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(pmodel_overview)=

# The Standard P Model

This page gives an overview of the calculations for the standard form of the P Model
{cite:p}`Prentice:2014bc,Wang:2017go` along with links to further details of the core
components of the model. It may be useful to read this alongside:

* The [worked examples](worked_examples) of using `pyrealm` to fit the Standard P Model.
* The overview of the [expected predictions](./envt_variation_outputs.md) of the model,
  given forcing data within normal bounds.
* The descriptions of the key Python classes used to fit the standard P Model:
  {class}`~pyrealm.pmodel.pmodel_environment.PModelEnvironment` and
  {class}`~pyrealm.pmodel.pmodel.PModel`.

:::{important}

The standard form of the P Model is intended to work with data on greater than weekly
timescales, where the plants are assumed to be roughly at equilibrium with their
environment. At faster time scales, the [subdaily form of the P
Model](../subdaily_details/subdaily_overview.md) accounts for more slowly responding
systems that introduce acclimation lags into the model calculations.

:::

The diagram below shows the workflow used for the calculation of the standard P Model:
the input forcing data is show in blue, internal variables are shown in green and the
core shared calculations are shown in red.

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In overview:

1. The forcing variables are used to calculate the [photosynthetic
  environment](../shared_components/photosynthetic_environment.md) for the model,
  including the:

    * photorespiratory compensation point ($\Gamma^*$),
    * Michaelis-Menten coefficient for photosynthesis ($K_{mm}$),
    * relative viscosity of water ($\eta^*$),
    * the partial pressure of $\ce{CO2}$ in ambient air ($c_a$), and
    * the absorbed irradiance ($I_{abs}$).

2. The photosynthetic environment is then used to calculate [optimal
   chi](../shared_components/optimal_chi.md), given the method set by the
   `method_optchi` argument, to calculate:

    * the ratio of internal to ambient $\ce{CO2}$ partial pressure ($\chi$),
    * the internal $\ce{CO2}$ partial pressure ($c_i$),
    * the $\ce{CO2}$ limitation factors to both light assimilation ($m_j$) and
      carboxylation ($m_c$) along with their ratio ($m_j / m_c$).

    The term $m_j$ is at the heart of the P model and describes the trade off between
    carbon dioxide capture and water loss in photosynthesis. Given the environmental
    conditions, a leaf will adjust its stomata to a value of $\chi$ that optimises this
    trade off. When $\chi$ is less than one, the partial pressure of $\ce{CO2}$ inside
    the leaf is lowered and $m_j$ captures the resulting loss in light use efficiency.

3. The photosynthetic environment is also used to calculate the [quantum
   yield](../shared_components/quantum_yield.md) of photosynthesis ($\phi_0$), given the
   method set by the `method_kphio` argument.

4. Theory suggests that $m_j$ and $m_c$ should be further limited ($J_max$ limitation),
   with different approaches proposed {cite}`Wang:2017go,Smith:2019dv`. The calculation
   of [limitation factors](../shared_components/jmax_limitation.md), given the
   method set by the `method_jmaxlim` argument, represents these alternative corrections
   as:

    * a limitation term on the electron transfer rate ($f_j$), and
    * a similar limitation term on the carboxylation capacity ($f_v$).

5. From these values, and the molar mass of carbon ($M_C$), the light use efficiency of
   photosynthesis can then be calculated as:

    $$
      \text{LUE} = M_C \cdot \phi_0 \cdot  m_j \cdot f_v
    $$

    The gross primary productivity (GPP) is then simply the product of LUE and the
    absorbed irradiation ($I_abs = f_{APAR} \cdot \text{PPFD}$).

6. The approach above is equivalent to directly calculating the minimum of the
   assimilation rates given limitation of the electron transfer pathway ($A_j$) and
   limitation of the carboxylation pathway ($A_c$). However the model does still
   estimate the maximum rates of electron transfer ($J_{max}$) and carboxylation
   capacity ($V_{cmax}$) and then uses [Arrhenius
   scaling](../shared_components/arrhenius.md) to estimate those values at standard
   temperature  ($J_{max25}$) and carboxylation
   capacity ($V_{cmax25}$).