Hello,

I am a plant ecologist and I am working with process based models. One of the aspects of process based models is that they are informed with parameters coming from empirical data. In my approach I am using brms hierarchical models to obtain a posterior distribution of each parameter and then inform the process based model by sampling from those distributions. However some parameters for the process based models are obtained by fitting non-linear equations. For instance, the parameters b and c from a weibull function to characterize plant’s xylem vulnerability to embolism:

K=K_{max}*e^-\left ( \frac{P}{b} \right )^{c}

```
# Fit the model in which the parameter(s) we are interested change as a function of elevation----
weibull_ML_formula = bf(formula = PLC ~ (100*(1-exp(-((-WP/b)^c)))),
b + c ~ 1 + (1|plot|elev_band),nl = TRUE)
# Prior Specification
weibull_ML_prior_pipo = c(set_prior(prior = "normal(3.8,0.3)",class = "b",nlpar = "b",lb = 2),#,lb = 1.8),#prior on PIPO b
set_prior(prior = "normal(2.5,0.2)",class = "b",nlpar = "c",lb = 1.2),#,lb = 1),
set_prior(prior = "normal(0,0.3)",class = "sd",coef ="Intercept",group = "elev_band",nlpar = "b"),
set_prior(prior = "normal(0,0.01)",class = "sd",coef ="Intercept",group = "elev_band",nlpar = "c"))#,lb = 1))
# Initial values
inits_l = list(inits = list(b_Intercept = 3.5, c_Intercept = 2,sd_elev_band__b_Intercept=0,sd_elev_band__c_Intercept=0.05),
inits = list(b_Intercept = 3.5, c_Intercept = 2,sd_elev_band__b_Intercept=0,sd_elev_band__c_Intercept=0.05),
inits = list(b_Intercept = 3.5, c_Intercept = 2,sd_elev_band__b_Intercept=0,sd_elev_band__c_Intercept=0.05),
inits = list(b_Intercept = 3.5, c_Intercept = 2,sd_elev_band__b_Intercept=0,sd_elev_band__c_Intercept=0.05))
# Fit model
vc_fits_pipo <- brm(formula = weibull_ML_formula,
data = vc_md[vc_md$SpeciesID %in% "PIPO",],family = gaussian(),
prior = weibull_ML_prior_pipo,
inits = inits_l,
iter = 30000,
warmup = 10000,
thin = 3,
cores = n_cores)
```

This model fits great, and the results are consistent with other approaches to fit the same function in a frequentist framework using non-linear squares. A more complicated non-linear model is the one describing photosynthesis in C3 plants. This model can be described by a set of non-linear equations used to predict carbon assimilation and then a function that minimizes the difference between observed and predicted carbon assimilation:

A_{n} = min\left ( A_{c}, A_{j} \right )-R_{d}

A_{c} = V_{cmax}\cdot \left [ \frac{C_{c}-\Gamma^{*}}{C_{c}+K_{c}\cdot \left ( 1+O/K_{o} \right )} \right ]-R_{d}

A_{j} = J\cdot \left [ \frac{C_{c}-\Gamma^{*}}{4C_{c}+8\Gamma^{*}} \right ]-R_{d}

C_{c} = C_{i}-\frac{A_{n}}{g_{m}}

**Measured data:**

A_{n}: net carbon assimilation rates

C_{i}: carbon dioxide concentration in the leaf

**Constants:**

K_{c}: Michaelis constant of Rubisco for carbon **27.238**

O: Partial pressure of oxygen at Rubisco **21** at sea level

K_{o}: inhibition constant of Rubisco for oxygen **16.582**

g_{m}: leaf resistance to carbon diffusion, can be measured but is also set constant to **0.3**

\Gamma^{*}: is the concentration at that oxygenation is twice the rate of carboxilation, in other words CO2 compensation point **3.743**

Non-linear parameters to be estimated:

R_{d}: mitochondrial respiration

V_{cmax}: Rubisco’s maximum rate of carboxilation

J: electron transport rate

The one challenge is that the non-linear functions are fitted in two different areas of the curve between A_{n} and C_{i}. This transition point used to be arbitrarily picked (https://doi.org/10.1111/j.1365-3040.2007.01710.x). Nowadays there is an R package that gives the option to automate this process (Plantecophys - An R Package for Analysing and Modelling Leaf Gas Exchange Data). I was wondering if fitting this kind of model would be possible at all using brms, as having the posterior distribution while accounting for the hierarchical structure of the data will be powerful. Otherwise I will just fit the curves and then model the non-linear parameters using a multilevel model. Some approaches have been developed in JAGS, but incorporating multiple levels is really challenging in JAGS. Attached you might find data on photosynthetic responses to Ci. The variables needed to fit the model are:

A_{n}: Photo

C_{i}: Ci_Pa

level 1: indiv

level 2: species

This is my first time posting a topic, apologize if I didn’t do it properly.

Thanks.

ACI_data_example.csv (50.7 KB)