A general method for calculating the optimal leaf longevity from the viewpoint of carbon economy

Abstract

According to the viewpoint of the optimal strategy theory, a tree is expected to shed its leaves when they no longer contribute to maximisation of net carbon gain. Several theoretical models have been proposed in which a tree was assumed to strategically shed an old deteriorated leaf to develop a new leaf. We mathematically refined an index used in a previous theoretical model [Kikuzawa (Am Nat 138:1250–1263, 1991)] so that the index is exactly proportional to a tree’s lifelong net carbon gain. We also incorporated a tree’s strategy that determines the timing of leaf expansion, and examined three kinds of strategies. Specifically, we assumed that a new leaf is expanded (1) immediately after shedding of an old leaf, (2) only at the beginning of spring, or (3) immediately after shedding of an old leaf if the shedding occurs during a non-winter season and at the beginning of spring otherwise. We derived a measure of optimal leaf longevity maximising the value of an appropriate index reflecting total net carbon gain and show that use of this index yielded results that are qualitatively consistent with empirical records. The model predicted that expanding a new leaf at the beginning of spring than immediately after shedding usually yields higher carbon gain, and combined strategy of the immediate replacement and the spring flushing earned the highest gain. In addition, our numerical analyses suggested that multiple flushing seen in a few species of subtropical zones can be explained in terms of carbon economy.

Appendices

Appendix A: Derivation of net gain afforded by the first leaf

The gain afforded by the first leaf, the continuous age of which is equal to the absolute time, is

$$\begin{aligned} G(\tau )=-C+\int _0^\tau \left( {p(s)\theta (s)-m(s)} \right) \mathrm{d}s . \end{aligned}$$

(26)

Substituting (4) into (26) and applying (5), we obtain

$$\begin{aligned} G(\tau )=-C+\left( {a-m} \right) \tau \left( {1-\frac{\tau }{2b}} \right) -a\int _0^\tau \left( {1-\theta (s)} \right) \left( {1-\frac{s}{b}} \right) \mathrm{d}s. \end{aligned}$$

(27)

Note that the third (negative) term on the right-hand side of (27), which we henceforth describe as the loss term, represents the carbon gain that the first leaf would have earned if no unfavourable period existed.

If $j \le \tau < j+f$, where $j$ is any non-negative integer and thus $j=\lfloor \tau \rfloor$, the first leaf experiences a total of $\lfloor \tau \rfloor$ unfavourable intervals. Except the case of 0 $\le \tau < f$ and thus $j =$ 0, the loss term is calculated as

$$\begin{aligned} a\sum _{i=1}^{\left\lfloor \tau \right\rfloor } {\left[ {\int _{i-1+f}^i {\left( {1-\frac{s}{b}} \right) \mathrm{d}s} } \right] }&= a\left( {1-f} \right) \sum _{i=1}^{\left\lfloor \tau \right\rfloor } {\left( {1+\frac{1-f}{2b}-\frac{i}{b}} \right) } \nonumber \\&= a\left( {1-f} \right) \left\lfloor \tau \right\rfloor \left( {1-\frac{\left\lfloor \tau \right\rfloor +f}{2b}} \right) \!. \end{aligned}$$

(28)

If 0 $\le \tau < f$, the loss term is obviously zero, and thus the term on the extreme right of (28) holds for this case. Consequently, we have

$$\begin{aligned} G(\tau )=-C+\left( {a-m} \right) \tau \left( {1-\frac{\tau }{2b}} \right) -a\left( {1-f} \right) \left\lfloor \tau \right\rfloor \left( {1-\frac{\left\lfloor \tau \right\rfloor +f}{2b}} \right) \!. \end{aligned}$$

(29)

If $j+f \le \tau < j$ + 1, where $j$ is any non-negative integer, the first leaf further experiences a part of an unfavourable period at the end of its life (i.e. from $j+f$ to $\tau$). The carbon gain that the first leaf earns during this period is

$$\begin{aligned} a\int _{\left\lfloor \tau \right\rfloor +f}^\tau {\left( {1-\frac{s}{b}} \right) \mathrm{d}s} =a\left[ {\tau \left( {1-\frac{\tau }{2b}} \right) -\left( {\left\lfloor \tau \right\rfloor +f} \right) \left( {1-\frac{\left\lfloor \tau \right\rfloor +f}{2b}} \right) } \right] \!. \end{aligned}$$

(30)

As the loss term can be calculated as the sum of (28) and (30), we may show (29) minus (30) as

$$\begin{aligned} G(\tau )&= -C-m\tau \left( {1-\frac{\tau }{2b}} \right) +af\left( {\left\lfloor \tau \right\rfloor +1} \right) \left( {1-\frac{\left\lfloor \tau \right\rfloor +f}{2b}} \right) \nonumber \\&= -C-m\tau \left( {1-\frac{\tau }{2b}} \right) +af\left\lceil \tau \right\rceil \left( {1-\frac{\left\lfloor \tau \right\rfloor +f}{2b}} \right) \!. \end{aligned}$$

(31)

Appendix B: Analytical results for Kikuzawa’s criterion

It can be shown, by substituting (6) into $g(\tau )$, that if $g(\tau )$ has at least one positive part, the parameter attains a maximum value at either one of the following two forms of $\tau$: $\tau =j+f$ and $\tau =t^{*}$ ($j \le t^{*} < j+f)$, where $j$ represents a non-negative integer and

$$\begin{aligned} t^{*}=\sqrt{\frac{2b}{a-m}\left[ {C+a\left( {1-f} \right) j\left( {1-\frac{j+f}{2b}} \right) } \right] }. \end{aligned}$$

(32)

To obtain (32), we use $\lfloor \tau \rfloor =j$ and $\mathrm{d}g(\tau ) / \mathrm{d}t = 0$. Note that $1 - (j+f) / (2b)$ in (32) is always positive because $2b > j+f$. Kikuzawa (1991) argued that, when $g(\tau )$ is maximised at a point other than $\tau =j+f$, which we have shown is definitely $\tau =t^{*}$, the truly optimal leaf longevity would be located near that point. For example, if $t^{*}$ is close to $j+f$, the ultimate leaf longevity would be $j+f$ and the tree should expand the second leaf at the beginning of the next favourable season (i.e. at $s=j + 1$). In other words, Kikuzawa (1991) indeed noted that it was not always possible to measure the optimal leaf longevity in a two-seasonal environment by simply maximising the criterion $g(\tau )$. We have shown that this statement holds true even when $g(\tau )$ is maximised at $\tau =j+f$.

Appendix C: Locally optimal leaf longevity of trees that replace leaves at the beginning of a favourable season

In this appendix, we obtain an integer by use of which the discrete function (13) is maximised when af $- m > 0$. Let us first consider (13) to be a continuous function of the real number $\tau$. By solving $\mathrm{d}\gamma (\varphi _{\mathrm{I}}, \tau ) / \mathrm{d}\tau = 0$ for $\tau$, we can show that the continuous function attains a maximum point at $\tau =t^{\ddagger }$, where $t^{\ddagger }$ is defined as (15). It follows that the original discrete function (13) assumes a maximum value at either $\tau =\lfloor t^{\ddagger } \rfloor$ or $\tau =\lceil t^{\ddagger }\rceil$. When $t^{\ddagger } \ge$ 1, we can simplify the condition $\gamma (\varphi _{\mathrm{I}}, \lfloor t^{\ddagger }\rfloor ) \ge \gamma (\varphi _{\mathrm{I}}, \lceil t^{\ddagger }\rceil )$ as

$$\begin{aligned} \gamma \left( \varphi _{\mathrm{I}} ,\left\lfloor {t^{{\ddagger }}} \right\rfloor \right)&\ge \gamma \left( \varphi _{\mathrm{I}} ,\left\lceil {t^{{\ddagger }}} \right\rceil \right) \Leftrightarrow -\frac{C}{\left\lfloor {t^{{\ddagger }}} \right\rfloor }+\frac{\left( {af-m} \right) \left\lfloor {t^{{\ddagger }}} \right\rfloor }{2b}\nonumber \\&\ge -\frac{C}{\left\lceil {t^{{\ddagger }}} \right\rceil }+\frac{\left( {af-m} \right) \left\lceil {t^{{\ddagger }}} \right\rceil }{2b}\Leftrightarrow t^{{\ddagger }}\le \sqrt{\left\lfloor {t^{{\ddagger }}} \right\rfloor \left\lceil {t^{{\ddagger }}} \right\rceil }, \end{aligned}$$

(33)

and vice versa.

In conclusion, the optimal integer leaf longevity for trees subject to the immediate replacement rule ($\varphi _{\mathrm{I}}$) is $\lfloor t^{\ddagger } \rfloor$ if (33) holds and $\lceil t^{\ddagger }\rceil$ otherwise, which can be also expressed as (14).

Appendix D: Locally optimal leaf longevity of deciduous trees subject to the combined expansion rule

This appendix focuses on deciduous trees subject to the combined expansion rule ($\varphi _{\mathrm{C}}$), and our argument has two parts. First, we show that leaf longevity written as $\tau =f / Q$ yields the largest total net gain within the range of

$$\begin{aligned} \frac{f}{Q}\le \tau <\min \left\{ {\frac{f}{Q-1},\frac{1}{Q}} \right\} \!. \end{aligned}$$

(34)

for any positive integer $Q$. In the second part, we identify the value of $Q$ by which the total net gain is maximised. In this two-step manner, we obtain a locally optimal leaf longevity for a deciduous tree subject to the $\varphi _{\mathrm{C}}$ rule. We exclude other values of $\tau$ associated with a deciduous character from consideration as the chosen value of $\tau _{\mathrm{C}}$.

Consider a tree expanding $Q$ leaves during a favourable period and shedding the $Q$-th leaf at a certain time during the following unfavourable period. In other words, consider a $\tau$ value satisfying ($Q - 1$) $\tau < f \le Q \tau < 1$, which is identical to (34). By definition, the tree is deciduous, and $N(\varphi _{\mathrm{C}}, \tau )=Q$ holds true. Using (23), $\gamma (\varphi _{\mathrm{C}}, \tau )$ within the range (34) is calculated as

$$\begin{aligned} \gamma (\varphi _\mathrm{C} ,\tau )&= -QC+\left( {Q-1} \right) \int _0^\tau {\left( {a-m} \right) \left( {1-\frac{t}{b}} \right) \mathrm{d}t}\nonumber \\&+\int _0^{f-\left( {Q-1} \right) \tau } {a\left( {1-\frac{t}{b}} \right) \mathrm{d}t} -\int _0^\tau {m\left( {1-\frac{t}{b}} \right) \mathrm{d}t} . \end{aligned}$$

(35)

The first term on the right-hand side of (35) represents construction cost of $Q$ leaves. The second term represents the net gain earned by the first $Q - 1$ leaves. The third and fourth terms represent the gain by the $Q$-th leaf during the remaining favourable period and the maintenance cost of the $Q$-th leaf, respectively. Differentiating (35) with respect to $\tau$ yields

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tau }\gamma (\varphi _{\mathrm{C}} ,\tau )=-\left( {Q-1} \right) \left( {Q\tau -f} \right) \frac{a}{b}-Qm\left( {1-\frac{\tau }{b}} \right) \!. \end{aligned}$$

(36)

As (36) is always non-positive, (35) is maximised at $\tau =f / Q$.

Substituting $\tau =f / Q$ into (35) yields

$$\begin{aligned} \gamma \left( \varphi _\mathrm{C} ,\frac{f}{Q}\right) =-QC+\left( {a-m} \right) f\left( {1-\frac{f}{2bQ}} \right) \!. \end{aligned}$$

(37)

Next, let us replace $Q$ of (37) with a continuous variable, $q$, and differentiate it with respect to $q$:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}q}\left[ {-qC+\left( {a-m} \right) f\left( {1-\frac{f}{2bq}} \right) } \right]&= -C+\frac{\left( {a-m} \right) f^{2}}{2bq^{2}}\nonumber \\&= -\frac{C}{q^{2}}\left( {q+q^{{\dagger }}} \right) \left( {q-q^{{\dagger }}} \right) \!, \end{aligned}$$

(38)

where $q^{\dagger }$ is defined as

$$\begin{aligned} t^{{\dagger }}=\frac{f}{q^{{\dagger }}}\Leftrightarrow q^{{\dagger }}=\frac{f}{t^{{\dagger }}}=f\sqrt{\frac{a-m}{2bC}}. \end{aligned}$$

(39)

Therefore, the continuous function of $q$ attains a maximum value at $q=q^{\dagger }$. It follows that the original discrete function (37) takes a maximum value either at $Q=\lfloor q^{\dagger } \rfloor$ or at $Q=\lceil q^{\dagger } \rceil$. When $q^{\dagger } \ge 1$, we can simplify the condition $\gamma (\varphi _{\mathrm{C}}, f / \lfloor q^{\dagger } \rfloor ) \ge \gamma (\varphi _{\mathrm{C}}, f / \lceil q^{\dagger } \rceil )$ to

$$\begin{aligned} q^{{\dagger }}\le \sqrt{\left\lfloor {q^{{\dagger }}} \right\rfloor \left\lceil {q^{{\dagger }}} \right\rceil }, \end{aligned}$$

(40)

and vice versa.

In conclusion, the optimal leaf longevity for deciduous trees subject to $\varphi _{\mathrm{C}}$ is $f / Q^{\dagger }$, where $Q^{\dagger }=\lfloor q^{\dagger } \rfloor$ if (40) holds and $Q^{\dagger }=\lceil q^{\dagger } \rceil$ otherwise, as shown in (24).