Ecological Archives M071-008-A5

P. R. Moorcroft, G. C. Hurtt, and S. W. Pacala. 2001. A method for scaling vegetation dynamics: the ecosystem demography model (ED). Ecological Monographs 71:557-585.

Appendix E. Growth and Reproduction.

In this section, we define the growth and fecundity functions $g_s({\bf z},{\bf x}, {\bf r},t)$, $g_a({\bf z},{\bf x}, {\bf r},t)$ (both in kgC yr$^{-1}$) and $f_{}({\bf z},{\bf x}, {\bf r},t)$ (yr$^{-1}$). Because the derivation of these functions requires specification of the complete carbon budget for a plant, we will also define five additional quantities: per plant production ( $\mbox{Prod}_{}({\bf z},{\bf x}, {\bf r},t)$), per plant nitrogen lost through decay of leaves and roots ( $N_{litter}({\bf z},{\bf x}, {\bf r},t)$), per plant nitrogen uptake ( $N_{up}({\bf z},{\bf x}, {\bf r},t)$), per plant water uptake ( $W_{up}({\bf z},{\bf x}, {\bf r},t)$), and per plant carbon lost through decay of leaves and roots ( $C_{litter}({\bf z},{\bf x}, {\bf r},t)$). For notational convenience, note that unless specified otherwise, in this and subsequent Appendices, we have dropped the functional dependencies $({\bf z},{\bf x}, {\bf r},t)$ of these per individual quantities. Note also that symbols not defined in this section have been defined in previous sections of the Appendix or the main text.

The total production by a plant's leaves, $A_n({\bf r},t,c)l({\bf x})B_l$, includes leaf respiration as well as photosynthesis (recall that $B_l=B_aq_a({\bf z},{\bf x}),B_r=B_aq_r({\bf z},{\bf x})$, and $B_{sw}=B_aq_{sw}({\bf z},{\bf x})$). We assume that 30% of leaf production is lost as growth respiration (Amthor 1984), both structural and sapwood respiration is negligible and that instantaneous root respiration (kgC yr$^{-1}$ per kgC roots) is given by

$$Resp=\frac{\epsilon(T_A\vert 1.0,3000)}{(1+e^{0.4(5.0-T_A)})(1+e^{0.4(T_A-45.0)})},$$ (E.1)

which has the same form of temperature dependence as leaf respiration. This function is integrated over each month and placed into a look-up-table of monthly integrated root respiration rates.

Plants also lose carbon by the decay of leaves and fine roots at rate ${1 \over x_2}$ for leaves (the leaf decay rate is simply the reciprocal of leaf longevity) and $\alpha_r$ for fine roots, both in units kgC yr$^{-1}$. For simplicity we assume that sapwood decay is negligible and that $\alpha_r= {1 \over x_2}$. Versions with a constant value of $\alpha_{r}$ behave similarly.

There are now four cases to consider. First, suppose net plant-level production is positive. That is: $\mbox{Prod}_{} > 0$, where

$$\mbox{Prod}_{} = B_a[A_n({\bf r},t,c^*)(1- \eta)l({\bf x})q_l({\bf z},{\bf x}) -q_... ... z},{\bf x})Resp - {1 \over x_2}(q_l({\bf z},{\bf x}) + q_r({\bf z},{\bf x}))].$$ (E.2)

and $\eta=0.3$ is the fraction lost as growth respiration.

Recall that all plants devote a fixed fraction $F_{}$ of positive net production to reproduction ($F_{}=0.3$) and $q_a({\bf z},{\bf x})$ of the remaining fraction to growth of $B_a$ and $1-q(a)$ to growth of $B_s$. Thus

$$g_a({\bf z},{\bf x},{\bf r},t) = \mbox{Prod}_{}(1-F)q_a({\bf z},{\bf x})$$ (E.3)

$$g_s({\bf z},{\bf x},{\bf r},t) = \mbox{Prod}_{}(1-F)(1-q_a({\bf z},{\bf x}))$$ (E.4)

$$f({\bf z},{\bf x},{\bf r},t) = F'\frac{\mbox{Prod}_{}}{z_{s0}+z_{a0}}$$ (E.5)

where $F_{}'$ is $F_{}$ times germination and seedling survivorship probability $(s_{0}=0.05)$ and $\left[z_{s0},z_{a0}\right]$ is the size of a seedling.

We define $\mbox{(C:N)}_{Prod}$ as the carbon-to-nitrogen ratio of an individual's new production:

$$(C:N)_{Prod} = \left[(1-F)q_a({\bf z},{\bf x}) + Fq_a({\bf z}_0,{\bf x})\right] (C:N)_a$$ (E.6)

$$+\left[(1-F)q_s({\bf z},{\bf x}) + Fq_s({\bf z}_0,{\bf x}) \right] (C:N)_s.$$

Then:

$$N_{up} = \frac{\mbox{Prod}_{}}{(C:N)_{Prod}}$$

$$C_{litter} = {1 \over x_2}B_a(q_l({\bf z},{\bf x})+q_r({\bf z},{\bf x}))$$

$$N_{litter} = \frac{C_{litter}}{(C:N)_a}$$

$$W_{up} = \Psi({\bf r},t,c^*)B_aq_l({\bf z},{\bf x})l({\bf x}).$$ (E.7)

Second, suppose that $\mbox{Prod}_{}<0$ and that soil moisture is above the critical threshold ($W_{crit}$) causing leaf drop. Because plants stop reproducing and producing structural stem if $\mbox{Prod}_{}<0$:

$$g_a({\bf z},{\bf x},{\bf r},t) = \mbox{Prod}_{}$$ (E.8)

$$g_s({\bf z},{\bf x},{\bf r},t) = 0$$ (E.9)

$$f({\bf z},{\bf x},{\bf r},t) = 0$$ (E.10)

and:

$$N_{up} = 0$$

$$C_{litter} = {1 \over x_2}B_a(q_l({\bf z},{\bf x})+q_r({\bf z},{\bf x}))$$

$$N_{litter} = \mbox{Prod}_{} \over{(C:N)_a}$$

$$W_{up} = \Psi({\bf r},t,c^*)B_a q_l({\bf z},{\bf x})l({\bf x})$$ (E.11)

Third, suppose that soil moisture is beneath the critical threshold for leaf drop ( $W(y,t)<W_{crit}$). Leaf carbon retained following leaf drop (see below) is held in a non-respiring, non-decaying pool but fine root respiration and decay continue. Thus:

$$g_a({\bf z},{\bf x},{\bf r},t) = -B_aq_r({\bf z},{\bf x})[Resp+ {1 \over x_2}]$$ (E.12)

$$g_s({\bf z},{\bf x},{\bf r},t) = 0$$ (E.13)

$$f({\bf z},{\bf x},{\bf r},t) = 0$$ (E.14)

and:

$$N_{up} = 0$$

$$C_{litter} = {1 \over x_2} B_aq_r({\bf z},{\bf x})$$

$$N_{litter} = \frac{g_a({\bf z},{\bf x},{\bf r},t)}{(C:N)_a}$$

$$W_{up} = 0$$ (E.15)

Finally, if $W(y,t) = W_{crit}$, then instantaneous leaf drop occurs. We reset $B_a$ to $B_a(1-\frac{q_l({\bf z},{\bf x})}{2})$ because plants re-translocate 50% of leaf carbon and nitrogen and an amount $B_a \frac{q_l({\bf z},{\bf x})}{2}$ is added to the fast litter pool (see Appendix F). This instantaneous transition results in a step change in $B_a$ when $W(y,t)$ falls below $W_{crit}$.

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