The rpmodel package implements the P-model as described in Stocker et
al. (2019) Geosci. Mod. Dev. The main function available
through the package is rpmodel(), which returns a list of
quantities (see ?rpmodel) for a given set of inputs. An
additional set of important functions that are used within
rpmodel() are also available through this package. Usage
examples are given below.
The P-model predicts how photosynthesis acclimates to a changing
environment, coordinating stomatal conductance, Vcmax and Jmax. This
yields a model that has the form of a light use efficiency model, where
gross primary production scales linearly with absorbed light, as
described in Stocker
et al. 2020. It is important to note that this implies that the
P-model is valid only for simulating responses to the environment that
evolve over the time scale at which the photosynthetic machinery (e.g.,
Rubisco) can be assumed to acclimate. Sensible choices are on the order
of a couple of weeks to a month. In other words, the arguments (climatic
forcing), provided to rpmodel() should represent typical
daytime mean values, averaged across a couple of weeks. The output is
then representative also for average values across the same time
scale.
Example P-model run
Let’s run the P-model, without \(J_{\text{max}}\) limitation (argument
method_jmaxlim = "none"), for one point. The set of inputs,
being temperature (tc), photosynthetic photon flux density
(ppfd), vapour pressure deficit (vpd), ambient
CO\(_2\) (co2), elevation
(elv), and fraction of absorbed photosynthetically active
radiation (fapar). The quantum yield efficiency parameter
is provided as an argument (kphio) and corresponds to \(\widehat{\varphi_0}\) in Stocker et
al. (2019) if the temperature-dependence of this parameter is ignored
(argument do_ftemp_kphio = FALSE, corresponding to
simulation setup ‘ORG’ in Stocker et al. (2019)), or to \(\widehat{c_L}\) if the
temperature-dependence of the quantum yield efficiency is included
(argument do_ftemp_kphio = TRUE, used in simulation setups
‘BRC’ and ‘FULL’ in Stocker et al. (2019)). By default the optional
argument do_soilmstress is set to FALSE,
meaning that the empirical soil moisture stress function is not
included. The unit cost ratio (\(\beta\) in Stocker et al. (2019)) is given
by argument beta.
To run the rpmodel() function we can do:
library(rpmodel)
out_pmodel <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
kphio = 0.049977, # quantum yield efficiency as calibrated for setup ORG by Stocker et al. 2020 GMD,
beta = 146, # unit cost ratio a/b,
c4 = FALSE,
method_jmaxlim = "wang17",
do_ftemp_kphio = FALSE, # corresponding to setup ORG
do_soilmstress = FALSE, # corresponding to setup ORG
verbose = TRUE
)
## Warning in rpmodel(tc = 20, vpd = 1000, co2 = 400, fapar = 1, ppfd = 30, : Atmospheric pressure (patm) not provided. Calculating it as a
## function of elevation (elv), assuming standard atmosphere
## (101325 Pa at sea level).
## $gpp
## [1] 7.119192
##
## $ca
## [1] 40.53
##
## $gammastar
## [1] 3.339251
##
## $kmm
## [1] 46.09928
##
## $ns_star
## [1] 1.125361
##
## $chi
## [1] 0.694352
##
## $xi
## [1] 63.3145
##
## $mj
## [1] 0.7123038
##
## $mc
## [1] 0.3340838
##
## $ci
## [1] 28.14209
##
## $iwue
## [1] 7.742446
##
## $gs
## [1] 0.04784805
##
## $vcmax
## [1] 1.774218
##
## $vcmax25
## [1] 2.78494
##
## $jmax
## [1] 4.001452
##
## $jmax25
## [1] 5.464979
##
## $rd
## [1] 0.02818463
Above, we specified the model paramters (arguments beta
and kphio). This overrides the defaults, where
rpmodel() uses the parameters as calibrated by Stocker et
al. (2019), depending on the choices for arguments
do_ftemp_kphio and do_soilmstress:
kphio = ifelse(do_ftemp_kphio, ifelse(do_soilmstress, 0.087182, 0.081785), 0.049977)
beta = 146.0
apar_soilm = 0.0
bpar_soilm = 0.73300
The function returns a list of variables (see also man page by
?rpmodel), including \(V_{\mathrm{cmax}}\), \(g_s\), and all the parameters of the
photosynthesis model (\(K\), \(\Gamma^{\ast}\)), which are all internally
consistent, as can be verified for…
\[
c_i = c_a - A / g_s = \chi c_a
\]
c_molmass <- 12.0107 # molecular mass of carbon
kphio <- 0.05 # quantum yield efficiency, value as used in the function call to rpmodel()
ppfd <- 30 # mol/m2/d, value as used in the function call to rpmodel()
fapar <- 1 # fraction, value as used in the function call to rpmodel()
print( out_pmodel$ci )
## [1] 28.14209
print( out_pmodel$ca - (out_pmodel$gpp / c_molmass) / out_pmodel$gs )
## [1] 28.14209
print( out_pmodel$ca * out_pmodel$chi )
## [1] 28.14209
Yes.
And for… \[
A = V_{\text{cmax}} \frac{c_i-\Gamma^{\ast}}{c_i + K} = \phi_0
I_{\text{abs}} \frac{c_i-\Gamma^{\ast}}{c_i + 2 \Gamma^{\ast}} = g_s
(c_a - c_i)
\]
print( out_pmodel$gpp / c_molmass )
## [1] 0.5927375
print( out_pmodel$vcmax * (out_pmodel$ci - out_pmodel$gammastar) / (out_pmodel$ci + out_pmodel$kmm ))
## [1] 0.5927375
print( out_pmodel$gs * (out_pmodel$ca - out_pmodel$ci) )
## [1] 0.5927375
print( kphio * ppfd * fapar * (out_pmodel$ci - out_pmodel$gammastar) / (out_pmodel$ci + 2 * out_pmodel$gammastar ))
## [1] 1.068456
Yes.
Elevation and pressure
Above, atmospheric pressure (patm) was not provided as
an argument, but elevation (elv) was. Hence the warning was
printed (only when verbose = TRUE), saying:
Atmospheric pressure (patm) not provided. Calculating it as a function of elevation (elv),
Assuming standard atmosphere (101325 Pa at sea level)..
Alternatively, we can provide atmospheric pressure (patm)
as input, which overrides the argument elv.
P-model for time series
The rpmodel() function can also be invoked for time
series, where tc, vpd, co2,
fapar, patm, and ppfd are
vectors.
set.seed(1982)
out_ts_pmodel <- rpmodel(
tc = 20 + rnorm(5, mean = 0, sd = 5),
vpd = 1000 + rnorm(5, mean = 0, sd = 50),
co2 = rep(400, 5),
fapar = rep(1, 5),
ppfd = 30 + rnorm(5, mean = 0, sd = 3),
elv = 0,
kphio = 0.049977,
beta = 146,
c4 = FALSE,
method_jmaxlim = "none",
do_ftemp_kphio = TRUE,
do_soilmstress = FALSE,
verbose = FALSE
)
## Warning in sqrt((1/fact_jmaxlim)^2 - 1): NaNs produced
## [1] 8.162522 7.967237 8.148933 7.468270 8.968635
Note that gpp (as well as all other returned variables)
are now vectors of the same length as the vectors provided as
inputs.
P-model in the tidyverse
We can create a data frame (in tidyverse this is a tibble) and
apply the rpmodel() function to each row.
library(dplyr)
library(purrr)
set.seed(1982)
df <- tibble(
tc = 20 + rnorm(5, mean = 0, sd = 5),
vpd = 1000 + rnorm(5, mean = 0, sd = 50),
co2 = rep(400, 5),
fapar = rep(1, 5),
ppfd = 30 + rnorm(5, mean = 0, sd = 3)
) %>%
mutate( out_pmodel = purrr::pmap(., rpmodel,
elv = 0,
kphio = 0.049977,
beta = 146,
c4 = FALSE,
method_jmaxlim = "none",
do_ftemp_kphio = FALSE
) )
print(df)
Note that the new column out_pmodel now contains the
list returned as output of the rpmodel() function applied
to each row separately. Additional (constant) arguments are just passed
to purrr::pmap as arguments.
If you prefer the elements of these lists to be in separate columns
of df, use tidyr to do:
library(tidyr)
df <- df %>%
mutate( out_pmodel = purrr::map(out_pmodel, ~as_tibble(.))) %>%
unnest(out_pmodel)
print(df)
C4 photosynthesis
Photosynthesis by C4 plants is simulated by the P-model based on the
assumption that the “CO2 limitation term” in the FvCB model (Eq. 11 in
Stocker et al., 2020) is 1 (no limitation), and based on a distinct
parametrisation of the quantum yield efficiency and its temperature
dependence \(\varphi_0\), following Cai
& Prentice, 2020. Under identical conditions, GPP of C3 and C4
photosynthesis are different:
out_c3 <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
c4 = FALSE
)
out_c4 <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
c4 = TRUE
)
## Warning in rpmodel(tc = 20, vpd = 1000, co2 = 400, fapar = 1, ppfd = 30, : rpmodel(): light and Rubisco-limited assimilation rates are not identical.
## Mean relative difference: 1.582923
## Warning in rpmodel(tc = 20, vpd = 1000, co2 = 400, fapar = 1, ppfd = 30, : rpmodel(): Assimilation and GPP are not identical.
## Mean relative difference: 1.582923
## [1] 7.642545
## [1] 126.2565
To accurately simulate C3 photosynthesis, a constant scalar was
calibrated by Stocker et al., 2020 to GPP from FLUXNET2015 data and
scaled in their ‘BRC’ and ‘FULL’ setup with a temperature dependence
factor (their Eq. 20). The implementation of C3 photosynthesis in
rpmodel uses kphio as a constant scalar, provided to
rpmodel() as an argument, and representing \((a_L b_L)/4\) in their Eq. 20. The
temprature dependence is calculated based on Bernacchi et al., 2003
using the function ftemp_kphio().
The implementation of C4 photosynthesis uses a different temperature
dependence of \(\varphi_0\) following
Eq. 5 in Cai & Prentice (2020), implemented by
ftemp_kphio(c4 = TRUE). The calibratable parameter
kphio doesn’t appear explicitly in Eq. 5 by Cai &
Prentice (2020), but is implemented the same way in rpmodel as for C3
photosynthesis. It should therefore be regarded as a “correction”
factor, where kphio = 1.0 means “no correction” and
reflects the parametrisation used by Cai & Prentice (2020).
The following shows \(\varphi(T)\),
corresponding to kphio * ftemp_kphio(), using parameters as
in Stocker et al., 2020 for C3 vegetation (their ‘BRC’ setup, black line
in the plot below), and as in Cai & Prentice (2020) for C4
vegetation (blue solid line).
Addendum: As of issue #19
(raised by David Orme), the values provided in Cai & Prentice (2020)
are erroneous. And instead, the temperature response function for C4
should be ftemp = -0.064 + 0.03 * tc - 0.000464 * tc^2.
This is now also implemented in rpmodel.
Auxiliary functions
A number of auxiliary functions, which are used within
rpmodel(), are available (public) through the package.
Instantaneous temperature scaling
Different instantaneous temperature scaling functions are applied for
\(V_\text{cmax}\) and dark respiration
(\(R_d\)).
ftemp_inst_vcmax() calculates the instantaneous
temperature response of \(V_\text{cmax}\). Let’s run the P-model for
tc = 10 (degrees C). The ratio of \(V_\text{cmax}/V_\text{cmax25}\) should
equal the instantaneous temperature scaling function for \(V_\text{cmax}\) at 10 degrees C (calculated
by ftemp_inst_vcmax(10)):
out_pmodel <- rpmodel(
tc = 10, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
kphio = 0.049977, # quantum yield efficiency as calibrated for setup ORG by Stocker et al. 2020 GMD,
beta = 146, # unit cost ratio a/b,
method_jmaxlim = "none",
do_ftemp_kphio = FALSE,
verbose = TRUE
)
## Warning in rpmodel(tc = 10, vpd = 1000, co2 = 400, fapar = 1, ppfd = 30, : Atmospheric pressure (patm) not provided. Calculating it as a
## function of elevation (elv), assuming standard atmosphere
## (101325 Pa at sea level).
print(paste("Ratio Vcmax/Vcmax25 :", out_pmodel$vcmax/out_pmodel$vcmax25))
## [1] "Ratio Vcmax/Vcmax25 : 0.260975632963417"
print(paste("ftemp_inst_vcmax(10):", ftemp_inst_vcmax(10)))
## [1] "ftemp_inst_vcmax(10): 0.260975632963417"
ftemp_arrh() Calculates the Arrhenius-type
temperature response and is used inside
ftemp_inst_vcmax().
ftemp_inst_rd() calculates the temperature response
of dark respiration (\(R_d\)), which is
slightly less steep than that for \(V_\text{cmax}\):
print(paste("ftemp_inst_rd(10):", ftemp_inst_rd(10)))
## [1] "ftemp_inst_rd(10): 0.284933345928884"
Parameters in the FvCB model
calc_gammastar() calculates the CO\(_2\) compensation point (\(\Gamma^\ast\)) in the Farquhar-von
Caemmerer-Berry model as a function of temperature (argument
tc) and atmospheric pressure (argument patm).
This is returned by the rpmodel() function and by the
separate auxiliary function calc_gammastar().
calc_gammastar() requires atmospheric pressure
(patm) to be given as an argument (in addition to
temperature). Corresponding to the rpmodel() call above,
let’s calculate this using the auxiliary function
calc_patm() with 0 metres above sea level, and assuming
standard atmospheric pressure (101325 Pa at 0 m a.s.l.):
print(paste("From rpmodel call :", out_pmodel$gammastar))
## [1] "From rpmodel call : 1.93016150706341"
print(paste("gammastar(10):", calc_gammastar(10, patm = calc_patm(elv = 0))))
## [1] "gammastar(10): 1.93016150706341"
calc_kmm() calculates the Michaelis Menten coefficient
for Rubisco-limited photosynthesis as a function of temperature
(argument tc) and atmospheric pressure (argument
patm). As above, calc_kmm() requires
atmospheric pressure to be given as an argument (in addition to
temperature). Corresponding to the rpmodel() call above,
let’s calculate this using the auxiliary function
calc_patm() with 0 metres above sea level, and assuming
standard atmospheric pressure (101325 Pa at 0 m a.s.l.):
print(paste("From rpmodel call:", out_pmodel$kmm))
## [1] "From rpmodel call: 19.6242174524746"
print(paste("kmm(10) :", calc_kmm(10, patm = calc_patm(elv = 0))))
## [1] "kmm(10) : 19.6242174524746"
Temperature dependence of quantum yield efficiency
The temperature dependence of quantum yield efficiency is modelled
following Bernacchi et al. (2003), if the argument to the
rpmodel() call do_ftemp_kphio = TRUE. This
affects several quantities returned by the rpmodel() call
(GPP, LUE, Vcmax), and can be calculated direction using
ftemp_kphio().
out_pmodel_ftemp_kphio_ON <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
do_ftemp_kphio = TRUE
)
out_pmodel_ftemp_kphio_OFF <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
do_ftemp_kphio = FALSE
)
print(paste("LUE ftemp_ON /LUE ftemp_OFF =", out_pmodel_ftemp_kphio_ON$lue / out_pmodel_ftemp_kphio_OFF$lue))
## [1] "LUE ftemp_ON /LUE ftemp_OFF = "
print(paste("GPP ftemp_ON /GPP ftemp_OFF =", out_pmodel_ftemp_kphio_ON$gpp / out_pmodel_ftemp_kphio_OFF$gpp))
## [1] "GPP ftemp_ON /GPP ftemp_OFF = 1.07351301598735"
print(paste("Vcmax ftemp_ON /Vcmax ftemp_OFF =", out_pmodel_ftemp_kphio_ON$vcmax / out_pmodel_ftemp_kphio_OFF$vcmax))
## [1] "Vcmax ftemp_ON /Vcmax ftemp_OFF = 1.07351301598735"
print(paste("ftemp_kphio(20) =", ftemp_kphio(20)))
## [1] "ftemp_kphio(20) = 0.656"
Soil moisture stress
The soil moisture stress function is available as a separate public
function.
vec_soilm <- seq(from = 1.0, to = 0.0, by = -0.05)
vec_soilmstress <- calc_soilmstress( vec_soilm, meanalpha = 1.0, apar_soilm = 0.0, bpar_soilm = 0.7330 )
plot(vec_soilm, vec_soilmstress)
Similar to above, the soil moisture dependence of LUE (and hence GPP,
and Vcmax) can be calculated directly using the function
calc_soilmstress() and affects several quantities returned
by the rpmodel() call (GPP, LUE, Vcmax):
out_pmodel_soilmstress_OFF <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
do_ftemp_kphio = FALSE,
do_soilmstress = FALSE
)
out_pmodel_soilmstress_ON <- rpmodel(
tc = 20, # temperature, deg C
vpd = 1000, # Pa,
co2 = 400, # ppm,
fapar = 1, # fraction ,
ppfd = 30, # mol/m2/d,
elv = 0, # m.a.s.l.,
do_ftemp_kphio = FALSE,
do_soilmstress = TRUE,
soilm = 0.2,
apar_soilm = 0.1,
bpar_soilm = 0.7,
meanalpha = 0.2
)
print(paste("LUE soilmstress_ON /LUE soilmstress_OFF =", out_pmodel_soilmstress_ON$lue / out_pmodel_soilmstress_OFF$lue))
## [1] "LUE soilmstress_ON /LUE soilmstress_OFF = "
print(paste("GPP soilmstress_ON /GPP soilmstress_OFF =", out_pmodel_soilmstress_ON$gpp / out_pmodel_soilmstress_OFF$gpp))
## [1] "GPP soilmstress_ON /GPP soilmstress_OFF = 0.662222222222222"
print(paste("Vcmax soilmstress_ON /Vcmax soilmstress_OFF =", out_pmodel_soilmstress_ON$vcmax / out_pmodel_soilmstress_OFF$vcmax))
## [1] "Vcmax soilmstress_ON /Vcmax soilmstress_OFF = 0.662222222222222"
print(paste("soilmstress(0.2, apar_soilm = 0.1, bpar_soilm = 0.7, meanalpha = 0.2) =", calc_soilmstress(0.2, apar_soilm = 0.1, bpar_soilm = 0.7, meanalpha = 0.2)))
## [1] "soilmstress(0.2, apar_soilm = 0.1, bpar_soilm = 0.7, meanalpha = 0.2) = 0.662222222222222"
ftemp_arrh() Calculates the Arrhenius-type temperature
response.
References
Bernacchi, C. J., Pimentel, C., and Long, S. P.: In vivo temperature
response func-tions of parameters required to model RuBP-limited
photosynthesis, Plant Cell Environ., 26, 1419–1430, 2003
Cai, W., and Prentice, I. C.: Recent trends in gross primary
production and their drivers: analysis and modelling at flux-site and
global scales, Environ. Res. Lett. 15 124050
https://doi.org/10.1088/1748-9326/abc64e, 2020
Stocker, B. D., Wang, H., Smith, N. G., Harrison, S. P., Keenan, T.
F., Sandoval, D., Davis, T., and Prentice, I. C.: P-model v1.0: An
optimality-based light use efficiency model for simulating ecosystem
gross primary production, Geosci. Model Dev. Discuss., https://doi.org/10.5194/gmd-2019-200, in review,
2019.