Abiotic environment

A voxel space, with a resolution of 1 \(m^3\), is used to explicitly model the abiotic environment. For each tree crown, leaf area density is calculated on tree geometry assuming a uniform distribution across voxels occupied by the crown. Leaf area density is computed within each voxel summing all tree crowns inside the voxel \(v\), and is denoted \(LAD(v)\) (leaf area per voxel in \(m².m^{-3}\)). The vertical sum of \(LAD\) from voxel \(v\) to the ground level defines \(LAI(v)\) (leaf area per ground area in \(m^2.m^{-2}\) commonly called leaf area index):

\(LAI(v) = \sum _{v'=v} ^\infty LAD(v')\)

Daily variations in light intensity (photosynthetic photon flux density PPFD in \(\mu mol_{photons}.m^{-2}.s^{-1}\)), temperature (T in degrees Celsius), and vapour pressure deficit (VPD in \(kPA\)) are computed to assess carbon assimilation within each voxel of the canopy and for a representative day per month (see Appendix 1 from @Li for further details). Variation of PPFD Within the canopy is calculated as a local Beer-Lambert extinction law:

\(PPFD_{max,month}(v) = PPFD_{top,max,month}*e^{-k*LAI(v)}\)

The daily maximum incident PPFD at the top of canopy \(PPFD_{top,max,month}\) is given as input. The extinction rate \(k\) is assumed as constant, besides its variation with zenith angle and species leaf inclination angle (Meir, 2000). Moreover only vertical light diffusion is considered ignoring lateral light diffusion, which can have an important role especially in logging gaps. Finally, intra-day variation at half hour time steps \(t\) for a representative day every month are used to compute \(PPFD_{month}(v,t)\), \(T_{month}(v,t)\) and \(VPD_{month}(v,t)\). Water and nutrient processes both in soil and inside trees are not simulated.

Photosynthesis

Autotrophic respiration

A large fraction of plants carbon uptake is actually used for plant maintenance and growth respiration. The autotrophic respiration can represent up to 65% of the gross primary productivity but varies strongly among species, sites, and environments.

TROLL uses Atkin et al. (2015) database of mature leaf dark respiration and associated leaf traits to compute leaf maintenance respiration:

\(R_{leaf-M} = 8.5431-0.1306*N-0.5670*P-0.0137*LMA+11.1*V_{cmax-M}+0.1876*N*P\)

\(R_{leaf-M}\) is the dark respiration rate per leaf dry mass at a temperature of \(25^\circ C\) (\(nmolCO_2.g^{-1}.s^{-1}\)). TROLL assumes leaf respiration during daylight to be 40% of leaf dark respiration, and computes total leaf respiration by accounting for the length of daylight.

TROLL model stem respiration (\(R_{stem}\) in \(\mu molC.s^{-1}\)) with a constant respiration rate per volume of sapwood:

\(R_{stem} = 39.6*\pi*ST*(dbh-ST)*(h-CD)\)

dbh, h, CD and ST are tree diameter at breast height, height, crown depth and sapwood thickness, respectively (\(m\)). TROLL assumes \(ST=0.04~m\) when \(dbh>30~cm\) and an increasing \(ST\) for lower \(dbh\).

Finally, TROLL computes both fine root maintenance respiration, as half of the leaf maintenance respiration. Whereas coarse root and branch maintenance respiration is computed as half of the stem respiration. And growth respiration (\(R_{growth}\)) is assumed to account for 25% of the gross primary productivity minus the sum of maintenance respiration.

Net carbon uptake

Net primary production of carbon for one individual \(NPP_{ind}\) (\(gC\)) is computed by the balance between gross primary production \(GPP_{ind}\) and respiration \(R\):

\(NPP_{ind} = GPP_{ind} - R_{maintenance} - R_{growth}\)

TROLL partitions individuals’ total leaf area \(LA\) into three pools for different leaf age classes corresponding to different photosynthesis efficiency (young, mature and old leaves with \(LA_{young}\), \(LA_{mature}\), and \(LA_{old}\) respectively). Consequently we can compute growth primary production for one individual as:

\(GPP_{ind} = 189.3 * \Delta t * \sum _{l= \lfloor h-CD \rfloor +1} ^{\lfloor h \rfloor} [A_l] * (\frac{LA_{young}}{2} + LA_{mature} + \frac{LA_{old}}{2})\)

h and CD are tree height and crown depth, respectively (\(m\)). \(\lfloor x \rfloor\) is the rounding function. \(\Delta t\) is the duration of a timestep (\(year\)).

Thus, TROLL can compute carbon allocation to wood into an increment of stem volume \(\Delta V\) (\(m^3\)):

\(\Delta V = 10^{-6} * \frac{f_{wood}*NPP_{ind}}{0.5*wsg}*Senesc(dbh)\)

\(f_{wood}\) is the fixed fraction of NPP allocated to stem and branches. \(wsg\) is the wood specific gravity (\(g.cm^{-3}\), see @ref(tab:traits)). TROLL assume large trees less efficient to convert NPP as growth by using a size-related growth decline with function \(Senesc\) after a specific diameter at breast height threshold \(dbh_{thresh}\):

\(Senesc(dbh) = max(0;3-2*\frac{dbh}{dbh_{thresh}})\)

Finally, TROLL can compute carbon allocation to canopy with canopy NPP fraction denoted \(f_{canopy}\) and decomposed into leaf, twig and fruit production. Carbon allocation to leaf results in a new young leaf pool, whereas other leaf pools are updated as follow:

\(\begin{array}{c} \\ \Delta LA_{young} = \frac{2*f_{leaves}*NPP_{ind}}{LMA}-\frac{LA_{young}}{\tau_{young}} \\ \Delta LA_{mature} = \frac{LA_{young}}{\tau_{young}} - \frac{LA_{mature}}{\tau_{mature}}\\ \Delta LA_{old} = \frac{LA_{mature}}{\tau_{mature}} - \frac{LA_{old}}{\tau_{old}} \end{array}\)

\(\tau_{young}\), \(\tau_{mature}\), and \(\tau_{old}\) are species’ residence times in each leaf pool (\(years\)). The sum of residency time thus defined the leaf lifespan \(LL = \tau_{young} + \tau_{mature} + \tau_{old}\) (\(years\)). \(\tau_{young}\) is set to one month and \(\tau_{mature}\) is set to a third of leaf lifespan \(LL\). Belowground carbon allocation is not simulated inside TROLL.

Tree growth

Once the increment of stem volume \(\Delta V\) is calculated, TROLL converts it into an increment of tree diameter at breast height denoted \(\Delta dbh\). TROLL infer tree height from \(dbh\) using a Michaelis-Menten equation:

\(h = h_{lim}*\frac{dbh}{dbh + a_h}\)

On the other hand, we have the trunk volume \(V = C * \pi * (\frac{dbh}{2})^2*h\), thus:

\(\begin{array}{c} \\ \Delta V = C*\frac{1}{2}*\pi*h*dbh*\Delta dbh + C * \pi * (\frac{dbh}{2})^2*h \\ \Delta V = V*\frac{\Delta dbh}{dbh}*(3-\frac{dbh}{dbh + ah})\end{array}\)

Next, TROLL used the new trunk dimension (\(dbh\) and \(h\)) to update tree crown geometry using allometric equations (Chave et al. 2005):

\(\begin{array}{c} \\ CR = 0.80 + 10.47*dbh - 3.33*dbh^2\\ CD = -0.48 + 0.26*h~;~CD = 0.13 + 0.17*h~(h<5~m)\end{array}\)

Finally, TROLL computes the mean leaf density within the crown (\(LD\) in \(m^2.m^{-3}\)) assuming a uniform distribution:

\(LD = \frac{LA_{young}+LA_{mature}+LA_{old}}{\pi*CR^2*CD}\)

Mortality

Mortality is partitioned in two factors inside TROLL: background death \(d_b\) and treefall death \(d_t\).

Chave et al. (2009) advocated for a wood economics spectrum opposing fast growing light wood species with high risk of mortality to slow growing dense wood species with reduced risk of mortality. Hence, background mortality is derived from wood specific gravity \(wsg\) inside TROLL:

\(d_b = m*(1-\frac{wsg}{wsg_{lim}})+d_n\)

\(m\) (\(events.year^{-1}\)) is the reference background death rate for lighter wood species (pioneers). \(d_n\) represents death by carbohydrates shortage. If the number of consecutive days with \(NPP_{ind} < 0\) is superior to tree leaf lifespan \(d_n\) is set to 1 and remains null in other cases.

Mortality by treefall inside TROLL depends on a specific stochastic threshold \(\theta\):

\(\theta = h_{max}*(1-v_T*|\zeta|)\)

\(h_{max}\) is the maximal tree height. \(v_T\) is the variance term set to 0.3. \(|\zeta|\) is the absolute value of a random centred and scaled Gaussian. If the tree height \(h\) is superior to \(\theta\) then the tree may fall with a probability \(1-\theta/h\) (Chave 1999). The treefall direction is random (drawn from a uniform law (\(\mathcal{U}[0,2\pi]\)). All trees in the trajectory of the falling tree will be hurted through a variable denoted \(hurt\), incremented by fallen tree height \(h\). If a tree height is inferior than its \(hurt\) values then it may die with a probability \(1-\frac{1}{2}\frac{h}{hurt}\). \(hurt\) variable is reset to null at each timestep (\(month\)).

Recruitment

Once the tree becomes fertile they will start to disperse seeds. TROLL consider tree as fertile after a specific height threshold \(h_{mature}\) (Wright et al. 2005):

\(h_{mature} = -11.47+0.90*h_{max}\)

But TROLL is not considering seed directly through a seedbank, instead seed might be interpreted as a seedling recruitment opportunity. The number of reproduction opportunities per mature tree is denoted \(n_s\) and set to 10 for all species. This assumption originates from a trade-off between seed number and seed size resulting in equivalent survival and recruitment probability. All \(n_s\) events are dispersed with a distance randomly drawn from a Gaussian distribution. Additionally, TROLL model consider external seedrain through \(n_{ext}\) events of seed immigration:

\(n_{ext} = N_{tot}*f_{reg}*n_{ha}\)

\(N_{tot}\) is the external seedrain per hectare (number of reproduction opportunities). \(f_{reg}\) is the species’ regional frequency. \(n_{ha}\) is the simulated plot size in \(ha\).

Finally, a bank of seedlings to be recruited is defined for each pixel. If the ground-level light reaches a species light compensation point \(LCP\) the species will be recruited:

\(LCP = \frac{R_{leaf}}{\phi}\)

\(R_{leaf}\) is the leaf respiration for maintenance. \(\phi\) is the quantum yield (\(\mu mol C.\mu mol~photon\)) set to 0.06. If several species reach their \(LCP\), one is picked at random. Seedlings are recruited with following initial geometry:

\(\begin{array}{c} \\ dbh = \frac{a_h}{h_{max} - 1}\\ h = 1~m\\ CR = 0.5~m\\ CD = 0.3~m\\ LD = 0.8~m^2.^{-3} \end{array}\)

References

Pacala, S.W., Canham, C.D., Saponara, J., Silander, J.A., Kobe, R.K. & Ribbens, E. (1996). Forest models defined by field measurements: estimation, error analysis and dynamics. Ecological Monographs, 66, 1–43.

Uriarte, M., Canham, C.D., Thompson, J., Zimmerman, J.K., Murphy, L., Sabat, A.M., Fetcher, N. & Haines, B.L. (2009). Natural disturbance and human land use as determinants of tropical forest dynamics: Results from a forest simulator. Ecological Monographs, 79, 423–443.

Köhler, P. & Huth, A. (1998). The effects of tree species grouping in tropical rainforest modelling: Simulations with the individual-based model FORMIND. Ecological Modelling, 109, 301–321.

Fischer, R., Bohn, F., Dantas de Paula, M., Dislich, C., Groeneveld, J., Gutierrez, A. G., … Huth, A. (2016). Lessons learned from applying a forest gap model to understand ecosystem and carbon dynamics of complex tropical forests. Ecological Modelling, 326, 124–133. https://doi.org/10.1016/j.ecolmodel.2015.11.018

Maréchaux, I., & Chave, J. (2017). An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs, 87(4), 632–664.

Baraloto, C., Paine, C.E.T., Poorter, L., Beauchene, J., Bonal, D., Domenach, A.M., Hérault, B., Patiño, S., Roggy, J.C. & Chave, J. (2010). Decoupled leaf and stem economics in rain forest trees. Ecology Letters, 13, 1338–1347.

Kattge, J., Diaz, S., Lavorel, S., Prentice, I. C., Leadley, P., Bonisch, G., … Wirth, C. (2011). TRY - a global database of plant traits. Global Change Biology, 17(9), 2905–2935. doi:10.1111

Meir, P., Grace, J. & Miranda, A.C. (2000). Photographic method to measure the vertical distribution of leaf area density in forests. Agricultural and Forest Meteorology, 102, 105–111.

Farquhar, G.D., Caemmerer, S. von & Berry, J.A. (1980). A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species. Planta, 149, 78–90.

Medlyn, B.E., Duursma, R.A., Eamus, D., Ellsworth, D.S., Prentice, I.C., Barton, C.V.M., Crous, K.Y., De Angelis, P., Freeman, M. & Wingate, L. (2011). Reconciling the optimal and empirical approaches to modelling stomatal conductance. Global Change Biology, 17, 2134–2144.

Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, T., Cornelissen, J.H.C., Diemer, M. & Others. (2004). The worldwide leaf economics spectrum. Nature, 428, 821–827.

Domingues, T.F., Meir, P., Feldpausch, T.R., Saiz, G., Veenendaal, E.M., Schrodt, F., Bird, M., Djagbletey, G., Hien, F., Compaore, H., Diallo, A., Grace, J. & Lloyd, J. (2010). Co-limitation of photosynthetic capacity by nitrogen and phosphorus in West Africa woodlands. Plant, Cell and Environment, 33, 959–980.

Bernacchi, C.J., Pimentel, C. & Long, S.P. (2003). In vivo temperature response functions of parameters required to model RuBP-limited photosynthesis. Plant, Cell and Environment, 26, 1419–1430.

Atkin, O.K., Bloomfield, K.J., Reich, P.B., Tjoelker, M.G., Asner, G.P., Bonal, D., Bonisch, G., Bradford, M.G., Cernusak, L.A., Cosio, E.G., Creek, D., Crous, K.Y., Domingues, T.F., Dukes, J.S., Egerton, J.J.G., Evans, J.R., Farquhar, G.D., Fyllas, N.M., Gauthier, P.P.G., Gloor, E., Gimeno, T.E., Griffin, K.L., Guerrieri, R., Heskel, M.A., Huntingford, C., Ishida, F.Y., Kattge, J., Lambers, H., Liddell, M.J., Lloyd, J., Lusk, C.H., Martin, R.E., Maksimov, A.P., Maximov, T.C., Malhi, Y., Medlyn, B.E., Meir, P., Mercado, L.M., Mirotchnick, N., Ng, D., Niinemets, ?., O’Sullivan, O.S., Phillips, O.L., Poorter, L., Poot, P., Prentice, I.C., Salinas, N., Rowland, L.M., Ryan, M.G., Sitch, S., Slot, M., Smith, N.G., Turnbull, M.H., Vanderwel, M.C., Valladares, F., Veneklaas, E.J., Weerasinghe, L.K., Wirth, C., Wright, I.J., Wythers, K.R., Xiang, J., Xiang, S. & Zaragoza-Castells, J. (2015). Global variability in leaf respiration in relation to climate, plant functional types and leaf traits. New Phytologist, 206, 614–636.

Chave, J., Andalo, C., Brown, S., Cairns, M.A., Chambers, J.Q., Eamus, D., Fölster, H., Fromard, F., Higuchi, N., Kira, T., Lescure, J.P., Nelson, B.W., Ogawa, H., Puig, H., Riéra, B. & Yamakura, T. (2005). Tree allometry and improved estimation of carbon stocks and balance in tropical forests. Oecologia, 145, 87–99.

Chave, J., Coomes, D., Jansen, S., Lewis, S.L., Swenson, N.G. & Zanne, A.E. (2009). Towards a worldwide wood economics spectrum. Ecology Letters, 12, 351–366.

Chave, J. (1999). Study of structural, successional and spatial patterns in tropical rain forests using TROLL, a spatially explicit forest model. Ecological Modelling, 124, 233–254.

Wright, S.J., Jaramillo, M.A., Pavon, J., Condit, R., Hubbell, S.P. & Foster, R.B. (2005). Reproductive size thresholds in tropical trees: variation among individuals, species and forests. Journal of Tropical Ecology, 21, 307–315.