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2.2. Model description
This protocol is extracted from research article:
Determinism of nonadditive litter mixture effect on decomposition: Role of the moisture content of litters
Ecol Evol, Jun 21, 2021;

Procedure

The time step of all the discrete equations is the day. The relationship between the different equations is illustrated in Figure 4. Equation 1 was used to model the decomposition of OM with time (discretization of the negative exponential decay model), with parameter k as the decomposition rate.

Mt +1 represents the remaining mass of litter at time “t + 1 day,” Mt is the mass of litter at time “t” and k is the decomposition rate. Concretely, k is the fraction of litter mass present at time t (Mt ) that was lost between time t and t + 1 day.

Schematic diagram showing the model relating litter water content to evaporation rate, the reaction rate to the litter water content, and the remaining mass to the change of reaction rate with time, with LWC: litter water content, k: decomposition rate, M: remaining mass, c, d, e: empirical parameters, and k max: maximum decomposition rate

To account for the effect of the properties of litter related to water content (maximum water content, varying water content with time) on the decomposition rate, the relationship between k and litter water content (LWC) as well as the evaporation rate was modeled. In a first step, two models of OM decomposition, which incorporate sensitivity to water content in different ways, were compared. Both models used two Michaelis–Menten equations to account for the simultaneous dependence of the biological processes on water and oxygen. The more water available, the less oxygen, and vice versa. However, the two models differ by their representation of the relationship to oxygen. The first one was derived from Moyano et al. (2013, Equation 2), which models soil OM and not specifically litter. This model assumes no biological activity at water saturation and uses oxygen as a substrate. Conversely, the Bunnell et al. (1977) model allows biological activity at water saturation. It derives from a Michaelis–Menten equation and LWC a coefficient, which describes how the gas exchange is facilitated. This coefficient is maximal at low LWC, but is not necessarily equal to zero at saturation.

with Oa represented as follows (Equation 4):

LWCmax is the maximum water content for a specific litter (water saturation) and a, b, c, and d are the Michaelis–Menten constants (the LWC when the reaction rates are equal to half of the k max). Oa is a coefficient calculated to allow the decomposition rate to be equal to zero at maximum LWC (when LWC = LWCmax, Oa  = 0 and thus k = 0). Finally, to represent LWC with time, we used a discrete negative exponential decay model, with parameter e as the rate at which water is lost (the fraction of water present at time t − 1 lost between time t − 1 and t, Equation 5).

At the start of each numerical experiment, the LWC of each litter was set at its maximum (LWCmax). At the beginning of each time step, LWC was allowed to change depending on the evaporation rate e (Equation5; Figure 4a), which was allowed to differ between species (Figure 1). The calculated LWC was then injected into Equation 3 (Figure 4b) and 4 to calculate k Bunnell and Oa , respectively, and then Oa was used to calculate k Moyano (Equation2). The a, b, c, and d parameters of the Equations 3 and 4 differed, or not, between the two litters. Each calculated k was finally injected into Equation 1 to obtain the remaining mass at the end of each time step for each litter and for each model type (Figure 4c). The parameters used in the “Model Comparison” and in “Rate Combinations” numerical experiments are listed in Table 1.

Values of the parameters used in the Model Comparison and Rate Combinations experiments (LWCmax = maximum litter water content in g of water g−1 of dry mass, k max and e in d−1, a, b, c, and d in g of water g−1 of dry mass, and n.a. = not applicable)

HIGH and low in brackets refer to the relative intensity of the different rates.

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