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Abstract and Figures

Size spectrum models have emerged from 40 years of basic research on how body size determines individual physiology and structures marine communities. They are based on commonly accepted assumptions and have a low parameter set, making them easy to deploy for strategic ecosystem-oriented impact assessment of fisheries. We describe the fundamental concepts in size-based models about food encounter and the bioenergetics budget of individuals. Within the general framework, three model types have emerged that differ in their degree of complexity: the food-web, the trait-based, and the community models. We demonstrate the differences between the models through examples of their response to fishing and their dynamic behavior. We review implementations of size spectrum models and describe important variations concerning the functional response, whether growth is food-dependent or fixed, and the density dependence imposed on the system. Finally, we discuss challenges and promising directions.
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1!
The$theoretical$foundations$for$size$spectrum$models$of$fish$
1!
communities$$
2!
Ken!H.!Andersen1,!Nis!S.!Jacobsen1!and!K.!D.!Farnsworth2!3!
!
4!
1Center!for!Ocean!Life,!National!Institute!of!Aquatic!Resources!(DTU-Aqua),!Technical!5!
University!of!Denmark,!Charlottenlund!Castle,!DK-2920,!Charlottenlund,!Denmark!6!
2!Institute!of!Global!Food!Security,!Queens!University!Belfast,!97!Lisburn!Road,!Belfast!7!
BT9!7BL,!Northern!Ireland,!UK!!
8!
!9!
Abstract(
10!
Size!spectrum!models!have!emerged!from!40!years!of!basic!research!on!how!body!size!11!
determines!individual!physiology!and!structures!marine!communities.!They!are!based!12!
on!commonly!accepted!assumptions!and!have!a!low!parameter!set,!which!make!them!
13!
easy!to!deploy!for!strategic!ecosystem!oriented!impact!assessment!of!fisheries.!!We!
14!
describe!the!fundamental!concepts!in!size-based!models!about!food!encounter!and!the!15!
bioenergetics!budget!of!individuals.!Within!the!general!framework!three!model!types!
16!
have!emerged!that!differs!in!their!degree!of!complexity:!the!food-web,!the!trait-based!
17!
and!the!community!model.!We!demonstrate!the!differences!between!the!models!
18!
through!examples!of!their!response!to!fishing!and!their!dynamic!behavior.!We!review!
19!
implementations!of!size!spectrum!models!and!describe!important!variations!concerning!20!
the!functional!response,!whether!growth!is!food-dependent!or!fixed,!and!the!density-
21!
dependence!imposed!on!the!system.!Finally!we!discuss!challenges!and!promising!
22!
directions.!23!
!
24!
Key(words:(Ecosystem!approach,!food-web,!ecosystem!based!fisheries!management! !
25!
!
2!
Introduction$26!
Marine!community!models!range!from!the!original!Lotka-Volterra!differential!equations!
27!
to!extremely!complicated!end-to-end!simulations!(Plagányi!2007;!Fulton!et!al.!2011).!In!
28!
the!middle!of!the!range!we!find!size!spectrum!models.!Size!spectrum!models!use!body!
29!
size!of!individuals!to!represent!the!entire!fish!community!as!a!size!distribution.!The!30!
reliance!of!body!size!simplify!the!description!of!predator-prey!interactions,!individual!31!
physiology!and!vulnerability!to!fishing!gear.!This!paper!highlights!one!of!the!important!32!
advantages!of!the!size!spectrum!approach:!a!well-founded!and!unifying!mechanistic!33!
basis!allowing!for!great!explanatory!power!and!parsimonious!use!of!data.!!34!
!35!
Size spectrum models are relevant to fisheries science in the context of!the!ecosystem!36!
approach!to!fisheries!management!(Pikitch!et!al.!2004).!While!single-species!stock!37!
assessments!and!impact!assessment!will!continue!to!be!important!management!tools,!38!
they!need!to!be!supplemented!by!strategic!impact!assessments!at!the!level!of!the!39!
ecosystem.!Such!impact!assessments!assist!the!development!and!implementation!of!40!
strategic!long-term!management!goals!for!the!ecosystem,!e.g.,!how!should!fishing!
41!
pressure!be!distributed!over!the!entire!ecosystem?!How!do!we!balance!exploitation!of!
42!
competing!fisheries!such!as!forage!fisheries!and!consumer!fisheries?!How!do!we!
43!
maximize!the!yield!(of!biomass!or!wealth)!of!the!entire!ecosystem!while!minimizing!risk!44!
of!failure!or!impoverish!components!of!the!system!under!environmental!change?!To!
45!
answer!these!questions,!we!need!to!quantitatively!understand!the!relationship!between!46!
fishing!practice!(what,!when!and!how!much)!and!the!abundance!of!species!and!sizes!of!47!
organisms!throughout!the!community.!!!
48!
!
49!
Size!spectrum!models!are!especially!suited!to!these!questions!because!they!resolve!the!
50!
most!import!aspects!of!fish!life!history!and!trophic!ecology.!A!key!characteristic!of!fish!is!
51!
that!individuals!grow!through!several!orders!of!magnitude!in!body!size!through!their!52!
!
3!
life.!This,!combined!with!the!strong!relationship!between!body-size!and!trophic!niche!53!
(Barnes!et!al.!2008;!Gilljam!et!al.!2011),!means!that!individuals!change!their!trophic!
54!
niche!throughout!ontogeny!(Werner!and!Gilliam!1984).!Such!ontogenetic!trophic!niche!
55!
shifts!makes!it!difficult!to!apply!the!conventional!food-web!approach,!where!each!
56!
species!is!described!by!a!single!metric!(abundance!or!biomass)!and!a!specific!trophic!57!
level,!to!fish!communities.!The!relation!between!body!size!and!trophic!niche!has!58!
prompted!the!hypothesis!that!individual!body!size!(rather!than!species!identity)!is!the!59!
primary!determinant!of!community!structure!(Jennings!et!al.!2001).!60!
!61!
Size!spectrum!models!are!based!upon!the!long!tradition!in!ecology!of!recognizing!body!62!
size!as!a!central!trait!to!describe!individuals!(Elton!1927;!Haldane!1928;!Andersen!et!al.!63!
2016a)!because!it!correlates!strongly!with:!metabolism!(Kleiber!1932;!Winberg!1956,!64!
Brown!et!al.!2004),!predator-prey!relations!(Ursin!1973;!Barnes!et!al.!2008),!encounter!65!
rates!(Acuña!et!al.!2011),!functional!responses!(Rall!et!al.!2012),!reproductive!effort,!and!66!
other!vital!rates!(Peters!1983).!For!application!to!fisheries,!body!size!furthermore!is!an!67!
excellent!descriptor!of!mesh-size!regulations!and!characterizes!the!value!of!a!catch!
68!
(Andersen!et!al.!2015).!Finally,!distributions!of!abundance!vs.!size!show!a!remarkable!
69!
regularity!(Sheldon!and!Prakash!1972;!Sheldon!et!al.!1977;!Boudreau!and!Dickie!1992)!
70!
and!deviations!from!this!regularity!has!been!used!to!characterize!ecosystem!level!71!
impact!of!fishing!(Rice!and!Gislason!1996;!Daan!et!al.!2005).!The!size!spectrum!
72!
modeling!paradigm!promises!a!“charmingly!simple”!(Pope!et!al.!2006)!set!of!tools!with!a!73!
low!to!intermediate!complexity!that!can!be!readily!deployed!for!a!given!system!and!74!
provide!quantitative!information!about!the!ecosystem!impact!of!fishing!(Collie!et!al.!
75!
2014).!This!makes!it!possible!to!apply!the!models!in!situations!where!more!complex!but!
76!
also!data-demanding!end-to-end!models!cannot!be!employed!either!because!of!lack!of!
77!
data!or!manpower!to!calibrate!and!run!them.!
78!
!79!
!
4!
The!various!types!of!size!spectrum!model!can!be!viewed!as!different!developments!of!80!
the!same!core!concepts.!We!review!the!common!basic!concepts!behind!size!spectrum!
81!
models!focusing!on!models!that!describe!an!entire!fish!community.!Among!them!we!
82!
recognize!three!broad!classes!of!decreasing!levels!of!complexity.!Most!complex!are!the!
83!
!""#$%&'’!models,!so!called!because!they!explicitly!represent!individual!populations!84!
with!species-specific!energy!budget!parameters!and!prey!preferences,!thereby!85!
quantifying!a!network!of!trophic!interactions!as!an!explicit!food-web.!These!are!86!
simplified!into!the!‘()*+($'*,&#’!models!by!reducing!differences!among!the!populations!87!
to!a!single!continuous!variable!representing!a!trait!(usually!maturation!size)!and!88!
simplifying!prey!selection!to!a!fixed!predator-prey!body!size!ratio.!Further!simplification!89!
produces!the!-."//01+(2-!size!spectrum!models,!so!called!because!they!ignore!90!
differences!among!populations!thereby!representing!the!community!as!a!single!91!
population!of!interacting!individuals!that!differ!only!in!their!body-size.!We!explain!how!92!
the!simpler!models!can!be!derived!from!the!more!complex,!starting!from!the!food-web!93!
and!ending!with!the!community!model.!Further!we!develop!analytical!“equilibrium”!94!
solutions!to!the!models.!We!illustrate!the!models’!behavior,!in!particular!their!response!
95!
to!fishing,!and!finally!discuss!challenges!and!open!issues!for!further!development.!
96!
Concepts$underlying$size$spectrum$models$
97!
Size!spectrum!models!are!founded!on!three!common!concepts:!First,!biomass!(and!98!
equivalent!energy)!is!conserved,!enabling!accountancy!of!energy!flows!at!the!
99!
community!level!based!on!individual!level!processes.!Second,!trophic!interactions!are!100!
the!main!determinant!of!community!structure!and!these!are!foremost!determined!by!101!
predator-prey!size!ratios.!!Third,!the!energy!budget!of!an!individual!is!allometrically!
102!
linked!to!body!size,!so!that!body!size!can!be!used!as!a!key!identifier!of!organisms!and!
103!
their!interactions!with!the!community.!The!three!main!ecological!processes!for!any!104!
organism!are!growth,!reproduction!and!mortality!and!all!three!can!be!linked!to!body!
105!
!
5!
size!in!this!modeling!framework.!This!simplification!has!great!strategic!value!as!it!106!
enables!ecological!measures!such!as!production!rate!and!size!structure!(which!are!107!
important!for!ecosystem!based!fisheries!management)!to!be!derived!from!relatively!108!
little!and!accessible!data!about!physiology!and!life!history!invariants.!The!equations!for!109!
the!models!used!in!the!examples!to!follow!are!provided!in!Table!1!and!parameters!in!110!
Table!2.!111!
3112!
45&3,+6&3,7&.()0/3113!
The!size!spectrum!represents!abundance!or!biomass!of!individuals!as!a!function!of!their!114!
body!size.!In!this!context!‘body!size’!usually!means!body!mass!because!it!is!the!natural!115!
metric!to!formulate!an!energy!budget.!116!
!117!
Three!size!spectrum!representations!are!common!in!the!literature!(Sprules!and!Barth,!118!
this!issue;!Andersen!and!Beyer!2006;!Rossberg!2012):!the!abundance!density!spectrum,!119!
the!biomass!density!spectrum!and!the!“Sheldon”!biomass!spectrum.!The!abundance!
120!
density!spectrum!!"#$!represents!the!number!of!individuals!in!the!body!mass!range!
121!
from!#%!to!#&!as! ! # '(#
)*
)+,!and!here!it!is!referred!to!as!the!size!spectrum!for!brevity.!
122!
It!can!be!constructed!from!observations!by!dividing!the!total!number!of!individuals!in!a!123!
size!class!by!the!width!of!the!size!class!and!therefore!has!dimensions!of!numbers!per!
124!
mass!(often!referred!to!as!the!“normalized!size!spectrum”;!Sprules!and!Barth,!this!
125!
issue).!The!biomass!density!spectrum!is!constructed!from!the!abundance!density!
126!
spectrum!by!multiplying!with!body!mass!! # #!(dimensions!biomass!per!mass).!The!127!
“Sheldon”!biomass!spectrum!(Sheldon!and!Parsons!1967)!is!the!biomass!in!128!
logarithmically!wide!classes,!i.e.,!the!biomass!in!the!range!#!to!,#!where!,!is!a!constant!
129!
larger!than!one!determining!the!width!of!the!size!class.!For!example,!the!“octave”!bin!
130!
used!by!Sheldon!implies!, - .,!and!normal!log10!base!implies!, - /01!If!we!assume!that!131!
the!abundance!density!spectrum!follows!a!power-law!! # - 2# 3!within!the!class!then!
132!
!
6!
the!biomass!spectrum!can!be!found!as:!133!
!134!
4567 # - ! 8 8'(8
9)
)
- 2 ,&:3 ; /
. < = #&:3 > #&! # ?!135!
!136!
where!8!is!a!dummy!variable!for!the!integration.!All!terms!except!#&:3 !are!independent!137!
of!size,!hence!the!biomass!spectrum!is!proportional!to!the!number!density!spectrum!138!
multiplied!by!the!body!mass!squared.!For!mathematical!analyses!the!density!spectra!are!139!
convenient!because!integrals!over!these!give!the!abundance!and!biomass.!For!140!
presentation!purposes!the!Sheldon!biomass!representation!#&!"#$!is!convenient!141!
because,!at!the!community!level,!it!shows!how!the!biomass!of!prey!is!distributed!with!142!
size!(assuming!that!the!size!range!of!preferred!prey!is!constant),!and!on!a!species!level!it!143!
is!proportional!to!the!cohort!biomass.3144!
33145!
8"1,&)9*(+"13&:0*(+"13146!
The!size!spectrum!is!calculated!by!considering!a!balance!between!mortality!and!growth!
147!
at!all!body!sizes! #.!Individuals!flow!into!size!classes!via!somatic!growth!whilst!some!
148!
are!lost!to!natural!and!fisheries!mortality.!This!balance!is!formalized!by!the!McKendric-
149!
von!Foerster!equation!(see!Silvert!and!Platt!(1978)!for!a!derivation):!150!
!
151!
@!A#
@B <@CA# !A#
@# - ;DA# !A# ?'''''"/$!
152!
!
153!
where!CA#!is!the!growth!rate!(mass!per!time)!and!DA# 'the!mortality!(per!time),!and!154!
!A#!is!the!size!spectrum!of!species!E.!Thus!eq.!(1)!scales!from!individual-level!155!
processes!of!growth!and!mortality!to!the!population-level!size!spectrum.!Recruitment!
156!
from!the!population!flows!into!the!size!spectrum!at!the!smallest!body!size!(typically!the!
157!
egg!size)!#F.!This!is!represented!as!a!boundary!condition:!158!
!
7!
!159!
CA#F!A#F- GA?'''".$!160!
!161!
where!GA!is!the!recruitment!(number!of!recruits!or!eggs!per!time).!The!above!two!162!
equations!are!mathematical!formalizations!of!a!mass!balance,!and!can!be!thought!of!as!163!
the!size-based!version!of!classic!survivor!analysis!used!in!age-based!models.!!164!
!165!
The!following!outlines!the!central!assumptions!in!the!models!about!how!growth!CA,!166!
mortality!DA'and!reproduction!(recruitment)!GA?!are!calculated.!With!the!partial!167!
exception!of!recruitment!these!are!all!calculated!from!individual!level!processes!of!168!
predator-prey!encounter!and!a!bioenergetic!budget!(Figure!1).!169!
!170!
45&3;1#&),&1$<),+13&1."01(&)3/"#&=3171!
The!key!process!in!the!models!is!predator-prey!encounters!between!individuals!172!
governed!by!a!formalization!of!the!general!rule,!bigger!fish!eat!smaller!fish!!(Andersen!
173!
and!Ursin!1977).!Individuals!prefer!prey!a!certain!fraction!smaller!than!themselves!(M1,!
174!
Table!1)!(Ursin!1973).!The!clearance!rate!(dimensions!time-1)!is!an!increasing!function!
175!
of!body!size!(larger!fish!clear!a!larger!volume!of!water!for!prey!per!time!than!small!fish)!176!
(M2).!The!combination!of!preference,!clearance!rate!and!prey!abundance!specifies!the!
177!
food!encounter!rate!(M3,!biomass!per!time).!Intake!upon!encounter!(“satiation”)!is!
178!
represented!with!a!type!II!functional!response!(M5)!as!the!“feeding!level”!H
A#,!i.e.,!the!
179!
ratio!between!consumption!and!maximum!consumption!(M4)!(dimensionless!number!180!
between!0!and!1).!181!
!
182!
>1#+9+#0*=3&1&)?23'0#?&(33
183!
The!energy!budget!describes!how!consumed!food!is!used!for!maintenance,!activity,!184!
growth!and!reproduction.!Consumed!food!is!assimilated!and!first!used!for!standard!
185!
!
8!
metabolism,!widely!recognized!to!be!an!allometric!function!of!body!mass:!IJ#K.!Juvenile!186!
individuals!use!the!remaining!available!energy!for!growth,!while!mature!individuals!187!
apportion!the!energy!between!growth!and!reproduction!(M6-M8).!The!exact!188!
specification!of!allocation!of!energy!between!growth!and!reproduction!is!not!crucial.!189!
The!one!used!here!ensures!that!when!the!feeding!level!is!constant,!size-at-age!curves!190!
resembles!a!von!Bertalanffy!curve!and!the!gonado-somatic!is!independent!of!body!size!191!
(Hartvig!et!al.!2011).3192!
!193!
@&7)"#0.(+"13*1#3)&.)0+(/&1(33194!
Reproduction!and!recruitment!represent!the!reproductive!output!from!the!entire!195!
population.!The!reproductive!output!GL1A!(numbers!per!time;!M9)!is!discounted!by!a!196!
reproductive!efficiency!M!to!represent!losses!due!to!egg!mortality!and!spawning!effort,!197!
and!used!to!calculate!the!recruitment!GA.!Within!this!context,!recruitment!refers!to!the!198!
rate!of!production!of!new!individuals!from!the!fertilized!egg!stage,!but!it!could!be!done!199!
at!a!later!stage!if!properly!discounted!(Andersen!and!Beyer!2015).!Ideally!the!
200!
recruitment!is!equal!to!the!reproductive!output,!but!many!models!apply!a!stock!
201!
recruitment!relationship!(M10).!The!density!dependence!imposed!by!the!stock!
202!
recruitment!relationship!avoids!the!competitive!exclusion!between!species!that!203!
otherwise!tends!to!occur!(Hartvig!and!Andersen!2013).!The!stock-recruitment!
204!
relationship!contains!two!essential!parameters:!the!“slope”!parameter!that!specifies!
205!
recruitment!at!low!population!sizes!and!the!maximum!recruitment!that!specifies!the!
206!
population!carrying!capacity.!The!slope!parameter!is!given!directly!by!the!egg!207!
production!of!the!population!(Andersen!and!Beyer!2015),!but!the!maximum!recruitment!208!
has!to!be!specified!separately.!This!parameter!represents!all!effects!on!the!population!
209!
that!are!not!explicitly!represented!in!the!model,!such!as!limitations!due!to!juvenile!
210!
habitat!size!that!is!known!to!limit!some!marine!populations!(Rijnsdorp!and!Leeuwen!211!
!
9!
1992).!The!maximum!recruitment!and!possibly!the!recruitment!efficiency!are!key!212!
parameters!for!calibrating!a!model!to!data!from!real!fish!stocks!(see!discussion).!213!
!214!
A")(*=+(2!215!
Three!categories!of!mortality!are!recognized.!First,!predation!mortality!(M13)!emerges!216!
from!the!trophic!dynamics!within!the!system;!second,!intrinsic!or!background!mortality!217!
(M12)!is!usually!represented!as!an!allometric!function!of!asymptotic!size!(Brown!et!al.!218!
2004),!though!starvation!mortality!can!be!explicitly!added!(e.g.,!Hartvig!et!al.!2011);!219!
third,!exogenous!sources!of!mortality!(especially!fishing)!are!often!added.!220!
!221!
@&,"0).&3222!
The!resource!spectrum!!N# 'represents!food!other!than!fish.!The!resource!is!needed!223!
for!the!smallest!individuals!who!are!not!yet!large!enough!to!be!piscivorous!but!it!can!224!
represent!any!kind!of!food:!a!single!size-group!of!small!zooplankton!prey!species,!a!size!225!
spectrum!of!zooplankton!prey!(as!in!Fig.!1),!or!a!size!distribution!including!larger!prey!,!
226!
e.g.!benthic!production.!The!resource!can!be!constant,!in!which!case!the!growth!rate!of!
227!
small!fish!is!fixed,!or!it!can!be!modeled!dynamically,!e.g.,!as!a!semi-chemostat!(M14).!The!
228!
semi-chemostat!formulation!is!convenient!because!it!leads!to!a!very!stable!dynamics!of!229!
the!resource.!Using!logistic!growth!results!in!a!more!pronounced!dynamical!response!of!
230!
the!resource!which!translate!into!stronger!dynamics!of!the!fish!part!of!the!model!(de!
231!
Roos!et!al.!2008).!
232!
Models$types$233!
We!now!briefly!describe!how!these!common!concepts!are!used!to!create!size!spectrum!234!
models!at!three!levels!of!complexity!and!demonstrate!an!approximate!analytical!
235!
solution!for!the!equilibrium.!!
236!
!237!
!
10!
B""#$%&'3/"#&=3238!
In!the!food-web!model,!the!processes!M1!–!M15!are!instantiated!with!all!parameters!239!
from!Table!2!being!species!specific!(either!representing!identified!species!or!240!
hypothetical!ones!matching!relevant!criteria),!but!in!practice!some!parameters!are!241!
usually!cross-species!constants,!such!as!the!exponents!O? P!and!Q.!Populations,!thus!242!
identified!as!different!species,!interact!through!predator-prey!relations!with!interaction!243!
strengths!specified!by!an!interaction!matrix,!which!represents!a!combination!of!species-244!
specific!preferences!and!encounter!probabilities.!This!matrix!could!be!populated!with!245!
empirical!interaction!coefficients!derived!from!stomach!content!analyses,!spatial!246!
overlap!(Blanchard!et!al.!2014),!or!it!may!represent!hypothetical!distributions!--!random!247!
and!uniform!(everything!eats!everything!else)!interaction!networks!are!popular!248!
hypotheses.!A!fully!specified!food-web!model!has!14!parameters!for!each!species,!plus!249!
an!interaction!matrix,!so!for!C!species!the!total!is!up!to!R < /SI < I&!parameters.!!250!
!251!
4)*+($'*,&#3/"#&=33
252!
The!trait-based!model!represents!differences!among!species!only!by!the!governing!trait!
253!
of!asymptotic!size!(Pope!et!al.!2006).!This!assumes!that!the!most!important!trait!is!the!
254!
asymptotic!size!(or,!equivalently,!size!at!maturation),!which!embodies!a!trade-off!255!
between!reproductive!output!and!asymptotic!size.!!The!trait-based!model!is!
256!
conceptually!derived!from!the!food-web!model!by!assuming!that!all!parameters!in!Table!
257!
2!are!cross-species!constants!and!by!using!theoretical!arguments!to!determine!GTUV!as!a!
258!
function!of!W!(appendix!A).!Feeding!interaction!are!solely!determined!by!individual!259!
size.!The!solution!is!the!trait!size!spectrum!!"#? W$!(dimensions!numbers!per!mass!per!260!
asymptotic!mass)!describing!the!joint!distribution!of!individual!and!asymptotic!sizes!
261!
(Andersen!and!Beyer!2006).!In!numerical!implementations!the!asymptotic!size!axis!is!
262!
discretized,!typically!into!logarithmic!‘bins’!grouping!species!in!asymptotic!size!classes.!263!
In!practice,!the!results!of!the!trait-based!model!are!effectively!independent!of!the!
264!
!
11!
number!of!simulated!asymptotic!size!classes!once!this!number!is!greater!than!10.!The!265!
trait-based!model!is!specified!with!the!18!parameters!in!Table!2.!!266!
!267!
8"//01+(23/"#&=33268!
The!community!model!ignores!all!differences!between!species!and!only!considers!269!
differences!in!size!(Benoît!and!Rochet!2004).!It!can!be!derived!from!the!trait-based!and!270!
food-web!models!by!integrating!over!all!trait-classes!or!summing!over!species!(Zhang!et!271!
al.!2012):!272!
!273!
!9# - ! #? W '(W
)
- !A"#$
AX%
1!274!
!275!
The!integral!only!runs!from!#!because!asymptotic!size!groups!with!# Y W!does!not!276!
contribute!to!the!community!spectrum!at!size!#.!The!community!model!only!resolves!277!
the!‘community!spectrum’!!9"#$.!Substantial!consequences!arise!because!this!model!is!278!
unable!to!represent!maturation,!reproduction!and!recruitment.!The!energy!budget!(M6-
279!
M8)!is!simplified!such!that!growth!is!solely!available!energy!multiplied!by!an!“average!
280!
growth!efficiency”!derived!from!equilibrium!theory!(M7b;!Appendix!A).!Because!
281!
energetic!losses!to!reproduction!are!not!explicitly!accounted!for,!the!model!will!not!282!
reproduce!von!Bertalanffy!growth.!Further,!most!implementations!in!the!literature!use!
283!
just!a!linear!functional!response,!ignore!standard!metabolism!and!use!a!fixed!resource!
284!
(Table!3),!however,!these!simplifications!are!not!significant!for!the!community!model.!In!
285!
its!most!comprehensive!form,!the!community!model!requires!only!11!parameters!(less!286!
without!the!functional!response!and!with!fixed!resource).!287!
!
288!
D:0+=+')+0/3,"=0(+"1,3
289!
!
12!
Analytical!solutions!to!the!trait-based!model!can!be!derived!under!the!assumption!that!290!
the!feeding!level!is!constant!H # - H
F!and!that!the!spectrum!is!infinitely!long,!i.e.!#F-291!
0!and!Z[\ W -
!(Andersen!and!Beyer!2006;!Hartvig!et!al.!2011).!This!results!in!an!292!
‘equilibrium!community!spectrum’:!293!
!294!
!9# - /
]
H
F
/ ; H
F
^
_#`a&ab '"R$'!295!
!296!
with!] - .cdeb a` f\g'h Q ; O &d&i.$j.!The!scaling!exponent!O ; . ; Q k ;.10l!is!in!297!
accordance!with!observations!(Boudreau!and!Dickie!1992).!The!equilibrium!solution!for!298!
each!species!spectrum!is:!!299!
! #? W > 2W&`abam:n #a`an / ; #
W
%a` ni"%a`$
'"S$3300!
!301!
with!the!“physiological!mortality”!o!given!in!(M17).!This!result!is!used!to!calculate!302!
expected!scaling!solutions!to!predation!mortality,!the!scaling!of!maximum!recruitment!303!
with!asymptotic!size!used!in!the!trait-based!models,!and!the!average!growth!efficiency!
304!
in!the!community!model!(Appendix!A).!Note!that!the!solution!in!eq.!(4)!does!not!fulfill!
305!
the!boundary!condition!(M9);!the!total!reproductive!output!calculated!from!(4)!will!lead!
306!
to!a!life-time!reproductive!output!larger!than!1!and!increasing!with!asymptotic!size!307!
(discussed!in!Hartvig!et!al.!2011!and!Rossberg!2012).!This!discrepancy!has!to!be!
308!
resolved!by!density-dependent!effects!within!each!population!not!accounted!for!in!(4),!
309!
and!it!has!been!used!to!relate!the!slope!parameter!in!stock-recruitments!relationship!to!
310!
asymptotic!size!(Andersen!and!Beyer!2015).!In!the!dynamical!models!the!emergent!311!
physiological!mortality!depends!on!asymptotic!size,!with!smaller!species!having!a!312!
smaller!o!than!larger!(Hartvig!et!al.!2011),!in!accordance!with!empirical!measurements!
313!
of!how!mortality!depends!on!asymptotic!size!(Gislason!et!al.!2010).!Even!though!(4)!is!
314!
!
13!
not!an!exact!solution!of!the!entire!model,!it!is!still!a!useful!approximation,!as!315!
demonstrated!by!its!ability!to!resolve!species!diversity!(Reuman!et!al.!2014).!316!
!317!
The!equilibrium!results!all!rely!on!the!metabolic!assumption!inherent!in!the!functional!318!
response!where!consumption!is!proportional!to!#`.!If!a!functional!response!is!not!used!319!
the!solution!for!the!exponent!of!the!community!spectrum!will!differ!from!eq.!(3).!In!320!
particular!it!will!not!depend!on!the!metabolic!exponent!O,!but!rather!on!the!preferred!321!
predator-prey!mass!ratio!e!(Benoît!and!Rochet!2004;!Datta!et!al.!2010;!Rossberg!2012).!322!
Such!solutions!will!result!in!consumption!rates!that!do!not!follow!metabolic!scaling.!!323!
!324!
E*)*/&(&),3325!
Parameters!are!either!determined!from!knowledge!about!the!specific!species!(for!the!326!
food-web!model),!or!from!cross-species!investigations!of!life-history!invariants!(Table!327!
2;!see!Hartvig!et!al.!(2011),!App!E.!for!a!detailed!discussion).!The!relatively!small!set!of!328!
parameter!facilitates!formal!investigations!of!model!behavior!under!varying!parameter!
329!
values!(Thorpe!et!al.!2015;!Zhang!et!al.!2015).!
330!
!
331!
>/7=&/&1(*(+"13332!
Size!spectrum!models!may!be!simulated!with!Mizer”,!a!reference!implementation!in!R!
333!
(Scott!et!al.!2014),!or!with!the!matlab!code!(see!online!supplementary).!For!the!food-
334!
web!model!we!have!used!the!parameterization!for!the!North!Sea!(Blanchard!et!al.!2014),!
335!
which!uses'O - .iR!as!is!customary!in!the!fisheries!literature.!For!the!trait-based!and!336!
community!models!we!have!used!O - RiS!to!conform!with!“metabolic”!theory!(West!et!337!
al.!2001).!The!results!are!qualitatively!sensitive!to!the!value!of!O!as!long!as!it!is!changed!
338!
in!all!the!relationships!in!Table!1.!The!community!model!has!been!implemented!as!a!
339!
trait-based!model!with!a!single!trait!group!having!a!very!large!asymptotic!size.!As!the!340!
individuals!in!the!trait!group!mature!their!growth!rate!declines!(Fig.!2d).!In!this!way!the!
341!
!
14!
average!growth!in!the!community!model!corresponds!to!the!average!growth!in!a!trait-342!
based!model.!Our!implementation!avoids!the!need!for!an!additional!senescent!mortality!343!
for!the!largest!individuals!as!is!common!practice!(Law!et!al.!2009;!Rochet!and!Benoît!344!
2012).!!345!
!346!
All!simulations!are!set!up!with!100!logarithmically!spaced!grid!points!on!the!mass!axis,!347!
with!the!first!grid!point!set!to!egg!size!w0.!Each!simulation!was!run!with!a!time!step!of!348!
0.25!year!until!convergence!(Appendix!B).!349!
Example$simulations$350!
All!three!models!predict!size!spectra,!growth!rates!and!mortality!(Figure!2).!In!the!351!
absence!of!fishing!and!top!predators!the!largest!size!groups!(# p /0!kg)!are!352!
superabundant,!i.e.,!that!part!of!the!spectrum!is!greater!than!predicted!by!the!353!
equilibrium!solution.!This!is!most!pronounced!in!the!food-web!model,!and!could!be!354!
changed!by!increasing!the!background!mortality!qF.!The!superabundance!results!in!355!
higher!predation!mortality!on!medium-sized!individuals,!which!triggers!a!trophic!
356!
cascade!and!associated!changes!in!growth!and!mortality!of!smaller!individuals!
357!
(Andersen!and!Pedersen!2010).!!
358!
!359!
@&,7"1,&,3("3!+,5+1?3
360!
The!behavior!of!the!models!is!illustrated!by!examining!the!response!of!the!time-
361!
averaged!solution!to!fishing!and!their!dynamical!behavior.!Fishing!using!size-selective!
362!
gear!is!represented!by!adding!a!fishing!mortality!that!depends!on!body!size,!and!363!
possibly!also!asymptotic!size!or!species.!!We!illustrate!fishing!through!two!scenarios:!a)!364!
community-wide!fishing!on!all!species!with!a!trawl-type!selectivity!pattern!having!50%!
365!
selectivity!at!010lW,!and!b)!a!bottom-up!perturbation!where!forage!fishery!is!removed!
366!
from!scenario!(a)!simulated!by!setting!fishing!mortality!zero!on!all!species!with!W Y367!
!
15!
.00!g!in!the!food-web!and!trait-based!models!and!on!individuals!with!# Y .00!g!in!the!368!
community!model!(Figure!3).!In!scenario!(a)!(community!wide!fishing)!the!reduction!of!369!
large!individuals!induces!a!trophic!cascade!throughout!the!community!seen!as!a!wave!in!370!
the!size!spectrum.!When!forage!fishing!is!removed!(b),!the!food-web!and!trait-based!371!
models!predict!an!increase!of!forage!fish!but!relatively!modest!effects!on!the!rest!of!the!372!
community.!The!forage!fish!have!a!higher!biomass!than!in!the!unfished!situation!due!to!373!
the!partial!release!from!predation!by!higher!trophic!levels!caused!by!fishing.!The!374!
response!of!the!community!model!is!different:!it!predicts!a!decline!in!the!size-range!375!
where!fishing!is!removed,!while!the!effects!on!the!rest!of!the!community!are!weak,!as!in!376!
the!other!models.!This!difference!in!the!community!models!stems!from!its!inability!to!377!
represent!fishing!(or,!in!this!case,!absence!of!fishing)!on!specific!species!or!life!histories,!378!
but!only!on!body!sizes.!!This!result!emphasizes!the!importance!of!representing!379!
individual!populations!in!fisheries!applications.!The!two!scenarios!illustrates!the!380!
relative!importance!of!mortality!and!growth!to!mediate!trophic!cascades:!in!scenario!(a)!381!
the!trophic!cascade!is!mainly!mediated!by!changes!in!predation!mortality,!which!leads!
382!
to!a!strong!cascade.!In!scenario!(b)!increases!in!forage!fish!abundance!has!two!effects!
383!
with!opposite!consequences:!1)!it!increases!growth!rates!of!larger!individuals,!but!2)!it!
384!
also!increases!competition!between!juvenile!individuals!of!larger!species!and!forage!fish!385!
leading!to!decreased!growth!rates.!The!end!result!is!a!modest!trophic!cascade!(Houle!et!
386!
al.!2013;!Jacobsen!et!al.!2015).!The!importance!of!competition!between!forage!fish!and!
387!
juvenile!predatory!fish!in!real!ecosystems!could!be!analyzed!by!comparing!stomach!
388!
contents!of!forage!fish!and!juvenile!predatory!fish.!In!summary:!the!food-web!and!trait-389!
based!models!predict!similar!response!to!selective!fishing,!while!the!community!model!390!
fails!to!resolve!effects!on!different!populations.!
391!
!
392!
F21*/+.3,"=0(+"1,3393!
!
16!
The!community!model!also!differs!from!the!food-web!and!trait-based!model!in!394!
dynamical!behavior,!i.e.,!how!the!solution!varies!over!time.!All!models!tend!to!be!395!
unstable!(oscillate!over!time)!if!the!trophic!overlap!is!small.!The!trophic!overlap!is!396!
determined!by!the!ratio!between!the!width!of!the!size!preference!function,!d,!and!the!397!
predator-prey!size!ratio!rst"e$:!the!smaller!the!value!of!dirst"e$,!the!smaller!the!398!
trophic!overlap,!and!the!more!unstable!the!solution!becomes!(larger!oscillations)!(Datta!399!
et!al.!2011;!Zhang!et!al.!2012)!(Figure!4).!Oscillations!in!the!trait-based!models!are!fairly!400!
modest,!but!the!community!model!is!prone!to!unrealistically!strong!non-linear!401!
dynamics:!the!solution!varies!by!up!to!10!orders!of!magnitude!(Figure!4b!and!d).!This!402!
means!that!some!parts!of!the!spectrum!alternate!between!being!completely!devoid!of!403!
fish!and!being!fully!populated.!It!is!therefore!evident!that!the!non-linear!properties!of!404!
the!community!model!are!fundamentally!different!from!the!models!with!life-history!405!
diversity,!such!as!the!trait-based!models!or!a!food-web!model.!!Even!if!the!community!406!
model!is!made!linearly!stable!with!a!high!trophic!overlap!or!a!diffusion!term!(Datta!et!407!
al.,!2011),!the!strong!dynamical!response!will!still!be!present!if!the!model!is!perturbed!
408!
away!from!the!equilibrium.!This!should!be!kept!in!mind!if!the!model!is!used!to!simulate!
409!
the!dynamical!behavior!of!marine!ecosystems!(Zhang!et!al.!2012;!Rossberg!2013!p.!273).!
410!
Challenges$and$open$issues$411!
Size!spectrum!models!distinguish!themselves!from!unstructured!models!by!resolving!
412!
individual!body!size!as!a!continuous!state!variable!(body!size).!To!what!extent!do!
413!
individual!body!size!and!species!identity!determine!the!ecological!outcomes!of!
414!
community!dynamics?!The!thrust!of!size!spectrum!modeling!has!become!an!emphasis!415!
on!the!former,!whilst!unstructured!models!have!emphasized!the!latter.!Some!size!416!
spectrum!models!represent!species!interactions,!and!some!attempts!have!been!made!to!
417!
find!a!common!understanding!between!the!two!perspectives.!Notably,!a!food-web!model!
418!
with!implicit!representation!of!intra-species!size!structure!was!obtained!(Rossberg!and!419!
!
17!
Farnsworth!2011)!to!(indirectly)!describe!interactions!among!species!of!different!sizes.!420!
The!importance!of!explicitly!resolving!the!size-structure!of!species,!or!where!it!can!421!
safely!be!ignored,!is!context!dependent!so!requires!specific!and!systematic!exploration!422!
(Jacobsen!et!al.!2015;!Woodworth-Jefcoats!et!al.!2015).!!423!
!424!
An!important!difference!between!implementations!of!size!spectrum!models!is!whether!425!
growth!is!fixed!or!food-dependent!(Table!3).!Fixing!growth!simplifies!model!setup!and!426!
calibration.!It!is!justified!by!the!modest!variations!in!growth!observed!in!marine!species.!427!
Fixing!growth,!however,!has!consequences!which!should!be!considered!when!the!model!428!
is!calibrated!and!results!are!interpreted.!A!model!with!fixed!growth!will!still!resolve!429!
trophic!cascades!mediated!by!mortality,!but!it!will!not!resolve!competition,!which!is!430!
crucial!to!describe!the!phenomenon!of!overcompensation!(De!Roos!and!Persson!2002)!431!
and!may!be!important!to!understand!the!response!of!the!spectrum!to,!e.g.!fishing!on!432!
‘forage’!species!(Houle!et!al.!2013;!Jacobsen!et!al.!2015).!More!importantly,!without!433!
food-dependent!growth,!the!mass!balancing!between!growth!and!predation!mortality!is!
434!
broken.!This!requires!that!care!is!taken!in!the!setup!to!ensure!that!predation!mortalities!
435!
are!in!the!correct!range!by!adjusting!the!“other!food”!compartment!(Thorpe!et!al.!2015).!
436!
Having!too!low!predation!mortalities!(as!in!Hall!et!al.!2006;!Worm!et!al.!2009;!Rochet!et!437!
al.!2011)!will!result!in!a!model!that!is!essentially!a!set!of!weakly!coupled!single-species!
438!
models!thus!defying!the!purpose!of!a!multi-species!model.!
439!
!!
440!
Size-based!models!have!been!characterized!as!“highly!unrealistic”!and!being!based!on!441!
“unrealistic!and!even!contradictory!assumptions”!(Froese!et!al.!2015;!Andersen!et!al.!442!
2016b).!We!hope!to!have!made!it!clear!that!the!basic!assumptions!are!realistic!and!
443!
internally!consistent.!Nevertheless,!while!the!size-based!models!have!matured!to!a!
444!
degree!where!they!can!be!applied!to!make!impact!assessments!of!fishing!on!marine!445!
ecosystems,!they!still!face!challenges!related!to!density-dependence,!life-history!trade-
446!
!
18!
offs,!termination!at!the!large!body!sizes!end,!calibration!procedure,!and!numerical!447!
implementation,!which!must!be!confronted:!each!will!be!briefly!discussed.!!448!
!449!
F&1,+(23#&7&1#&1.&3450!
All!food-web!models!of!real!ecosystems,!i.e.!with!specific!species,!require!some!form!of!451!
density!dependent!regulation!of!the!abundance!of!each!species!to!avoid!competitive!452!
exclusion.!Not!much!is!known!about!the!exact!mechanism!of!the!regulation!and!how!453!
different!mechanisms!affect!model!results.!The!size!based!interactions!and!trait!454!
differences!among!species!in!size!spectrum!models!provide!insufficient!niche!455!
differentiation!to!avoid!competitive!exclusion!(Hartvig!and!Andersen!2013).!Additional!456!
niche!differentiation!may!be!represented!by!a!random!species-specific!interaction!457!
matrix,!which!can!support!coexistence!(Hartvig!2011;!Hartvig!et!al.!2011).!Other!458!
commonly!used!mechanisms!are!(Table!3):!stock-recruitment!relationships;!fixed!459!
recruitment;!predator-dependent!functional!responses,!whereby!intake!depends!on!the!460!
density!of!competitors!as!well!as!prey!(also!used!in!Ecosim)!(Abrams!2014);!and!prey!
461!
switching!(Maury!and!Poggiale!2013),!whereby!rare!prey!are!not!attacked!(leading!to!an!
462!
emergent!type!III!functional!response).!Which!of!these!mechanisms!is!the!most!correct!
463!
representation!of!effects!in!real!ecosystem!is!currently!unknown:!stock-recruitment!464!
relations!and!fixed!recruitment!are!in!line!with!standard!practice!in!fisheries!science,!
465!
but!have!little!theoretical!support;!predator-dependent!functional!responses!and!prey!
466!
switching!certainly!occur!to!some!extent,!but!the!understanding!is!currently!too!weak!to!
467!
make!general!statements!of!the!strength!of!the!processes.!Within!structured!models,!468!
such!as!size!spectrum!models,!the!type!of!density!dependent!regulation!may!have!a!469!
profound!impact!on!the!solution,!both!the!size!spectrum!of!the!individual!species!and!
470!
the!relative!abundances!of!species!(compare!Fig.!4!in!Maury!and!Poggiale!(2013)!with!
471!
Figure!2B).!As!an!example,!we!tested!the!prediction!from!the!trait-based!model!against!472!
empirical!observations!by!comparing!the!asymptotic!size!distribution!with!observations!
473!
!
19!
from!three!trawl!surveys!in!the!North!Sea!(Daan!et!al.!2005)!(Figure!5).!Even!though!the!474!
comparison!does!not!reject!the!modeled!distribution,!more!comparisons!with!similar!475!
data!from!other!systems!are!needed!to!build!confidence!in!the!predicted!asymptotic!size!476!
distributions.!477!
!478!
In!addition!to!ensuring!coexistence!of!species,!the!imposed!density!dependence!also!acts!479!
as!a!carrying!capacity.!In!the!food-web!model!the!carrying!capacity!(GTUV$!is!found!by!480!
calibrating!to!observed!biomasses.!The!trait-based!model,!however,!relies!on!theoretical!481!
results!from!the!equilibrium!theory.!That!theory!compares!favorably!to!the!calibrated!482!
results!(Figure!6).!Nevertheless,!the!use!of!a!stock-recruitment!relationship!is!483!
unsatisfactory!as!it!introduces!a!dominating!external!regulation!on!the!biomass!of!484!
species.!This!may!bias!the!response!time!of!community!size!structure!(Fung!et!al.!2013),!485!
which!is!of!interest!in!conservation.!Further,!the!stock-recruitment!relationship!means!486!
that!a!large!amount!of!spawned!biomass!is!simply!lost!to!unspecified!density-dependent!487!
processes.!The!stock-recruitment!relationship!therefore!breaks!the!mass-balancing!
488!
which!is!carefully!observed!in!the!other!processes!in!the!model!(Persson!et!al.!2014).!!
489!
Since!there!is!no!generally!accepted!solution!to!the!problem!of!maintaining!coexistence,!
490!
results!should!be!interpreted!in!light!of!the!assumptions!used!to!represent!density!491!
dependent!regulation.!It!must!be!emphasized!that!this!problem!is!common!to!all!food-
492!
web!models!and!not!unique!to!size!spectrum!models.!
493!
!
494!
4)*+(,3*1#3()*#&$"!!,!495!
The!trait-based!model!assumes!that!the!most!important!trait!is!the!asymptotic!size.!Fish,!496!
however,!vary!in!other!traits!than!asymptotic!size.!The!questions!are!then:!which!other!
497!
trait(s)!should!be!included!in!a!model!to!represent!observed!variation?!And!how!can!
498!
suitable!trade-offs!be!formulated!and!parameterized?!An!obvious!trait!is!activity.!499!
Increased!activity!causes!higher!prey!encounter!rates!(higher!value!of!the!clearance!rate!
500!
!
20!
constant,!_).!On!the!other!hand,!higher!activity!results!in!increased!metabolic!rates!and!501!
increased!vulnerability!due!to!higher!exposure!to!predators.!It!is!possible!that!inclusion!502!
of!an!activity!trait!would!make!it!possible!to!distinguish!sedentary!from!active!species!503!
with!the!same!asymptotic!size,!such!as!anglerfish!and!scombroids.!Such!a!trait!may!not!504!
solve!the!problem!of!competitive!exclusion!because!it!does!not!lead!to!sufficient!niche!505!
differentiation.!A!trait!which!would!lead!to!niche!differentiation!could!be!related!to!506!
habitat!(Hartvig!2011;!Zhang!et!al.!2013),!i.e.,!pelagic!vs.!benthic!(Blanchard!et!al.!2011).!507!
In!both!cases!more!theoretical!investigations!are!needed!but!also!empirical!work!to!508!
establish!and!parameterize!the!trade-offs.!509!
!510!
8=",0)&3"!3(5&3,7&.()0/3*(3=*)?&3'"#23,+6&,3511!
An!overlooked!issue!is!the!termination!(closure)!of!the!model!spectrum!at!the!largest!512!
body!sizes.!Closure!is!usually!achieved!(rather!arbitrarily)!by!choosing!a!maximum!body!513!
size!and!enforcing!some!background!mortality!to!kill!of!the!largest!individuals.!514!
However,!the!size!of!this!mortality!clearly!influences!the!results,!in!particular!in!the!un-
515!
fished!situation.!In!the!simulations!presented!here!(Figure!2),!this!background!mortality!
516!
is!relatively!low,!leading!to!the!superabundance!of!large!individuals!compared!to!the!
517!
equilibrium!solution.!However,!we!do!not!know!the!real!abundance!of!the!largest!518!
individuals!in!an!unfished!system,!because!most!systems!are!heavily!perturbed.!!
519!
Further,!what!is!the!theoretical!largest!size!of!a!fish?!Why!are!teleost!fish!not!larger!than!
520!
a!few!hundred!kg?!There!is!no!physiological!mechanism!in!the!model!to!limit!the!
521!
asymptotic!size,!and!current!theoretical!understanding!can!only!guess!at!an!answer!to!522!
this!question!(Freedman!and!Noakes!2002;!Andersen!et!al.!2008;!Andersen!et!al.!2016a).!523!
A!satisfactory!theoretical!understanding!of!the!factors!limiting!the!upper!size!of!fish!is!
524!
needed!to!bolster!the!consistency!of!the!models.!
525!
!526!
8*=+')*(+"13("3)&*=3,2,(&/,3
527!
!
21!
Size!based!models!can!be!calibrated!to!real!ecosystems,!for!instance!on!the!scale!of!a!528!
continental!shelf!(in!smaller!systems!immigration!/!emigration!violate!the!assumed!529!
population!closure).!!In!the!trait-based!models!calibration!is!achieved!by!varying!some!530!
of!the!crucial!parameters,!such!as!the!growth!rate!parameter!^!and!the!carrying!capacity!531!
of!the!resource!to!reproduce!observed!average!growth!rates!of!individuals!and!eco-532!
system!level!catch!rates!of!the!fishery!(Pope!et!al.!2006;!Kolding!et!al.!2015).!The!next!533!
level!of!sophistication!is!to!match!modeled!species!to!known!species!characteristics!534!
(Jacobsen!et!al.!2015),!and!further!to!include!a!species!interaction!matrix!(Hall!et!al.!535!
2006;!Blanchard!et!al.!2014).!In!these!cases!the!biomass!of!each!species!has!to!be!536!
calibrated!by!adjusting!GTUV.!!Another!important!parameter!that!hitherto!has!been!537!
ignored!is!the!reproductive!efficiency!u.!This!parameter!represents!egg!survival,!which!538!
is!likely!to!vary!substantially!between!species.!Introducing!yet!another!calibration!539!
parameter,!however,!requires!more!data.!Currently,!the!calibration!methods!applied!are!540!
statistically!simple.!A!more!sophisticated!approach!acknowledges!uncertainty!by!541!
creating!an!ensemble!of!plausible!models,!requiring!them!to!fulfill!general!criteria!
542!
(Thorpe!et!al.!2015).!A!possible!future!direction!could!be!introduction!of!a!full!statistical!
543!
framework!where!distributions!of!parameter!values!are!derived!from!observations!of!
544!
biomasses!and!stomach!content!by!maximizing!a!likelihood!function!(Lewy!and!Vinther!545!
2004;!Spence!et!al.!2015).!Finally,!it!should!be!kept!in!mind!that!even!though!a!model!
546!
may!be!well!calibrated!to!current!situations!there!is!no!guarantee!that!it!will!reliably!
547!
predict!the!future.!
548!
!549!
G0/&)+.*=3,"=0(+"137)".&#0)&3550!
Gains!in!accuracy!and!speed!of!the!numerical!solution!may!be!achieved!by!moving!to!
551!
more!advanced!methods.!The!standard!method!is!a!first-order!semi-implicit!upwind!
552!
scheme!that!is!simple!to!implement!(Appendix!B).!The!drawback!of!this!method!is!that!it!553!
has!numerical!diffusivity,!is!not!very!efficient,!possibly!inaccurate!for!dynamics,!and!is!
554!
!
22!
unable!to!resolve!“cohort!cycles”!(de!Roos!and!Persson!2001)!with!discontinuities!in!the!555!
solution!in!the!form!of!“shocks”.!To!move!forward,!we!recommend!looking!in!the!rich!556!
literature!from!computational!fluid!mechanics!for!inspiration,!in!particular!towards!557!
higher!order!finite-volume!techniques!with!limiters,!which!maintains!positivity!of!the!558!
solution!(Zijlema!1996),!or!spectral!methods!(Rossberg!2012).!Enhancements!to!the!559!
numerical!scheme!could!be!implemented!to!common!benefit!in!the!(open!access)!560!
reference!implementation!“Mizer”!(Scott!et!al.!2014).!!561!
!562!
80))&1(3*1#3!0(0)&3*77=+.*(+"1,3563!
We!have!shown!how!size!spectrum!models!can!be!used!to!simulate!how!fishing!of!one!564!
group!of!species!affects!the!entire!system!and!that!is!very!difficult!to!achieve!with!565!
alternative!models.!Despite!several!open!issues!in!size-based!modeling,!as!discussed!566!
above,!the!model!framework!has!already!shown!its!use!to:!illustrate!how!fishing!drives!567!
trophic!cascades!(Andersen!and!Pedersen!2010),!simulate!the!impact!of!rising!568!
temperatures!(Maury!et!al.!2007;!Pope!et!al.!2009),!explore!the!potential!impacts!of!
569!
climate!change!scenarios!on!fish!production!(Blanchard!et!al.!2012;!Woodworth-Jefcoats!
570!
et!al.!2013;!Barange!et!al.!2014;!Lefort!et!al.!2014),!describe!the!indirect!effect!of!
571!
ecosystem!recovery!strategies!(Andersen!and!Rice!2010),!quantify!the!interaction!572!
between!forage!and!consumer!fishery!fleets!(Engelhard!et!al.!2013;!Houle!et!al.!2013)!or!
573!
between!fisheries!and!marine!mammals!(Houle!et!al.!2015),!evaluate!ecosystem!fishing!
574!
strategies!and!indicators!(Houle!et!al.!2012;!Blanchard!et!al.!2014;!Jennings!and!
575!
Collingridge!2015;!Spence!et!al.!2015;!Thorpe!et!al.!2015),!evaluate!balanced!harvesting!576!
(Rochet!et!al.!2011;!Jacobsen!et!al.!2014;!Law!et!al.!2014)!and!describe!the!ecosystem!577!
level!yield!(Worm!et!al.!2009;!Andersen!et!al.!2015).!Other!obvious!uses!would!be!as!
578!
operating!models!in!management!strategy!evaluations,!as!the!basis!for!bio-economic!
579!
evaluations!of!fishing!on!the!entire!community!(Andersen!et!al.!2015),!and!to!further!580!
developing!our!basic!understanding!of!fish!community!functioning.!In!our!view,!the!
581!
!
23!
community!model!is!best!limited!to!theoretical!work!examining!the!steady-state!582!
solutions.!The!trait-based!model!can!be!quickly!deployed!in!data-poor!situations!and!583!
makes!a!flexible!tool!for!exploration!of!community-level!fishery!interactions.!The!food-584!
web!types!of!models!are!well!suited!to!more!specific!fisheries!questions!where!a!higher!585!
level!of!species!identity!is!needed!than!provided!by!the!trait-based!model.!Besides!these!586!
strategic!applications,!it!is!tempting!to!deploy!size!spectrum!models!for!tactical!587!
ecosystem!based!management,!e.g.,!for!providing!advice!on!specific!species.!We!are!588!
reluctant!to!endorse!such!uses!because!we!find!the!purely!process-oriented!framework!589!
too!rigid!to!provide!precise!quantitative!information!on!the!species!level.!In!conclusion:!590!
the!small!number!of!parameters,!the!low!computational!requirements!and!the!solid!591!
mechanistically!basis!provide!a!framework!of!low!to!intermediate!complexity,!highly!592!
suited!to!the!strategic!impact!assessment!of!pressures!such!as!fishing!and!593!
environmental!change!on!marine!ecosystems.!594!
! !595!
!
24!
!596!
Acknowledgements(597!
We!thank!Julia!L.!Blanchard!and!Axel!Rossberg!for!comments!on!the!manuscript.!This!598!
work!was!supported!by!the!Centre!for!Ocean!Life,!a!VKR!Centre!of!Excellence!supported!599!
by!the!Villum!Foundation.!600!
!601!
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Neuheimer, A.B., Hartvig, M., Heuschele, J., Hylander, S., Kiørboe, T., Olsson, K., 757!
Sainmont, J., and Andersen, K.H. 2015. Adult and offspring size in the ocean
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over 17 orders of magnitude follows two life-history strategies. Ecology.
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Persson, L., van Leeuwen, A., and de Roos, A.M. 2014. The ecological foundation for
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ecosystem-based management of fisheries: mechanistic linkages between the 761!
individual-, population-, and community-level dynamics. ICES J. Mar. Sci. 762!
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Peters, R.H. 1983. The ecological implications of body size. Cambridge University
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29!
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867!
scheme with application to turbulent flows in general domains. Int. J. Numer.
868!
Methods Fluids 22: 619–641.
869!
!870!
! 871!
!
32!
872!
873!
4*'=&3HI3J"9&)1+1?3&:0*(+"1,3"!3(5&3!0==3!""#$%&'3/"#&=I33K0',.)+7(3L+M3)&!&),3("3,7&.+&,310/'&)I!874!
D1."01(&)3*1#3."1,0/7(+"13
!
!!Prey!size!selection!
v#Lwxy
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&
i".dA
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M1!
!!Clearance!rate!
{
A# - _A#b''|}~•''_A-H
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F.c2dA
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M2!
!!Encountered!food!
A# - {
A"#$ ‚
ƒ
v#Lwxy
#!ƒ#Lwxy #Lwxy(#Lwxy
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M3!
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rate!
TUV1A"#$ - ^A#`!
M4!
!!Feeding!level!
H
A# - A"#$
A# < „TUV1A"#$!
M5!
J)"%(53*1#3)&7)"#0.(+"13
!!Maturation!function!
# - / < #
AW
A
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W
A
%a`
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M6!
!!Somatic!growth!
CA# - ‡H
A# „TUV1A ; IA#K"/ ; … # $!
M7a!
!
C # - ˆ ‡H # „TUV ; I#K"/ ; … # $!
M7b!
!!Egg!production!
C# - ‡H
A# „TUV1A ; IA#K… # !
M8a!
!
C# - ˆ ‡H # „TUV ; I#K… # !
M8b!
N"01#*)23."1#+(+"13
!!Population!egg!
production!
GL1A -u
.#F
!A# C# '(#
Š
)
!
M9!
!!Recruitment!
GA- GTUV1A
GL1A
GTUV1A < GL1A
!
M10!
!!Boundary!condition!
!A#FC #F- GA!
M11!
!
33!
A")(*=+(23
!!Background!mortality!
DF- qFW
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M12!
!!Predation!mortality!
DL1A #Lwxy - v #Lwxy
#/ ; H
ƒ# {
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)
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M13!
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@&,"0).&3,7&.()0/3
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(!w"#$
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M14!
!!Carrying!capacity!
2 # - ' 2w#a3!
M15!
4)*+($'*,&#3/"#&=3
3
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!!Maximum!
recruitment!
GTUV1A - 2N2 ‡H
F^#F
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M16!
!!Physiological!
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o - H
F^
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.!
M17!
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4*'=&3OI3K2/'"=,3*1#39*=0&,3"!3,7&.+&,$+1#&7&1#&1(37*)*/&(&),3+13(5&3()*+($'*,&#3*1#3."//01+(23/"#&=,I33879!
!880!
Symbol!
Explanation!
Value!and!unita!
#!
Body!weight!
g!
W!
Asymptotic!body!weight!
g!
!"#$!
Abundance!density!spectrum!
numbers/gram!b!
e!
Preferred!predator-prey!mass!ratio!
100!c!
d!
Width!of!size!preference!
1.3!d!
!
Species!preference!
1!
Q!
Exponent!for!clearance!rate!
0.8!!e!
H
F!
Initial!feeding!level!
0.6!!f!
!
Assimilation!efficiency!
0.6!
^!
Factor!for!maximum!consumption!
20!g0.25yr-1!!!!g!
O!
Exponent!for!maximum!consumption!
¾!!h!
IJ!
Factor!for!standard!metabolism!
2.4!g0.25yr-1!
P!
Exponent!for!standard!metabolism!
¾!!h!
3
Ratio!between!size!at!maturation!and!W!
0.25!(0.01)!!i!
2N!
Constant!for!max.!recruitment!
1.7!!!!g!
u!
Efficiency!of!reproduction!
0.1!!!j!
#F!
Egg!size!
1!mg!!!k!
qF!
Factor!for!background!mortality!
2!g0.25yr-1!!!!l!
!
Exponent!for!background!mortality!
-0.25!!!!m!
ŽF!
Resource!productivity!
4!g0.25yr-1!!!!n!
#‘’“!
Maximum!size!of!resource!
1!g!!o!
2!
Resource!carrying!capacity!
1012!g-1!!!b,!g!
!
35!
aValues!in!parentheses!refers!to!the!community!model!
bThe!units!of!the!abundance!density!spectrum!could!also!be!expressed!as!a!concentration,!i.e.,!as!
numbers/gram/volume!or!numbers/gram/area.!In!that!case!the!units!of!the!resource!carrying!capacity!
should!also!be!changed!accordingly.!Further,!the!units!of!the!clearance!rate!would!become!volume!per!time!
or!area!per!time!respectively.!
c!Ursin!1973;!Jennings!et!al.!2001.!
d!Ursin!(1973)!finds!d - /,!but!here!d!is!increased!to!represent!cross-species!variation.!In!the!food-web!
model'd - /!except!if!specific!knowledge!about!a!species!exists.!
e!Andersen!and!Beyer,!2006.!
f!!Assumes!that!fish!in!general!are!not!satiated!(H
FY /$!while!also!have!a!signifant!surplus!after!assimilation!
and!standard!metabolism,!i.e,!larger!than!IJi"‡^$ - 01.1'Setting!H
F!in!the!middle!of!the!range!0.2…1!gives!
0.6.!See!also!Hartvig!et!al.!(2011),!App.!E.!
g!!Adjusted!to!give!similar!results!to!the!North!Sea!model!(Blanchard!et!al.!2014),!despite!the!use!of!different!
exponent!for!O;!see!text!and!Figure!2.!
h!!See!text.!
i!!Beverton,!1992.!
j!!Andersen!and!Beyer!(2013).!Note!that!this!differs!from!the!North!Sea!model!(Blanchard!et!al.!2014),!
where!a!value!of!u - /!was!used.!This!will!(in!the!North!Sea!model)!lead!to!overestimations!of!
T•y!for!
individual!species.!
k!!!Neuheimer!et!al.,!2015.!
l!!!This!value!leads!to!a!background!mortality!on!a!10!kg!individual!of!0.2!yr-1.!These!individuals!will!have!
little!predation!mortality!in!the!model,!so!background!mortality!is!the!largest!part.!See!also!discussion.!
m!!!!Standard!“metabolic”!assumption!(Brown!et!al.!2004).!
n!!!Hartvig!et!al.,!2011.!
o!!!Set!to!include!mesoplankton,!such!as!shrimps.!
!881!
3 3882!
!
36!
3883!
4*'=&3P3Implementations!of!size!spectrum!models.!References!are!given!to!where!a!particular!884!
model!was!first!formulated!or!calibrated!to!specific!system,!but!not!to!applications!of!the!models.!885!
The!column!“growth”!refers!to!the!functional!response!used!to!calculate!growth!rate.!886!
Reference!
Growth!
Reproduction!
Resource!
Density!dependence!
B""#$%&'3/"#&=,3
!
!
!
!
Hall!et!al.!2006,!
Rochet!et!al.!2011!
Fixed!
Fixed!
N.a.!
Stock-recruitment!
Hartvig!2011,!Hartvig!
et!al.!2011!
Type!II!
Dynamic!
Dynamic!
Emergent!
Houle!et!al.!2012,!
Blanchard!et!al.!2014!
Type!II!
Dynamic!
Dynamic!
Stock-recruitment!
Rossberg!et!al.!2013!
Type!II!
Dynamic!
Dynamic!
Emergent!
4)*+($'*,&#3/"#&=,3
!
!
!
!
Pope!et!al.!2006,!2009!
Fixed!
Fixed!
N.a.!
Stock-recruitment!
relationship!
Andersen!and!
Pedersen!2010!
Type!II!
Fixed!
Dynamic!
Fixed!recruitment!
Houle!et!al.!2013;!
Jacobsen!et!al.!2014!
Type!II!
Dynamic!
Dynamic!
Stock-recruitment!
relationship!
Maury!and!Poggiale!
2013!
Type!II!
Dynamic!
Dynamic!
Switching!
8"//01+(23/"#&=,3
!
!
!
!
Benoît!and!Rochet!
2004;!Blanchard!et!al.!
2009;!Law!et!al.!2009!
QHR!
!
Type!I!
N.a.!
Fixed!
Fixed!boundary!
condition!
Blanchard!et!al.!2011!
Type!I!&!
Dynamic!
Fixed!
Fixed!boundary!
!
37!
3887!
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"(5&)3/"#&=,I3889!
!3890!
II!
condition!
Maury!et!al.!2007!!
Type!II!
Dynamic!
Dynamic!
!
This!article!
Type!II!
N.a.!!
Dynamic!
Fixed!boundary!
condition!
!
38!
!891!
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,*/&37)&#*(+"13/")(*=+(23,+1.&3!&&#+1?3+,3"1=23'*,&#3"13,+6&I3909!
! !910!
Community model
108
109
1010
1011
1012
1013
Sheldon spectra (g)
(a)
10-1
101
103
105
Growth (g/year)
(d)
10-3 10-1 10110 310 5
0
2
4
6
Mortality (year-1 )
(g)
Trait-based model
(b)
(e)
10-3 10-1 10110 310 5
Body mass (g)
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a)
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0.1
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b)
Body mass (g)
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Relative spectra
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Sheldon spectrum (g)
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10
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7
10
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Predator spectrum (g
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!939!
W1 (g)
10210310410 5
Numbers
106
107
108
109
1010
1011
1012
W1 (g)
10110210310 410 5
Rmax (numbers/yr)
109
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!
... β i is the preferred predator-prey mass ratio, and σ i is the width of the size selection function. Details of other equations (Hartvig et al., 2011;Andersen et al., 2016b) are listed in Table 2. ...
... However, our results may only reflect part of pelagic ecosystem dynamics because most of our species are top predators caught by pelagic longlines. In addition, we validated the SSBs and size-at-age data of nine species, but that may not guarantee a reliable projection of the whole ecosystem (Andersen et al., 2016b). ...
Article
Catch, bycatch and discard information is important for the assessment and management of fisheries. Using Chinese pelagic tuna longline observer data from 2010 to 2018, we studied the catch composition in the Chinese pelagic tuna longline fisheries in Atlantic targeting bigeye tuna (Thunnus obesus) and bluefin tuna (Thunnus thynnus), and analyzed the survival status and discard rates of common bycatch species. A total of 55 species, including tunas, billfishes, sharks, sea turtles, cetaceans, seabirds, and other pelagic species, were observed. The results indicated that the catch composition of the Chinese pelagic tuna longline fishery targeting bigeye tuna was significantly different from that targeting bluefin tuna. The annual discard rates of common species decreased over this period. Discard rate by length and discard mortality for common species were varied among species. This is the first study to estimate catch, bycatch, and discard using Chinese pelagic tuna longline observer data in the Atlantic Ocean, which is important for the management of Chinese tuna longline fisheries in Atlantic Ocean.
... β i is the preferred predator-prey mass ratio, and σ i is the width of the size selection function. Details of other equations (Hartvig et al., 2011;Andersen et al., 2016b) are listed in Table 2. ...
... However, our results may only reflect part of pelagic ecosystem dynamics because most of our species are top predators caught by pelagic longlines. In addition, we validated the SSBs and size-at-age data of nine species, but that may not guarantee a reliable projection of the whole ecosystem (Andersen et al., 2016b). ...
Article
Full-text available
The size-spectrum model has been considered a useful tool for understanding the structures of marine ecosystems and examining management implications for fisheries. Based on Chinese tuna longline observer data from the central and eastern tropical Pacific Ocean and published data, we developed and calibrated a multispecies size-spectrum model of twenty common and commercially important species in this area. We then use the model to project the status of the species from 2016 to 2050 under five constant-fishing-mortality management scenarios: (1) F=0; (2) F=Frecent, the average fishing mortality from 2013 to 2015; (3) F=0.5Frecent; (4) F=2Frecent and (5) F=3Frecent. Several ecological indicators were used to track the dynamics of the community structure under different levels of fishing, including the mean body weight, slope of community size spectra (Slope), and total biomass. The validation demonstrated that size-at-age data of nine main catch species between our model predictions and those empirical data from assessments by the Western and Central Pacific Fisheries Commission matched well, with the R2>0.9. The direct effect of fishing was the decreasing abundance of large-sized individuals. The mean body weight in the community decreased by ∼1 500 g (21%) by 2050 when F doubled from Frecent to 2Frecent. The higher the fishing mortality, the steeper the Slope was. The projection also indicated that fishing impacts reflected by the total biomass did not increase proportionally with the increasing fishing mortality. The biomass of the main target tuna species was still abundant over the projection period under the recent fishing mortality, except Albacore tuna (Thunnus alalunga). For sharks and billfishes, their biomass remained at relatively higher levels only under the F=0 scenario. The results can serve as a scientific reference for alternative management strategies in the tropical Pacific Ocean.
... Fish and mobile invertebrate dynamics are captured using size spectrum theory (Andersen et al., 2016;Guiet et al., 2016) and based on an established model for coral reef ecosystems (Rogers et al., 2014(Rogers et al., , 2018. Processes including feeding, growth, and predation (Dunic & Baum, 2017) tend to scale well with body size (Brown et al., 2004), and this model has been shown to accurately capture changes in the biomass, productivity, and size structure of coral reef fish assemblages in response to habitat degradation. ...
Article
Full-text available
Ocean warming is already causing widespread changes to coral reef ecosystems worldwide. Warming is having direct and indirect impacts on food webs, but their interaction is unclear. Warming directly affects fishes and invertebrates by increasing their metabolic rate, resulting in changes to demographic processes such as growth rates. Indirect effects involve a loss of reef habitat quality as coral bleaching reduces the availability of refuges. We used a size‐structured dynamic energy budget model of fishes and invertebrates, coupled to a spatially explicit model of coral and algae, to explore potential changes to ecosystem function with warming. Modeled changes in biomass for +3°C of warming were found to be controlled predominantly by the direct effects of warming on growth rates, rather than by indirect effects via the changed coral habitat. Crucially for fisheries, the biomass of predators decreased by at least 50% with +3°C of warming, and productivity of predators decreased by at least 60%.
... Similarly, the retention efficiency of bivalves as a function of food particle size (Riisgard, 1988;Strohmeier et al., 2012;Sonier et al., 2016), coupled with the strong feeding pressure exerted in aquaculture settings, could affect the pelagic community size distribution at a system scale. According to size-spectrum theory (Sheldon et al., 1972;Andersen et al., 2016), bivalve culture could then affect the way energy propagates through trophic levels. Further work is required to better understand how and how much these cascading effects might change coastal ecosystem functioning and the services they provide, and to examine the balance between these negative effects and the positive effects that accrue from added food production and nutrient loading mitigation from bivalve aquaculture. ...
Article
Full-text available
Bivalve aquaculture may provide a variety of ecosystem services including nitrogen extraction from estuaries, which are often subject to excess nutrient loading from various land activities, causing eutrophication. This nitrogen extraction may be affected by a combination of various non-linear interactions between the cultured organisms and the receiving ecosystem. The present study used a coupled hydro-biogeochemical model to examine the interactive effects of various factors on the degree of estuarine nutrient mitigation by farmed bivalves. These factors included bay geomorphology (leaky, restricted and choked systems), river size (small and large rivers leading to moderate (105.9 Mt N yr-1) and high (529.6 Mt N yr-1) nutrient discharges), bivalve species (blue mussel (Mytilus edulis) and eastern oyster (Crassostrea virginica)), farmed bivalve area (0, 10, 25 and 40% of estuarine surface area) and climate change (water temperature, sea level and precipitation reflecting either present or future (Horizon 2050) conditions). Model outputs indicated that bivalve culture was associated with the retention of nitrogen within estuaries, but that this alteration of nitrogen exchange between estuaries and the open ocean was not uniform across all tested variables and it depended on the nature of their interaction with the bivalves as well as their own dynamics. When nitrogen extraction resulting from harvest was factored in, however, bivalve culture was shown to provide a net nitrogen removal in the majority of the tested model scenarios. Mussels provided more nutrient mitigation than oysters, open systems were more resilient to change than closed ones, and mitigation potential was shown to generally increase with increasing bivalve biomass. Under projected future temperature conditions, nutrient mitigation from mussel farms was predicted to increase, while interactions with the oyster reproductive cycle led to both reduced harvested biomass and nutrient mitigation potential. This study presents the first quantification of the effects of various biological, physical, geomorphological and hydrodynamical processes on nutrient mitigation by bivalve aquaculture and will be critical in addressing questions related to eutrophication mitigation by bivalves and prediction of possible nutrient trading credits.
... Data Availability Statement 10 et al., 2014) and a Baltic Sea fish assemblage of 3 interacting pelagic and demersal species (Jacobsen et al., 2017). Species are defined by species-specific life-history parameters, and allometric scaling rules are used to scale individual processes (growth and mortality) to population-and community-level size structure (Andersen et al., 2016;Jacobsen et al., 2014). Energy flux is accounted for between all individual processes for a full energy budget. ...
Article
Full-text available
Wild-caught fish are a bioavailable source of nutritious food that, if managed strategically , could enhance diet quality for billions of people. However, optimising nutrient production from the sea has not been a priority, hindering development of nutrition-sensitive policies. With fisheries management increasingly effective at rebuilding stocks and regulating sustainable fishing, we can now begin to integrate nutritional outcomes within existing management frameworks. Here, we develop a conceptual foundation for managing fisheries for multispecies Maximum Nutrient Yield (mMNY). We empirically test our approach using size-based models of North Sea and Baltic Sea fisheries and show that mMNY is predicted by the relative contribution of nutritious species to total catch and their vulnerability to fishing, leading to trade-offs between catch and specific nutrients. Simulated nutrient yield curves suggest that vitamin D, which is deficient in Northern European diets, was underfished at fishing levels that returned maximum catch weights. Analysis of global catch data shows there is scope for nutrient yields from most of the world's marine fisheries to be enhanced through nutrient-sensitive fisheries management. With nutrient composition data now widely available, we expect our mMNY framework to motivate development of nutrient-based reference points in specific contexts, such as data-limited fisheries. Managing for mMNY alongside policies that promote access to fish could help close nutrient gaps for coastal populations, maximising the contribution of wild-caught fish to global food and nutrition security. K E Y W O R D S fisheries management, food security, nutrition, overfishing, seafood, sustainable fisheries 2 | ROBINSON et al.
... The size spectra of fish communities may also reveal differences in ecosystem productivity, with communities in areas of high productivity exhibiting steep size spectra slopes due to a high relative abundance of small, planktivorous fishes (Emmrich et al. 2011;Secor 2015). Since size is thought to be the primary determinant of many biological processes in marine organisms (Andersen et al. 2016), size-based community metrics, such as the slope of the size spectrum, are attractive for application in Ecosystem-Based Fisheries Management Shin et al. 2005). ...
Article
Assessing marine fish community size spectra with hydroacoustics is challenging, as communities are diverse, schooling and swim bladder-less fishes are common, and fish orientation is variable. We developed an approach to examine these challenges and applied it to data from 51 optic-acoustic surveys of fishes at petroleum platforms throughout the U.S. Gulf of Mexico. When in situ target strength (TS; dB re 1 m2) distributions were used to calculate the density (and subsequently abundance) of schooling fishes, fish lengths and size spectra slopes were significantly smaller than in simulated communities and fish community censuses at platforms (i.e., reference datasets). However, acoustic slopes were comparable to reference slopes when simulated TS values (based on species composition) were used to calculate schooling fish abundance. These findings held regardless of whether specific or general models were used to convert TS to length. Fish orientation was not a useful predictor of TS or slope, but may explain why in situ TS measurements from small groups of fishes around schools were unsuitable for abundance calculations. By examining the challenges associated with assessing size spectra with acoustics, this study aids progress towards using acoustic size spectra metrics for ecological inferences.
... Fish growth is also a quantitative trait that is easy to measure, and as such, can be used to assess physiological and demographic responses to human-mediated environmental changes (Denechaud et al. 2020). In addition, pressures from capture fisheries have produced one of the fastest rates of phenotypic change observed in wild populations (Oke et al. 2020), with effects not only to targeted species but also to associated communities (Andersen et al. 2016;Dijoux and Boukal 2021). However, size-selective harvesting and climatic effects could be species and system specific (Denechaud et al. 2020), and despite increasing concerns, the evolutionary consequences have so far been investigated in only a limited number of species. ...
Article
Growth is one of the most important traits of an organism. For exploited species, this trait has ecological and evolutionary consequences as well as economical and conservation significance. Rapid changes in growth rate associated with anthropogenic stressors have been reported for several marine fishes, but little is known about the genetic basis of growth traits in teleosts. We used reduced genome representation data and genome-wide association approaches to identify growth-related genetic variation in the commercially, recreationally, and culturally important Australian snapper (Chrysophrys auratus, Sparidae). Based on 17,490 high-quality SNPs and 363 individuals representing extreme growth phenotypes from 15,000 fish of the same age and reared under identical conditions in a sea pen, we identified 100 unique candidates that were annotated to 51 proteins. We documented a complex polygenic nature of growth in the species that included several loci with small effects and a few loci with larger effects. Overall heritability was high (75.7%), reflected in the high accuracy of the genomic prediction for the phenotype (small vs. large). Although the SNPs were distributed across the genome, most candidates (60%) clustered on chromosome 16, which also explains the largest proportion of heritability (16.4%). This study demonstrates that reduced genome representation SNPs and the right bioinformatic tools provide a cost-efficient approach to identify growth-related loci and to describe genomic architectures of complex quantitative traits. Our results help to inform captive aquaculture breeding programmes and are of relevance to monitor growth-related evolutionary shifts in wild populations in response to anthropogenic pressures.
Article
Inland fisheries have a significant cultural and economic value around the globe, providing dietary protein, income, and recreation. Consequently, methods for monitoring and managing these important fisheries are continually being refined. In marine systems, multispecies size spectrum models have been increasingly used to explore management scenarios of important fish stocks within an ecosystem‐based fisheries management framework; however, these models have not been applied as extensively in freshwater systems. In this study, we developed a multispecies size spectrum model for the fish community of Lake Nipissing, a large, productive lake in Ontario, Canada. To the best of our knowledge, this is the first fully calibrated multispecies size spectrum model for an inland fishery. Using this model, we explored the impacts of potential fishing regimes and management scenarios on fish community dynamics while taking species interactions into account. Specifically, we examined how changes in fishing mortality affect (1) species biomass, (2) community size structure, and (3) stock recovery times. We found that community dynamics following changes in fishing mortality were driven by complex interactions among species, including competition and predation. The greatest changes in biomass and community size structure were observed following changes in fishing mortality of top predators, with community size structure most strongly influenced by changes in the mortality of the largest species. Counter to predictions based on generation time, the smallest species in our model exhibited the longest time to recovery due to strong competition and predation. Our results demonstrate the importance of considering species interactions in the management of inland fisheries and highlight the potential of size spectrum model use in freshwater systems.
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Full-text available
Climate change and fisheries exploitation are dramatically changing the abundances, species composition, and size spectra of fish communities. We explore whether variation in 'abundance size spectra', a widely studied ecosystem feature, is influenced by a parameter theorized to govern the shape of size-structured ecosystems-the relationship between the sizes of predators and their prey (predator-prey mass ratios, or PPMRs). PPMR estimates are lacking for avast number of fish species, including at the scale of trophic guilds. Using measurements of 8128 prey items in gut contents of 97 reef fish species, we established predator-prey mass ratios (PPMRs) for four major trophic guilds (piscivores, invertivores, planktivores, and herbivores) using linear mixed effects models. To assess the theoretical predictions that higher community-level PPMRs leads to shallower size spectrum slopes, we compared observations of both ecosystem metrics for ~15,000 coastal reef sites distributed around Australia. PPMRs of individual fishes were remarkably high (median ~71,000), with significant variation between different trophic guilds (~890 for piscivores; ~83,000 for planktivores), and ~8700 for whole communities. Community-level PPMRs were positively related to size spectrum slopes, broadly consistent with theory, however, this pattern was also influenced by the latitudinal temperature gradient. Tropical reefs showed a stronger relationship between community-level PPMRs and community size spectrum slopes than temperate reefs. The extent that these patterns apply outside Australia and consequences for community structure and dynamics are key areas for future investigation.
Preprint
Full-text available
Climate change and fisheries exploitation are dramatically changing the species composition, abundances, and size spectra of fish communities. We explore whether variation in abundance-size spectra, a widely studied ecosystem feature, is influenced by a critical parameter thought to govern the shape of size-structured ecosystems—the relationship between the sizes of predators and their prey (predator-prey mass ratios, or PPMRs). PPMR estimates are lacking for vast numbers of fish species, including at the broader trophic guild scale. Using measurements of 8,128 prey items in gut contents of 97 reef fish species, we established PPMRs for four major trophic guilds (piscivores, invertivores, planktivores and herbivores) using linear mixed effects models. To assess theoretical predictions that higher mean community-level PPMR leads to shallower size spectrum slopes, we compared observations of mean community-level PPMR with size spectrum slopes for coastal reef sites distributed around Australia. PPMRs of individual fishes were remarkably high (median ~71,000), with significant variation between different trophic guilds (~890 for piscivores; ~83,000 for planktivores), and ~8,700 for whole communities. Community-level PPMRs were positively related to size spectrum slopes, broadly consistent with theory, however, this pattern was also influenced by the latitudinal temperature gradient. Tropical reefs showed a stronger relationship between community-level PPMRs and community size spectrum slopes than temperate reefs. The extent that these patterns apply outside Australia, and consequences for community structure and dynamics, are key areas for future investigation.
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Full-text available
The advent of an ecosystem-based approach dramatically expanded the scope of fisheries management, creating a critical need for new kinds of data and quantitative approaches that could be integrated into the management system. Ecosystem models are needed to codify the relationships among drivers, pressures and resulting states, and to quantify the trade-offs between conflicting objectives. Incorporating ecosystem considerations requires moving from the single-species models used in stock assessments, to more complex models that include species interactions, environmental drivers and human consequences. With this increasing model complexity, model fit can improve, but parameter uncertainty increases. At intermediate levels of complexity, there is a ‘sweet spot’ at which the uncertainty in policy indicators is at a minimum. Finding the sweet spot in models requires compromises: for example, to include additional component species, the models of each species have in some cases been simplified from age-structured to logistic or bioenergetic models. In this paper, we illuminate the characteristics, capabilities and short-comings of the various modelling approaches being proposed for ecosystem-based fisheries management. We identify key ecosystem needs in fisheries management and indicate which types of models can meet these needs. Ecosystem models have been playing strategic roles by providing an ecosystem context for single-species management decisions. However, conventional stock assessments are being increasingly challenged by changing natural mortality rates and environmentally driven changes in productivity that are observed in many fish stocks. Thus, there is a need for more tactical ecosystem models that can respond dynamically to changing ecological and environmental conditions.
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The majority of higher organisms in the marine environment display indeterminate growth; that is, they continue to grow throughout their life, limited by an asymptotic size. We derive the abundance of species as a function of their asymptotic size. The derivation is based on size‐spectrum theory, where population structure is derived from physiology and simple arguments regarding the predator‐prey interaction. Using a hypothesis of constant satiation, which states that the average degree of satiation is independent of the size of an organism, the number of individuals with a given size is found to be proportional to the weight raised to the power −2.05, independent of the predator/prey size ratio. This is the first time the spectrum exponent has been derived solely on the basis of processes at the individual level. The theory furthermore predicts that the parameters in the von Bertalanffy growth function are related as \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $K\propto L^{-1}_{\infty }$ \end{document} .
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Book
Food webs have now been addressed in empirical and theoretical research for more than 50 years. Yet, even elementary foundational issues are still hotly debated. One difficulty is that a multitude of processes need to be taken into account to understand the patterns found empirically in the structure of food webs and communities. Food Webs and Biodiversity develops a fresh, comprehensive perspective on food webs. Mechanistic explanations for several known macroecological patterns are derived from a few fundamental concepts, which are quantitatively linked to field-observables. An argument is developed that food webs will often be the key to understanding patterns of biodiversity at community level. Key Features: Predicts generic characteristics of ecological communities in invasion-extirpation equilibrium. Generalizes the theory of competition to food webs with arbitrary topologies. Presents a new, testable quantitative theory for the mechanisms determining species richness in food webs, and other new results. Written by an internationally respected expert in the field. With global warming and other pressures on ecosystems rising, understanding and protecting biodiversity is a cause of international concern. This highly topical book will be of interest to a wide ranging audience, including not only graduate students and practitioners in community and conservation ecology but also the complex-systems research community as well as mathematicians and physicists interested in the theory of networks. "This is a comprehensive work outlining a large array of very novel and potentially game-changing ideas in food web ecology." Ken Haste Andersen, Technical University of Denmark "I believe that this will be a landmark book in community ecology ... it presents a well-established and consistent mathematical theory of food-webs. It is testable in many ways and the author finds remarkable agreements between predictions and reality." Géza Meszéna, Eötvös University, Budapest
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Abstract: Charles Elton introduced the “pyramid of numbers” in the late 1920s, but this remarkable insight into body-size dependent patterns in natural communities lay fallow until the theory of the biomass size spectrum was introduced by aquatic ecologists in the mid-1960s. They noticed that the summed biomass concentration of individual aquatic organisms was roughly constant across equal logarithmic intervals of body size from bacteria to the largest predators. These observations formed the basis for a theory of aquatic ecosystems, based on the body size of individual organisms, that revealed new insights into constraints on the structure of biological communities. In this review, we discuss the history of the biomass spectrum and the development of underlying theories. We indicate how to construct biomass spectra from sample data, explain the mathematical relations among them, show empirical examples of their various forms, and give details on how to statistically fit the most robust linear and nonlinear models to biomass spectra. We finish by giving examples of biomass spectrum applications to production and fisheries ecology and offering recommendations to help standardize use of the biomass spectrum in aquatic ecology.
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A recent publication about balanced harvesting (Froese et al., ICES Journal of Marine Science; doi:10.1093/icesjms/fsv122) contains several erroneous statements about size-spectrum models. We refute the statements by showing that the assumptions pertaining to size-spectrum models discussed by Froese et al. are realistic and consistent. We further show that the assumption about density-dependence being described by a stock recruitment relationship is responsible for determining whether a peak in the cohort biomass of a population occurs late or early in life. Finally, we argue that there is indeed a constructive role for a wide suite of ecosystem models to evaluate fishing strategies in an ecosystem context.