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 consumertoward 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).