`FF16`

growth strategyAbstract

This document outlines the`FF16`

physiological strategy model used in the `plant`

package.
This document outlines the core physiological model used in the `plant`

package. This model has primarily been developed elsewhere, in particular in Falster, Brännström, Dieckmann, & Westoby (2011). The model’s equations are presented here not as original findings, but rather so that users can understand the full system of equations being solved within `plant`

.

The purpose of a physiological model in `plant`

is to take a plant’s current size, light environment, and physiological parameters as inputs, and return its growth, mortality, and fecundity rates. In the `FF16`

physiological model, these vital rates are all derived from the rate at which living biomass is produced by the plant, which in turn is calculated based on well-understood physiology Fig. 1. Various physiological parameters influence demographic outcomes. Varying these parameters allows accounting for species differences, potentially via traits (see last section). Tables 1, 3, 4 summarize the units and definitions of all variables, parameters, and hyper-parameters used in the material below.

where \(q(z, H)\) is the vertical distribution of leaf area with respect to height \(z\) (Eq. (13)).

Here, \(M, r\), and \(k\) refer to the mass, maintenance respiration rate, and turnover rate of different tissues, denoted by subscripts \(l\) = leaves, \(b\) = bark, \(s\) = sapwood, and \(r\) = roots. \(\bar{p}\) is the assimilation rate of CO\(_2\) per unit leaf area, \(\alpha_{\rm y}\) is yield (i.e., the fraction of assimilated carbon fixed in biomass, with the remaining fraction being lost as growth respiration; this comes in addition to the costs of maintenance respiration), and \(\alpha_{\rm bio}\) is the amount of biomass per unit carbon fixed. Gross photosynthetic production is proportional to leaf area, \(A_{\rm l} = M_{\rm l} / \phi\), where \(\phi\) is leaf mass per area. The total mass of living tissues is \(M_{\rm a} = M_{\rm l} + M_{\rm b} + M_{\rm s} + M_{\rm r}.\)

The first factor, \({\rm d}H / {\rm d}A_{\rm l}\), is the growth in plant height per unit growth in total leaf area – accounting for the architectural strategy of the plant. Some species tend to leaf out more than grow tall, while other species emphasise vertical extension.

The second factor, \({\rm d}A_{\rm l} / {\rm d}M_{\rm a}\), accounts for the marginal cost of deploying an additional unit of leaf area, including construction of the leaf itself and various support structures. As such, \({\rm d}A_{\rm l} / {\rm d}M_{\rm a}\) can itself be expressed as a sum of construction costs per unit leaf area, \[\begin{equation} \frac{{\rm d}A_{\rm l}}{{\rm d}M_{\rm a}} = \bigg( \frac{{\rm d}M_{\rm l}}{{\rm d}A_{\rm l}} + \frac{{\rm d}M_{\rm s}}{{\rm d}A_{\rm l}} + \frac{{\rm d}M_{\rm b}}{{\rm d}A_{\rm l}} + \frac{{\rm d}M_{\rm r}}{{\rm d}A_{\rm l}}\bigg)^{-1}. \tag{5} \end{equation}\] The third factor, \({\rm d}M_{\rm a} / {\rm d}B\), is the fraction of net biomass production (Eq. (3)) that is allocated to growth rather than reproduction or storage. In the`FF16`

physiological model, we let this growth fraction decrease with height according to the function
\[\begin{equation}
\frac{{\rm d}M_{\rm a}}{{\rm d}B}(H) = 1 -
\frac{\alpha_{{\rm f1}}} {1 + \exp\left(\alpha_{{\rm f2}} \left(1 - H / H_{\rm mat}\right)\right)},
\tag{6}
\end{equation}\]
where \(\alpha_{{\rm f1}}\) is the maximum possible allocation (\(0-1\)) and \(\alpha_{{\rm f2}}\) determines the sharpness of the transition (Falster et al., 2011).

Here we describe an allometric model linking to a plant’s height its various other size dimensions required by most ecologically realistic vegetation models (i.e., the masses of leaves, sapwood, bark, and fine roots). This approach allows us to track only the plant’s height, while still accounting for the mass needs to build leaves, roots, and stems. The growth rates of various tissues can then also be derived (Table 2).

This relationship is normalised around a leaf area of 1m\({^2}\).

The pipe model is assumed to hold within individual plants, as well as across plants of different size. It follows that,

\[\begin{equation} Q(z, H) = \left(1-\left(\frac{z}{H}\right)^\eta\right)^2. \tag{11} \end{equation}\] Differentiating with respect to \(z\) then yields a solution for the probability density of leaf area as a function of height, \[\begin{equation} q(z, H) = 2\frac{\eta}{H}\left(1-\left(\frac{z}{H}\right)^{\eta}\right) \left(\frac{z}{H}\right)^{\eta-1}. \tag{13} \end{equation}\]Even though nitrogen and water uptake are not modelled explicitly, imposing a fixed ratio of root mass to leaf area ensures that approximate costs of root production are included in calculations of carbon budget.

where \(\omega\) is the mass of the seed and \(\alpha_{{\rm f3}}\) is the cost per seed of accessories, such as fruits, flowers, and dispersal structures. The function \(\frac{{\rm d}M_{\rm a}}{{\rm d}B}\) is the fraction of \(\frac{{\rm d}B}{{\rm d}t}\) that is allocated to growth (from Eq. (6), while \(1-\frac{{\rm d}M_{\rm a}}{{\rm d}B}\) gives the fraction allocated to reproduction).

where \(X = {\rm d}B/{\rm d}t / A_{\rm l}\). This relationship allows for plants to increase in mortality as their growth rate approaches zero, while allowing for species to differ in the parameters \(\alpha_{{\rm dG1}}\) and \(\alpha_{{\rm dG2}}\).

We also require a function \(S_{\rm G} (x^\prime, H_0, E_{{\rm a}0})\) for plant survival through germination. For the demographic model to behave smoothly, \(S_{\rm G} (x^\prime, H_0, E_{{\rm a}0}) / g(x, H_0, E_{{\rm a}0})\) should approach zero as \(g(x, H_0, E_{{\rm a}0})\) approaches zero. Following Falster et al. (2011), we use the function \[\begin{equation} S_{\rm G} (x^\prime, H_0, E_{{\rm a}0}) = \frac1{1 + X^2}, \tag{21} \end{equation}\]where \(X = \alpha_{{\rm d0}} \frac{A_{\rm l}}{{\rm d}B / {\rm d}t}\) and \(\alpha_{{\rm d0}}\) is a constant. Eq. (21) is consistent with Eq. (20), as both cause survival to decline with mass production.

The `FF16`

physiological model includes default values for all needed parameters (Table 3). Species are known to vary considerably in many of these parameters, such as \(\phi\), \(\rho\), \(\nu\), and \(\omega\); so by varying parameters one can account for species differences. When altering a parameter in the model, however, one must also consider whether there are trade-offs linking parameters.

`plant`

allows for the hyper-parameterisation of the `FF16`

physiological model via plant functional traits: this enables simultaneous variation in multiple parameters in accordance with an assumed trade-off. In the `FF16`

physiological model, we implement the relationships described below. For more details, see `make_FF16_hyperpar.R`

.

This relationship is normalised around \(\phi_0\), the global mean of \(\phi\). This allows us to vary \(\beta_{{\rm kl1}}\) and \(\beta_{{\rm kl2}}\) without displacing the relationship from the observed mean.

We also vary the mass-based leaf respiration rate so that it stays constant per unit leaf area and varies with \(\phi\) and nitrogen per unit leaf area \(\nu\), as empirically observed by Wright et al. (2004), \[\begin{equation} r_{\rm l} = \frac{\beta_{{\rm lf4}}\, \nu}{\phi}. \tag{23} \end{equation}\]As for \(\phi\), these relationships are normalized around \(\rho_0\), the global mean of \(\rho\). By default, \(\beta_{{\rm kI2}}\) and \(\beta_{{\rm ks2}}\) are set to zero, so these linkages only become present when these parameters are set to something other than their default values.

The rate of sapwood respiration per unit volume is assumed to be constant, so sapwood respiration per unit mass varies as \[\begin{equation} r_{\rm s} = \frac{\beta_{{\rm rs1}}}{\rho}, \tag{26} \end{equation}\] where \(\beta_{{\rm rs1}}\) is a default rate per volume of sapwood. Similarly, the rate of bark respiration per unit mass varies as \[\begin{equation} r_{\rm b} = \frac{\beta_{{\rm rb1}}}{\rho}, \tag{27} \end{equation}\]with \(\beta_{{\rm rb1}} = 2 \beta_{{\rm rs1}}\).

as empirically observed (Henery & Westoby, 2001).

where

- \(A_{\rm max}\) is the maximum photosynthetic capacity of the leaf,
- \(\beta_{{\rm lf2}}\) is the curvature of the light response curve,
- \(Y(t) = \beta_{{\rm lf3}} I(t)\) is the initial yield of the light response curve, with \(\beta_{{\rm lf3}}\) being the quantum yield parameter,
- \(I(t) = k_{\rm I} \, I_0(t)\, E\) is the intensity of light on the leaf surface, and
- \(I_0(t)\) is light incident on a surface perpendicular to the sun’s rays directly above the canopy at time \(t\).

The profile of \(I_0(t)\) is given by a solar model adapted from Ter Steege (1997).

We allow for the maximum photosynthetic capacity of the leaf to vary with leaf nitrogen per unit area, as \[\begin{equation} A_{\rm max} = \beta_{{\rm lf1}} \, \left(\frac{\nu}{\nu_0}\right)^{\beta_{{\rm lf5}}}, \tag{30} \end{equation}\]where \(\beta_{{\rm lf1}}\) and \(\beta_{{\rm lf5}}\) are constants. The relationships is normalized around \(\nu_0\), the global mean of leaf nitrogen per unit area.

Values of \(p(\nu, E)\) are calculated across a range of values of \(E\), and then an expression of the form in Eq. (1) is fitted to extract the parameters \(\alpha_{{\rm p1}}\) and \(\alpha_{{\rm p2}}\), such that these become functions of \(\nu\).

Symbol | Unit | Description |
---|---|---|

Plant construction |
||

\(x\) | Vector of traits for a species | |

\(H_0\) | m | Height of a seedling after germination |

\(H\) | m | Height of a plant |

\(B\) | kg | Biomass originating from parent plant |

\(M_i\) | kg | Mass of tissue type \(i\) retained on plant |

\(A_i\) | m\(^2\) | Surface area or area of cross-section of tissue type \(i\) |

\(q(z, H)\) | m\(^{-1}\) | Vertical distribution of leaf area across heights \(z\) for a plant with height \(H\) |

\(Q(z, H)\) | Fraction of leaf area above height \(z\) for a plant with height \(H\)Mass production | |

Mass production |
||

\(p,\bar{p}\) | mol yr\(^{-1}\) m\(^{-2}\) | Photosynthetic rate per unit area |

\(r_i\) | mol yr\(^{-1}\) kg\(^{-1}\) | Respiration rate per unit mass of tissue type \(i\) |

\(k_i\) | yr\(^{-1}\) | Turnover rate for tissue type \(i\) |

Environment |
||

\(a\) | yr | Patch age |

\(E_{a}\) | Profile of canopy openness within a patch of age \(a\) | |

\(E_{a}(z)\) | Canopy openness at height \(z\) within a patch of age \(a\) | |

Demographic outcomes |
||

\(g(x, H, E_{a})\) | m yr\(^{-1}\) | Height growth rate of a plant with traits \(x\) and height \(H\) in the light environment \(E_a\) in a patch of age \(a\) |

\(f(x, H, E_{a})\) | yr\(^{-1}\) | Seed production rate of a plant with traits \(x\) and height \(H\) in the light environment \(E_a\) in a patch of age \(a\) |

\(d(x, H, E_{a})\) | yr\(^{-1}\) | Instantaneous mortality rate of a plant with traits \(x\) and height \(H\) in the light environment \(E_a\) in a patch of age \(a\) |

\(S_{\rm G} (x, H_0, E_{{\rm a}0})\) | Probability that a seed germinates successfully |

Variable | Function | Allocation | Growth.rate |
---|---|---|---|

(a) Assumed |
|||

Height | \(H = \alpha_{{\rm l1}} \, \left(A_{\rm l} \, {\rm m^{-2}} \right)^{\alpha_{{\rm l2}}}\) | \(\frac{{\rm d}H}{{\rm d}A_{\rm l}} = \alpha_{{\rm l2}} \alpha_{{\rm l1}} \left(A_{\rm l} \, {\rm m^{-2}} \right)^{\alpha_{{\rm l2}}-1}\) | \(\frac{{\rm d}H}{{\rm d}t} = \frac{{\rm d}H}{{\rm d}A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Sapwoood area | \(A_{\rm s} = \theta \, A_{\rm l}\) | \(\frac{{\rm d}A_{\rm s}}{{\rm d} A_{\rm l}} = \theta\) | \(\frac{{\rm d}A_{\rm s}}{{\rm d}t} = \frac{{\rm d}A_{\rm s}}{{\rm d} A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Bark area | \(A_{\rm b} = \alpha_{{\rm b1}} \, \theta \, A_{\rm l}\) | \(\frac{{\rm d}A_{\rm b}}{{\rm d} A_{\rm l}} = \alpha_{{\rm b1}} \, \theta\) | \(\frac{{\rm d}A_{\rm b}}{{\rm d}t} = \frac{{\rm d}A_{\rm b}}{{\rm d} A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

(b) Derived |
|||

Leaf mass | \(M_{\rm l} = \phi \, A_{\rm l}\) | \(\frac{{\rm d}M_{\rm l}}{{\rm d}A_{\rm l}} = \phi\) | \(\frac{{\rm d}M_{\rm l}}{{\rm d}t} = \frac{{\rm d}M_{\rm l}}{{\rm d}A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Sapwood mass | \(M_{\rm s} = \rho \, \theta \, \eta_{\rm c} \, A_{\rm l} \, H\) | \(\frac{{\rm d}M_{\rm s}}{{\rm d}A_{\rm l}} = \rho \, \theta \, \eta_{\rm c} \, \big(H + A_{\rm l} \, \frac{{\rm d}H}{{\rm d}A_{\rm l}} \big)\) | \(\frac{{\rm d}M_{\rm s}}{{\rm d}t} = \frac{{\rm d}M_{\rm s}}{{\rm d}A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Bark mass | \(M_{\rm b} = \alpha_{{\rm b1}} \, \rho \, \theta \, \eta_{\rm c} \, A_{\rm l} \, H\) | \(\frac{{\rm d}M_{\rm b}}{{\rm d}A_{\rm l}} = \alpha_{{\rm b1}} \, \rho \, \theta \, \eta_{\rm c}\big(H + A_{\rm l} \, \frac{{\rm d}H}{{\rm d}A_{\rm l}} \big)\) | \(\frac{{\rm d}M_{\rm b}}{{\rm d}t} = \frac{{\rm d}M_{\rm b}}{{\rm d}A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Root mass | \(M_{\rm r} = \alpha_{{\rm r1}} \, A_{\rm l}\) | \(\frac{{\rm d}M_{\rm r}}{{\rm d}A_{\rm l}} = \alpha_{{\rm r1}}\) | \(\frac{{\rm d}M_{\rm r}}{{\rm d}t} = \frac{{\rm d}M_{\rm r}}{{\rm d}A_{\rm l}} \, \frac{{\rm d}A_{\rm l}}{{\rm d}t}\) |

Description | Symbol | Unit | Code | Value |
---|---|---|---|---|

Plant construction |
||||

Crown-shape parameter | \(\eta\) | eta | 12 | |

Leaf mass per area | \(\phi\) | kg m\(^{-2}\) | lma | 0.1978791 |

Wood density | \(\rho\) | kg m\(^{-3}\) | rho | 608 |

Sapwood area per unit leaf area | \(\theta\) | theta | 0.0002141786 | |

Height of plant with leaf area of 1m\(^2\) | \(\alpha_{\rm l1}\) | m | a_l1 | 5.44 |

Exponent of relationship between height and leaf area | \(\alpha_{\rm l2}\) | a_l2 | 0.306 | |

Root mass per unit leaf area | \(\alpha_{\rm r1}\) | kg m\(^{-2}\) | a_r1 | 0.07 |

Ratio of bark area to sapwood area | \(\alpha_{\rm b1}\) | a_b1 | 0.17 | |

Production |
||||

Leaf photosynthesis per area | \(\alpha_{\rm p1}\) | mol yr\(^{-1}\) m\(^{-2}\) | a_p1 | 151.1778 |

Saturation of leaf photosynthesis per area | \(\alpha_{\rm p2}\) | a_p2 | 0.2047162 | |

Yield = fraction of carbon fixed converted into mass | \(\alpha_{\rm y}\) | a_y | 0.7 | |

Biomass per mol carbon | \(\alpha_{\rm bio}\) | kg mol\(^{-1}\) | a_bio | 0.0245 |

Leaf respiration per mass | \(r_{\rm l}\) | mol yr\(^{-1}\) kg\(^{-1}\) | r_l | 198.4545 |

Fine-root respiration per mass | \(r_{\rm r}\) | mol yr\(^{-1}\) kg\(^{-1}\) | r_r | 217 |

Sapwood respiration per mass | \(r_{\rm s}\) | mol yr\(^{-1}\) kg\(^{-1}\) | r_s | 6.598684 |

Bark respiration per mass | \(r_{\rm b}\) | mol yr\(^{-1}\) kg\(^{-1}\) | r_b | 13.19737 |

Turnover rate for leaves | \(k_{\rm l}\) | yr\(^{-1}\) | k_l | 0.4565855 |

Turnover rate for sapwood | \(k_{\rm s}\) | yr\(^{-1}\) | k_s | 0.2 |

Turnover rate for bark | \(k_{\rm b}\) | yr\(^{-1}\) | k_b | 0.2 |

Turnover rate for fine roots | \(k_{\rm r}\) | yr\(^{-1}\) | k_r | 1 |

Fecundity |
||||

Seed mass | \(\omega\) | kg | omega | 0.000038 |

Height at maturation | \(H_{{\rm mat}}\) | m | hmat | 16.59587 |

Maximum allocation to reproduction | \(\alpha_{\rm f1}\) | a_f1 | 1 | |

Parameter determining rate of change in\(r(x, m_\textrm{l})\) around\(H_{\rm mat}\) | \(\alpha_{\rm f2}\) | a_f2 | 50 | |

Accessory cost per seed | \(\alpha_{\rm f3}\) | kg | a_f3 | 0.000114 |

Mortality |
||||

Survival probability during dispersal | \(S_{\rm D}\) | S_D | 0.25 | |

Parameter influencing survival through germination | \(\alpha_{\rm d0}\) | kg yr\(^{-1}\) m\(^{-2}\) | a_d0 | 0.1 |

Intrinsic or growth-independent mortality | \(d_{\rm I}\) | yr\(^{-1}\) | d_I | 0.01 |

Baseline rate for growth-dependent mortality | \(\alpha_{\rm dG1}\) | yr\(^{-1}\) | a_dG1 | 5.5 |

Risk coefficient for dry-mass production per unit leaf area in growth-dependent mortality | \(\alpha_{\rm dG2}\) | yr m^\({2}\) kg\(^{-1}\) | a_dG2 | 20 |

Description | Symbol | Unit | Code | Value |
---|---|---|---|---|

Leaf turnover |
||||

Global average leaf mass per area | \(\phi_0\) | kg m\(^{-2}\) | lma_0 | 0.1978791 |

Rate of leaf turnover at average leaf mass per unit leaf area, \(\phi_0\) | \(\beta_{\rm kl1}\) | yr\(^{-1}\) | B_kl1 | 0.4565855 |

Scaling exponent for \(\phi\) in leaf turnover | \(\beta_{\rm kl2}\) | B_kl2 | 1.71 | |

Sapwood turnover |
||||

Global average wood density | \(\rho_0\) | kg m\(^{-3}\) | rho_0 | 608 |

Rate of sapwood turnover at average wood density, \(\rho_0\) | \(\beta_{\rm ks1}\) | yr\(^{-1}\) | B_ks1 | 0.2 |

Scaling exponent for \(\rho\) in sapwood turnover | \(\beta_{\rm ks2}\) | B_ks2 | 0 | |

Growth-independent mortality |
||||

Rate of instantaneous mortality at average wood density, \(\rho_0\) | \(\beta_{\rm dI1}\) | yr\(^{-1}\) | B_dI1 | 0.01 |

Scaling exponent for wood density in intrinsic mortality | \(\beta_{\rm dI2}\) | B_dI2 | 0 | |

Photosynthesis |
||||

Leaf nitrogen per unit leaf area | \(\nu\) | kg m\(^{-2}\) | narea | 0.00187 |

Global average nitrogen per unit leaf area | \(\nu_0\) | kg m\(^{-2}\) | narea_0 | 0.00187 |

Potential CO\(_2\) photosynthesis at average leaf nitrogen, \(\nu_0\) | \(\beta_{\rm lf1}\) | mol d\(^{-1}\) m\(^{-2}\) | B_lf1 | 0.8273474 |

Curvature of light response curve | \(\beta_{\rm lf2}\) | B_lf2 | 0.5 | |

Quantum yield of leaf photosynthesis (CO\(_2\) per unit photosynthetically active radiation) | \(\beta_{\rm lf3}\) | B_lf3 | 0.04 | |

Scaling exponent for leaf nitrogen in maximum leaf photosynthesis | \(\beta_{\rm lf5}\) | B_lf5 | 1 | |

Respiration |
||||

CO\(_2\) respiration per unit leaf nitrogen | \(\beta_{\rm lf4}\) | mol yr\(^{-1}\) kg\(^{-1}\) | B_lf4 | 21000 |

CO\(_2\) respiration per unit sapwood volume | \(\beta_{\rm rs1}\) | mol yr\(^{-1}\) m\(^{-3}\) | B_rs1 | 4012 |

CO\(_2\)respiration per unit bark volume | \(\beta_{\rm rb1}\) | mol yr\(^{-1}\) m\(^{-3}\) | B_rb1 | 8024 |

Reproduction |
||||

Cost of seed accessories per unit seed mass | \(\beta_{\rm f1}\) | B_f1 | 3 |

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