Appendix.tex

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\title[Appendix]{{\helveticaitalic{Appendix}}:
\\ \helvetica{A dynamic optimality principle for water use strategies explains isohydric to anisohydric plant responses to drought}} %Please insert the title of your article here


\maketitle


\section*{Appendix 1: Deriving the Euler-Lagrange equations}

Section 2.6 presents the Euler-Lagrange equations. These are the equations to be solved for the optimal time variation in stomatal aperture in terms of carbon gain. These emerge from maximizing the objective function shown in section 2.3 ($J_{WUS}$). The starting premise of the maximization of $J_{WUS}$ is that if one perturbs $J_{WUS}$ in any direction away from its optimum $J_{WUS,opt}$, one would get: $J_{WUS} < J_{WUS,opt}$. This assumes that the optimum is a maximum. To ensure this condition, perturbations about the variables of the augmented Lagrangian $L$ are introduced:
\begin{enumerate}
    \item $\widetilde{x}(t) = x_{opt}(t) + \epsilon h(t)$, where the tilde is the perturbed value of the soil moisture $x$, $\epsilon$ is a very small real number, $x_opt$ is the optimal value of x whose existence is at this point only assumed, and $h$ is a perturbation around the the optimal value of $x$ and is also a function of time.
    \item $\widetilde{x'}(t) = x'_{opt}(t) + \epsilon h'(t)$, where the prime is the time derivative of the concerned function.
    \item $\widetilde{\lambda}(t) = \lambda_{opt}(t) + \epsilon \gamma(t)$, where $\gamma(t)$ is the perturbation around the optimum $\lambda(t)$.
    \item $\widetilde{g_s}(t) = g_{s,opt}(t) + \epsilon G(t)$, where $G(t)$ is the perturbation around the optimum $g_s(t)$.
\end{enumerate}
Expanding $J_{WUS}$ as a Taylor series with $\epsilon \rightarrow 0$:
\begin{equation}
    \label{eqn:objective_taylor}
    \widetilde{J}_{WUS}(t) = J_{WUS,opt}(t) + \epsilon \frac{d\, J_{WUS}}{d \epsilon}(t) |_{\epsilon = 0} + O(\epsilon^2),
\end{equation}
where, in the second term after equality in equation \ref{eqn:objective_taylor}, $\epsilon$ factors the 'first variation' $\frac{d\, J_{WUS}}{d \epsilon}(t) |_{\epsilon = 0}$ which equals zero at the optimum.
\begin{equation}
    \label{eqn:first_variation}
    \frac{d\, J_{WUS}}{d \epsilon}(t) = \int_0^T \Big(\frac{\partial L}{\partial x} h + \frac{\partial L}{\partial x'} h' + \frac{\partial L}{\partial \lambda} \gamma + \frac{\partial L}{\partial g_s} G \Big) dt + \frac{\partial J_T}{\partial x}|_{t=T} h(T),
\end{equation}
where the perturbation about the terminal gain $J_T$ only exists as a perturbation about $x$ because of the expression in equation 4 of the main text. However, different expressions include might include perturbations around the other variables. As a rule, one avoids the time derivative of perturbations such as $h'(t)$ in equation \ref{eqn:first_variation} by integrating by parts:
\begin{equation}
    \label{eqn:IBP}
    \int_0^T \frac{\partial L}{\partial x'} h' dt = \frac{\partial L}{\partial x'} h |_{t=0}^{t=T} - \int_0^T \frac{d(\partial L / \partial x')}{dt} h dt.
\end{equation}
Since the boundary condition $x(0)$ is a fixed number determined through measurement of the soil water content at the beginning of drought, $h(0)=0$ which means that there is no perturbation around $x$ at $t=0$. So one ends up with the following expression involving $h(t)$ from equation \ref{eqn:first_variation},
\begin{equation}
    \frac{\partial L}{\partial x'} h(t) = \frac{\partial J_T}{\partial x}|_{t=T} h(T),
\end{equation}
from which one obtains the boundary conditions (equation 17 of the main text). The last piece of information required is the fundamental theorem of the calculus of variations. It states that for a general function y(x), if $\int_a^b y(x)p(x) = 0$, $\forall p(x)$ and $p(x)$ is the perturbation, then it is necessary that $y(x)=0$ when $a<x<b$. Applying this theorem to equation \ref{eqn:first_variation}, one gets:
\begin{itemize}
    \item $\frac{\partial L}{\partial x} - \frac{d(\partial L / \partial x')}{dt} = 0$ using equation \ref{eqn:IBP}, this gives equation 16 of the main text.
    \item  $\frac{\partial L}{\partial \lambda} = 0$ which gives equation 8 of the main text.
    \item  $\frac{\partial L}{\partial g_s} = 0$ which gives equation 15 of the main text.
\end{itemize}

\section*{Appendix 2: Deriving the soil-root-atmosphere conductance}

The soil-root-atmosphere conductance include both the conductances of soil to root and root to leaf in series. It is defined as the marginal increase in transpiration rate $E$ with an increase in leaf water potential $\psi_l$. One can write:
\begin{equation}
    \label{eqn:g_sl}
    g_{sl} = \frac{\partial E}{\partial \psi_l},
\end{equation}
where $g_{sl}$ is the soil to leaf water conductance. Because of mass conservation, $E$ is equal to water flow from soil to root ($E_{sr}$) and from root to leaf ($E_{rl}$) (equation 13 of the main text).
Taking the derivative of $E_{sr}$ with respect to $\psi_l$:
\begin{equation}
    \label{eqn:Esr_psil}
    g_{sl} (\psi_l) = - g_{sr} (\psi_x) \frac{\partial \psi_r}{\partial \psi_l} (\psi_l).
\end{equation}
Taking the derivative of $E_{rl}$ with respect to $\psi_l$:
\begin{equation}
    \label{eqn:Erl_psil}
    g_{sl} (\psi_l) = g_{rl} (\psi_r) \frac{\partial \psi_r}{\partial \psi_l} (\psi_l) - g_{rl} (\psi_l).
\end{equation}
Equating $g_{sl}$ from both equations \ref{eqn:Esr_psil}, \ref{eqn:Erl_psil}:
\begin{equation}
    \label{eqn:psir_psil}
    \frac{\partial \psi_r}{\partial \psi_l} (\psi_l) = \frac{g_{rl} (\psi_l)}{g_{rl} (\psi_r) + g_{rl} (\psi_r)}.
\end{equation}
One now obtains a closed form relation for $g_{sl}$:
\begin{equation}
    \label{eqn:gsl_complete}
    g_{sl} (\psi_l) = \frac{g_{rl} (\psi_l) g_{sr} (\psi_x)}{g_{rl} (\psi_r) + g_{sr} (\psi_x)}.
\end{equation}
One can also find the maximum possible soil to leaf conductance at different soil moisture values by replacing all water potential in equation \ref{eqn:gsl_complete} with $\psi_x$ as follows:
\begin{equation}
    \label{eqn:gsl_max}
    g_{sl,max} = \frac{g_{rl} (\psi_x) g_{sr} (\psi_x)}{g_{rl} (\psi_x) + g_{sr} (\psi_x)}.
\end{equation}
Now, the critical leaf water potential $\psi_{l,crit}$ is found by equating $g_{sl}(\psi_l) = 0.05 g_{sl,max}$ from equations \ref{eqn:gsl_complete}, \ref{eqn:gsl_max}.



\section*{Appendix 3: the effect of competitive plant transpiration on $\lambda$}
Equation 24 of the main text expresses $d \lambda / dt$ as a function of $\partial E_{comp} / \partial x$ where $E_{comp}$ is the competitive plant's water use from the modeled plant's rooting zone water. As explained in the paragraph following equation 23 of the main text, $E_{comp}=0.2E$ based on the assumption that both modeled plant and competitive plant have the same VC and have the same rooting zone water content. Therefore, we can write:
\begin{equation}
    \label{eqn:E_comp}
    E_{comp} = g_{sr,comp}(\psi_x)(\psi_x - \psi_{r,comp}),
\end{equation}
where $\psi_x$ corresponds to the water potential in the vicinity of the modeled plant's rooting zone and $\psi_{r,comp}$ is the effective root water potential of the competitive plant. $g_{sr,comp} = 0.2 g_{sr}$ because competitive RAI is less than modeled plant RAI in the rooting zone of interest (equation 11 of the main text) such that the competitive plant only has access to 20\% of the modeled plant's rooting zone water (equations 23, 24 of the main text). Since $\psi_{r,comp}$ responds to the competitive plant's rooting zone water content and not on the model plant's, we write $\partial \psi_{r,comp} / \partial x = 0$ because $x$ is the modeled plant's rooting zone water content. Therefore:
\begin{equation}
    \label{eqn:Ecomp_x}
    \frac{\partial E_{comp}}{\partial x} = 0.2 \frac{\partial \psi_x}{\partial x} \Bigg[ \frac{\partial g_{sr}}{\partial \psi_x} (\psi_x - \psi_{r,comp}) + g_{sr}(\psi_x) \Bigg].
\end{equation}

Note how the result of equation \ref{eqn:Ecomp_x} is different from $\partial E / \partial x = 0$ where $E$ is the modeled plant's transpiration rate.
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