatch,butoneofthecountries(in ourcasetheone withthehighestcarryingcapacitydensity,zone 3)maygetlesslandingswhen collaboratingwithother countries thanifitminimisesitsfishingcosts.Thus,theconcernedcountry, inourcasestudySenegal,mayhavepositiveincentivestoact non-cooperativelyinstead ofparticipating to thecooperative manage-ment.Ishimuraetal.(2013)demonstratedthatthisproblemcanbe solvedusingside-payments,whicharepositiveincentivesgivenby thecountriesthatmaygetbenefitofcooperative managementto thecountrythatmayhavemoreinteresttoactasafreeplayeri.e. non-cooperatively. Adaptedto ourcase studyit meansthat Zone 1and2should provideaside paymenttozone3to reducetheir fishingefforts.Becausethe totalcatchincrease incaseof cooper-ativemanagement,zone 1 and2would still increase their bene-fitfromfisheriesincomparisonwiththenon-cooperative manage-mentsituation.However, asexplained inthe previous paragraph, ourdynamicmodelshowsthat thefishing agreementsshould be constantlytunedovertimeinordertomaintainthesystemaround thevery particularcase of optimalequilibrium. Also, taking into account the spatialvariability of carryingcapacity densityinside theEEZ maychangethe results; inparticularthe very high con-centrationoffisheriesobserved off CapBlancinMauritanian wa-tersseems toindicateaparticularlyhighcarryingcapacityinthis area.Furtherrefinement ofthe carryingcapacityspatialand sea-sonal distribution will be neededfor more informative manage-mentinformation.

3.1.2. Fullycooperativeinternationalfleet

Inthecaseofthecooperativemodel,withtheparameterslisted inTable 1 andthe cost per unit offishing effortsatisfies B∗<K, thenthereexistsasinglepositiveequilibrium(B∗,e∗).This equilib-riumisstable(

σ

>0,seeSection2.2),thusstartingatanypositive initialconditions,alltrajectoriesaretendingtothisunique equilib-rium(Fig.5).Thisisagoodsituationbecauseitallowsmaintaining adurablefisheryinthelongterm.Adesirablecooperativestrategy would be also to avoid the case of a stable limit cycle because itwould generate importantfluctuations ofthe stockand ofthe fleetsinthelongterm. Forinstance,atsomeperiodsoftime, the fishstockwouldbevery low,andthiscould leadto fishcollapse asaresultofanystochasticfluctuations.Suchstochasticprocesses are not considered in our deterministic models butoccur inthe realworld.Asaconsequence,thebestcooperativestrategywould be to maintain the fishery at the positive equilibrium (B∗, e∗) whenitisgloballyasymptotically stableinthepositive quadrant, i.e.when B∗<K and

σ

>0.Furthermore,itwouldalsobe needed tochooseparametersinordertomaintainthefisheryatits

Max-imum Sustainable Yield (MSY) by fixing the costs at c=c∗. This canbeachievedbythethreenationsuponacommonagreement. Because we have several parameters that could be selected by fishery regulation, such as taxes, costs and maybe catchability (size net...), it would be possible to find a parameter set that maintainsthefisheryatitsMSY.Suchinternationalfleetisdifficult toimplementinourcasestudyasasplitofobjectivestakesplace atnationallevel.IndeedtheSenegaleseneedthisfishresourcefor their foodsecurity ratherthan increasetheir GDPandMauritania target the opposite. Neverthelessa smart agreementcould target maximumsustainable yield goalproviding inkindandcash con-tributions according to each national priority. The use of foreign distant fleet should be avoided to conciliate all parties seeking compromise fromall sides. Finally, our result underline that the cooperative model is the optimal management scenario because thecorrespondingmodelispermanent.Indeed,thereisaglobally asymptotically stableequilibriumforthetotalfisherythat can be reachedby fixing CUFEin a rather large set ofvalues. This con-trastswiththecompetitivemodelforwhichitisneededtofixthe CUFEineachEEZatasinglepointinthewholeparameterspace.

3.2. Towardanoperationalvirtualrepresentationoftransboundary fishstock

The present model may evolve toward an operational model forthemanagementofsharedfisheries.Forthatpurpose,accurate dataisneededaboutthenatureofthefisheriesinthedifferent ar-easandthecharacteristics (speciesgrowthparameters,stock spa-tialdistribution,andenvironmentalcarryingcapacity)forthemain fishpopulations exploited.Thisworkstresses onemoretime that whatever the methodological approach, operational tool in fish-eriesmanagementrequestsmonitoringprocedureofthebiological andanthropicfeaturesofthesystem.

ComparedtothegrossestimateofthecrudeCUFEwithout in-tegrating anytax orsubsidy, (300 US$ per prospected km2_{) and}

assuming that a similar fishing fleet operates inthe three zones, settingthelocalCUFEaccordingtotheoptimalcasesuggestedby themodel(Table3)wouldimplicate theimplementationof addi-tional tax on fisheries in zone 3 and 2 (optimal CUFE of ∼ 400 and∼ 360US$/km2_{),but,oppositely,subsidiesforfisheriesinzone}

1 (optimal CUFE of ∼ 260 US$/km2_{). However, actually the local}

fisheriesinthethreezonesareofverydifferentnature,which im-pliesdifferentcrudeCUFE.Indeed,thefishingcostinSenegal(zone 1) may be lower than in Mauritania (zone2) ifwe assume that smallscalefisheries (canoe)getlower costthanindustrial fishing (vessel), whichappears asan acceptable assumption. Small scale fisheriesalsooperateinMauritaniawatersbutstartingoftenfrom Senegal andthusget highercostthan intheir home zone (Sene-gal).MoreoverinMauritaniathereismoreindustrialfishing activ-ity(foreign fleet)thanin Senegalwhereindustrial fishing isvery low (<5vessels, CRODT,2013). InSouthern Morocco/Western Sa-hara most of the small pelagic fish comes from semi industrial fishing as found in Laayoune the main fishing harbour forsmall pelagic landing in Morocco(FAO, 2012). Thus, according tothese qualitativecriteriatheactual crudeCUFEshouldberankedas fol-lows:thelower costmayoccurinSenegal,(mainlylocalfisheries, dominated by local small scale canoes), intermediate costs may occur in Morocco/Western Sahara (local fisheries, semi industrial boats),andthehighestcostmayoccurinMauritania(fisheries al-mostall foreign,mixedsmall scaleandindustrial fleets). Accord-ing to our results, such CUFE ranking may exacerbate the dise-quilibrium alreadyfound assuming equal CUFE among the three zones(Fig.2a),thusincreasingtheadvantageofzone3. Neverthe-less,theaverageCUFEinSenegalmaybeincreasedifwealsotake intoaccountthedistantwaterfisheriesDWF(Sumailaand Vascon-cellos,2000)andIUU(illegal,unreportedandunregulated),which

Fig. 5. Numerical simulations of the complete cooperative model, phase portraits with parameters values are in Table 1 . Coexistence of the 3 fisheries when the costs per unit of effort among the three zones remain uniform. (a) The costs per unit of effort are similar to Fig. 2 a: c 1 = c 2 = c 3 = 300 . Blue, red and dotted lines correspond with the orbit of zone 1, 2 and 3, respectively using the complete model (5) . Initial values are B 1 = 1 . 8 , B 2 = 1 . 2 , B 3 = 0 . 9 , e 1 = 90 , e 2 = 80 , e 3 = 74 . (b): Total fishing effort and total fish biomass of three zones, using the aggregated model (6) and the optimal cost per unit of effort c 1 = c 2 = c 3 = c ∗= 322 . 7449099 . The aggregated model has a single positive equilibrium which is globally asymptotically stable for different initial values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

probablyoccur morein zone3 thanin thetwo other zones, due tolowregulationandmonitoringcapabilities(Agnewetal.,2009). Finally, a precise estimate of the CUFEaccording to each fishery isnecessarybeforesettingeventualtaxesorsubsidiesonthe fish-ingefforttobeimplementedtotendtowardoptimalmanagement. Withinagivenzone,suchpolicymustbeadjustedtoeachfishery typeaccordingtoitscrudeCUFE.

As in all models dealing withfisheries issues, the hypotheses ontheecologicalpropertiesoftheexploitedspeciesaresimplified toimplementthemodel.Amongthesehypotheseswe canreport, thegravity centreofthespatialdistributionofdifferentexploited fish speciesmaydiffer, aswell astheir growthrateandcarrying capacity.Also, theresultsmaybesensibletotheexistence ofhot spotsoffish densitywithin agivenzone, asforexamplethe Cap Blanc area in Mauritania (Braham et al., 2014). Further develop-ment ofthe methodologypresented inour work could includea realisticdistributionofthecarryingcapacitywithineachzone(i.e. non uniform) based onphysical andbio-geochemical simulations of theenvironment (e.g. Auger etal., 2016). The time dependent carryingcapacityprovidedbythe simulationsoftheenvironment couldbeincludedintheapproachbyusingnon-autonomous mod-els.

The model might be easily generalized to the case of more thanthreecountriessharingthesamefishstock.Indeed,the Gam-bia could get differentpolitical goal than Senegal, Guinea-Bissau could be included inthe process (Fig. 2) even if atthe opposite Regional FisheriesOrganisation(RFO)cansimplifyinteractions i.e. in our case the Sub Regional Fisheries Commission (SRFC) could harmonizetherulesforMauritania,Senegal,TheGambiaandeven Guinea-Bissau.

A tricky point that happens in a large partof the world, es-pecially in the third world, is the presence of illegal and unre-ported fishing activity that crosses the borders without fishing agreement (known as illegal, unreported, and unregulated (IUU) Fishing (Agnew etal.,2009)),followingthe fishmigrationor dis-placement.At afirstglance, inourmodelthismaycausethe ex-tinction offisheries inzone 1 and2.This mightimpact the

effi-ciencyofthefishingmanagementproposedhereandits equitabil-ityamongthe country.Onemainperspective oftheworkwillbe tostudytheimpactofthepresenceofIUUactingamongdifferent countrieswherelocalfisheries are exploitingthe samemigratory fish population,taking asexample the caseof fisheries targeting sardinellainNorth–WestAfrica.

Finally,thisapproachprovidesapracticalframeworkfor build-ing a reflection on the international management of shared ex-ploited fish species i.e. a common issue, sometimes a matter of contention,forseveralRFOasinternationalfisheriescommissions, non governmental organisation(NGO) and national agencies.The modelconsiders the fishing effortregulation through thecontrol ofthecostper unitoffishing effort(including taxorfishing per-mit), its analytical interpretation allows one to provide original fisheries managementrecommendations, whichshould encourage deeperstudiesforoperationaluses.

Acknowledgements

Theauthorsaregratefultotworeviewersfortheirconstructive comments.EricMachu (IRD)provided theROMS-PISCESplankton biomassusedforthecarryingcapacityestimates.

Thesecond,thethirdandthefifthauthorshavebeensupported by the project AWA (BMBF - IRD; 01DG12073E) "Ecosystem ap-proach to the management of fisheries and the marine environ-ment in West African waters" and the European Preface project (DGEnv.,FP7)underGrantAgreementnumber603521.

Thispaper was partially done while the first author was vis-itingattheVietnamInstitute forAdvancedStudy inMathematics (VIASM).TheauthorisgratefultoVIASMforthehospitality.

AppendixA

Proof of Proposition 2.1. Put R4

+1=

{

x∈R4:x1>0andxm≥

0,m=2,4

}

.AtpointA,weconsidertheLyapunovfunction VA=p



B− K− KlogB K



+c1

φ

e1+ c2

φ

e2+ c3

φ

e3.

Then,VAispositivedefinite,radiallyunboundedinR4₊₁and ˙ VA =− pr K

(

B− K

)

2₋ 3 i=1



ci− K pq

α

i Si



ei=− pr K

(

B− K

)

2 − 3 i=1 pq

α

i Si

(

Bi− K

)

ei.

SinceB_{i≥ K for}alli₌1_,3soV˙A isnegativesemi-definiteandthe

non-negativeequilibrium pointsof system (2) only includeO, A. ByLaSalle’sinvarianceprinciple,pointAisgloballyasymptotically stable.Hence,limt→∞ei

(

t

)

=0,i=1,3. 

ProofofTheorem2.2. a):Forfixed i=1,3,correspond toPiwe

considertheLyapunovfunction

VPi =p



B− Bi− Bilog B Bi

+ci

φ



ei− rSi

(

K− Bi

)

Kq

α

i − rSi

(

K− Bi

)

Kq

α

i log eiKq

α

i rSi

(

K− Bi

)

+cj

φ

ej+ ck

φ

ek

with j, k =i. Then VPi is positive definite, radially unbounded in

R4 +1∩R4+,i+1and ˙ VPi=− pr K

(

B− Bi

)

2₋pq

α

j Sj

(

Bj− Bi

)

ej− pq

α

k Sk

(

Bk− Bi

)

ek.

If B_i is the smallest then V˙P_i is negative semi-definite on R4₊₁∩

R4

+,i+1. Moreover, remaining equilibrium points will be

unsta-ble follow linearized principle. Therefore, by LaSalle’s invariance principle then P_i is globally asymptotically stable. This yields to limt→∞ek

(

t

)

=0, k =i.

b):Ife∗_i =0ore∗_j=0thentheequilibriumpointisPi,Pjinthe

case1,respectively.Withe∗_i,e∗_j>0weconsidertheLyapunov func-tion VMi =p



B− Bi− Bilog B Bi

+ci

φ



ei− e∗i− e∗ilog ei e∗ i

+cj

φ



ej− e∗j− e∗jlog ej e∗_j

+ck

φ

ek.

ThenVMi ispositivedefiniteand

˙ VMi=− pr K

(

B− Bi

)

2₋pq

α

k Sk

(

Bk− Bi

)

ek.

Argumentasinproofofassertions(a),ifBkisthesmallestthenPk

isgloballyasymptotically stableandthe equilibriumpointsin



_i areunstable.Otherwise,Pkisunstableand



iisattractionsetfor

allsolutiontrajectoriesofthesystem.Thus,thereexistsisuchthat lim_t→∞e_i

(

t

)

₌0.

c): Ifthereexists e∗_i=0withi=1,3thenthe Lyapunov func-tion of these type points is constructed as in the case 2. With

e∗_1,e∗_2,e∗_3>0weconsiderthenLyapunovfunction VM= p



B− B∗− B∗log B B∗



+c1

φ



e1− e∗1− e∗1log e1 e∗₁

+c2

φ



e2− e∗2− e∗2log e2 e∗ 2

+c3

φ



e3− e∗3− e∗3log e3 e∗₃

.

ThenVMispositivedefinite,radiallyunboundedinR4++ and

˙ VM=− pr K

(

B− B ∗

)

2_{≤ 0.}

ByLaSalle’sinvarianceprinciple,



isattractionsetforallsolution trajectories of the system. The next, we will show that solution ofthesystemwithinitialconditionM0

(

B0,e10,e20,e30

)

∈R4++will

convergetoequilibriumpointin



_∩R4

++.Weconsideran

equilib-riumpointM∈



∩R4

++,put

c=

{

x∈R4₊:VM

(

x

)

≤ c

}

.Then

cis

compact set, invariant with respect to the system (2) and

c⊂

R4

++. Denote E=

{

x∈

c:V˙M

(

x

)

=0

}

, then the biggest invariant

set with respect to the system (2) in E is L=E∩



⊂ R4 ++. By

LaSalle’s invariance principle, we conclude that every trajectory starting in

c approachesL. Now, choice c is large enough such

thatM0∈

c. 

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