One may assume that the general dynamics of the coupled C and N cycling in terrestrial ecosystems can be described by a small list of processes (functions). The simplest CN-model (“sCN-model”) embodies the following relationships:

  • The plant (divided into an above and a belowground compartment) balances the acquisition of C and N in order to satisfy its stoichiometric requirements (fixed: \(r_{C:N}\)), imposed by new growth (balanced growth condition). It tends to achieve a balance between supply of N through acquisition pathways and the demand imposed by new growth and its respective C:N ratio. \[ {N_{\text{acq}}}(C_r) = y/{r_{\text{C:N}}}[ P(C_l) - r(C_l+C_r) ] \; \; \; \; \; \; (1) \] \(P\) is the C assimilation rate and is a function of leaf mass (\(C_l\)), \(y\) is the growth efficiency, \(r\) a respiration coefficient, implying that plant respiration scales with its size, \(C_l\) and \(C_r\) are the leaf carbon and root pool sizes respectively, and \(N_{\text{acq}}\) is the the N acquisition flux and is a function of the root mass (\(C_r\)).

  • The size of the above-ground pool \(C_l\) (each pool consists of C and N) determines the assimilation rate of C (\(P\)) and has declining marginal returns towards increasing \(C_l\). \[ P(C_l) = I \varepsilon \left(1 - e^{-k_b \sigma C_{\text{l}}} \right) \] Here, \(I\) is PPFD, \(\varepsilon\) is the light use efficiency, \(k_b\) is the light extinction parameter, and \(\sigma\) is the specific leaf area. To simplify the mathematics, I’ve tried to model this with Michaelis-Menten as an alternative: \[ P(C_l) = I \varepsilon \frac{C_l}{C_l + K_P} \]

  • The size of the below-ground pool (\(C_r\)) determines the fraction of net N mineralisation \({N_{\text{min}}}\) that is acquired by the plant (\({N_{\text{acq}}}\)). As for \(P\), the function \(f(C_r)\) represents declining marginal returns in \({N_{\text{acq}}}\) with increasing \(C_r\): \[ {N_{\text{acq}}}= f(C_r) {N_{\text{min}}}\\ f(C_r) = (1 - u) (1 - e^{-k_r C_r } ) \] Here, \(u\) is the unaccessible fraction of net mineralisation, i.e. N losses that are not subject to plant control. Again, in order to simplify mathematics, I’ve tried to model this with Michaelis-Menten as an alternative: \[ f(C_r) = (1-u) \frac{C_r}{C_r + K_N} \]

Example

For given \(I\), \(\varepsilon\), \(N_{\text{min}}\), a balance of above-to-belowground pool size \(a = C_l:C_r\) can be found that satisfies Eq. (1). The imbalance can be expressed as \[ f((1-a)C){N_{\text{min}}}= y/{r_{\text{C:N}}}\; [ P(aC) - rC ] \] Here, \(C=C_l+C_r\) and \(a=C_l/C\). Below is an example plot for the imbalance term as a function of \(a\).

Steady-state solution

In steady state, the total stock of N in the soil is constant. This implies that net mineralisation equals total N inputs through litterfall and N deposition (assuming deposited N bypasses plants). For the sake of simplicity, N fixation and resorption are ignored here. \[ {N_{\text{min}}}= \frac{C}{{r_{\text{C:N}}}\tau} + {N_{\text{in}}}\] Here, \(\tau\) is the residence time of C and N in the plant pool; \({r_{\text{C:N}}}\) is the C:N ratio of plant biomass; and \({N_{\text{in}}}\) is the external input of N, subsuming atmospheric deposition and weathering. Using this, the steady-state solution of the coupled soil-plant system can be found, without relying on prescribed net mineralisation rates, with the balanced-growth condition from above: \[ f((1-a)C)(\frac{C}{{r_{\text{C:N}}}\tau} +{N_{\text{in}}}) = y / {r_{\text{C:N}}}\; [ P(aC) - rC ]\; \; \; \;\; \; \; \;(2) \] and the plant C blance (C assimilation equals respiration plus new growth, replacing litterfall in steady-state): \[ P(aC) = (\tau/y + r)\;C \; \; \; \; \; \; \; \;(3) \] This set of two equations should in theory allow us to cancel \(C\) and solve for \(a\) as a function of environmental conditions: \(a = f(\varepsilon, I, {N_{\text{in}}})\). Note that \(\varepsilon\) is not treated here as a parameter because it’s itself a function of environmental conditions.

My problem

Something is apparently not right here. I’ve tried to solve this system of two non-linear equations numerically but the solution I get is nonsense. A problem might be that I cannot constrain the solutions to a (phyically meaningful) interval, \(0<a<1\) in this case. But the nature of the problem shouldn’t require this anyways. So I am asking myself if the second equation above (Eq. 3) is appropriate or if something else is wrong.

So far, I haven’t attempted to analytically solve this. Feel free if you like…

Dynamic CN-model

Above equations can also be implemented in a simple dynamical model that additionally accounts for soil C and N turnover. The results from a spinup to equilibrium look like this (suggesting that there is a physically meaningful solution). In this case (last plot at the very bottom), the ratio of \(C_l/C\) is around 0.83.