Appendix 6. Equations used in algal primary production and biomass submodel.

To represent the complex space-time pattern of algal biomass and production that is likely to result from diurnal production/dessication/turbidity interactions, we predict monthly average algal biomass per area, B(r,d), for a set of 2-foot depth slices (contours), d (d = 1 represents the bottom area from the deepest point to 2 foot depth across the channel, etc.) in each reach, r. Mean production per area Pr is then assumed to be proportional to the area-weighted mean of these biomass densities: Pr = kΣdB(r,d)W(r,d)/ ΣdW(r,d), where W(r,d) is the cross-sectional width of depth slice d for a typical cross-section in reach r, and k is a production/biomass scaling parameter. On monthly time scales, we assume that average biomass will satisfy the balance relation production = mortality, but we assume that (1) mortality is inversely related to the proportion of time H(r,d) that site r,d is flooded (H(r,d) is the hours per day that site r,d is flooded/24), and (2) production can be limited by biomass at low biomasses, according to a saturating relationship of the form

(A6.1)

where Pmax(r,d) is the maximum net production per unit biomass per day at site r,d; PA is a maximum primary production per unit area in sites where B(r,d) is high enough so that production rate is not limited by algal biomass. Taking the rate of biomass change to be

(A6.2)

Where M is a base mortality rate per biomass per day for sites that are permanently flooded, we can predict mean B when dB/dt = 0 by setting P(r,d) = M/H(r,d):

(A6.3)

Note here that the M/H mortality assumption causes mortality to double for each halving of H, e.g., mortality is 2M if dessication occurs for 12 h/d, 4H if it occurs for 18 h/d, etc. For low H or P, Eq. A6.3 predicts negative average biomass (daily mortality exceeds daily production for all B); in this case, we set B(r,d) = 0.

To apply the algal biomass prediction model (Eq. A6.3), we need to estimate proportions of time flooded H(r,d) and maximum daily production per biomass Pmax(r,d). The hydrology submodel provides hourly stages S(r,h) for each reach r and hour h of the day (h = 1…24), using results from the Wiele-Smith one-dimensional wave propagation model, and the sediment transport submodel provides mean 24-h turbidities by reach, from which we can calculate light extinction coefficients λr by reach. For each depth slice d, we estimate H(r,d) simply by summing the hours for which S(r,h)>d. Daily maximum primary production rate Pmax is taken to be proportional to the sum over daylight hours of depth-corrected hourly rates, including in the sum only those hours for which S(r,h)>d:

(A6.4)
Here, K represents maximum daily P/B for shallow water when the photoperiod is 12 h and the hour-summing indices "dawn" and "dusk" are adjusted seasonally for changes in photoperiod. Note that we need to evaluate Eq. A6.4 for a potentially large number of reaches r and depth slices d, at least once per month during simulations. To avoid unnecessary and potentially massive computational costs, Eq. A6.4 is evaluated for each r each month only for those depth slices d such that the light extinction effect exp(-X) > 0.01; for very turbid downstream reaches, this means that the Eq. A6.4 sum only needs to be done each month for a few depth slices d. In Eq. A6.4, the light extinction coefficient lr is assumed to be proportional to total turbidity Tr, as λr = λ1 +λ2Tr..

Note that the basic form of the biomass prediction relationship Eq. A6.3 is robust to alternative assumptions about how biomass is limited. For example, suppose we start with the assumption that P/B depends only on environmental factors (Eq. A6.4), such that there is no production/area limit, but that mortality rate increases with biomass. Then we would use the equilibrium of the rate equation dB/dt = PmaxB-MB2/H to provide an approximation to the mortality rate effect. At equilibrium for this model, PmaxB = MB2/H, implying that B = PmaxH/M, which is similar to Eq. A6.3 (B is proportional to PmaxH and inversely proportional to M), except that it does not predict a threshhold minimum for P,H below which B would be zero.

As with other trophic components of the model, a key issue is whether primary production exerts bottom-up control on production of herbivores, or whether, instead, it can be controlled through top-down grazing effects. We assume that persistent high biomass (mats) of forms like Cladophora implies bottom-up control, i.e., the benthic algal mat must not be subject to very high grazing rates (perhaps because of chemical defenses of the algae, or because grazing insects are prevented from feeding on open surfaces by high predation risk from fish). Extensive direct grazing would be obvious, in the form of patchiness in algal biomass at the scale of a few centimeters ("worm trails" and other grazing impact signals). Without such signals, we assume that algal production moves through the food chain mainly via (1) consumption of epiphytic diatoms growing on the macroalgae, and (2) detritus production and consumption of this detritus by insects that spend most of their time in protected microhabitats (e.g., web spinners under rocks). Detritus production per unit area is assumed to be proportional to primary production, and "instantaneous" downstream profiles of detritus concentration are predicted each week, using an exponential profiling procedure based on Lagrange tracking of water masses downstream while accounting for additions from the bottom (and side terrestrial sources), decomposition, and grazing removal rates.