Author: Beni Stocker

# Running the P-model in R

#### Beni Stocker

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.

Important note:

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).

    print(out_pmodel)

    ## $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

    print(out_ts_pmodel$gpp)

    ## [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

    print(out_c3$gpp)

    ## [1] 7.642545

    print(out_c4$gpp)

    ## [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.