Janisch II model for fitting thermal performance curves

Usage

janisch2_1925(temp, m, a, b, topt)

Arguments

temp: temperature in degrees centigrade
m: scale parameter (controlling the height of the curve)
a: shape parameter (controlling the shape of the rising part of the curve)
b: shape parameter (controlling the shape of the falling part of the curve)
topt: temperature of max performance (ºC)

Value

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

Details

Equation: $$rate = \frac{1}{\frac{m}{2} \cdot \left[a^{T-T_{\text{opt}}}+b^{-(T-T_{\text{opt}})}\right]}$$

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

Note

Generally we found this model easy to fit.

References

Janisch, E. Über die Temperaturabhängigkeit biologischer Vorgänge und ihre kurvenmäßige Analyse. Pflüger's Arch. Physiol. 209, 414–436 (1925).

Author

Francis Windram

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 = 'janisch2_1925')
# fit model
mod <- nls.multstart::nls_multstart(rate~janisch2_1925(temp = temp, m, a, b, topt),
data = d,
iter = 200,
start_lower = start_vals - 10,
start_upper = start_vals + 10,
lower = get_lower_lims(d$temp, d$rate, model_name = 'janisch2_1925'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'janisch2_1925'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ janisch2_1925(temp = temp, m, a, b, topt)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> m     0.91049    0.11327   8.038 4.22e-05 ***
#> a     4.52758    6.14681   0.737    0.482    
#> b     1.07555    0.01398  76.952 9.06e-13 ***
#> topt 43.53056    0.54446  79.951 6.68e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.2038 on 8 degrees of freedom
#> 
#> Number of iterations to convergence: 33 
#> 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()