Logistic kinetics

Function describing exponential decline from a defined starting value, with an increasing rate constant, supposedly caused by microbial growth

logistic.solution(t, parent.0, kmax, k0, r)

Arguments

t	Time.
parent.0	Starting value for the response variable at time zero.
kmax	Maximum rate constant.
k0	Minumum rate constant effective at time zero.
r	Growth rate of the increase in the rate constant.

Note

The solution of the logistic model reduces to the SFO.solution if k0 is equal to kmax.

Value

The value of the response variable at time t.

References

FOCUS (2014) “Generic guidance for Estimating Persistence and Degradation Kinetics from Environmental Fate Studies on Pesticides in EU Registration” Report of the FOCUS Work Group on Degradation Kinetics, Version 1.1, 18 December 2014 http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics

Examples

  # Reproduce the plot on page 57 of FOCUS (2014)
  plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.2),
       from = 0, to = 100, ylim = c(0, 100),
       xlab = "Time", ylab = "Residue")
  plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.4),
       from = 0, to = 100, add = TRUE, lty = 2, col = 2)
  plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.8),
       from = 0, to = 100, add = TRUE, lty = 3, col = 3)
  plot(function(x) logistic.solution(x, 100, 0.08, 0.001, 0.2),
       from = 0, to = 100, add = TRUE, lty = 4, col = 4)
  plot(function(x) logistic.solution(x, 100, 0.08, 0.08, 0.2),
       from = 0, to = 100, add = TRUE, lty = 5, col = 5)
  legend("topright", inset = 0.05,
         legend = paste0("k0 = ", c(0.0001, 0.0001, 0.0001, 0.001, 0.08),
                         ", r = ", c(0.2, 0.4, 0.8, 0.2, 0.2)),
         lty = 1:5, col = 1:5)

  # Fit with synthetic data
  logistic <- mkinmod(parent = mkinsub("logistic"))

  sampling_times = c(0, 1, 3, 7, 14, 28, 60, 90, 120)
  parms_logistic <- c(kmax = 0.08, k0 = 0.0001, r = 0.2)
  d_logistic <- mkinpredict(logistic,
    parms_logistic, c(parent = 100),
    sampling_times)
  d_2_1 <- add_err(d_logistic,
    sdfunc = function(x) sigma_twocomp(x, 0.5, 0.07),
    n = 1, reps = 2, digits = 5, LOD = 0.1, seed = 123456)[[1]]

  m <- mkinfit("logistic", d_2_1, quiet = TRUE)
  plot_sep(m)
  summary(m)$bpar
#>              Estimate   se_notrans    t value       Pr(>t)        Lower
#> parent_0 1.057896e+02 2.3743105248 44.5559374 6.656664e-16 1.006602e+02
#> kmax     6.398190e-02 0.0193490291  3.3067243 2.836921e-03 3.329058e-02
#> k0       1.612775e-04 0.0009640761  0.1672871 4.348592e-01 3.972250e-10
#> r        2.263946e-01 0.2822811886  0.8020181 2.184792e-01 1.531165e-02
#>                Upper
#> parent_0 110.9190170
#> kmax       0.1229682
#> k0        65.4803698
#> r          3.3474197
  endpoints(m)$distimes
#>            DT50     DT90  DT50_k0 DT50_kmax
#> parent 36.86533 62.41511 4297.854  10.83349