Coupling stage-structured food webs and biomass dynamics in the
allometric trophic network (ATN) framework
To simulate deterministic population dynamics of the species, we
employed a bioenergetic model in the allometric trophic network (ATN)
framework developed by Brose et al. (2006) and expanded by Bland et al.
(2019) to food webs with stage-structured fishes (see Brose et al. 2006
for a complete description). Consequently, we used many parameter values
and sub-models used in their work.
Body mass: In this framework, body mass plays integral
part in determining bioenergetic parameter values . More specifically,
the rates of metabolism and maximum consumption are approximated by
means of body-mass scaling relationships (Yodzis & Innes 1992). We
calculated relative masses of the taxa based on the short-weighted
trophic position \((T)\) in accordance with the theory of allometric
predator-prey body mass ratio (Brose et al. 2006). We set the body mass
ratio (\(Z\)) of fish predators and their prey to
102.6 and of invertebrate predators and their prey to
101.15 (; Table 1b). The function, body
mass,\(\ M=\ Z^{T-1}\), was used to define the body masses of
invertebrates and the terminal stages of stage-structured fishes. Hence,
the body masses were relative to those of autotrophs whose body masses
were defined to be equal to 1 (Brose et al. 2006; Bland et al. 2019). As
in Bland et al. (2019), to model the well-known pattern of fish growth
with time, we used a von Bertalanffy isometric growth curve to define
the body masses of lower stages (Table 2). We assumed that the
individuals of terminal stages reach 90% of their asymptotic weight
(Bland et al. 2019). Although body masses in lower stages no longer
strictly conformed to the allometric body mass ratios, the median ratios
from our model fell near the modes of the empirical distributions (Fig.
3 in Brose et al. 2006; Figure A2).
Dynamical model: The population dynamics within the food
webs were formulated as a multispecies consumer-resource model (Yodzis
& Innes 1992; Williams & Martinez 2004; Brose et al. 2006;
Bland et al. 2019). They were described by a set of ordinary
differential equations (ODE)
\begin{equation}
\frac{dB_{i}}{\text{dt}}=\overset{\text{logistic\ growth\ of\ autotrophs}}{\overbrace{g_{i}\left(1-\sum_{j\in autotrophs}\frac{B_{j}}{K}\right)B_{i}}}-\overset{\text{loss\ to\ grazing}}{\overbrace{\sum_{j\in consumers}{x_{j}y_{\text{ji}}B_{j}\frac{F_{\text{ji}}}{e_{\text{ji}}}}}}\nonumber \\
\end{equation}\begin{equation}
\frac{dB_{i}}{\text{dt}}\underset{\text{metabolic\ loss}}{}+\underset{\text{dietary\ intake}}{}-\underset{\text{loss\ to\ predation}}{}\nonumber \\
\end{equation}where \(g_{i}\) was the intrinsic growth rate of autotroph \(i\), \(K\)was the carrying capacity, \(x_{i}\) was the metabolic rate of consumer\(i\), \(y_{\text{ij}}\) was the maximum consumption rate relative to
metabolic rate, \(e_{\text{ij}}\) was the assimilation efficiency of
predator \(i\) eating prey \(j\), \(f_{m}\) was the fraction of
assimilated carbon lost for maintenance, and \(f_{a}\) was the fraction
of assimilated carbon that contributes to biomass growth (see Table 1b
for parameter values). The model deterministically simulated the biomass
dynamics during growing seasons. \(F_{\text{ij}}\) was the functional
response of consumer \(i\) when dealing with prey \(j\)
\begin{equation}
F_{\text{ij}}=\frac{\frac{\omega_{\text{ij}}}{\sum_{l\in resources}\omega_{\text{il}}}B_{j}^{q}}{B_{0_{\text{ij}}}^{q}+\sum_{k\in consumer}{\left(c_{\text{kj}}p_{\text{ik}}B_{k}B_{0_{\text{ij}}}^{h}\right)+\sum_{l\in resources}\left(\frac{\omega_{\text{ij}}}{\sum_{l\in resources}\omega_{\text{il}}}B_{l}^{q}\right)}}\nonumber \\
\end{equation}where \(\omega_{\text{ij}}\ \) was the preference of consumeri toward prey \(j\), \(B_{0_{\text{ij}}}\) was the half
saturation density for consumer \(i\) eating prey \(j\),\(c_{\text{kj}}\) was the predator interference competition coefficient
of \(k\) eating \(j\), and \(p_{\text{ik}}\) was the fraction of
resources of consumer \(i\) shared with consumer \(k\). The values
of\(\ B_{0_{\text{ij}}}\) and \(c_{\text{kj}}\) varied among taxa and
were taken from and Bland et al. (2019, their Figure 1) with
modifications (Table 1; also see ). The parameters for interspecific or
between-stage interference competition were set to zero (i.e.,\(c_{\text{kj}}=0\) for \(k\neq i\)) for simplicity (sensitivity to
these assumptions were checked in the sensitivity analysis). Previous
studies that used the ATN framework for aquatic systems (Brose et al.
2006, Boit et al. 2012, Bland et al. 2019) differentiated the
assimilation rates of consumers between non-basal and basal species
only. We added a rate for fish prey because fish is highly effective
food for fish growth (Table 1, and lowered the assimilation rate for
non-basal species (i.e., invertebrates) to have the average of the two
rates remain the same.
We added an ecologically plausible assumption that fishes preferred to
feed on fish over invertebrates and on invertebrates over autotrophs, if
they were included in their diets, to quickly grow beyond a size
vulnerable to predation and for higher fecundity. To achieve these
preferences in the absence of such empirical data, we set the parameter\(\omega_{\text{ij}}\ \)such that fishes whose diets included both
autotrophs and animals fed almost exclusively on fish, to a lesser
extent on invertebrates, but not much on autotrophs (Table 1b).
Similarly, we assumed that invertebrates preferred invertebrates the
most, followed by fish, to autotrophs. Growth of fish depends on the
quantity and quality of food they eat, and shifting to piscivory
invariably increases fish growth rate . As fish grow, piscivory could be
necessary to meet energetic demands (Juanes et al. 2002). Also, because
optimal morphologies for different diets (e.g., planktivory, benthivory,
piscivory) are quite different, tradeoffs often arise and a diet
specializing on the most profitable is likely preferred (Persson 2002).
Herbivory by fish occurs mostly in tropics and is much less common above
55° latitude because the enzyme to digest plant material is not active
at low temperatures . If we assumed no preference of fish for prey items
(consumption proportional to relative availability), the majority of
fish would consume high proportions of autotrophs due to their high
abundance, an unlikely scenario in temperate and northern systems. If
prey taxa went extinct (< 10-6), they were
removed from preference calculation.
The Hill exponent \(q\) of the functional response was set to 1.8,
higher than the value commonly used in previous ATN models (1.2–1.5),
to ensure sufficient dynamical stability in large food webs (see Fig. A5
for sensitivity analysis; Williams & Martinez 2004). The high value of
the exponent was desired especially because food preferences of
consumers increased energy flow higher up in the food web and reduced
stability of the food webs in the model
(Martinez et al. 2006). Higher values of \(q\) effectively
converted the functional response closer to Holling type III\(\left(q=2\right)\), which implicitly incorporates prey refugia,
other evasive behavior, or adaptive foraging .
Growth and reproduction: Growth and reproduction from
surplus energy (dietary intake – metabolic loss – loss to predation;
Eqn. 1) were accounted for at the end of the growing season when the ODE
model was paused, which implicitly assumed that fishes all reproduced at
the beginning of each growing season . The fraction of mature fish in
each stage was determined by using a logistic function (Table 2). We
assumed that 50% of individuals were mature halfway through to the
terminal stage. For example, if the taxon had five stages, about 50% of
individuals were mature in Stage 3. We further assumed that fish in
immature stages invested all their surplus energy in somatic growth,
while mature fish allocated surplus energy to both growth and
reproduction (Kuparinen et al. 2016). The allocation to reproduction
(\(I\)) linearly increased with stage, and the terminal stage allocated
20% of surplus energy to reproduction (Table 2). Therefore, the biomass
of the first stage class produced through reproduction was surplus
energy multiplied by the probability of being mature and reproductive
investment. We used the Leslie matrix to shift somatic biomass to the
stage above via growth and to convert it to new recruitment (Table 2).
The model allowed phenotypic variability within a stage such that some
individuals did not grow enough during the preceding growing season to
be recruited to the higher stage. We assumed that fish in the terminal
stage reproduced without having energy surplus in exchange for somatic
mass (Wootton 1998). Each column added up to 1 in this formulation;
therefore, there was no loss of biomass between consecutive growing
seasons (i.e., fish did not gain or lose mass or die during winter).