Ecology Letters Supporting Information Understanding patterns and processes in models of trophic cascades Michael R. Heath1, Douglas C. Speirs1, John H. Steele2   1University of Strathclyde, Departmentof Mathematics andStatistics, Livingstone Tower, Glasgow, G11XP, UK. E-mail: m.heath@strath.ac.uk; d.c.speirs@strath.ac.uk 2Marine Policy Centre, Woods Hole Oceanographic Institution, Woods Hole, MA02543, USA.E-mail: jsteele@whoi.edu Appendix S2. North Sea food web model 2.1 Model summary description (reproduced from Heath (2012)) The model simulated fluxes of a single nutrient element (nitrogen), between bulk mass state variables representing classes of detritus, dissolved nutrient, phytoplankton, benthos, zooplankton, fish, and top-predators. The state variables were further resolved to represent two depth layers in the water column, and a seabed sedimentlayer (Fig. S3). Rates of exchange between the mass compartments were described by a set of ordinary differential equations. Key features of the model were: • External sources of nitrogen were fromphytoplankton, detritus, nitrate and ammonia advected into the model domain by ocean currents, atmospheric deposition and river inputs of nitrate and ammonia. • Exports of nitrogen from the model (sinks) were phytoplankton, detritus, nitrate and ammonia advected out of the model domain, nitrogen gas produced by denitrification, and fishery landings. • Phytoplankton, detritus, nitrate and ammonia in the water columnwere subject to vertical exchange between depth layers (by sinking of particulate material and mixing). Detritus, ammonia and nitrate were subject to vertical exchange between the water column and sediment. • Uptake of dissolved ammonia and nitrate by phytoplankton was confined to the surface layer and modeled according to a Type–II function scaled by the depth averaged daily irradiance. Maximumuptake rate was temperature dependent according to a Q10function, but the half-saturation coefficient was temperature independent. • A constant proportion of phytoplankton per day was converted to detritus. • Uptake of prey by all classes of predators was described by Type-II functions with prey and temperature-dependent maximum uptake rates. The half-saturation coefficient was independent of both temperature and prey. • Fixed proportions of food ingested by predators were assimilated, excreted to ammonia, and defaecated to detritus. • Predators excreted a temperature-dependentproportion of their body mass per day to ammonia, according to a Q10function. • Predators were subjected to a density-dependent mortality rate, which created a flux to corpses.   1• A proportion of corpse mass was converted to detritus per day, and similarly detritus to ammonia, ammonia to nitrate, and nitrate to nitrogen gas (denitrification). The proportions were temperature dependent according to a Q10function. • Fish were resolved into two demographic stages – larvae (including eggs), and adults. Adults shed a fixed proportion of their mass per day to larvae during prescribed time intervals each year. A fixed proportion of larvae were promoted to adults per day during a different prescribed interval. • Adult fish and benthos categories were subject to harvesting which removed a proportion of their mass per day as catch. A fraction of the catch was returned to the food web as fishery discards, the remainder was regarded as landings which were a sink. • Benthos categories additionally suffered a by-catch mortality which was a fixed fraction of the demersal fish harvesting rate. This by-catch was passed directly to fishery discards. • The proportion of catch discarded was constant for pelagic fish and benthos, but scaled with adult abundance for demersal fish, to caricature the shift in fish size distribution towards smaller individuals with declining abundance in demersal fish communities. A full technical description of the model is given in Heath (2012). The model was implemented in the R statistical environment version 2.11.1 (R Development Core Team 2005), and used the lsoda routine in the package odesolve to solve the differential equations and output values of the state variables and fluxes at daily time intervals. 2.2 Default model run Model parameters were optimized by simulated annealing toidentifythe set providing the best fit ofthe stationary state model to a suite of observations on biomasses and fluxes in North Sea averaged over the period 1970-1999, whilst being driven by a repeating annual cycle of 1970-1999 monthly averaged driving variables. We refer to this as the ‘default model run’, and averaged the state variable abundances over a stationary annual cycle. 2.3 Scenario model runs We defined 4 scenario runs of the model based on halving or doubling of various external driving time series: a) halving or doubling of the default nitrate and ammonia concentrations in inflowing river waters; b). halving or doubling of the default harvesting rate applied to demersal fish. For each model state variable X (averaged over a stationary annual cycle), the difference between default and scenario states was expressedas ΔX = log2(Xscenario/Xdefault). Hence ΔX = 0 corresponds to no difference between scenario and default, ΔX = 1 to a doubling of abundance in the scenario run, and ΔX = -1 to a halving. 2.4 Comparison of scenario and default model results The results showed several key features: 1) Halving and doubling of demersal harvesting rates led to inverse responses throughout the water column food web (Fig.S4). However, this was not universally the case in the recycling food web, with water column ammonia, corpses, and carnivorous/scavenging benthos all showing a positive response to both halving and doubling of demersal harvesting rate. This occurred because corpses, which constituted a significant portion of the diet of carnivorous/scavenging benthos, were produced by a combination of fishery discards, and quadratic density dependent mortality terms applied to upper trophic components of the food web. Corpse   2production showed a U-shaped response to demersal harvesting rates (Heath 2012). Carnivorous/scavenging benthos abundance was close to a local minimum with respect to fish harvesting rates in the default model run. 2) The conceptual top-down alternating responses were evident along some trophic pathways, e.g. demersal fish, pelagic fish, omnivorous zooplankton, phytoplankton, nitrate. However, carnivorous zooplankton were anomalous in this respect, always responding in the same direction as their main prey (omnivorous zooplankton). In the model, carnivorous zooplankton were both predators on fish larvae (pelagic and demersal), and prey of adult fish. Hence the connectivity of carnivorous zooplankton in the food web was complex. 3) Changes in demersal fish landings responded negatively to both a doubling and halving of the harvesting rate. This was because landing under the default harvesting rate were close to the maximumsustainable yield, so any changein harvesting rate was guaranteed to produce a decrease in landings. Conversely, pelagic landings responded positively and negativelyto a doubling and halving of demersal harvesting rate respectively due to connectivity between demersal and pelagic fish in the food web. 4) In marked contrast to the patterns above, doubling of river nutrient concentrations produced an increase in stationary annual average abundances throughout the entire web (Fig. S5). Conversely, halving river nutrients produced a decrease throughout the web. References Heath, M.R. (2012). Ecosystemlimits to food web fluxes and fisheries yields in the North Sea simulated with an end-to-end food web model. Prog. Oceanogr., 102, 42-66.   3Figure S3Schematic showing the North Sea food web model components and driving variables, inrelation to physical structures (surface water layer, deep water layer, and sediments). Rivers and atmosphere: Nitrate & ammonia FisherylandingsNitrogengas (denitrification) Ammonia Nitrate Phytoplankton Suspended detritus Suspended detritus Ammonia Nitrate Phytoplankton Sediment Ammonia Surface Deep OCEAN INPUTS Ammonia Nitrate Suspended detritus Phytoplankton PHYSICAL DRIVERS Irradiance Turbidity Temperature Vertical mixing Transport fluxes River discharges Zooplankton, fishand top predators Birds & mammals Pelagic fish larvae Pelagic fish adults Demersal fishlarvae Demersal fish adults Carnivorous zooplankton Omnivorous zooplankton Fishery discards Nitrate Detritus Corpses Benthos Filter & deposit feeding fauna Carnivorous & scavenge feeding fauna FISHERY DRIVERS Harvest rates of pelagic & demersal fish and benthos. By-catch rates and discard fractions   4Figure S4Changes in stationary annual average abundances of ecosystem components in the North Sea food web model parameterized against observations collected during 1970-1999 (Heath, 2012), as a result of doubling (left) and halving (right) the demersal (benthivorous and piscivorous) fish harvesting rate. Numerical values of the change are indicated by a log2 scale, so that +1 represents a doubling of abundance relative to the 1970-1999 model, and -1 indicates a halving of abundance. Boxes with dashed outlines represent fishery landings and hence export fluxes out of the system. Black arrows indicate uptake fluxes, green arrows indicate fishery harvests. Fluxes due to excretion and death are not shown for clarity. 0 +1 +2 -1 -2 -3 +3 Doubled demersal harvest rate Pelagicfish +0.53 Demersal fish -2.44 Carn. zoop. -0.65 Phytoplankton +0.04 WC. detritus +0.010 WCnitrate -0.003 WC ammonia +0.003 Sed. nitrate -0.002 Bird /mammal +1.28 Corpses +0.021 Discards +0.12 Demersal landings -3.33 Pelagic landings +0.41 Susp/dep. benthos +0.05 N2 Sed. ammonia -0.011 Sed.detritus -0.024 Omniv. zoop. -0.05 Carn/scav. benthos +0.08 Shellfishlandings +0.05 0 +1 +2 -1 -2 -3 +3 Halveddemersal harvest rate Pelagic fish -1.07 Demersal fish +0.63 Carn. zoop. +0.53 Phytoplankton -0.06 WC. detritus -0.027 WCnitrate +0.006 WC ammonia +0.006 Sed. nitrate +0.004 Bird/mammal -2.81 Corpses +0.40 Discards -1.09 Demersal landings -0.07 Pelagic landings -1.05 Susp/dep.benthos -0.05 N2 Sed. ammonia +0.026 Sed. detritus +0.042 Omniv. zoop. +0.07 Carn/scav. benthos +0.09 Shellfishlandings -0.041   5Figure S5As Fig. S4 but showing the response to doubling and halving the concentrations of nitrate and ammonia in rivers flowing into the North Sea. 0 +1 +2 -1 -2 -3 +3 Double river nutrient concentrations Pelagic fish +0.101 Demersal fish +0.143 Carn. zoop. +0.052 Phytoplankton +0.059 WC. detritus +0.035 WCnitrate +0.081 WC ammonia +0.097 Sed. nitrate +0.080 Bird /mammal +0.654 Corpses +0.145 Discards +0.037 Demersal landings +0.222 Pelagic landings +0.091 Susp/dep. benthos +0.134 N2 Sed.ammonia +0.045 Sed.detritus +0.011 Omniv.zoop. +0.064 Carn/scav. benthos +0.056 Shellfish landings +0.131 0 +1 +2 -1 -2 -3 +3 Halved river nutrient concentrations Pelagic fish -0.064 Demersal fish -0.075 Carn. zoop. -0.025 Phytoplankton -0.054 WC. detritus -0.019 WCnitrate -0.042 WC ammonia -0.052 Sed. nitrate -0.042 Bird /mammal -0.413 Corpses -0.081 Discards -0.026 Demersal landings -0.119 Pelagic landings -0.058 Susp/dep. benthos -0.072 N2 Sed.ammonia -0.022 Sed.detritus -0.005 Omniv.zoop. -0.033 Carn/scav. benthos -0.031 Shellfish landings -0.071   6