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# $J_{max}$ limitation

The worked example from this page {download}`can be downloaded as Jupyter notebook here </_static/JmaxLimitation.ipynb>`.

```{code-cell} ipython3
:tags: [hide-input]

import matplotlib.pyplot as plt
import numpy as np
from pyrealm.pmodel.pmodel import PModel
from pyrealm.pmodel import PModelEnvironment

%matplotlib inline

# Set the resolution of examples
n_pts = 201

# Create a range of representative values for key inputs.
tc_1d = np.linspace(-25, 50, n_pts)
```

Environmental conditions can also lead to limitation of both the electron transfer rate
($J_{max}$) and the carboxylation capacity ($V_{cmax}$) of leaves. The
{class}`~pyrealm.pmodel.jmax_limitation` module implements three alternative approaches
to the calculation of $J_{max}$ and $V_{cmax}$ and these are specified when fitting a P
Model using the argument `method_jmaxlim`.

These options implement alternative calculations of two factor ($f_j$ and $f_v$) which
are applied to the calculation of $J_{max}$ and $V_{cmax}$. The options for this setting
are:

* `none`: This approach implements $f_j  = f_v = 1$ and hence no limitation.
* `wang17`: This is the default setting for `method_jmaxlim` and applies the
  calculations describe in  {cite:t}`Wang:2017go`. The calculation details can be
  seen in the {class}`~pyrealm.pmodel.jmax_limitation.JmaxLimitationWang17` method.

* `smith19`: This is an alternate calculation for optimal values of $J_{max}$
  and $V_{cmax}$ described in {cite:t}`Smith:2019dv`. The calculation details can be
  seen in the {class}`~pyrealm.pmodel.jmax_limitation.JmaxLimitationSmith19` method.

The plot below shows the effects of each method on the light use efficienct across a
temperature gradient. The other forcing variables are fixed ($P=101325.0 , \ce{CO2}= 400
\text{ppm}, \text{VPD}=820$) and $\phi_0$ is also fixed ($\phi_0=0.08$).

```{code-cell} ipython3
:tags: [hide-input]

# Calculate variation in m_jlim with temperature - no estimation of GPP needed
# so set fapar and ppfd to unity
env = PModelEnvironment(tc=tc_1d, patm=101325, vpd=820, co2=400, fapar=1, ppfd=1)

model_jmax_simple = PModel(
    env, method_jmaxlim="none", method_kphio="fixed", reference_kphio=0.08
)
model_jmax_wang17 = PModel(
    env, method_jmaxlim="wang17", method_kphio="fixed", reference_kphio=0.08
)
model_jmax_smith19 = PModel(
    env, method_jmaxlim="smith19", method_kphio="fixed", reference_kphio=0.08
)

# Create a line plot of the resulting values of m_j
plt.plot(tc_1d, model_jmax_simple.lue, label="simple")
plt.plot(tc_1d, model_jmax_wang17.lue, label="wang17")
plt.plot(tc_1d, model_jmax_smith19.lue, label="smith19")

plt.title("Effects of J_max limitation")
plt.xlabel("Temperature °C")
plt.ylabel("Light Use Efficiency (g C mol-1)")
plt.legend()
plt.show()
```