Beta model for fitting thermal performance curves
Value
a numeric vector of rate values based on the temperatures and parameter values provided to the function
Details
Equation: $$rate=\frac{a\left(\frac{temp-b+\frac{c(d-1)}{d+e-2}}{c}\right)^{d-1} \cdot \left(1-\frac{temp-b+\frac{c(d-1)}{d+e-2}}{c}\right)^{e-1}}{{\left(\frac{d-1}{d+e-2}\right)}^{d-1}\cdot \left(\frac{e-1}{d+e-2}\right)^{e-1}}$$
Start values in get_start_vals are derived from the data or sensible values from the literature.
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.
References
Niehaus, Amanda C., et al. Predicting the physiological performance of ectotherms in fluctuating thermal environments. Journal of Experimental Biology 215.4: 694-701 (2012)
Examples
# \donttest{
# load in ggplot
library(ggplot2)
# 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 = 'beta_2012')
# fit model
mod <- nls.multstart::nls_multstart(rate~beta_2012(temp = temp, a, b, c, d, e),
data = d,
iter = c(7,7,7,7,7),
start_lower = start_vals - 10,
start_upper = start_vals + 10,
lower = get_lower_lims(d$temp, d$rate, model_name = 'beta_2012'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'beta_2012'),
supp_errors = 'Y',
convergence_count = FALSE)
# look at model fit
summary(mod)
#>
#> Formula: rate ~ beta_2012(temp = temp, a, b, c, d, e)
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> a 1.3704 0.2076 6.603 0.000304 ***
#> b 38.0183 2.1181 17.949 4.12e-07 ***
#> c 39.8317 26.8210 1.485 0.181099
#> d 3.5728 4.4135 0.810 0.444851
#> e 1.9794 0.7894 2.507 0.040558 *
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.3508 on 7 degrees of freedom
#>
#> Number of iterations till stop: 95
#> Achieved convergence tolerance: 1.49e-08
#> Reason stopped: Number of calls to `fcn' has reached or exceeded `maxfev' == 600.
#>
# 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()
# }