1
Supporting information SI. Description of the model
The model framework is based on two central assumptions and a number of lesser
standard assumptions. The first central assumption is that an individual can be
characterised by its mass 𝑚 and asymptotic mass 𝑀 only. The aim of the model is to
calculate the size- and trait-spectrum 𝑁(𝑚, 𝑀) which is the density of individuals
such that 𝑁(𝑚, 𝑀)d𝑚 d𝑀 is the number of individuals in the interval [𝑚: 𝑚 + d𝑚]
and [𝑀: 𝑀 + d𝑀]. In practice the trait-direction is discretised in a number of size
spectra which each represent individuals with a range of trait values 𝑁𝑖 (𝑚, 𝑀𝑖 ). The
dimension of the size spectrum is numbers per mass per volume. Scaling from
individual-level processes of growth and mortality to the size spectrum of each trait
group is achieved by means of the M’Kendrick-von Foerster equation which is simply
a conservation equation [equation (A1)] where individual growth 𝑔(𝑚, 𝑀) and
mortality 𝜇(𝑚, 𝑀) are both determined by the availability of food and predation from
the community size spectrum, which is the sum of all the size spectra [equation (A2)].
The conservation equation is supplemented by a boundary condition at the right
boundary at mass 𝑚0 where the flux of individuals (numbers per time) 𝑔(𝑚0 )𝑁𝑖 (𝑚0 )
is determined by the recruitment of offspring by mature individuals in the population
𝑅BH,𝑖 [equation (A9)].
The second central assumption is that the preference of food is only
determined by individual mass, not by the trait-value or species identity of prey. The
preference for prey mass is described by the log-normal selection model (Ursin, 1973)
which describes prey preference in terms of the ratio between the mass of predators
and prey of mass 𝑚 and 𝑚𝑝 respectively [equation (A4)] where 𝛽 is the preferred
predator-prey mass ratio and 𝜎 the width of the mass selection function. The rest of
the formulation of the model rests on a number of “standard” assumptions from
ecology and fisheries science about how encounters between predators and prey leads
to growth 𝑔(𝑚, 𝑀) and recruitment 𝑅𝑖 of the predators, and mortality of the prey
𝜇(𝑚). The available food (mass per volume) for a predator of mass 𝑚 is determined
by integrating over the community size spectrum weighted by the size selection
function [equation (A4)]: ∫ ∑𝑁𝑖 (𝑚) 𝜙(𝑚𝑝 𝑚−1 )𝑚 d𝑚𝑝 . The food actually
encountered 𝐸𝑒 (mass per time) depends on the search rate (volume per time) which is
assumed to scale with individual mass as 𝛾𝑚𝑞 [equation (A5)]. Note that encounters
between predators and prey are only determined by the relative individual masses, not
by the trait 𝑀. This means that a 100 g cod will consume the same food as a herring
of 100 g. The encountered food is consumed subject to a standard Holling type II
functional response to represent satiation. This determines the feeding level 𝑓(𝑚),
which is a dimensionless number between 0 (no food) and 1 (fully satiated) [equation
(A6)], where ℎ𝑚𝑛 is the maximum consumption rate. The consumed food 𝑓(𝑚)ℎ𝑚𝑛
is assimilated with an efficiency 𝛼 and used to fuel the requirements of standard
metabolism and activity 𝑘𝑠 𝑚 𝑚𝑝 . The remaining available energy, 𝛼𝑓(𝑚)ℎ𝑚𝑛 −
𝑘𝑠 𝑚𝑝 , is divided between growth and reproduction by a function of mass changing
between zero around the mass of maturation to one at the asymptotic mass where all
available energy is used for reproduction [equation (A7)], leading to an equation for
growth [equation (A8)]. The form of the allocation function is chosen such that the
growth curve approximates a von Bertalanffy growth curve if the feeding level is
constant (see Hartvig et al., 2011 for details of the derivation). The actual emerging
growth curves from the model depend on the amount of food available.
2
The recruitment of new fish is determined from food-dependent egg
production and density-dependent regulation. The total production of eggs 𝑅𝑝
(numbers per time) is found by integrating the energy allocated to reproduction over
all individuals [equation (A8)]. Density dependence of recruitment is modelled as
compensation on the egg production such that the recruitment flux 𝑅BH,𝑖 (numbers per
time) is lowered towards a maximum recruitment as egg production increases.
Compensation is modelled as a “Beverton-Holt” type stock-recruitment function
[equation (A9)] where the maximum recruitment flux 𝑅max,𝑖 is determined from
equilibrium size spectrum theory (Andersen & Beyer, 2006) [equations (A10) and
(A11)].
The mortality rate of an individual 𝜇(𝑚) has two sources: predation mortality
𝜇𝑝 (𝑚) and a constant background mortality 𝜇𝑏.𝑖 (𝑀). Predation mortality is calculated
such that all that is eaten translates into corresponding predation mortalities on the
ingested prey individuals (equation A12; see Hartvig et al., 2011, App. A for
derivation). When food supply does not cover metabolic requirements 𝑘𝑠 𝑚𝑝 , growth
stops, i.e. no negative growth occurs, and the individual is subjected to starvation
mortality. Starvation mortality is assumed proportional to the energy deficiency
𝑘𝑠 𝑚𝑝 − 𝛼𝑓(𝑚)ℎ 𝑚𝑛 , and inversely proportional to lipid reserves, which are assumed
proportional to body mass. Starvation is not an important process in the simulations
presented here. Mortality from sources other than predation is assumed constant
within a species and inversely proportional to generation time (Peters, 1983) [equation
(A13)]. The background mortality is needed to remove the largest individuals which
do not experience predation mortality. Fishing mortality 𝐹𝑖 is included in the total
mortality rate applied to an individual: 𝜇(𝑚) = 𝜇𝑝 (𝑚) + 𝜇𝑏.𝑖 (𝑀) + 𝐹𝑖 (𝑚, 𝑀).
Food items for the smallest individuals (smaller than 𝛽𝑚𝑜 ) are represented by
a resource spectrum 𝑁𝑅 (𝑚). The temporal evolution of each size group in the
resource spectrum is described using semi-chemostatic growth [equation (A14)]
where 𝑟0 𝑚𝑝−1 is the population regeneration rate (Fenchel, 1974; Savage et al., 2004)
and 𝜅𝑅 𝑚−𝜆 = 𝜅𝑅 𝑚−2−𝑞+𝑛 the carrying capacity.
3
Table S1: Model equations
Scaling:
𝜕𝑁𝑖 𝜕𝑔𝑁𝑖
+
= −𝜇𝑖 𝑁𝑖
𝜕𝑡
𝜕𝑚
𝑁𝑐 (𝑚) = 𝑁𝑅 (𝑚) + ∑ 𝑁𝑖 (𝑚)
(A1)
(A2)
𝑖
Food encounter and consumption:
2
𝜙(𝑚𝑝 𝑚−1) = exp[−(ln(𝑚 𝑚𝑝 −1 𝛽 −1 )) (2𝜎 2 )−1 ]
𝐸𝑒 (𝑚) = 𝛾𝑚𝑞 ∫ 𝑁𝑐 (𝑚)𝜙(𝑚𝑝 𝑚−1 )𝑚 d𝑚𝑝
𝑓(𝑚) = 𝐸𝑒 (𝐸𝑒 + ℎ𝑚𝑛 )−1
Growth:
𝜓(𝑚 𝑀−1 ) = [1 + (𝑚 𝜂 −1 𝑀−1 )−10 ]−1 (𝑚 𝑀−1 )1−𝑛
𝑔(𝑚, 𝑀) = (𝛼𝑓(𝑚)ℎ𝑚𝑛 − 𝑘𝑠 𝑚𝑝 )(1 − 𝜓(𝑚 𝑀−1 ))
Reproduction:
𝑅𝑝,𝑖 = 0.5 𝜖 𝑚0 −1 ∫ 𝑁𝑖 (𝑚)(𝛼𝑓(𝑚)ℎ𝑚𝑛 − 𝑘𝑠 𝑚𝑝 )𝜓(𝑚 𝑀−1 ) d𝑚
−1 −1
𝑅BH,𝑖 = 𝑅𝑝,𝑖 (1 + 𝑅𝑝,𝑖 (𝜅𝑅max,𝑖 ) )
𝑅max,𝑖 = (𝛼𝑓0 ℎ𝑚0𝑛 − 𝑘𝑠 𝑚0𝑝 )𝑀𝑖2𝑛−𝑞−3+𝑎 𝛥𝑀𝑖
𝑎 = 𝑓0 ℎ𝛽 2𝑛−𝑞−1 exp[0.5(2𝑛(𝑞 − 1) − 𝑞 2 + 1)𝜎 2 ] (𝛼𝑓0 ℎ − 𝑘𝑠 )−1
Mortality:
𝜇𝑝 (𝑚𝑝 ) = ∫ 𝜙(𝑚 𝑚𝑝 −1 ) (1 − 𝑓(𝑚))𝛾𝑚𝑞 𝑁𝑐 (𝑚)d𝑚
𝜇𝑏.𝑖 = 𝜇0 𝑀𝑛−1
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
(A9)
(A10)
(A11)
(A12)
(A13)
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Table S2. Parameters and their units
Resource spectrum
𝜅𝑅
5
g 𝜆−1 m−3 Magnitude of the resource spectrum
× 10−3
Exponent of resource spectrum (= 2 − 𝑛 − 𝑞)
𝜆
2.05
1−𝑝
−1
4
𝑟0
g
year Constant for regeneration rate of resource
1
g
Upper mass limit of resource spectrum
𝑤𝑐𝑢𝑡
Individual growth
0.6
Initial feeding level
𝑓0
0.6
Assimilation efficiency
𝛼
1−𝑛
−1 Constant for max. consumption
20
ℎ
g
year
n
0.75
Exponent for max. consumption
1−𝑝
−1 Constant for std. metabolism and activity
2.4
𝑘𝑠
g
year
0.75
Exponent of standard metabolism
𝑝
Food encounter
100
Preferred predator-prey mass ratio
𝛽
1.3
Width of size selection function
𝜎
−𝑞
−1
1459.5
Factor for volumetric search ratea
𝛾
g year
0.8
Exponent for volumetric search rate
𝑞
Mortality
0.1
Fraction of body mass containing reserves
𝜉
1−𝑛
−1
𝜇0 2
g
year Constant for background mortality
Reproduction and recruitment
mg
Offspring mass
𝑚0 1
0.25
Mass at maturation divided by 𝑀
𝜂
0.1
Efficiency of offspring production
𝜖
50
Constant for maximum recruitment
𝜅
a
Calculated from the other parameters (Hartvig et al. 2011; eq. D1).
5
Supporting information. References
Fenchel, T. (1974). Intrinsic rate of natural increase: the relationship with body size.
Oecologica 14, 317–326. doi:10.1007/BF00384576
Savage. V. M., Gillooly, J. F., Brown, J. H., West, G. B., Charnov, E. L. (2004).
Effects of body size and temperature on population growth. The American
Naturalist 163, 429–441. doi:10.1086/381872