Lobry model for fitting thermal performance curves

Lobry model for fitting thermal performance curves

Usage

lobry_1991(temp, rmax, topt, tmin, tmax)

Arguments

temp

temperature in degrees centigrade

rmax

the maximum rate

topt

optimum temperature (ºC) at which rates are maximal

tmin

low temperature (ºC) at which rates become negative

tmax

high temperature (ºC) at which rates become negative

Value

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

Details

Equation: $$rate = rmax \cdot (1 - \frac{(temp - topt)^2)}{(temp - topt)^2 + temp \cdot (tmax + tmin - temp) - tmax \cdot tmin}$$

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 derived from the data or based extreme values that are unlikely to occur in ecological settings.

Note

Generally we found this model easy to fit.

References

Lobry, J. R., Rosso, L., & Flandrois, J. P. (1991). A FORTRAN subroutine for the determination of parameter confidence limits in non-linear models. Binary, 3(86-93), 25.

Author

Daniel Padfield

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 = 'lobry_1991')
# fit model
mod <- nls.multstart::nls_multstart(rate~lobry_1991(temp = temp, rmax, topt, tmin, tmax),
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 = 'lobry_1991'),
upper = get_upper_lims(d$temp, d$rate, model_name = 'lobry_1991'),
supp_errors = 'Y',
convergence_count = FALSE)

# look at model fit
summary(mod)
#> 
#> Formula: rate ~ lobry_1991(temp = temp, rmax, topt, tmin, tmax)
#> 
#> Parameters:
#>       Estimate Std. Error   t value Pr(>|t|)    
#> rmax 1.272e+00  3.462e-01 3.675e+00  0.00626 ** 
#> topt 3.803e+01  2.721e+00 1.398e+01 6.65e-07 ***
#> tmin 1.600e+01  6.369e-18 2.512e+18  < 2e-16 ***
#> tmax 4.902e+01  1.331e+00 3.684e+01 3.23e-10 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.3619 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()