Modified gaussian model for fitting thermal performance curves

Usage

modifiedgaussian_2006(temp, rmax, topt, a, b)

Arguments

temp: temperature in degrees centigrade
rmax: maximum rate at optimum temperature
topt: optimum temperature
a: related to full curve width
b: allows for asymmetry in the curve fit

Value

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

Details

Equation: $$rate = r_{max} \cdot exp^{\bigg[-0.5 \left(\frac{|temp-t_{opt}|}{a}\right)^b\bigg]}$$

Start values in get_start_vals are derived from the data and gaussian_1987

Limits in get_lower_lims and get_upper_lims are based on extreme values that are unlikely to occur in ecological settings.

Note

Generally we found this model difficult to fit.

References

Angilletta Jr, M. J. (2006). Estimating and comparing thermal performance curves. Journal of Thermal Biology, 31(7), 541-545.

Examples

# 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 = 'modifiedgaussian_2006')
# fit model
mod <- nls.multstart::nls_multstart(rate~modifiedgaussian_2006(temp = temp, rmax, topt, a, b),
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 = 'modifiedgaussian_2006'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'modifiedgaussian_2006'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ modifiedgaussian_2006(temp = temp, rmax, topt, a, b)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> rmax   1.0016     2.0113   0.498    0.632    
#> topt  37.0000     2.6631  13.893 6.97e-07 ***
#> a     23.0120    91.1292   0.253    0.807    
#> b      0.7888     7.1276   0.111    0.915    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.5851 on 8 degrees of freedom
#> 
#> Number of iterations to convergence: 20 
#> 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()