Identifying limit reference points for robust harvest control rules in fisheries management

Iden i ying limi e e ence poin s
o obus ha es con ol ules
in fishe ies managemen
Jose
´-Ma ı
´a Da-Rocha
1
, Ja ie Ga cı
´a-Cu ı
´n
1
and Ma ı
´a-Jose
´Gu ie
´ ez
2
*
1
Facul ade de Econo
´micas de Vigo, Uni e sidade de Vigo, Vigo, Spain,
2
Depa men o Economic
Analysis, Facul y o Economics and Business, Uni e si y o he Basque Coun y (UPV/EHU),
Bilbao, Spain
Risk and unce ain y a e in insic cha ac e is ics o na u al esou ces ha mus be
aken in o accoun in hei managemen . Ha es con ol ules (HCR) used o be
he cen al managemen ool o con ol s ock fishe ies in an unce ain con ex . A
ypical HCR de e mines fishing mo ali y as a linea ela ionship o he biomass
binding only when he biomass is abo e a c i ical isk alue. Choosing he linea
ela ionship and he isk alue is a complex ask when he e is unce ain y because
i equi es a high le el o da a and an in-deep knowledge o he s ock. This pape
ully cha ac e izes obus HCRs ha explici ly include scien ific unce ain y using
he obus con ol heo y app oach. Ou heo e ical findings show ha unde
unce ain y: i) Cons an HCRs a e no obus ; ii) Robus HCRs show a s eepe linea
ela ionship be ween fishing mo ali y and biomass and a highe alue o biomass
o be conside a isk han non- obus HCRs. F om he implemen a ion iewpoin ,
we assume a h ee-sigma ule and show ha obus ness is achie ed by selec ing a
fishing mo ali y such ha i s de ia ion om he fishing mo ali y a ge is wice he
de ia ion o he biomass om he biomass a ge , and he c i ical alue o he
biomass ( he poin below which fishing should cease, o become as close o ze o
as possible) is hal o he biomass associa ed wi h he maximum sus ainable yield
when his is he a ge .
KEYWORDS
ha es con ol ules, limi e e ence poin s, obus ness, unce ain y, obus con ol
heo y, fishe ies managemen
1 In oduc ion
P ese a ion o na u al esou ces equi es managemen o conside he isks and
unce ain y inhe en o his ype o goods (Gollie e al., 2004;Williams, 2011). Changes in
en i onmen al condi ions, unp edic able changes in esou ce demand, echnological
ad ancemen s, o e en geopoli ical e en s a e po en ial sou ces o unce ain y ha may
impac he success ul and sus ainable use o he esou ce leading i o isk si ua ions o
o e use, deg ada ion, o deple ion o esou ces.
F on ie s in Ma ine Science on ie sin.o g01
OPEN ACCESS
EDITED BY
Ma ia G azia Pennino,
Spanish Ins i u e o Oceanog aphy (IEO),
Spain
REVIEWED BY
Go ka Me ino,
Technology Cen e Expe in Ma ine and
Food Inno a ion (AZTI), Spain
And e E ic Pun ,
Uni e si y o Washing on, Uni ed S a es
Pamela Mace,
Minis y o P ima y Indus ies, New Zealand
*CORRESPONDENCE
Ma ı
´a-Jose
´Gu ie
´ ez
[email p o ec ed]
RECEIVED 30 Janua y 2024
ACCEPTED 07 Augus 2024
PUBLISHED 23 Sep embe 2024
CITATION
Da-Rocha J-M, Ga cı
´a-Cu ı
´n J and
Gu ie
´ ez M-J (2024) Iden i ying limi
e e ence poin s o obus ha es con ol
ules in fishe ies managemen .
F on . Ma . Sci. 11:1379068.
doi: 10.3389/ ma s.2024.1379068
COPYRIGHT
© 2024 Da-Rocha, Ga cı
´a-Cu ı
´nand
Gu ie
´ ez. This is an open-access a icle
dis ibu ed unde he e ms o he C ea i e
Commons A ibu ion License (CC BY). The
use, dis ibu ion o ep oduc ion in o he
o ums is pe mi ed, p o ided he o iginal
au ho (s) and he copy igh owne (s) a e
c edi ed and ha he o iginal publica ion in
his jou nal is ci ed, in acco dance wi h
accep ed academic p ac ice. No use,
dis ibu ion o ep oduc ion is pe mi ed
which does no comply wi h hese e ms.
TYPE O iginal Resea ch
PUBLISHED 23 Sep embe 2024
DOI 10.3389/ ma s.2024.1379068
Fishe ies a e one o he na u al esou ces subjec o unce ain y
and isk ac o s ha a e h ea ened om a sus ainabili y pe spec i e
(F ancis and Sho on, 1997;FAO, 2022). The abundance and
dis ibu ion o fish s ocks can be influenced by unp edic able and
a iable ac o s such as changes in ocean condi ions, o e fishing,
na u al disas e s, and socio-economic and poli ical isks associa ed
wi h he fishing indus y, such as ma ke fluc ua ions and ade
dispu es, and changes in fishing egula ions. These unce ain ies
and isks can impac he sus ainabili y and iabili y o fish s ocks
and he fishing indus y, making e ec i e managemen and decision-
making challenging (Ga cia, 2000). Mos fishe ies managemen
agencies ake hese unce ain ies and isks in o accoun unde he
p ecau iona y p inciple amewo k despi e i s limi a ion om he
economic e ficiency poin o iew (Gollie and T eich, 2003). This
p inciple ecognizes he po en ial nega i e consequences associa ed
wi h high unce ain y and ad oca es among o he s o he use o
p edefined decision ules and conse a i e managemen ac ions
(FAO, 1995;Mildenbe ge e al., 2022).
F om he managemen pe spec i e, he use o he Maximum
Sus ainable Yield (MSY) as he benchma k o assessing he s a e o
fishe ies has become he p ima y ool o fishe ies managemen
since i was accep ed as a goal by he Uni ed Na ions Con en ion on
he Law o he Sea [UNCLOS A icle 61, UN (1982)]. The Wo ld
Summi on Sus ainable De elopmen (WSSD, 2002) u ged s a es o
main ain o es o e deple ed fish s ocks un il hey can p oduce he
MSY. This demand was ecognized by, among o he s, he Eu opean
Union, which es ablished he ope a ional objec i e o ebuilding o
main aining s ocks abo e he biomass le els ha could p oduce he
MSY in i s Common Fishe ies Policy (CFP) (A icle 2, EU (2013)).
The o iginal basis o limi e e ence poin s was ac ually yield
maximiza ion a he han conse a ion ( hink abou he sloped
con ol ule as a less ex eme e sion o he bang-bang con ol ule).
Howe e , e e ence poin s such as he biomass needed o
p oduce he MSY a e a ge s and do no explici ly ecognize
h ea s o he s ock. In his sense, al hough he o iginal basis o
hese e e ence poin s was yield maximiza ion, hey a e now mo e
closely associa ed wi h conse a ion. Hence, s ock size “limi
e e ence poin s”a e usually defined and in e p e ed as he s ock
biomass below which ec ui men becomes subs an ially educed
(Bedding on e al., 2007). In ac ual p ac ice, hese limi e e ence
poin s a e ex ensi ely used by fishe ies manage s o se simple
ha es con ol ules (HCRs) ha link he s a e o he biomass o
con ol a iables such as fishing mo ali y, e o , and ca ches. The
use o limi e e ence poin s o biomass in ha es con ol ules
implici ly ecognizes ha he e a e s ock sizes below which
ec ui men may be impai ed (Pun e al., 2014). Fishe ies
managemen ypically consis s o compa ing he e ec i eness o
al e na i e HCRs o a a ie y o assump ions abou he dynamics o
fish and fishe ies [e.g., Mildenbe ge e al. (2022) and Rosa
e al. (2022)].
The main objec i e o his pape is o p o ide heo e ical
suppo o se ing HCRs ha accoun o scien ific unce ain y.
In fishe ies managemen , scien ific unce ain y ela es o
unce ain ies associa ed wi h na u al s a es and p ocesses (p ocess
unce ain y), he measu emen he eo (obse a ion unce ain y),
he s uc u e o he es ima ion model (model unce ain y), he
e ospec i e biases eflec ed in longi udinal ex ension o da a
(s uc u al unce ain y) and he applica ion o managemen
s a egies and policies (implemen a ion unce ain y) (F ancis and
Sho on, 1997;Pun and Dono an, 2007;Mildenbe ge e al., 2022;
Bi e al., 2023). This aim ames wi hin he FAO guidelines
ad oca ing fishe ies scien is s and manage s should es cu en
and al e na i e con ol ules and associa ed e e ence poin s o
de e mine obus ness o p edominan sou ces o unce ain y and
esponsi eness o he desi ed cha ac e is ics o pe o mance. Failu e
o e ec i ely accoun o unce ain y can lead o o e shoo ing
managemen a ge s, ailing o ebuild deple ed s ocks, and
missing oppo uni ies o ake ad an age o sus ainable fishing
oppo uni ies (Schwaab, 2015).
In his con ex , we design model-based HCRs ha explici ly
include scien ific unce ain y unde he obus con ol heo y
amewo k. In pa icula , we assume ha manage s unde s and
ha he pe cei ed dynamics a e an app oxima ion o he eal model
which is no ully known. Following he ideas o Fellne (1965) and
Hansen and Sa gen (2011), we cha ac e ize obus HCR by
dis o ing he pe cei ed dynamics up o a p e-specified wo s case
le el o e o be ween he u h and pe cei ed eali y. Figu e 1
summa izes his idea ha will be explained in de ail in Sec ion 2.3.
Unde his amewo k, a obus p ecau iona y HCR is cha ac e ized
by sol ing an ex emiza ion p oblem: Manage s maximize he
fishe y’s pe o mance, assuming ha a hypo he ical male olen
na u e chooses he le el o scien ific unce ain y – he dis o ion o
he eal model– o minimize fishe y pe o mance. Fo simplici y,
he analysis is ca ied ou assuming a simple age-s uc u e wi h a
Be e onHol popula ion ame (Be e on and Hol , 1957), simila
FIGURE 1
Adap ed om Hansen and Sa gen (2011) (Figu e 1.7.1). Se o nea by
models, ep esen ing he “ eal wo ld”, o which he decision ule
will wo k well using he pe cei ed model. R and P s and o eal and
pe cei ed, espec i ely and his he maximum dis ance be ween he
wo models ep esen ing he maximum scien ific unce ain y
accep ed by manage s.
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g02
o Hannesson (1975), whe e he ec ui men is he only sou ce o
unce ain y which is assumed o be au oco ela ed.
The analysis e eals ha when manage s a e conce ned abou
scien ific unce ain y, hey know ha a ac ion o he ola ili y
obse ed in he da a is gene a ed by he obse e ’s igno ance. Fo
he sake o simplici y, he sou ce o unce ain y in ou amewo k is
conside ed o a ec ec ui men , which is cha ac e ized as a
pe sis ence model a ec ed by shocks. In his con ex , manage s
in e ha he eal model –which is gene a ing he pe cei ed
ec ui men shocks–is mo e pe sis en han he pe cei ed model.
As a esul , obus HCRs ha e o be designed assuming a mo e
pe sis en fishe y dynamics p ocess han he one pe cei ed in he
da a. In pa icula , ou analysis shows ha a obus HCR always has
a highe limi e e ence poin o he p ecau iona y biomass. Thus,
o he ange o biomass be ween he p ecau iona y and he a ge
alues, he linea ela ionship es ablished by he s anda d HCR
becomes s eepe in he obus con ex han in he non-
obus se ing.
Finally, we show ha HCRs ha use hal o he biomass le el
associa ed wi h MSY (0.5B
MSY
) as he limi e e ence poin o he
p ecau iona y biomass a e consis en wi h ou heo e ical esul s. In
his sense, ou esul s can be said o be aligned wi h p ac ices such as
hose implemen ed by he Aus alian and New Zealand fishe ies
au ho i ies (Rayns, 2007;New Zealand Minis y o Fishe ies, 2008)
o wi h he p oposal by F oese e al. (2011) o Eu opean
fishe ies managemen .
The es o he pape is o ganized as ollows: Sec ion 2 desc ibes
he assump ions unde he model and he cha ac e iza ion o obus
HCRs. Sec ion 3 shows he heo e ical findings o he analysis and
de i es a ule o humb o fixing c i ical alues o he biomass.
Sec ion 4 concludes by discussing he esul s.
2 Me hods
2.1 Ha es ing con ol ules and limi
e e ence poin s
De e mining op imal fishing mo ali y may no be su ficien ly
help ul om an ope a ional iewpoin . Di e en managemen
app oaches, including wha is poli ically easible, lead o fishe ies
managemen being implemen ed h ough di e en ools (e.g., o al
allowable ca ch (TAC) limi s, limi s on he amoun o fishing e o ,
es ic ions on he gea , seasonal closu es), some imes in a
combined way and wi h di e en deg ees o success (Da-Rocha
and Gu ie ez, 2012;Selig e al., 2017).
A well-managed fishe y equi es he design o explici ha es
s a egies ha indica e how much ca ch should be a emp ed o be
ha es ed unde wha ci cums ances (Hilbo n and Wal e s, 1992,
Chap e 15). In p ac ice, simple ules, known as ha es con ol
ules (HCRs), a e used by many fishe ies manage s o se a a ge
le el o fishing mo ali y. Howe e , any specific design o HCR
depends on he quan i y and quali y o da a a ailable o he fishe y
(Smi h e al., 2009;Pun , 2010). Fo hose s ocks wi h he highes
quali y in o ma ion a ailable, a model-based HCR design may be
app op ia e. In hese cases, as Eikese e al. (2013) poin ou , an
HCR can be unde s ood as an algo i hm ha ela es s a e a iables
ha show he biological in o ma ion o he fishe y (e.g., biomass,
spawning biomass, e c.) o he con ol a iables o he fishe y ha
eflec he managemen in o ma ion (e.g., fishing mo ali y, e o ,
ca ches, e c.). In hose cases wi h poo o limi ed da a, “empi ical”
HCR can be p oposed (Pun , 2010). Fo ins ance, when a ge s and
limi s a e based on his o ical s anda dized ca ch a e, a cpue-based
HCR is mo e app op ia e (Li le e al., 2011;Ja dim e al., 2015).
This ype o HCR has p o ed o be pa icula ly e ec i e in mixed
and mul i-specificfishe ies (Canales e al., 2024).
HCRs ake di e en o ms in di e en se ings in eal p ac ice
(K amsdal e al., 2016;F ee e al., 2023). Fo he pu pose o his
s udy, we ocus on s anda d model-based HCRs ha ad ise on
fishing mo ali y ha a ge s MSY (F
MSY
)byconside ing
p ecau iona y biomass h esholds B
lim
and B
igge
. The h eshold
B
lim
ep esen s he poin below which i is belie ed ha he
ep oduc i e capaci y o he s ock may be a isk, so he HCR
p ohibi s fishing when biomass d ops below i (i.e., fishing mo ali y
is se o ze o). In his sense B
lim
can be unde s ood as a
p ecau iona y h eshold ha can be imposed e en i he e is no
comple e in o ma ion on he ep oduc i e capaci y o he s ock
below ha h eshold. The h eshold B
igge
ep esen s he lowe
bound compa ible wi h a gi en biomass a ge and he HCR consis s
o se ing a cons an fishing mo ali y consis en wi h ha biomass
le el whene e he biomass is abo e ha h eshold. When he
biomass is in he ange (B
lim
,B
igge
), he HCR es ablishes a linea
ela ionship be ween fishing mo ali y and biomass. Figu e 2 shows
his ype o HCR, which is e e ed o as “p o ec i e”by Mackinson
e al. (2018) ega ding he No h Sea mul i-annual plan (Eu opean
Commission, 2016). No e ha when a ge s a e gi en by fishe ies
policy make s, a cha ac e iza ion o HCR consis s in selec ing
app op ia e B
lim
acco ding o some c i e ia.
In ac ual p ac ice, he concep o p ecau iona y biomass is mo e
complex. Fo ins ance, he Ha es S a egy S anda d o New
Zealand Fishe ies es ablishes wo ypes o limi e e ence poin s
associa ed wi h di e en managemen ac ions called”so limi s”
and “ha d limi s”(Mace e al., 2013). The so limi is a biomass
le el below which a s ock mus be subjec ed o a o mal, ime-
cons ained ebuilding plan o ebuild i back o he B
MSY
(usually 1/2
B
MSY
o 20% o he biomass in he absence o fishing -B
0
oB
F=0
-,
FIGURE 2
A“p o ec i e”HCR based on biomass h esholds B
lim
and B
igge
wi h MSY as he a ge (Mackinson e al., 2018,Figu e 4B).
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g03
whiche e is highe ). The ha d limi is a biomass le el below which all
fishing ac i i y on he s ock in ques ion should cease (usually 1/4
B
MSY
o 10% B
0
, whiche e is highe ). The e a e a ia ions on his
heme. Fo he US Na ional 1 S anda d Guidelines, o example, a
minimum s ock size h eshold (MSST, usually equi alen o abou 1/2
B
MSY
) is specified below which a simila ype o o mal, ime-
cons ained ebuilding plan is equi ed o ebuild he s ock back o
B
MSY
. Howe e , he guidelines do no speci y a biomass limi below
which all fishing o a s ock mus cease, al hough some US
ju isdic ions o indi idual US fishe ies do so (US Na ional Ma ine
Fishe ies Se ice, 2016). The Aus alian Ha es S a egy Policy
(DAW, 2018)se sB
lim
a 0.2B
0
which, based on hei o he
defini ions, would be consis en wi h 1/2B
MSY
. When his poin is
eached, all di ec ed fishe ies o he s ock in ques ion should be
closed, bu byca ch fishe ies may emain open wi hin limi s.
In any case, he design o any HCR also equi es a ela i ely high
le el o da a and knowledge o he dynamics o he s ocks conce ned.
In he ype o HCR e e ed o he e, i is necessa y o know he alues
o he a ge e e ence poin s (e.g., hose associa ed wi h MSY) and
he biomass h esholds, B
igge
and B
lim
. To es ima e hese alues
accu a ely, comple e knowledge o he biological, economic, and
ecologic models behind he s ock popula ion dynamics is needed. I
is no always possible o apply he HCR because o he lack o
in o ma ion. Fo example, in 2005, i was only possible o use his
ype o HCR on 22 o he 80 species managed by he Pacific Fishe y
Managemen Council (Pun and Dono an, 2007).
An addi ional p oblem is ha in mos cases, he design o HCRs
does no explici ly include a way o deal wi h unce ain y (Pun and
Dono an, 2007;De oba and Bence, 2008). In he case o Eu opean
fishe ies s ocks, when he da a and knowledge equi emen s a e no
ulfilled, ICES some imes ad ises a lowe fishing mo ali y han
F
MSY
e en when he s ock is in good condi ions; o ins ance, F
0.1
,
ins ead o F
MSY
, which is he mo ali y a e whe e he yield pe
ec ui slope is 10% o he maximum yield pe ec ui slope.
2.2 The popula ion model
We build up he popula ion model as in Da-Rocha and Ma o-
Amboage (2016), whe e a s ochas ic e sion o he fishe y o
Hannesson (1975) is conside ed wi h wo age classes: ju eniles
and adul s. Le N
,1
, and N
,2
be he popula ions o ju eniles and
adul s in pe iod , espec i ely. Each yea , , a s ochas ic exogenous
numbe o ju enile fish a e bo n, N
,1
=exp(z
), whe e z
is a andom
a iable ha de e mines he ec ui men o he fishe y. This is he
only sou ce o unce ain y a ec ing he model and i is pe cei ed by
he manage s as ollowing an AR(1) p ocess:
z +1 = z +~
e +1, (1)
whe e ~
e +1 is a Gaussian i.i.d. p ocess wi h ze o mean and a iance
s2
e, and
jj
∈½0, 1) is he au oco ela ion coe ficien .
Manage s see his pe cei ed model as an app oxima ion o he
eal model, ep esen ing he “ eal wo ld”. Following Hansen and
Sa gen (2011), his scien ific unce ain y is ep esen ed wi h a se o
al e na i e models o he o m
z +1 = z +e +1 +w +1, (2)
whe e e +1 is ano he Gaussian i.i.d. p ocess wi h ze o mean and
a iance s2
e,andw +1 is a ec o o pe u ba ions in he mean o ~
e +1,
ha can eed back in o he his o y o he s a e, z.No e ha since
p ocess ep esen ed in Equa ion 2 is he model ha gene a es he
da a, i is as hough he e o s ~
e +1in he pe cei ed model (1) we e
condi ionally dis ibu ed as N(w +1,s2
e) a he han as  N(0, s2
e).
Mo eo e , i is also impo an o highligh ha pe u ba ions w +1
a ec he eal pe sis ence o ec ui men . Since w +1 eeds back in o
he his o y o he s a e, z,pa ame e does no ep esen pe sis ence
in he eal model (2). We show bellow wha his eal pe sis ence is
when manage s selec op imal HCRs.
The dynamics o he adul age g oup is hen gi en by N +1,2 =
N ,1e−F −m,, whe e F ep esen s he fishing mo ali y applied in
pe iod and mis he na u al mo ali y a e, which o he sake o
simplici y is assumed o be cons an o e ime. Finally, o his
p oo -o concep a icle we assume, as Da-Rocha and Ma o-
Amboage (2016), ha he spawning s ock biomass o he fishe y
is defined as B = lnN ,2. This ela ion implies ha he spawning
s ock biomass is an inc easing unc ion o he numbe o adul s in
he popula ion and ha only a non-cons an ac ion o adul s
a e spawne s
1
.
This popula ion ep esen a ion enables policymake s’cons ain s
o be modeled as a linea -quad a ic p oblem, which is essen ial o apply
he obus con ol heo y (Hansen and Sa gen , 2011).
2.3 Robus p ecau iona y HCRs
Unce ain y is modeled ollowing he mul iplie p e e ence
app oach based on he obus con ol heo y p oposed by Hansen
and Sa gen (Hansen and Sa gen , 2001,2011). Unde his
amewo k, op imal ( obus ) policies a e selec ed among all
possible dis ibu ions consis en wi h wha is known and
obse ed, by adding a penal y e m ha is in e sely ela ed o he
dis ance o any gi en dis ibu ion om he bes guess.
We s a by assuming ha fishe y manage s wan o design a
obus p ecau iona y HCR ha linea ly ela es fishing mo ali y and
biomass so ha exogenous a ge s (B a ,F a ) a e achie ed while
a oiding he isk o he s ock alling below le el B which would
esul in he fishe y being conside ed as no longe sus ainable om
he biological iewpoin . No e ha we a e assuming ha B a and F a
1
The applica ion o his model o a specific s ock would equi e empi ical
es ing o he ela ionship be ween spawning s ock biomass and he adul
popula ion and e en aking in o accoun o he a iables ha accu a ely
measu e ep oduc i e po en ial (Kell e al., 2016).
2
In eal fishe ies, howe e , hese a iables a e selec ed endogenously
acco ding o managemen c i e ia ha ypically allow o long- e m
sus ainable and e ficien exploi a ion o he s ocks (Yagi and
Yamakawa, 2020).
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g04
a e exogenously gi en by he manage s
2
. A ypical ICES HCR conside s
B a =MSYB igge and F a =FMSY (Mackinson e al., 2018).
The objec i e unc ion o manage s is cha ac e ized in e ms o
dis ances o fishing mo ali y and biomass om hei espec i e
a ge poin s as in Da-Rocha and Ma o-Amboage (2016). They aim
o s abilize he esou ce a ound he desi ed poin s. This idea is
o malized wi h a loss unc ion ha ep esen s he ne p esen
weigh ed sum o he squa ed dis ance o fishing mo ali y, F , and
biomass, B , om hei espec i e a ge s
E0o
∞
=0
b +1½(F −F a )2+l(B −B a )2, (3)
whe e 0 < b< 1 ep esen s he subjec i e discoun a e, E
0
is he
ma hema ical expec a ion condi ioned on he in o ma ion a ailable
a he ime o decision-making and lis a pa ame e ha ep esen s
he weigh o biomass de ia ion ela i e o fishing mo ali y
de ia ion. The e a e h ee no ewo hy ema ks ega ding he
manage s’loss unc ion (3): Fi s , Fis, by defini ion, a a ia ion
a e (mo ali y a e), and i s de ia ion om i s a ge is also a a e. In
addi ion, he de ia ion o he B om i s a ge mus also be seen also
as a a ia ion a e since bo h a iables a e defined in loga i hms.
Hence, he wo sums o he loss unc ions a e a ios wi h no
measu emen uni s. Second, i penalizes bo h de ia ions abo e o
below he desi ed alues (hence he squa e o he dis ance). Thi d, i
conside s he dynamic na u e o he esou ce, enabling long- un
de ia ions o be o se by mo e mino de ia ions in he sho un.
How much p esen de ia ions can o se u u e de ia ion depends
on he discoun ac o , b: La ge discoun ac o s mean small
discoun a es, ha is manage s ca e as much abou u u e
changes as i hey occu ed in he cu en yea
3
.
An HCR can be unde s ood as he esul o minimizing he loss
unc ion (3), aking in o accoun he popula ion model. This
in e p e a ion means ha an HCR is cha ac e ized by he
pa ame e l. No e ha wi h l= 0 he ule is independen o he
biomass, and he ins umen is cons an o e di e en biomass
le els. Wi h lapp oaching infini y, he ule is (equi alen ly) linea
in he biomass le el, and a bang-bang o mos apid app oach pa h
solu ion eme ges. A posi i e, fini e lambda dic a es a ade-o
be ween fishing mo ali y and biomass and may be associa ed
wi h a posi i e Blim.Figu e 3 illus a es hese cases, which shows
a e somewha eminiscen o classical hinking as embodied by
(Hilbo n and Wal e s, 1992, Chap e 15).
An HCR ha ollows he p ecau iona y p inciple is sough he e,
so he ule needs o ensu e a mos a % p obabili y o he biomass
alling below he limi poin , B; ha is, he HCR has o sa is y he
equi emen ha
P (B≤B) ≤ , (4)
whe e is gi en by he manage s. Equa ion 4 specifies a
p ecau iona y HCR which gua an ees ha he s ock is abo e he
limi poin B wi h a leas a 1 − % p obabili y (e.g., = 0.05 o
ICES ad ice).
In addi ion, manage s know ha he pe cei ed model (1) is an
app oxima ion o he eal model (2), so hey a e awa e o he
dynamic misspecifica ion o he model ( he scien ific unce ain y).
The e o e a scien ific unce ain y le el his conside ed, i.e.
E0o
∞
=0
b +1w2
+1 ≤h:(5)
The le -hand side o Equa ion 5,E0o∞
=0b +1w2
+1,isan
in e empo al measu e o he size o model misspecifica ion called
condi ional ela i e en opy. This cons ain is used o measu e he
s a is ical disc epancy be ween he pe cei ed model (1) and he eal
model (2), which di e only in he w e m [see Hansen and Sa gen
(2011)]. The e o e, Equa ion 5 exp esses he idea ha manage s
know ha he eal model can be any nea by model a ound he
pe cei ed model (see Figu e 1). The pa ame e hmeasu es he se o
models su ounding he pe cei ed model o which manage s hink
he decision ule will wo k well. Mo e significan scien ific
unce ain y implies a mo e ex ensi e se o al e na i e models
ha may ep esen he “ eal wo ld” o be compa ed wi h he
pe cei ed model. So in his con ex , hcan be unde s ood as a
measu e o scien ific unce ain y accep ed by manage s. Fo mally,
he condi ional ela i e en opy is cons ained o be lowe han o
equal o an exogenous scien ific unce ain y le el hwhich he
manage s p o ide.
(Hansen and Sa gen 2011, chap e 9) p opose using Bayesian
de ec ion e o p obabili y o es ima e hwhen guiding he choice o
he se o models agains which he pe cei ed model is compa ed.
This app oach assumes ha he models on and inside he ball In
Figu e 1 a e di ficul o dis inguish s a is ically om he eal model
wi h he amoun o da a a hand. In essence, i akes a neu al s ance
on whe he he ue da a-gene a ing p ocess is ep esen ed by he
pe cei ed model o by he wo s -case model (a he bounda y o he
ball). The me hod in ol es calcula ing he likelihood a io es s o
choose be ween hese wo models unde bo h hypo heses, bases on
in-sample fi , o a sample o a gi en size. Then a p obabili y o a
de ec ion e o is calcula ed on he basis o a la ge numbe o
simula ions by gi ing equal p obabili y o he wo models o being
he ue model.
4
When he model assumes no obus ness
(maximum h), he de ec ion e o p obabili y es 50%. Hansen
and Sa gen (2011) sugges choosing hsuch ha he ange o he
de ec ion e o ange is be ween 10% and 20%. Sample size also
plays an impo an ole. The la ge he a ailable sample is, he lowe
his chosen because he unce ain y is less o a conce n.
Conside ing all hese specifica ions, in looking o a obus
p ecau iona y HCR, he manage s’p oblem consis s o minimizing
he loss unc ion exp essed in Equa ion 3, aking in o accoun he
3 Discoun is equen ly in oduced in o fishe y economics using he
discoun a e, , ins ead o a discoun ac o , b; he o me is usually applied
in con inuous ime amewo ks, whe eas he la e is mo e commonly used in
disc e e se ups. The in e se ela ionship be ween he wo is gi en by b=
(1 + )−1, which is comp essed be ween 0 and 1 (Da Rocha e al., 2010). The e
is e en some li e a u e in which discoun ing he u u e is included as a ime-
a ying ac o (Da-Rocha e al., 2016).
4 Technically, he de ec ion e o p obabili ies can be calcula ed using
p og am de ec ion2.m in Ma lab.
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g05

popula ion model and he p ecau iona y and misspecifica ion side
cons ain s Equa ions 4,5, espec i ely).
Technically, he obus p ecau iona y HCR is cha ac e ized in
wo s eps. Fi s , o a gi en HCR (a gi en l) and an admi ed
unce ain y le el h, manage s seek o maximize hei in e empo al
a ge gap while a hypo he ical male olen na u e minimizes ha
same a ge by selec ing he wo s pe u ba ion p ocess, gi en he
popula ion dynamics. Fo mally, he ollowing ex emiza ion
p oblem is sol ed:
max
F ,B +1
g
∞
=0
 min
w +1
g
∞
=0
E0o
∞
=0
b −(F −F a )2−l(B −B a )2+bqw2
+1

,
s: : B +1 =z −F −m,
z +1 = z +e +1 +w +1 :
(
(6)
whe e he mul iplie q ep esen s he penal y o de ia ing om he
eal model in he unc ion o be op imized. No e ha since he
p oblem seeks o minimize o he pe u ba ion w, a e y low q
allows he na u e o w eak ha oc, while q→∞co esponds o a
ze o penal y o he de ia ion.
Second, gi en he a ge pa hs solu ions om he ex emiza ion
p oblem (Equa ion 6), he HCR, l, and he mul iplie qassocia ed
wi h he scien ific unce ain y le el, h, ha sa is y he p ecau iona y
and misspecifica ion cons ain s (Equa ions 4,5, e ospec i ely)
a e ound. To his espec , i should be no ed ha since he unc ion
o op imize is mono onous and conca e in h, he e is a nega i e
bijec i e unc ion om h o he mul iplie q(Gio dani and
Söde lind, 2004).
Appendix A.1 p o es ha o a gi en scien ific unce ain y le el
hand a p ecau iona y p obabili y o biomass alling below he limi
poin , , he obus p ecau iona y HCR is gi en by
l=1
b
see −1(2 −1) ffiffiffi
2
p
(B−B a )(1 −^
2)1=2−1

, (7)
whe e e is he Gaussian e o unc ion and ^
is gi en by
h=b
1−b
(^
− )2s2
e
(1 −^
2):(8)
Pa ame e ^
can be in e p e ed as he ac ual pe sis ence o he
eal model. This pa ame e is unknown o manage s bu is
endogenously de e mined by Equa ion 8 o a gi en scien ific
unce ain y. Two ac s ha eme ge om Equa ion 8 a e wo h
no ing: On he one hand, i manage s a e no conce ned abou
scien ific unce ain y (h→0), hen ^
= . Howe e , when
manage s a e highly conce ned abou scien ific unce ain y
(h→∞), ^
> meaning ha he (in e ed) eal model is mo e
pe sis en han he pe cei ed one. On he o he hand, e en o
unco ela ed pe cei ed p ocesses ( = 0), a obus HCR would ha e
o conside he exis ence o some pe sis ence, i.e. ^
2=h
b
1−b

s2
e+h>0.
Finally, he comple e cha ac e iza ion o he obus HCRs gi en
by Equa ions 7,8shows unambiguously ha he mo e significan
he scien ific unce ain y (h) is, he la ge ^
2and la e.
3 Resul s
3.1 Theo e ical findings
Two heo e ical conclusions can be highligh ed om he
cha ac e iza ion o obus p ecau iona y HCRs (Equa ions 7,8).
Fi s , an HCR o keeping fishing mo ali y cons an a he a ge
le el canno be a p ecau iona y obus ule.
P oposi ion 1. A cons an e o ule,F =F a is no a obus
p ecau iona y HCR. The e o e, a obus limi e e ence poin o
biomass is g ea e han ze o.
P oo : See Appendix A.2
Unde scien ific unce ain y, i can be in e ed ha he eal
p ocess is co ela ed, e en when he s ochas ic p ocess ob ained
om he pe cei ed model is no . Robus ness implies he use o
biomass-based HCRs, l>0 (see a nume ical example in Da-Rocha
and Ma o-Amboage (2016)).
The logic behind he esul ha a cons an e o HCR is no
obus unde scien ific unce ain y can be illus a ed wi h he
ollowing easoning. Suppose ha a nai e manage conside s bo h
ha he pe cei ed model (1) is he one ha gene a es he da a (bu i
is no ) and he p ocess is pe cei ed as unco ela ed, = 0. Unde
his assump ion, a cons an e o HCR, lNR = 0, is expec ed o
gene a e a a iance ~
se=(B−B a )
e −1(2 −1) ffiffi2
p( om Equa ion 7 when l=0
and = 0). This esul means ha he expec ed biomass ola ili y
-based on nai e expec a ions- is s2
B=~
s2
e. Howe e , da a is
gene a ed no by he pe cei ed model (1) bu by he eal model
FIGURE 3
l= 0 is associa ed wi h a cons an e o HCR; l> 0 is associa ed wi h a biomass-based HCR; l→∞is associa ed wi h a cons an o fixed
escapemen ule.
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g06
(2), which includes he pe u ba ion w +1. The e o e, he ola ili y o
he biomass is ac ually gi en by s2
B=s2
e
1−^
2.
5
Manage s conce ned wi h obus ness seek eliable HCR o all
close eal models (2) in he se shown in Figu e 1. This is equi alen
o designing an HCR ha akes in o accoun ha ^
> = 0. This
obus HCR is gi en by Equa ion 7 and o an unco ela ed p ocess
is lR=1
b
1
(1−^
2)1=2−1
hi
>0. I gene a es a isk measu e o
P (B≤B) = 1
2½1 + e ((1 + lRb)(1 −^
2)1=2e −1(2 −1)):(9)
Figu e 4 shows how nai e HCR pe o mance de e io a es mo e
quickly han obus HCR ulesasscien ific unce ain y ( he
co ela ion gene a ed by he pe u ba ion p ocess, ^
) inc eases.
When he nai e cons an e o ule, lNR = 0, is applied he
p ecau iona y cons ain is iola ed, i.e.
P (B≤B) = 1
2½1 + e ((1 −^
2)1=2e −1(2 −1))> :(10)
In gene al, HCR educes p ecau iona y le els when he
pe cei ed model is co ec . Howe e , he pe o mance becomes
mo e p ecau iona y as scien ific unce ain y inc eases.
Second, how much as e fishing mo ali y is educed when a
s ock is assessed o be below he a ge biomass depends on he le el
o scien ific unce ain y. Ou esul s show ha o he same
unce ain y conce n and p ecau iona y c i e ia (gi en by hand
), a obus HCR selec s a highe biomass limi e e ence poin (Blim)
han non- obus HCR. Figu e 5 illus a es his esul showing ha
BR
lim >B
NR
lim. Gi en his, he linea ela ionship be ween fishing
mo ali y and biomass in he ange (Blim,B a ) becomes s eepe
wi h l( )>l(^
). P oposi ion 2 es ablishes hese esul s o mally.
P oposi ion 2. G ea e scien ific unce ain y le els imply: i) a
s eepe ela ionship be ween biomass and obus fishing mo ali y in
he obus HCR, and ii) a highe limi e e ence poin o biomass,
which is gi en by Blim =B a −F a =bl(^
).
P oo : See Appendix A.3.
3.2 A obus ule o humb o HCR
Acco ding o ou modeling, cha ac e izing he obus
p ecau iona y HCR o a pa icula s ock would equi e ime
se ies da a o compu e se, , and B. Howe e , i he idea is only
o explo e he impac o scien ific unce ain y – o he gi en le els
o – he e is no need o compu e hese s a is ics.
To see his mo e clea ly, assume ha he s ock has been assessed
as abo e B igge , and he ICES MSY (cons an e o ) ad ice ule has
been applied. In ha case, (see Appendix A.4), he obus
p ecau iona y HCR has o sa is y he equi emen ha
bl=1− 2
1−^
2

1=2
−1=
^
s−s
s,
whe e sand ^
s ep esen he s anda d de ia ion o he
ec ui men p ocess in he pe cei ed model (1) and in he eal
model (2), espec i ely. This esul means ha obus ness is
p opo ional o he di e ence be ween he s anda d de ia ion o
he pe cei ed and he eal models.
Following he h ee-sigma ule, ^
s=3s, which gua an ees ha
99.7% o andom e en s lie a ound he mean o i s no mal
dis ibu ion (see Pukelsheim (1994)), he obus HCR is o se bl
as 2. This esul implies ha whene e he biomass is abo e he limi
e e ence poin o he biomass, Blim, he obus HCR se s fishing
mo ali y such ha i s de ia ion om he a ge is wice he
de ia ion o he biomass om i s a ge . Mo eo e , he obus
HCR also endogenously de e mines he biomass e e ence poin as
hal o he biomass a ge , i.e., Blim =0:5BMSY.
In sho , he limi e e ence poin used by Aus alian and New
Zealand fishe ies au ho i ies (New Zealand Minis y o Fishe ies,
2008;Sainsbu y, 2008) and p oposed by F oese e al. (2011) o
Eu opean fishe ies managemen is he endogenous obus limi
poin associa ed wi h a obus HCR whe e fishing mo ali y
de ia ion is wice he biomass de ia ion when a s ock is assessed
using a h ee-sigma ule.
4 Discussion and conclusions
Ma ine esou ce managemen p ocedu es ha ake accoun o
unce ain y include specifica ion o he da a o be collec ed and how
hese da a will be used o p o ide managemen ad ice ha
inco po a es eedback mechanism in he o m o decision ules o
HCRs (Pun and Dono an, 2007). This pape shows ha scien ific
unce ain y can be ea ed analy ically using he obus con ol
heo y p oposed by Hansen and Sa gen (Hansen and Sa gen ,
2001,2011). In pa icula , obus model-based HCRs a e
heo e ically cha ac e ized in closed o m using his app oach.
This esul is a no el y wi h espec o o he pape s ha also
s udy obus con ol in na u al esou ce managemen (Va das
and Xepapadeas, 2010;A hanassoglou and Xepapadeas, 2012;
Xepapadeas and Rose a-Palma, 2013).
This heo e ical cha ac e iza ion o obus HCRs allows
es ablishing wo no el poin s o be made om a fishe ies
managemen pe spec i e. Fi s , i can be s a ed heo e ically ha
cons an HCRs a e no obus unde scien ific unce ain y. This
esul p o ides heo e ical suppo o app oaches sugges ing ha ,
in he p esence o unce ain y, biomass-based h eshold HCRs,
which indica e ha fishing mo ali y should dec ease wi h biomass,
a e mo e app op ia e han cons an fishing mo ali y HCRs
(De oba and Bence, 2008;Pun , 2010). Second, obus HCRs se
highe biomass p ecau iona y e e ence poin s han hose o non-
obus HCRs. Mo eo e , hese p ecau iona y le els a e defined in
e ms o a ge e e ence poin s [as in F oese e al. (2011)]. These
esul s a e aligned wi h he idea ha sou ces o unce ain y can be
educed by defining in e als a ound he limi e e ence poin s
(Rindo e al., 2016;Da-Rocha e al., 2017;Rindo e al., 2017a,b)
o by choosing app op ia e me hods o es ima ing hem ( an
Deu s e al., 2021;Bi e al., 2023).
5
When ~
e +1 is pe u bed, biomass e ol es ollowing DB +1 =^
DB +1
1+bl e .
See Equa ion 17 in Appendix A.1.2 o u he de ails.
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g07
This esea ch also show ha hese heo e ical findings can be
easily implemen ed by designing HCRs ha use 0.5B
MSY
as he limi
e e ence poin o biomass. In his sense, ou esul s can be said o
be aligned wi h eal p ac ices. The Aus alian Ha es S a egy
Policy (DAW, 2018) iden ifies 20% o he biomass in absence o
fishing (0.2B
0
) as he s anda d limi e e ence poin because i is
conside ed a sui able p oxy ha a oids ec ui men o e fishing o
p oduc i e s ocks (Sainsbu y, 2008). Fo less p oduc i e s ocks,
mo e conse a i e limi e e ence poin s a e p oposed (e.g. 0.3B
0
).
Addi ionally, he Aus alian Ha es S a egy conside s ha i B
MSY
can be eliably es ima ed and i is abo e 0.4B
0
hen 0.5B
MSY
is an
app op ia e al e na i e as a limi e e ence poin (Rayns, 2007).
New Zealand uses 0.5B
MSY
as a limi below which a o mal
ebuilding plan is equi ed (New Zealand Minis y o Fishe ies,
2008). F oese e al. (2011) p opose using his limi e e ence poin o
design HCR o manage Eu opean fishe ies. Mo e ecen ly, F oese
e al. (2018) use his e e ence o asses Eu opean s ocks and ound
51% o hem o be ou side sa e biological limi s.
In his ega d, i is wo h men ioning ha i has been common
p ac ice in he In e na ional Council o he Explo a ion o he Sea
(ICES) in he las ew yea s o se managemen e e ence poin s based
on a de e minis ic equilib ium ela ionship be ween yield, fishing
mo ali y, and biomass (ICES, 2017). In pa icula , o s ocks o
which he e is no app op ia e popula ion in o ma ion, he
de e minis ic e sion o he su plus p oduc ion Schae e model
(Schae e , 1954) is used and in mos cases 1/3B
MSY
se as a p oxy
o Blim This de e minis ic app oach is no app op ia e because, in
gene al e ms, de e minis ic e e ence poin s o e es ima e fishing
mo ali y and he biomass equi ed o suppo MSY. Fo example, he
US, Aus alia and New Zealand all ha e he de aul assump ion ha
BMSY =0:4B0, which is way highe han he de e minis ic B
MSY
in
mos cases. In any case, no ice ha his selec ion is no incompa ible
wi h ou esul s. We find ha wi h unce ain y, he le el o
p ecau iona y biomass (below which fishing is banned) should be
highe han in de e minis ic cases. In his con ex , i is wo h s udying
whe he i is ad isable o inc ease Blim om 1/3B
MSY
o 0.5B
MSY
.
On he o he hand, ICES has ecen ly s a ed using he SPiCT
model o hei MSY ad ice. SPiCT is a su plus p oduc ion model ha
akes unce ain y in ca ches and biomass indexes in o accoun
(Pede sen and Be g, 2017). This amewo k enables p ecau iona y
limi s o fishing mo ali y (Fp−05) o be se such ha he p obabili y o
he p edic ed biomass being below an ag eed lowe limi (Blim)is5%o
less (Rindo e al., 2016). F om his pe spec i e, ou cha ac e iza ion o
he p ecau iona y biomass associa ed wi h he obus HCR (BR
lim in
Figu e 5) is a concep simila o he p edic ed biomass implied o he
p ecau iona y fishing mo ali y (Fp−05)inRindo e al. (2016).The
no el y o ou esul is i suppo s his idea o s ocks whose popula ion
can be desc ibed in a e y simple way om age coho s.
F om he poin o iew o he s ock modeling, i should be
emphasized ha in his s udy i has been se up in he simples way
possible. In pa icula , i has been assumed ha he s ock is di ided
in o wo age g oups (ju eniles and adul s) wi hou conside ing he
exis ence o a plus g oup and he spawning s ock biomass ollows a
loga i hmic ela ionship wi h he adul popula ion. Fu he mo e,
he only sou ce o unce ain y conside ed is ec ui men , which is
assumed o be au oco ec ed o o de 1. All hese simplifica ions
FIGURE 4
Risk o biomass d opping below B o a nai e HCR, l=0(Equa ion 10) and a obus HCR, l>0(Equa ion 9). The obus HCR was designed by
assuming ^
o be 0.5. No ice ha when he co ela ion gene a ed by he eal model is 0.5 he p obabili y o he biomass d opping below B is exac ly
= 0.05.
FIGURE 5
Robus HCR e sus non- obus HCR. Robus design o HCR leads o
a highe limi e e ence poin o he biomass. BNR
lim and BR
lim s and o
non- obus and obus biomass limi e e ence poin s, espec i ely.
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g08
may seem a emo ed om he eali y obse ed in he popula ion
dynamics o mos fish s ocks. Howe e , his simplified modeling is
use ul (and necessa y) o ob ain analy ical solu ions ha help
in e p e he esul s in simple scena ios. Applica ion o he model
o specific popula ions would equi e adap a ion o accoun o
hei in insic biological cha ac e is ics.
The simplifica ion o he biological model o a o m wi h only one
sou ce o unce ain y sugges s ha his s udy is a kind o p oo o concep
o a single sou ce o unce ain y. Howe e , one o he a ac ions o he
me hodology used in his s udy is ha he class o dis u bances used o
cap u e unce ain y may be mo e gene al han he simplici y o he
model analyzed appa en ly shows. Wi h Hansen and Sa gen (2011)
app oach, he unce ain y conside ed may include unknown pa ame e
alues and misspecfica ion o highe momen s o he e o dis ibu ion as
long as he decision make ’s objec i e unc ion is quad a ic and his
app oxima ing model is linea wi h Gaussian e o s (see chap e s 1.13, 3
and 7). E en mo e s uc u ed kinds o unce ain y can be accommoda ed
by sligh ly ein e p e ing he decision make ’s objec i e unc ion (see
chap e 19). In his sense, his app oach can be ex ended o all he sou ces
o unce ain y classified by F ancis and Sho on (1997), i.e., obse a ion,
model s uc u e, p ocess e o , and implemen a ion e o s. The e o e,
ou findings can be applied in case s udies ha use he simula ion
modeling app oach o assess di e en managemen s a egies (MSE)
unde a ious sou ces o unce ain y. E en in hose fishe ies wi h poo
da a, he p oposed me hodology can be used o design obus HCR ha
link abundance indices such as CPUE o su ey da a wi h ca ch limi s.
This s udy ocuses on HCRs ha aim o main ain s ock biomass a
le els consis en wi h MSY, elying on a limi ed numbe o biological
e e ence poin s (F a ,B a and Blim), whe e ed as a biomass limi
e e ence poin ha indica es he closed/open s a us o he s ock o he
fishing ac i i y. This simplifica ion can be seen as a limi a ion because i
does no ake in o accoun ha fishe ies managemen may ha e
mul iple objec i es (maximizing ca ch, minimizing isk o he
esou ce, and maximizing indus ial s abili y) ha may conflic wi h
each o he . To mi iga e hese possible conflic s in he p esence o
unce ain y, se e al ypes o measu es ha e been ecommended; among
o he s, he defini ion o op imal HCRs based on a la ge numbe o
biological e e ence poin s (Yagi and Yamakawa, 2020), he e alua ion
o he heo e ical and applied impac o spa io- empo al measu es (Da-
Rocha e al., 2012;DAW, 2018), he alua ion o fishe y esou ces wi h
non-cons an discoun ac o s (Da-Rocha e al., 2016). The obus ness
o such solu ions could also be e alua ed in he con ex o obus
con ol heo y, ex ending he applicabili y o ac ual p ac ice.
Da a a ailabili y s a emen
The o iginal con ibu ions p esen ed in he s udy a e included
in he a icle/supplemen a y ma e ial. Fu he inqui ies can be
di ec ed o he co esponding au ho .
Au ho con ibu ions
J-MD-R: W i ing – e iew & edi ing, W i ing –o iginal d a ,
Visualiza ion, Valida ion, Supe ision, So wa e, Resou ces, P ojec
adminis a ion, Me hodology, In es iga ion, Funding acquisi ion,
Fo mal analysis, Da a cu a ion, Concep ualiza ion. JG-C: W i ing –
e iew & edi ing, W i ing –o iginal d a , Visualiza ion, Valida ion,
Supe ision, So wa e, Resou ces, P ojec adminis a ion,
Me hodology, In es iga ion, Funding acquisi ion, Fo mal analysis,
Da a cu a ion, Concep ualiza ion. M-JG: W i ing – e iew &
edi ing, W i ing –o iginal d a , Visualiza ion, Valida ion,
Supe ision, So wa e, Resou ces, P ojec adminis a ion,
Me hodology, In es iga ion, Funding acquisi ion, Fo mal analysis,
Da a cu a ion, Concep ualiza ion.
Funding
The au ho (s) decla e financial suppo was ecei ed o he
esea ch, au ho ship, and/o publica ion o his a icle. J-MD-R and
JG-C g a e ully acknowledge financial suppo om Xun a de
Galicia (ED431B 2022/03). M-JG also acknowledges financial
suppo om he Basque Go e nmen (BiRTE IT-1461-22) and
he Uni e si y o he Basque Coun y (PES20/44).
Acknowledgmen s
The au ho s hank And e Pun , Pamela Mace and Go ka
Me ino who ha e conside ably imp o ed he a icle.
Conflic o in e es
The au ho s decla e ha he esea ch was conduc ed in he
absence o any comme cial o financial ela ionships ha could be
cons ued as a po en ial conflic o in e es .
Publishe ’s no e
All claims exp essed in his a icle a e solely hose o he au ho s
and do no necessa ily ep esen hose o hei a filia ed o ganiza ions,
o hose o he publishe , he edi o s and he e iewe s. Any p oduc
ha may be e alua ed in his a icle, o claim ha may be made by i s
manu ac u e , is no gua an eed o endo sed by he publishe .
Da-Rocha e al. 10.3389/ ma s.2024.1379068
F on ie s in Ma ine Science on ie sin.o g09