Modified deutsch model for fitting thermal performance curves

Usage

deutsch_2008(temp, rmax, topt, ctmax, a)

Arguments

temp: temperature in degrees centigrade
rmax: maximum rate at optimum temperature
topt: optimum temperature (ºC)
ctmax: critical thermal maximum (ºC)
a: related to the full curve width

Value

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

Details

Equation: $$\textrm{if} \quad temp < t_{opt}: rate = r_{max} \cdot exp^{-\bigg(\frac{temp-t_{opt}}{2a}\bigg)^2}$$ $$\textrm{if} \quad temp > t_{opt}: rate = r_{max} \cdot \left(1 - \bigg(\frac{temp - t_{opt}}{t_{opt} - ct_{max}}\bigg)^2\right)$$

Start values in get_start_vals are derived from the data.

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 easy to fit.

References

Deutsch, C. A., Tewksbury, J. J., Huey, R. B., Sheldon, K. S., Ghalambor, C. K., Haak, D. C., & Martin, P. R. Impacts of climate warming on terrestrial ectotherms across latitude. Proceedings of the National Academy of Sciences, 105(18), 6668-6672. (2008)

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 = 'deutsch_2008')
# fit model
mod <- nls.multstart::nls_multstart(rate~deutsch_2008(temp = temp, rmax, topt, ctmax, a),
data = d,
iter = c(4,4,4,4),
start_lower = start_vals - 10,
start_upper = start_vals + 10,
lower = get_lower_lims(d$temp, d$rate, model_name = 'deutsch_2008'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'deutsch_2008'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ deutsch_2008(temp = temp, rmax, topt, ctmax, a)
#> 
#> Parameters:
#>       Estimate Std. Error t value Pr(>|t|)    
#> rmax    1.4363     0.1639   8.765 2.25e-05 ***
#> topt   38.2306     2.7887  13.709 7.73e-07 ***
#> ctmax  48.7175     0.9798  49.722 2.96e-11 ***
#> a       6.7950     1.9474   3.489  0.00821 ** 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.3105 on 8 degrees of freedom
#> 
#> Number of iterations to convergence: 27 
#> 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()