Thomas model (2012) for fitting thermal performance curves

Thomas model (2012) for fitting thermal performance curves

Usage

thomas_2012(temp, a, b, c, tref)

Arguments

temp

temperature in degrees centigrade

a

arbitrary constant

b

arbitrary constant

c

the range of temperatures over which growth rate is positive, or the thermal niche width (ºC)

tref

determines the location of the maximum of the quadratic portion of this function. When b = 0, tref would equal topt

Value

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

Details

Equation: $$rate = a \cdot exp^{b \cdot temp} \bigg(1-\bigg(\frac{temp - t_{ref}}{c/2}\bigg)^2\bigg)$$

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 on extreme values that are unlikely to occur in ecological settings.

Note

Generally we found this model easy to fit.

References

Thomas, Mridul K., et al. A global pattern of thermal adaptation in marine phytoplankton. Science 338.6110, 1085-1088 (2012)

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 = 'thomas_2012')
# fit model
mod <- nls.multstart::nls_multstart(rate~thomas_2012(temp = temp, a, b, c, tref),
data = d,
iter = c(4,4,4,4),
start_lower = start_vals - 1,
start_upper = start_vals + 2,
lower = get_lower_lims(d$temp, d$rate, model_name = 'thomas_2012'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'thomas_2012'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ thomas_2012(temp = temp, a, b, c, tref)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> a     0.20859    0.20466   1.019 0.337937    
#> b     0.05296    0.02782   1.904 0.093414 .  
#> c    33.30540    4.79114   6.951 0.000118 ***
#> tref 32.08003    2.47486  12.962 1.19e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.3351 on 8 degrees of freedom
#> 
#> Number of iterations to convergence: 22 
#> 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()