Benthic insect biomass dynamics are represented by monthly difference equations at the reach spatial scale; no attempt is made to simulate biomasses by depth slices within reaches, because of the likelihood that movement processes (crawling, drifting, etc.) among slices are fast enough to prevent substantial slice-scale variation from developing over time. Also, benthos sampling data do not indicate clear trends in biomass with depth, except for substantial biomass reduction in diurnally dried varial zones (we assume zero production in these zones) and lower average biomass in deeper sites because more of these sites have unstable sand substrates.

For each invertebrate type s (grazer, detritovore) in each reach r, we update monthly (m) biomass density (g/m² ash-free dry mass) B_sr(m) using the difference equation

where g_sr(F_sr(m)) is a biomass growth function dependent on food density F_sr(m); and m_sr(m) is a mortality rate dependent on temperature, predator abundances, diurnal dessication, and (in extreme situations) turbidity. Also, between months of rising mean stage, B is "diluted" by assuming that animals rapidly spread over the wider river width available Between months of falling stage, animals along the drying river margin are assumed to die (rather than contribute to increases in biomass density B by moving to remaining wetted areas).

The biomass growth function g_sr(F_sr(m)) was derived by thinking initially about filter feeders (dominant in most river reaches), and assuming that animals compete locally within and upon the substrate for limited delivery rates of detritus to that substrate. We assume that similar local "delivery" rate mechanisms apply to production of the epiphytic algae (diatoms) that appear to be the main food supply of grazers (macroalgae like Cladophora and Oscillatoria appear not to be grazed directly, or at least not enough to cause noticeable effects such as hedging or patchiness). Within the limited "grazing arena" at the substrate, we suppose that g has the product form efc, where c is the effective food concentration in the arena; f is the filtering/scraping rate (volume or area/time); and e is food conversion efficiency. The problem then is to predict c as a function of overall food concentration F_sr(m) in the environment and of exchange/production/loss processes between c and F. We assume that mixing processes result in dc/dt = MF - rMF - fB/V, i.e., to have input rate proportional to F with instantaneous rate M, nongrazing loss rate proportional to F with rate rM (for detritus, r represents the "resuspension" rate relative to the "sinking" rate M), and feeding loss rate fB/V, where f has units of volume per biomass per time, and V is the effective volume of the grazing arena. We then assume that c reaches equilibrium quickly with respect to B and F (on time scales of hours to days), so that we can solve for c by setting dc/dt = 0. Substituting this variable speed-splitting solution for c into the rate product efc finally results in the overall relationship:

Here, the hyperbolic decreasing term Fsr(m))/[rs + fBsr(m)/V] represents the fast variable solution for c (density of food locally available to animals immediately at the substrate where they are competing). Note that in the case of s = grazers, we think of F_sr(m) as proportional to macroalgae biomass density, so that it represents algal concentration on macrophyte growth sites not accessible to grazing due to predation risk or other factors.

To simplify the estimation of parameters in (A9.2), we directly specify only e, r, M, baseline equilibrium values F₀, B₀, and mortality rate m, and maximum biomass growth rate r₀ at low densities (this growth rate is r₀ = efF₀/r - m). Substituting these "knowns" into (A9.1) and (A9.2), with B(t +1) = B(t) results in estimates of effective f and V. In fact, this method allows us to make arbitrary choices for M as well (only product MV appears at equilibrium, so changing M just rescales estimate of effective arena volume V), and to think of the "resuspension" fraction r as a way of representing concentrating mechanisms such as quiet backwaters that support some insect production even where overall water column concentrations of detritus F (or bottom concentrations of algae) would be too low to support any production.

Why assume the relatively complex competition function (A9.2) rather than assuming growth rates simply proportional to overall detritus or algae concentrations in or below the water column? First, all of the available data suggest that food supply is indeed important to benthic insect abundance (strong downriver gradients, etc.). However, if we simply assume that growth rate is proportional to overall food density F_sr(m), we end up predicting that g should be either large enough to always predict B(m + 1)>B(m), or always low enough so that B(m + 1)<B(m), i.e., we end up predicting either unlimited population growth or decline. In simple food limitation models, increase in B results in decrease in F and. hence, population-limiting feedback. But especially for F = detritus in a large, fast-flowing river, the relatively low benthic biomass simply cannot affect overall food concentrations significantly, even over substantial downstream distances. The "grazing arena" density effect (reduced food density in the immediate environment where animals feed) is actually a very parsimonious way to introduce food-mediated density dependence (and, hence, population regulation) into the model.

Mortality rate m_sr(m) is assumed to consist of two components, m₀ + m₁. We assume that the baseline "natural" rate m₀ would occur in the absence of fish and bird predation, due to various physiological problems and to maturation/emergence. The predation rate m₁ is assumed to vary on interannual time scales with abundances of fish, and is parameterized by assigning each vertebrate predator species a baseline proportional contribution to m (i.e., a component of m₁) when the vertebrate species is at baseline (1993) abundance, such that m₀ consists of overall m times whatever proportional contribution is not accounted for by modeled predators. Then m₁ is varied over time in proportion to ratios of predator abundance to baseline predator abundance. This approach allows us to easily examine various hypotheses about top-down vs. bottom-up control of predation effects (to effect bottom-up control, we set the relative predator contributions to total m at low values to model the idea that predation rate has little additive effect on m; to model top-down control, we set the proportions so that most of m is directly accounted for by predators).