The theoretical foundations for size spectrum models of fish communities

Abstract

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.

Full text

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The$theoretical$foundations$for$size$spectrum$models$of$fish$
1!
communities$$
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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!
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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!
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868!
Methods Fluids 22: 619–641.
869!
!870!
! 871!

!
32!
872!
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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
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!!Prey!size!selection!
v#Lwxy
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H
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A
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CA# - ‡H
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M7a!
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C # - ˆ ‡H # „TUV ; I#K"/ ; … # $!
M7b!
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C€# - ‡H
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M8a!
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M8b!
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GL1A -u
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!
M9!
!!Recruitment!
GA- GTUV1A
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M10!
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M11!

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A")(*=+(23
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DF- qFW
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DL1A #Lwxy - v #Lwxy
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!!Carrying!capacity!
2 # - ' 2w#a3!
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GTUV1A - 2N2 ‡H
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!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!
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! !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)
(h)
Food-web model
(c)
(f)
10-3 10-1 10110 310 5
(i)

!
40!
!911!
!912!
B+?0)&3PI3@&,7"1,&3("3!+,5+1?3"13(5&3."//01+(23!")3(5&3G")(53K&*3!""#$%&'3/"#&=3Q?)&2RV3(5&3()*+($'*,&#3/"#&=3913!
Q,"=+#3'=*.CR3*1#3(5&3."//01+(23/"#&=3Q#*,5&#3'=*.CRI3K+6&3,7&.()*3*)&3,5"%13)&=*(+9&3("3(5&301!+,5&#3,+6&3914!
,7&.()*3!)"/3B+?0)&3OI3*R3@&,7"1,&3("3!+,5+1?3*==3,7&.+&,3%+(53*3()*%=$(27&3,&=&.(+9+(237*((&)1I3K&=&.(+"13"13&*.53915!
,7&.+&,3,(*)(,3*(3010lW*1#3(5&3!+,5+1?3/")(*=+(23+,3TI\32)$HI3>13(5&3."//01+(23/"#&=3*==3+1#+9+#0*=,3=*)?&)3(5*13916!
HT3?3*)&3!+,5&#I3'R385*1?&,3("3(5&3!+,5&#3."//01+(23+137*1&=3*R3%5&13!")*?&3!+,5+1?3+,3)&/"9&#V3+I&I3!+,5+1?3"13917!
,7&.+&,3%+(53*13*,2/7("(+.3,+6&3,/*==&)3(5*13OTT3?3Q")3+1#+9+#0*=,3,/*==&)3(5*13OTT3?3!")3(5&3."//01+(23918!
/"#&=RI33919!
! !920!
a)
Relative spectra
0.1
0.2
0.5
1
2
5
10
b)
Body mass (g)
10-3 10-1 10110 310 5
Relative spectra
0.2
0.5
1
2
5

!
41!
!921!
B+?0)&3]I3>==0,()*(+"13"!3(5&3(+/&$#&7&1#&1(3,"=0(+"13+13(5&3."//01+(23/"#&=3Q(%"3=&!($/",(3."=0/1,R3*1#3(5&3922!
()*+($'*,&#3/"#&=3Q)+?5($/",(3."=0/1RI34"73)"%X3*9&)*?&3'+"/*,,3,7&.()*3Q,"=+#R3*1#3)*1?&3"!39*)+*(+"13Q?)&23923!
7*(.5RI3N"(("/3)"%X3(5&375*,&$7=*1&3'&(%&&13*37)&23"!3HT3?3*1#3*37)&#*(")3%+(53*3,+6&3"1&37)&#*(")$7)&23/*,,3924!
=*)?&)3Qe/0'C$3+==0,()*(&#3%+(539&)(+.*=3#*,5&#3=+1&,3+13(5&3("73)"%I3S&!(3."=0/1X3."//01+(23/"#&=3%+(53%+#(53925!
"!3,+6&37)&!&)&1.&3d - /1RI3A+##=&3."=0/1X3."//01+(23/"#&=3%+(53d - /I3@+?5(3."=0/1X3()*+($'*,&#3/"#&=3%+(53926!
d - /I3^(5&)37*)*/&(&),3*,3+13B+?,I3O3*1#3PV3&Z.&7(3(5*(3(5&3=*)?&,(3,+6&3"!3(5&3")?*1+,/,3+,3HTTT3C?I33927!
! !928!
10
-3
10
-1
10
1
10
3
10
5
10
0
10
2
10
4
10
6
10
8
10
10
10
12
Sheldon spectrum (g)
(a)
10
3
10
5
10
7
10
9
10
11
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Predator spectrum (g
-1
)
(d)
10
-3
10
-1
10
1
10
3
10
5
Body mass (g)
(b)
10
3
10
5
10
7
10
9
10
11
Prey spectrum (g
-1
)
(d)
10
-3
10
-1
10
1
10
3
10
5
(c)
10
3
10
5
10
7
10
9
10
11
(f)

!
42!
!929!
B+?0)&3\I3;,2/7("(+.3%&+?5(3,7&.()0/3Q&Z(&1#&#3K5&=#"13527"(5&,+,RV3,5"%+1?3("(*=310/'&),3"!3+1#+9+#0*=,3+13930!
="?*)+(5/+.*==23,7*.&#3*,2/7("(+.3%&+?5(3?)"07,3+13*1301!+,5&#3Q?)&2R3*1#3!+,5&#3,+(0*(+"13Q'=*.CR3!)"/3(5&3931!
()*+($'*,&#3/"#&=I345&3,+/0=*(+"13)&,0=(,3*)&3."/7*)&#3("3#*(*3!)"/3(5)&&3#+!!&)&1(3()*%=3,0)9&2,3(Daan!et!al.!932!
2005)I3>(3+,3*,,0/&#3(5*(3(5&3()*%=3,0)9&2,3"1=23)&(*+13+1#+9+#0*=,3=*)?&)3(5*13HT3./I3933!
3934!
!935!
B+?0)&3_I3A*Z+/0/3)&.)0+(/&1(V3G–n— V3!")3(5&3G")(53K&*3/"#&=3Q.+).=&,R3*1#3(5&3()*+($'*,&#3/"#&=3%+(53="?$936!
,7*.&,3*,2/7("(+.3,+6&3?)"07,3Q=+1&R3*1#3%+(53*,2/7("(+.3,+6&3?)"07,3%+(53(5&3,*/&3*,2/7("(+.3,+6&,3*,3+13(5&3937!
G")(53K&*3/"#&=3Q.)",,&,RI3938!
!939!
W1 (g)
10210310410 5
Numbers
106
107
108
109
1010
1011
1012
W1 (g)
10110210310 410 5
Rmax (numbers/yr)
109
1010
1011
1012
1013
1014
1015
!