MODEL DESCRIPTION

This very simple box model of the terrestrial carbon cycle combines logistic plant growth with decomposing soil organic mater in a series of three cascading pools. The idea is to enhance plant growth rates according to CO2 fertilization and increase the carrying capacity according to nutrient availability, which grows over time due to nitrogen deposition. Plant mortality (conversion to dead litter) is in turn enhanced by deforestation or disturbance. Dead carbon in litter and soil pools is metabolized by microbes whose respiration efficiency is adjustable and depends on temperature.

We track four carbon pools: plants (P), litter (L), “fast” soil (F), and “slow” soil (S). The amount of carbon stored in each pool is determined by four predictive model equations:

\[ \frac{dP}{dt} = NPP - MORT \] \[ \frac{dL}{dt} = MORT - \frac{L}{\tau_L}Q_{10} ^{\frac{T-T_0}{10}} \] \[ \frac{dF}{dt} = \left[ \frac{L}{\tau_L}(1-\epsilon) - \frac{F}{\tau_F} \right] Q_{10} ^{\frac{T-T_0}{10}} \] \[ \frac{dS}{dt} = \left[ \frac{F}{\tau_F}(1-\epsilon) - \frac{S}{\tau_S} \right] Q_{10} ^{\frac{T-T_0}{10}} \]

Symbols used: * NPP = Net primary production (photosynthesis minus plant respiration, in GtC/yr) * MORT = plant mortality (transfers carbon from live plants to dead litter, GtC/yr) * \(\tau_L\), \(\tau_F\), \(\tau_S\) = turnover time for carbon in litter, fast, and slow soil pools (years) * \(\epsilon\) = efficiency of microbial respiration (fraction of what they “eat”" that turns into CO2) * \(Q_{10}\) = fractional increase in decomposition rates per 10 degrees of warming * T = temperature (Celsius), prescribed from a simple climate model using the IPCC SRES A2 emissions scenaro.

Plant growth (NPP) is modeled assuming Logistic growth as widely used in biology and ecology to describe populations growth under environmental constraints. Essentially this is a modification of exponential growth with a growth rate modified (reduced) as the population approaches limits set by nutrients or other environmental conditions.

\[ NPP = g P \left(1 - \frac{P}{NK} \right) \]

It’s helpful to think of \(g(1-\frac{P}{NK})\) as a modified or constrained growth rate, which is the probability of reproduction for each individual in a population per timestep. Here \(N\) is a “nutrient status” which multiplies \(K\), the “carrying capacity” or nutrient-limited equilibrium size of the plant carbon pool. As the plant pool \(P\) approaches the carrying capacity \(NK\), the modified growth rate \(g(1-\frac{P}{NK})\) approaches zero. When \(P\) is very small, the plant pool grows exponentially.

The nutrient status \(N\) multiplies the original (unfertilized) carrying capacity according to a prescribed function that ramps up following an acrtangent curve from 1.0 in the year 1800 to 1.2 in 2150. This factor can be adjusted with a “Nitrogen fertilization” slider in the left-hand control panel of the website.

The growth rate \(g\) of the plants is also enhanced by increasing atmospheric CO2 (CO2 fertilization). This is represented in the model as

\[g = g_0 \left(1 + \beta ln \frac{CO2}{CO2_{init}} \right)\]

where \(g_0\) is the equilibrium growth rate at the initial CO2 concentration \(CO2_{init}\), and \(\beta\) represents the strength of the CO2 fertilization effect on NPP. Analyses of forest inventory data suggest that \(\beta \approx 0.36\) for modern temperate forests. This value corresponds to an increase of NPP by 25% per doubling of atmospheric CO2. The strength of this effect can be adjusted using another model control slider. CO2 is prescribed from a simple climate model using the IPCC SRES A2 emissions scenaro.

Plant mortality \(MORT\) is modeled as a fraction of the living plant carbon pool, plus a prescribed disturbance of the plant pool due to deforestation:

\[ MORT = death {P} + disturbance = \frac{g_0}{\lambda} P + disturbance\]

where \(\lambda\) is the average lifetime of plants in years.

The disturbance rate due to historical deforestation is prescribed in the model using a very simple (1+cosine) function that ramps up from zero in 1825 to a maximum of 2 GtC/yr in 1975, and then back down to zero in 2125. The magnitude of this term can be scaled up or down with yet another model input slider on the website.

The model is initialized in the year 1800 by setting values of the initial plant pool (\(P_{eq}\), default value is 500 GtC) and global NPP (\(NPP_{eq}\), default value is 60 GtC/yr). Carrying capacity in this undisturbed state is given by

\[ K = P_{eq} \left(1 - \frac{1}{\lambda} \right)\].

The equilibrium growth rate of the plant pool for the initial condition is given by

\[ g_0 = \frac{NPP_{eq}}{P_{eq} \left( 1 - \frac{P_{eq}}{K} \right)}\].

The program that does the calculation is very simple, and the code that controls this website is surprisingly simple too! It is all written in the programming language R, using a web programming package called shiny. You can read all about it on the “Website Code” tab to the right.