Ratkowsky model for fitting thermal performance curves

Usage

ratkowsky_1983(temp, tmin, tmax, a, b)

Arguments

temp: temperature in degrees centigrade
tmin: low temperature (ºC) at which rates become negative
tmax: high temperature (ºC) at which rates become negative
a: parameter defined as sqrt(rate)/(temp - tmin)
b: empirical parameter needed to fit the data for temperatures beyond the optimum temperature

Value

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

Details

Equation: $$rate = (a \cdot (temp-t_{min}))^2 \cdot (1-exp(b \cdot (temp-t_{max})))^2$$

Start values in get_start_vals are derived from the data and previous values in the literature.

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

Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N., Chandler, R.E., Model for bacterial growth rate throughout the entire biokinetic temperature range. J. Bacteriol. 154: 1222–1226 (1983)

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 = 'ratkowsky_1983')
# fit model
mod <- nls.multstart::nls_multstart(rate~ratkowsky_1983(temp = temp, tmin, tmax, a, b),
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 = 'ratkowsky_1983'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'ratkowsky_1983'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ ratkowsky_1983(temp = temp, tmin, tmax, a, b)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> tmin  9.51995    4.60886   2.066  0.07273 .  
#> tmax 47.93852    0.32095 149.366 4.51e-15 ***
#> a     0.04841    0.01163   4.162  0.00316 ** 
#> b     0.22857    0.06129   3.729  0.00579 ** 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.2244 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()