Lobry-Rosso-Flandros (LRF) model for fitting thermal performance curves

Usage

lrf_1991(temp, rmax, topt, tmin, tmax)

Arguments

temp: temperature in degrees centigrade
rmax: maximum rate at optimum temperature
topt: optimum temperature (ºC)
tmin: low temperature (ºC) at which rates become negative
tmax: high temperature (ºC) at which rates become negative

Value

a numeric vector of rate values based on the temperatures and parameter values provided to the function

Details

Equation: $$rate= rmax \cdot \frac{(temp - t_{max}) \cdot (temp - t_{min})^2}{(t_{opt} - t_{min}) \cdot ((t_{opt} - t_{min}) \cdot (temp - t_{opt}) - (t_{opt} - t_{max}) \cdot (t_{opt} + t_{min} - 2 \cdot temp))}$$

Start values in get_start_vals are derived from the data.

Limits in get_lower_lims and get_upper_lims are derived from the data or based extreme values that are unlikely to occur in ecological settings.

Note

Generally we found this model easy to fit.

References

Rosso, L., Lobry, J. R., & Flandrois, J. P. An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model. Journal of Theoretical Biology, 162(4), 447-463. (1993)

Author

Daniel Padfield

Examples

# load in ggplot
library(ggplot2)
library(nls.multstart)

# subset for the first TPC curve
data('chlorella_tpc')
d <- subset(chlorella_tpc, curve_id == 1)

# get start values and fit model
start_vals <- get_start_vals(d$temp, d$rate, model_name = 'lrf_1991')
# fit model
mod <- nls_multstart(rate~lrf_1991(temp = temp, rmax, topt, tmin, tmax),
data = d,
iter = c(3,3,3,3),
start_lower = start_vals - 10,
start_upper = start_vals + 10,
lower = get_lower_lims(d$temp, d$rate, model_name = 'lrf_1991'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'lrf_1991'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ lrf_1991(temp = temp, rmax, topt, tmin, tmax)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> rmax   1.3962     0.1607   8.688 2.40e-05 ***
#> topt  37.7416     2.0042  18.831 6.54e-08 ***
#> tmin  14.0262     5.8327   2.405   0.0429 *  
#> tmax  48.7822     0.9827  49.643 3.00e-11 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.3253 on 8 degrees of freedom
#> 
#> Number of iterations to convergence: 46 
#> Achieved convergence tolerance: 1.49e-08
#> 

# get predictions
preds <- data.frame(temp = seq(min(d$temp), max(d$temp), length.out = 100))
preds <- broom::augment(mod, newdata = preds)

# plot
ggplot(preds) +
geom_point(aes(temp, rate), d) +
geom_line(aes(temp, .fitted), col = 'blue') +
theme_bw()