Supplementary Text
This document is a supplementary methods section for “Coupled energy pathways and
the resilience of size-structured ecosystems.” by Julia L. Blanchard, Richard Law,
Matthew D. Castle & Simon Jennings
Dynamics of coupled size-structured communities
The analyses of food web resilience at steady state and responses to perturbations were
contingent on the development of a size-based model that allowed us to couple the
dynamics of the "pelagic predators" and "benthic detritivores". The underlying size-based
model was developed from the coupled dynamic size spectra model proposed by
Blanchard et al. (2009) but incorporated a dynamic reproductive process, which is a
consequence of the energy density and energy budgets of individual organisms within the
size spectra. All model equations and parameter values are shown in Tables S1 and S2.
Model structure
The coupling in the model consists of predation and production linkages between two
size-structured communities: “pelagic predators” and “benthic detritivores”. In both
communities, we are concerned with the continuous function N (m,t) (m-3 g-1) which
gives the density per unit mass per unit volume for organisms of mass m at time t. The
continuous processes of growth G and mortality D that arise from organisms
encountering and eating available and suitable food govern the temporal dynamics and
lead to a partial differential equation for each size spectrum i  {P=pelagic predators, B
=benthic detritivores} (Table S1).
In the pelagic community there is a background “primary producer” spectrum that
spans the size range 10-12 to 10-3 g which forms part of the food for “pelagic predators”.
For simplicity we do not model the dynamics of the primary producer size spectrum and
hold the numerical density as a constant function of body mass through time Npp=c·m-1.
Feeding
The feeding rate FPi (m,t) of a given size pelagic predator is a function of the
preference for prey i in spectrum i, the volume of water searched per unit time AP m P
(m-3 yr-1), and the amount of suitably sized food available in spectrum i,
  m m' N (m' , t ) m' dm' (Table S1) . The probability of a predator of size m eating an
i
encountered prey of size m' is given by the lognormal probability density function:
  ln m m'   2 
 1  2
2 2


 m, m'  exp 


(S.1)
when m > m' and  m, m' =0 otherwise, where  is the logarithm of the preferred
predator: prey mass ratio and  the width. The assumption of a linear predator
functional response is relaxed later.
Benthic consumers compete for an unstructured shared pool of food. For
simplicity, we call the latter group detritivores since in most benthic invertebrate
communities detritus forms the bulk of their food (but it could also be supplemented by
living phytoplankton). Detritus (BD) is produced by sinking phyto- and zooplankton,
faeces from the pelagic predator size spectrum, and other dead organic matter from both
communities. The feeding rate of a given size detritivore FB (m,t) depends on the volume
of water either searched or filtered per unit time AB m B (m-3 yr-1) and the available
biomass density of detritus BD (t ) (g m-3).
Assimilation, growth and reproduction
Of the food consumed only a fraction is assimilated. Assimilated food is allocated
to growth K and reproduction R after accounting for losses associated with meeting
metabolic requirements and activity (SM). We assume the fraction of each type of food
that is assimilated, 1-Ei, reflects food quality (Table S2). Ki is the net growth conversion
efficiency, the fraction of each type of assimilated food that is converted to production of
new somatic tissue with the surplus fraction for reproduction given by Ri = 1 – (SMi +
Ki). We assume that lower food quality is associated with lower assimilation efficiency
and higher metabolic costs (hence lower conversion efficiencies to growth and
reproduction). The combinations of conversion efficiencies for each food type meet the
requirements of a simple energy budget (Table S2). Multiplying feeding rates by
conversion efficiencies for each food type gives rise to the size-specific growth Gi(m,t)
and reproductive ri(m,t) rates in each size spectrum i  {P=pelagic predators, B =benthic
detritivores} shown in Table S1.
The reproductive investment per unit mass of an individual declines with body
mass in each community, consistent with the levels of reproductive investment reported
for marine teleost fishes and bivalves (Gunderson 1997, Van der Veer et al. 2006). The
total biomass allocated to reproduction per unit time by all individuals in the size
mi ,max
spectrum larger than the egg size
 r (m, t ) N (m, t ) dm
i
mi ,min  dm
i
divided by the mass of an egg
gives the rate at which new offspring enter each size spectrum. We assume that both
females and males invest the same energy to support the production of fertilized eggs.
The renewal term or rate of change in the numerical density at the smallest (egg)
size, mi ,m in , in each size spectrum is given by the rate that new individuals (eggs) enter
the spectrum minus the rates that individuals either grow larger or die (S.2). The renewal
function is similar to that given in Maury et al. (2007):
dN i (mi ,min , t )
dt
  Gi (mi ,min , t ) N i (mi ,min , t )
 Di (mi ,min , t )mi ,min N i (mi ,min , t )
1
 mi ,min 
(S.2)
mi , max
 r (m, t ) N (m, t ) dm
i
mi , min  dm
i
Mortality
The overall mortality rate Di in each size spectrum i  {P, B} arises from feeding by
predators (as described above) and other sources of mortality (disease, senescence). Other
mortality DiO includes an intrinsic term that decreases as a function of body size, and
senescent mortality that increases with body size at the size ms.
Detritus
The biomass density of detritus BD changes from the fluxes of biomass being added and
removed through time. Flux into the detritus pool, ID, is the total rate at which mass is
egested and dead mass that is generated by non-predation mortality; a term for dead
plankton is also included (Table S1). The proportion of detritus from the pelagic zone
reaching the benthic zone is S. Within the benthic community, detritus is derived from
dead biomass but not from egested material (which is assumed instead to become part of
the sediment). The flux out of the detritus pool OD, is the biomass density consumed per
unit time by all detritivores.
Numerical integration method
Prior to the stability analyses, numerical integrations were carried out to determine
whether the model communities would tend to a steady state. For mathematical
convenience and because size spectra usually span many orders of magnitude in mass, we
carried out our numerical analyses on a log10 mass scale (see Blanchard et al. 2009). The
numerical technique used to solve these equations was implicit time upwind finite
differencing (Press et al. 1992). All simulations were run for a period of 50 years (with a
daily time step and 0.1 log10 gram body mass intervals).
Initial conditions
Initial conditions for the predator and benthic detritivore size spectra comprised a low
density of individuals in small sizes (< 1 g). The constant intercept for the density of the
primary producer spectrum c was 0.006 m-3 g-1 and the initial detritus biomass density
was 0.6 g m-3. These values represent average annual levels of productivity for the North
Sea (Blanchard et al. 2009).
Ecosystem-level properties at steady state
1. Total biomass density for each size spectrum (i) is expressed as:
mmax
 N (m) m dm
with units g m-3.
i
(S.3)
mi min
The detritus pool, BD, is expressed as a biomass density and has units of g m-3.
2. Total throughput is the sum of all consumption rates (or input rate in the case of the
detritus pool) and is expressed as:
F
PP
(m) N P m dm   FPB (m) N P m dm   FB (m) N B m d m  I D
(S.4)
with units: g m-3 yr-1.
3. Production was assumed to be the total of all energy fluxes out (due to mortality) for
each size spectrum (Allen 1971):
 D (m) N (m)m dm
i
i
with units: g m-3 yr-1
(S.5)
and the production:biomass ratio was equal to the production divided by the total
biomass of each size spectrum at steady state :
 D (m) N (m)m dm
i
i
 N (m) m dm
i
with units: yr-1
(S.6)
In order to examine production:biomass ratio for each resource (primary producers and
detritus) and each consumer (pelagic predators and benthic detritivores) separately, the
pelagic size spectrum needed to be separated into primary producer and pelagic predator
size ranges. The above integral was therefore evaluated across the range [mmin, mP,min) for
primary producers and [mP,min, mmax] for pelagic predators.The equivalent measure of
production:biomass ratio for the non-living detritus pool is flux out of the detritus pool,
OD (g m-3 yr-1) divided by biomass density of detritus, BD (g m-3).
Predator functional response
The feeding rates described in Table S2 do not include satiation and individuals
simply feed in proportion to the suitably sized encountered prey (equivalent to a linear
Lotka-Volterra functional response). We also tested the effects of incorporating a
saturating Type II (hyperbolic) functional response on our results. We incorporated a
handling time that was constant across predator size where h  5.7 E  7 years per gram
(Ware 1978). The realised feeding rate of an individual predator at size m feeding on prey
of size m' in each size spectrum i  {P, B} then became:
i AP m
  m m' N (m' , t ) m' dm'
P
i
FPi (m, t ) 
P
1  h i AP m
  m m' N (m' , t ) m' dm'
(S.7)
i
Similarly, the realised feeding rates of an individual benthic detritivore was changed to:
FB (m, t ) 
AB m B BD (t )
1  h AB m B BD (t )
(S.8)
The predation mortality rate inflicted on an individual prey at size m in each size
spectrum i by all size m' predators became:
DiP (m, t ) 

i AP m'  m' m N P (m' , t )
P
P
1  h i AP m'
  m' m N (m, t ) m dm
dm'
(S.9)
i
The biomass flux per unit time out of the detritus pool from detritivore feeding became:
OD (t ) =
N
B
(m) FB (m, t ) dm
(S.10)
Using a Type II functional response did not affect our overall results. For local
stability analyses the max values were marginally closer to zero when a Type II
functional response was used, but there were no discernible differences in the patterns of
how coupling affected local stability, steady state size spectra or returns following large
perturbations. This is likely due to the fact that the levels of food availability in our
model simulations were lower that the levels at which feeding rates would saturate.
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