Electronic Supplementary Material: Thermodynamic Analysis of Food Web Models

In this Electronic Supplementary Material, we provide the mathematical details of the model simulations that are presented in the main article. We detail the model construction, underlying assumptions, and constitutive expressions for flows for the three food web models that are explored. In addition, we provide the analytical solutions for the steady state, and the results of the linear stability analysis of these steady state solutions. 1 Model parametrization: physical parameters As shown in Meysman & Bruers (2007), the transport coefficients αR and αW can be directly calculated from the dominant physical transport processes that are operating in the sediment, i.e., burial of solids and pore water diffusion. αR = ω L (1) αW = 3D L2 (2) In these expressions, ω denotes the burial velocity, D is the pore water diffusion coefficient of CO2, and L denotes the thickness of the sediment layer that is modeled. Typical values for a deep sea sediment are ω ≤ 0.1 cm yr−1 (Middelburg et al. 1997), L ≈ 10 cm (Boudreau 1998), and D = 330 cm2 yr−1 (Boudreau 1997). This provides αR ≤ 0.01 yr−1 and αW = 10 yr−1. In other words, the exchange rate coefficient of resource and waste differs at least by two orders of magnitude, and so, the slow exchange of resource will be rate limiting. Consequently, a simplification that can use is that the exchange rate between the internal and external waste reservoirs is very large (αW → ∞). Under this condition, we can approximate CW ∼ C0 W . This assumption greatly simplifies the resulting expressions, but won’t affect our conclusions. We also need to provide suitable ranges for the concentrations C0 R and C 0 W that are imposed as external boundary conditions on the ecosystem model. The total carbonate concentration is relatively constant across the ocean, and so we adopt the typical value C0 W = 2.5 mol C m−3 as a fixed parameter (Sarmiento & Gruber 2007). In contrast, the flux of organic carbon FOM that reaches the ocean floor varies significantly between locations, and ranges in the ocean between 0 and 200 mol C m−2 yr−1 (Middelburg et al. 1997; Andersson et al. 2003). Using the relation FOM = αRC R, this provides typical range for C 0 R from 0 up to 20 kmol C m−3. This is also the range we will use in the simulations. For reference, a typical value for the organic matter concentration in deep-sea sediments is about 1 kmol C m−3 or 0.6% dry weight (Middelburg et al 1997; Seiter et al. 2004).