the New Caledonia Abore Reef - ENPC

A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve caseM. De Lara: ENPC, CermicsL. Doyen: CNRS - MNHNJ. Ferraris: IRD, UR CoReUsD. Pelletier: IFREMER, EMHA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 1

To begin with. . .A quite honest talkonthe multiple difficulties of a multidisciplinary workwith multiple contributors, multiple species,multiple objectives!ora MISSION IMPOSSIBLE fromA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 2

to. . .////////////////////////////////////////////////////// DATA AND PARAMETER IDENTIFICATION// CORAL////////////////////////////////////////////////////cor_max=0.8;// maximum covering = 80 %cor_min=0.05;// minimal covering = 5 %// R_c tested from 1.001 to 1.01 by dichotomyR_c=1.002;K_c = cor_max * R_c / (R_c -1) ;function Xp=logistic(t,X)Xp= X .* (R_c * (1 - X/K_c) );// on force les Densités à être positivesendfunctiontime=0:(9*365);A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 3

Scientific contextAppel à propositions de recherche biodiversité etchangement globalInstitut français de la biodiversitéMinistère de l’Ecologie et du Développement durableAccepted project (2004):Modèles pour une gestion durable de labiodiversité sous incertitude et dynamique globalesA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 4

Problem statement• Protected area: a relevant tool for sustainablemanagement of renewable resource?• What is a protected area (PA) effect?- on stocks?- on catches?• Which sustainable management through PA?• Size• PlacementA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 5

Modelling requirements• The context:• Population dynamics: nonlinearity,complexity, age-structured, spatial• Decision : size and location of the PA• Uncertainties: catches, stocks, processes, etc.• The issues:• Multi-criteria• Ecology: conservation• Economy: fishing income• Intergenerational equity: both short and longterm horizonA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 6

PLAN1. The Abore reef reserve, New Caledonia2. A dynamic state model3. The difficulties of parameter estimation from data4. Measuring the reserve effect by stochasticviability5. Simulations (skipped because not yet discussedwith biologists)6. DiscussionA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 7

Some facts and figuresAssessment of the Impact of Removing MarineReserve Status on Demersal and Benthic FishCommunities : a Comprehensive ApproachJ. Ferraris (1), D. Pelletier (2), M. Kulbicki (1),C. Chauvet (3)(1) Unité de Recherche CoRéUs, IRD, Nouméa,Nouvelle-Calédonie(2) Laboratoire MAERHA, IFREMER, Nantes,France(3) Laboratoire LERVEM, Université, Nouméa,Nouvelle-CalédonieA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 8

The Noumea lagoon, New CaledoniaThe Noumea lagoon, located in South-Western NewCaledonia, South Pacific (Figure 1) is a large coralreef ecosystem where several marine reserves wereestablished in the 1980’s in view of protecting thecoral reef ecosystem from damage due to fishing andother human activities.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 9

The Abore reef reserveThe Abore Reef reserve is located on a 25 km longbarrier reef representing an area of ca. 15 000 ha.Fishing was banned from the whole reef from 1990 to1993, and allowed again on 2/3 of the reef fromAugust 1993 for a fishing experiment in theperspective of adaptive management.This opening was monitored by the Natural ResourceDepartment of the South Province and by LERVEM(New Caledonia University)A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 10

Expected reserve effectsThe main effects expected from the establishment ofreserves are• increased abundances and biomasses ofspawning stocks and recruitment inside theprotected area and in surrounding areas throughspillover• rebuilding of ecosystems through protection ofhabitat from fishing gearsA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 11

A strong fishing pressureIn the Pacific islands, the reef and lagoon subsistencefisheries represent about 80 % of total coastal catch.Right after the reserve opening in August 1993:• the number of boats and fish yield during the2 weeks after the opening reached the levelspreviously observed for a whole year;• monitoring of fishing effort and catch ratesshowed that, in the open area, benefits from the1990-1993 closure were dissipated within a fewweeks.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 12

Fishing experimentThe whole reef was finally closed to fishing fromAugust 1995.In 1995, 2/3 of the reef had been closed for 3 years(1990-1993) and fished for 2 years (1993-1995),while the remaining had been permanently closedduring the 5 years (1990-1995).The area open to fishing from August 1993 is theimpact area (area B in Figure 2), while the referencearea has been permanently closed (area A in Figure 2).A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 13

Survey / Before After Control Impact designA survey was conducted in July 1993 and July 1995,respectively right before the opening and before thefinal closure.A scientific evaluation of how reserve is likely toaffect fish community, e.g. increased densities, largerfish, modified interspecific relationships.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 14

A glimpse at complexity and uncertaintyThe experimental design rests on a stratification of thereef into three morphological zones : reef flat, innerslope and lagoon (Figure 2), delineated on aerialphotographs.The structure and functioning of communities areoften poorly known, in particular because coral reefecosystems exhibit a very high diversity of fishspecies, generally linked with live coral cover.For demersal and benthic species, spatial distributionsof populations mostly depend on habitat preferences.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 15

A glimpse at biodiversity: species richnessA total of 374 species were identified during thesurvey. As a consequence of the high diversity of thereef community, the number of species observed at agiven transect often exceeded 100, most species beingencountered only at a few stations.The criteria used for partitioning the fish communityinto species groups were mobility, taxonomy andfeeding habits.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 16

Four groups of species defined for mobility• territorial species living in a very restricted range(usually less than 10 m 2 );• sedentary species with a restricted range betweenten and a few hundred square meters;• weakly mobile species not restricted to a specificrange(up to several thousands m 2 );• highly mobile species usually foraging over verylarge areas; they are not restricted to a given reef overa short period of time.These categories form a continuum, and some specieswere difficult to assign to a given category.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 17

Feeding habits / dietSpecies at a high trophic level were generally thosetargeted by fishermen, and were thus likely to besensitive to the reserve status.Feeding habits were expressed in percentage of foodtypes in diet. Food types were categorized as nekton,macroinvertebrates, macroalgae, microinvertebrates,microalgae, zooplankton, other plankton, coral anddetritus.The analysis of species diets yielded 7 clusters, eachcluster forming a trophic group (Table 2). Note that ineach group, the mean diet included several food items.Groups were named on the basis of their mean dietcomposition.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 19

DensitiesTotal fish density per transect ranged between 0.95and 114 individuals/m 2 , with a mean of 7.6 ind/m 2 .The largest densities observed occurred at one or twostations with large concentrations of Clupeidae. Whenthis family was excluded from computations, totaldensity per transect over the two years dropped to 2.9ind/m 2 , with a mean of 4.0 ind/m 2 in1993 and1.7 ind/m 2 in 1995.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 20

A brief summary• Area: 15 000 ha• Biodiversity: some 374 species ; 41 families ;7 trophic groups ; few functional groups but a lot ofspecies within each group• Catch pressure: very strong ; the biomass surplusaccumulated in 3 years (from 1990 to 1993) has beenfished in. . . 15 days• A climatic impact on coral: the coral is a refugefor (small) fishes ; affected by cyclones destructions• Issues:- Is there a reserve effect?- How does climate change modify the previousassertions?A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 21

A DYNAMIC STATE MODELA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 22

Highly delicate modelling options• Restriction to . . . 4 trophic groups + habitat• One “typical” species is selected within eachgroupThese basic options lead to very difficultdiscussions between marine biologists andmathematiciansA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 23

State: densities• Coral/Habitat x 0 (t): percentage of covering;• Piscivors x 1 (t): predators of fishes, target offishermen (length 77 cm on average);• Macrocarnivores x 2 (t): predators ofmacroinvertebrates and of a few fishes (length38 cm on average);• Herbivors x 3 (t): some are targets of fishermen(length from 24 to 39 cm);• Other fishes (small) x 4 (t): sedentary andterritorial organisms, microcarnivores (17 cm),coralivores (16 cm), zooplanctonophages(13 cm);A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 24

“Typical” species: a highly delicate choice. . .• big grouper• small grouper• parrot fish• damsel fishA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 25

A logistic habitat covering growth model(x 0 (t + 1) = x 0 (t)(R 0 1 − x ) )0(t)K 0The time unit ∆t = 1 is to be fixed later.• R 0 is the intrinsic growth rate (for low covering).• K 0 is related to the so called carrying capacity x ♯ 0solution of1 = R 0(1 − x♯ 0K 0)Cyclonic events occur randomly with probability p atevery time step and bring the coral covering to 30 %of its last value.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 26

A Lotka-Volterra modelx 1 (t + 1) = x 1 (t) R 1 + α 1 2`x0 (t)´(E1 1 − 1)x 1(t) + E2 1α1 2 (x 0(t))x 2 (t) + · · · + E4 1α1 4 (x 0(t))x 4 (!x 2 (t + 1) = x 2 (t) R 2 − α 1 2 (x 0(t))x 1 (t) + α 2 2 (x 0(t))(E 2 2 − 1)x 2(t) + · · · + E 2 4 α2 4 (x 0(t))x 4 (t).x 4 (t + 1) = x 4 (t) R 4 − α 1 4 (x 0(t))x 1 (t) − α 2 4 (x 0(t))x 2 (t) − · · · + α 4 4 (x 0(t))(E 4 4 − 1)x 4(t)!.The time unit ∆t = 1 is to be fixed later.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 27

Details for trophic group 2x 2 (t + 1)x 2 (t)= R 2 intrinsic growth rate− α 1 2(x 0 (t))x 1 (t)predator 1’s catches rate− α 2 2(x 0 (t))x 2 (t)predator 2’s catches rate+ E 2 2α 2 2(x 0 (t))x 2 (t) conversion of predator 2’scatches rate of prey 2.+ E 2 4α 2 4(x 0 (t))x 4 (t) conversion of predator 2’scatches rate of prey 4A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 28

• Intrinsic growth rate R k includes mortality andrecruitment of trophic group k due to externaltrophic groups.• α i j (x 0)x i is the proportion of prey j captured byx i predators i in one time unit. It depends oncoral covering through a refuge mechanism:• E i jα i j(x 0 ) = γ(x 0 )̂α i j = e (β−λx 0)̂α i jstands for the conversion factor of one unit ofprey j density into growth of predator i.• The matrices α and E are upper triangular.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 29

To summarize the dynamics of the trophic groups,except the coral, we write in a matrix formx k (t+1) = x k (t)(R k + ˜S(x 0 (t))x(t)) k , k = 1, . . . , N ,with⎛⎞˜S =⎜⎝α 1 1E 1 1 − α 1 1 α 1 2E 1 2 · · · α 1 4E 1 4−α 1 2 α 2 2E 2 2 − α 2 2 · · · α 2 4E 2 4...⎟⎠−α 1 4 −α 2 4 · · · α 4 4E 4 4 − α 4 4A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 30

THE DIFFICULTIES OFPARAMETER ESTIMATION FROM DATAA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 31

The coral datax 0 (t+1) =⎧⎪⎨⎪⎩( )x 0 (t)(R ) 0 1 − x 0(t)K 0prob. (1 − p)0.3 × x 0 (t) prob. pWe identify the maximal value of 80 % with thedef= R 0−1carrying capacity x ♯ 0 R 0K 0 = 0.8.After a cyclonic event, the coral grows by 10 % a yearbut not linearly: it takes 8 to 10 years to reach theinitial covering.Probability occurence p corresponds to 1 cycloneevery 5 to 6 years.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 32

The case of coralSimulations give R 0 = 1.002 for ∆t = 1 day.Coral growth over 9 years with R_0=1.002 and K_0=400.80.80.70.60.50.40.30.20.10.00 400 800 1200 1600 2000 2400 2800 3200 3600A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 33

Characteristic figures of the organismsThe diet composition Stom i j is the proportion ofprey j in the stomach of predator i:Stom i j =biomass of prey jtotal biomass of preys in stomach ofiThe matrix Stom is upper triangular.Theoretically, all lines sum up to 1 in the matrix Stom( ∑ preys j Stomi j = 1) but, as the predators do not onlyeat organisms mentioned in this model, the sums oflines may be less than 1.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 34

Characteristic figures of the organismsThe stomachal capacity B k is the maximum biomassthat the stomach of the organism k can contain: wesuppose that it represents 30 % of the organism mass.The average mass W k of the organism k is supposedto be given byW = (0.5 0.5 0.7 0.1) kgwith typical speciesbig groupersmall grouperparrot fishdamsel fishA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 35

Field data: abundances, densitiesIn 1995, outside the Abore reserve, hence underfishing pressure, the mean data were (except for coral)(x 1 x 2 x 3 x 4 ) = (0.04 0.48 1.17 0.49) ind/m 2In the sequel, we shall assume that this vectorrepresents densities at equilibrium, denoted by(x ⋆ 1 x ⋆ 2 x ⋆ 3 x ⋆ 4).Problem of coherency because data should not beunder fishing pressure. This is a consequence offrequent changes in modelling options.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 36

Diet composition estimationThe diet composition is evaluated to be Stom =! 0.01 0.17 0.41 0.17 !! 0. 0.02 0.05 0.02 !! 0. 0. 0. 0. !! 0. 0. 0. 0. !The lines should sum up to 1, but the predators eatother organisms like invertebrates, zooplankton,algae, etc.Line 2: macrocarnivores eat mostly invertebratesLines 3: herbivors eat mostly algaeLines 4: other fishes (small) eat mostly zooplanktonor coralA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 37

Piscivors and macrocarnivors eat fish. . . but the groupis unknown, so that the above matrix is the fruit ofdifferent assumptions:• piscivors’ diet is composed of 77 % fish, of allspecies, especially small ones, with canibalism• macrocarnivors’ diet is composed of 10 % fish,the rest being mostly invertebrates• the proportions of diet composition by group issupposed to be the ambient proportions (perfectmixing and “opportunistic” behaviourassumptions)A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 38

A "stomachal cycle"Suppose that the densities x ⋆ j are at equilibrium.During a "stomachal cycle", we have1. Stom i j: proportion of biomass of prey j in thestomach of predator i.2. Stom i j × B i : biomass of prey j in the stomach ofpredator i. (should be a volume times a volumic mass)3. Stom i j × B i × x ⋆ i : biomass of preys j in x⋆ istomachs of predator i.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 39

Identification of α i jDuring a time unit, we have1. α i j x⋆ j x⋆ i : number of preys j caught by x⋆ ipredators i;2. α i j x⋆ j x⋆ i × W j: biomass of preys j in x ⋆ i stomachsof predator i.If the time unit coincides with one "stomachal cycle":Stom i jB i x ⋆ i = α i jx ⋆ jx ⋆ i W j⇐⇒ α i j = 1 x ⋆ jStom i jB iW j∆t = 1 dayA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 40

Identification of α! 0.106 0.106 0.076 0.53 !! 0. 0.014 0.01 0.069 !! 0. 0. 0. 0. !! 0. 0. 0. 0. !A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 41

Identification of conversion factors E i jE i j =number of individuals i producednumber of idividuals j consumed .Figures in the litterature (Arias Gonzales) indicate thatẼ i j =biomass i producedbiomass j consumed ≤ 0.3with values of 0.12, 0.13 for piscivores andmacrocarnivores, whatever the prey:Ẽ i j = 0.125 , j = 1, 2 , i = 1, . . . , 4 .A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 42

We deduce thatẼ i j = number of individuals i produced × W inumber of individuals j consumed × W jand thusE i j = Ẽi j × W jW i≈ 0.125 W jW i=! 0.125 0.125 0.175 0.025 !! 0. 0.125 0.175 0.025 !! 0. 0. 0. 0. !! 0. 0. 0. 0. !A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 43

Identification of ˜S˜S =⎛⎜⎝α1E 1 1 1 − α1 1 α2E 1 2 1 α3E 1 3 1 α4E 1 41−α2 1 α2E 2 2 2 − α2 2 α3E 2 3 2 α4E 2 42−α3 1 −α3 2 0 0−α4 1 −α4 2 0 0⎞⎟⎠ =! - 0.093 0.013 0.013 0.013 !! - 0.106 - 0.012 0.002 0.002 !! - 0.076 - 0.01 0. 0. !! - 0.53 - 0.069 0. 0. !A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 44

Identification of growth ratesBy writing that the densities x ⋆ i are at equilibrium, weobtain the intrinsic growth ratesR i = 1 −! 0.975 !! 1.007 !! 1.008 !! 1.054 !4∑j=1˜S j i x⋆ j , i = 1, . . . , 4A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 45

Trajectories (1)1.61.41.21.00.80.60.40.200 400 800 1200 1600 2000 2400 2800 3200 3600 4000Piscivor x1(t)Carnivor x2(t)Herbivor x3(t)Others x4(t)Coral x5(t)A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 46

Trajectories (2)1.21.00.80.60.40.20.00 400 800 1200 1600 2000 2400 2800 3200 3600 4000Piscivor x1(t)Carnivor x2(t)Herbivor x3(t)Others x4(t)Coral x5(t)A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 47

A MATHEMATICAL MEASURE OF THEPROTECTED AREA EFFECTA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 48

Catches• Catches on the piscivors x 1 (t), macrocarnivorsx 2 (t) and herbivors x 3 (t):C i = e i x i with e 4 = 0• Uncertainty scenarios for exploitation ratese i (ω(t)) ∈ [ e i − σ i , e i + σ i ] ⊂ [0, 1]A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 49

Exploited dynamicsx 0 (t + 1) = (1 −+ θ(ω(t))} {{ }θ=1: cyclone.θ=0: no cyclone.{ }} {θ(ω(t)))x 0 (t)0.3 × x 0 (t)} {{ }30 %reduction)x i (t) = x e i(R (t) + S(x 0 (t), ω(t))x e (t)x e i (t + 1) = x i(t)(1 − e i (ω(t)))logistic{ }} {( (R 0 1 − x ) )0(t)K 0catchesiA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 50

Exploited dynamics with reserve• Model not spatially explicit:scalar 1 − A measures the size of protected areaPA(A = 0: full reserve; A = 1: no reserve)• A fixed proportion A ∈ [0, 1] is open toharvesting so that catches are given byC i (t) = e i (ω(t))Ax i (t)• Exploited dynamics with a protected area:idem but x e → x A andx A i (t + 1) = x i (t)(1 − Ae i (ω(t))) , i = 1, . . . , 4A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 51

Sources of uncertaintiesScenarios ω(0), ω(1), . . . comprise uncertainties from• cyclonic events impacting coral:• catch effort:θ(ω(t)) ∼ Bernoulli (1, p)e i (ω(t)) ∼ Uniform [ e i − σ i , e i + σ i ]Assuming statistical independence, this gives aprobability P on scenarios ω ∈ ΩA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 52

Catches economic value• Utility function of catches:()U C 1 (t), C 2 (t), C 3 (t)• Substitutable and essential factors:U(c 1 , c 2 , c 3 ) = c 0.51 × c 0.52 × c 0.53A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 53

Stochastic intertemporal decision• Optimal discounted[ (utility approach:)]∑Tmax A E ω t=0 ρt U C 1 (t), C 2 (t), C 3 (t).• Maximin approach:max A E ω[min t=0,...,T U()]C 1 (t), C 2 (t), C 3 (t)• Viability and effectiveness approaches:.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 54

Conservation requirements: existence values• Implicit Conservation with viability:U(x(t)) ≥ U(C(t)) ≥ U ♭=⇒x 1 (t) > 0, x 2 (t) > 0, x 3 (t) > 0• Explicit conservation:x 3 (t) ≥ x 3,♭A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 55

A stochastic effectiveness analysisAn indicator of sustainability at confidence levelβ ∈ [0, 1] is the sustainable kernel Sust β (A) def{( ( )=)x(0)∣ P ω U C 1 (t), C 2 (t) ≥ U ♭ for all times≥ β}.It consists of initial states x(0) = (x 0 (0), . . . , x 4 (0))such that the probability that a random trajectorystarting from x(0) provides a utility from catchesgreater than U ♭ is at least β.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 56

Global reserve effect definitionA particular case is the robust kernel Sust 1 = Sust:whatever the scenarios, a minimal utility U ♭ isensured.We say that global reserve effect holds if there existsA < 1 such that Sust β (1) Sust β (A)(Global reserve effect holds true if the kernel isenlarged when fishing is restricted)A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 57

Optimal sustainable reserve sizeGiven an initial condition x = x(0):• Maximal guaranteed use values is, for sizeA ∈ [0, 1]:U♭ ⋆ def(A, x, β) = max(U ♭ , x ∈ Sust β (A))The largest utility which can be ensured withprobability β.• Optimal reserve size isA ⋆ β(x) def= arg max U ♭ ⋆ (A, x, β)A∈[0,1]The corresponding reserve sizeA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 58

A reserve effect measure• Reserve effect index:PAI β (x) def= U♭ ⋆(A⋆ β (x), x) − U ♭ ⋆ (1, x)Compares difference of utility witout and with(optimal) reserve• Definition: reserve effect holds true for state x ifPAI β (x) > 0A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 59

Contribution values• Contribution of trophic group i:compare the sustainability kernel Sust in twocases1. Sust i without i, namely x i (t) ≡ 02. Sust with i, namely any x i (t) > 0• Contribution index: the differenceI idef= Sust\Sust i• A particular case: Sust i = ∅A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 60

CONCLUSIONSA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 61

Conclusion• Methodology: invariance, co-viability- The role of constraints- Multi-criteria: conservation and efficiency- Intergenerational equity• Reserve problem:- Formalization of reserve effect• Aboré reserve: a model- with calibrated trophic interactions- account for the climatic change through coralA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 62

Perspectives• Robustness with respect to parameter changes• Indicator of refuge function of coral• Spatially explicit• Biodiversity measures• Non cooperative harvesting agents• Age structureA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 63

To end with. . .NEVER PREPARE A TALK FOR A CONFERENCEIN SEPTEMBERAT THE OTHER END OF THE WORLDWHEN YOUR SCIENTIFIC PARTNERSARE IN AUGUST VACATIONS!A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 64

Contribution value for herbivorsIf herbivors x 3 collapse, the whole ecosystem candisappear?Conjecture 1 If x 3 = 0, the sustainability kernel isempty: Sust 3 (A) = ∅=⇒ Strong contribution value of x 3I 3 = Sust\Sust 3 = SustA mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 65

A co-viability analysis without global changes• Assumption 1: No damage for coral: p = 0. Coralat equilibrium 80%• Assumption 2: Robust approach: Confident rateβ = 1• A reserve effect with moderate harvesting:Conjecture 2 There exists a fishing effort thresholde ♯ such that• For any e ∈]0; e ♯ ], a PA effect.• For any e > e ♯ , no PA effect.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 66

MPAEI(e)MPA Index0.00240.00200.00160.00120.00080.00040.00000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9eFigure 4: Reserve Effect Index PAI(u) at x(0) =(0.26; 1.7; 6.84; 0.8). Uncertainty σ = 5%.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 67

A reserve effect with moderate harvestingGuaranteed utility of captures U*(A)Guaranteed utility of captures for e=10%0.00140.00120.00100.00080.00060.00040.00020.0000Harvesting size A0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0(a) Guaranteed utility of captures U ⋆ ♭ (A)Figure 5: A reserve effect at x(0) =A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 68(0.26; 1.7; 6.84; 0.8) for e = 10%. Uncertainty

A reserve effect with moderate harvestingX(t)Densities Trajectories for MPA=0%and e=10%X(t)Densities paths for maximal MPA=94%and e=10%14812710658463422100 200 400 600 800 1000 1200 1400 1600 1800 2000Piscivor x1(t)Coral x4(t)Carnivor x2(t)Herbivor x3(t)t00 200 400 600 800 1000 1200 1400 1600 1800 2000Piscivor x1(t)Coral x4(t)Carnivor x2(t)Herbivor x3(t)t(a) x(t) without reserve MPA=0%(b) x(t) with optimal MPA = 94%Figure 6: A reserve effect at x(0) =A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 69

No reserve effect for large exploitation ratesGuaranteed utility of captures U*(A)Guaranteed utility of captures for e=80%2.4e-0052.0e-0051.6e-0051.2e-0058.0e-0064.0e-0060.0e+000Harvesting size A0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0(a) Guaranteed utility of captures U ⋆ ♭ (A)Figure 7: No reserve effect at x 0 =A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 70(0.26; 1.7; 6.84; 0.8) for a large exploitation e = 80%.

No reserve effect for large exploitation rateX(t)Densities Trajectories for MPA=0%and e=80%X(t)Densities paths for maximal MPA=98%and e=80%90098008700760065005400430032002100100 200 400 600 800 1000 1200 1400 1600 1800 2000Piscivor x1(t)Coral x4(t)Carnivor x2(t)Herbivor x3(t)t00 200 400 600 800 1000 1200 1400 1600 1800 2000Piscivor x1(t)Coral x4(t)Carnivor x2(t)Herbivor x3(t)t(a) x(t) without reserve MPA=0%(b) x(t) with optimal MPA= 98%Figure 8: No reserve effect at x 0 =A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 71

A co-viability analysis with certain climaticchanges• Assumption: Certain damage for coral: p = 1.• A stronger reserve effect !!!!Conjecture 3 There exists a threshold e ♯ such that• For any e ∈]0; e ♯ ], a PA effect.• For any e > e ♯ , no PA effect.A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 72

MPAEI(e)MPA Index0.00320.00280.00240.00200.00160.00120.00080.00040.00000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9eFigure 9: With climatic change: Reserve Effect IndexPAI(u).A mathematical approach toviable management offisheries and biodiversitythrough protected areas:the Abore reef reserve case – p. 73